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http://mathoverflow.net/questions/127423/how-many-vertices-can-a-convex-polytope-have/127426 | # How many vertices can a convex polytope have?
One has an $n$-dimensional convex polytope $P$ represented by an intersection of half-spaces:
$$H_i = \{ (x_1,x_2, \ldots,x_n) \in \mathbb{R}^n \mid \sum_{j=1}^n a_{ij} x_j \ge a_{i0}, \ a_{ij} \in \mathbb{R} \}, \ i = \overline{1,k}$$ $$P = \{ X \in \mathbb{R}^n \mid X \in \bigcap_{i=1}^k H_i \}$$
Then let's say that $(v_1,v_2,\ldots,v_n) \in P$ is a vertex if we can find at least $n$ subspaces (out of $k$) such that $$\sum_{j=1}^n a_{\tilde i j} v_j = a_{\tilde i 0}, \quad \text{ for some } \tilde i \in \{ 1,2, \ldots , k\}$$
Now let's get to the point. It is known that we must have at least $k = n+1$ subspaces in order to construct a closed convex polytope. In this initial case it'd be a simplex having $n+1=|V|$ vertices (here $V$ is a set of vertices).
We can obviously introduce any number of somewhat trivial additional subspaces and stay at $|V|=n+1$ vertices, but I am interested in the maximum number of vertices for any given number of subspaces $k>n$ in $n$-dimensional space.
$\mathbb{R}^2$ case: since restrictions are geometrically half-planes one can draw lines on paper and see what happens. The simplest closed figure is a triangle, it has 3 vertices and needs a minimum of 3 restrictions to exist. Now every additional restriction can add no more than 1 additional vertice (this is obvious in 2-dimensional space). This means that for $k \ge 3$ restrictions there can exist no more than $k$ vertices: $\max |V| = k.$
$\mathbb{R}^3$ case: similarly the simplest closed figure is a triangular pyramid, it has 4 vertices and needs a minimum of 4 restrictions to exist. Now it gets a little more complicated with additional restrictions, but what follows from fiddling with Euler characteristic is that for $k \ge 4$ restrictions there can exist no more than $(2k-4)$ vertices: $\max |V| = 2k-4.$
$\mathbb{R}^n$ case: I only understand clearly that $\max_{k = n+1} |V| = n+1.$ What happens after introducing additional restrictions is hard to conceive.
How can I generalize it? I hoped to draw some conclusions from Dehn–Sommerville equations but I am not quite experienced enough and maybe it's a completely wrong way to look at this problem...
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Using $k$ half-spaces, the polytope has at most $k$ facets. For a fixed number of facets, the number of vertices is maximized, for example, by the dual polytope of the cyclic polytope with $k$ vertices. More generally, by the dual polytope of any neighborly polytope with $k$ vertices. This maximum number of vertices is equal to $${k-\lceil n/2 \rceil \choose \lfloor n/2 \rfloor} + {k-\lfloor n/2 \rfloor - 1 \choose \lceil n/2 \rceil - 1}.$$
What you call $n$ in your posting is often called $d$ in the literature, but I will stick with your notation. So $n$ is the dimension. Let $V$ be the number of vertices, and $k$ the number of facets. Then $V = \Theta( k ^ {\lfloor n/2 \rfloor} )$. More precisely, the maximum $V$ is given by McMullen's Upper Bound Theorem, realized by duals of cyclic polytopes. Cyclic polytopes maximize the number of facets for a fixed number of vertices, so their duals maximize the number of vertices for a fixed number of facets. | {"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": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.8607276082038879, "perplexity": 103.92219322038828}, "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/1454701161946.96/warc/CC-MAIN-20160205193921-00231-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.arxiv-vanity.com/papers/math.PR/0112234/ | ###### Abstract
This paper proves that the scaling limit of loop-erased random walk in a simply connected domain is equal to the radial path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that is a simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc , is the chordal path in joining the endpoints of . A by-product of this result is that is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.
\date\title
Conformal invariance of planar loop-erased random walks and uniform spanning trees \authorGregory F. LawlerDuke University and Cornell University; partially supported by the National Science Foundation and the Mittag-Leffler Institute.Duke University and Cornell University; partially supported by the National Science Foundation and the Mittag-Leffler Institute.\andOded SchrammMicrosoft Research.Microsoft Research.\andWendelin WernerUniversité Paris-Sud and IUF.
## 1 Introduction
### 1.1 Motivation from statistical physics
One of the main goals of both probability theory and statistical physics is to understand the asymptotic behavior of random systems when the number of microscopic random inputs goes to infinity. These random inputs can be independent, such as a sequence of independent random variables, or dependent, as in the Ising model. Often, one wishes to understand these systems via some relevant “observables” that can be of geometric or analytic nature. In order to understand this asymptotic behavior, one can attempt to prove convergence towards a suitable continuous model. The simplest and most important example of such random continuous models is Brownian motion, which is the scaling limit of random walks. In particular, simple random walk on any lattice in converges to (a linear image of) Brownian motion in the scaling limit.
Physicists and chemists have observed that critical systems (i.e., systems at their phase transition point) can exhibit macroscopic randomness. Hence, various quantities related to the corresponding lattice models should converge as the mesh refines. In fact, one of the important starting points for theoretical physicists working on two-dimensional critical models is the assumption that the continuous limit is independent of the lattice and, furthermore, displays conformal invariance. This assumption has enabled them to develop and use techniques from conformal field theory to predict exact values of certain critical exponents. Until very recently, the existence of the limit, its conformal invariance, and the derivation of the exponents assuming conformal invariance remained beyond mathematical justification for the basic lattice models in critical phenomena, such as percolation, the Ising model, and random-cluster measures. Although there are many interesting questions about higher dimensional systems, we will limit our discussion to two dimensions where conformal invariance plays an essential role.
### 1.2 Recent progress
In [Sch00], a one-parameter family of random growth processes (loosely speaking, random curves) in two dimensions was introduced. The growth process is based on Loewner’s differential equation where the driving term is time-scaled one-dimensional Brownian motion, and is therefore called stochastic Loewner evolution, or . The parameter of is the time scaling constant for the driving Brownian motion. It was conjectured that the scaling limit of the loop-erased random walk (LERW) is , and this conjecture was proved to be equivalent to the conformal invariance of the LERW scaling limit [Sch00]. The argument given was quite general and shows that a conformally invariant random path satisfying a mild Markovian property, which will be described below, must be . On this basis, it was also conjectured there that the scaling limits of the critical percolation interface and the uniform spanning tree Peano curve are the paths of , and , respectively, and it was claimed that conformal invariance is sufficient to establish these conjectures. (For additional conjectures regarding curves tending to , including the interfaces in critical random cluster models — also called FK percolation models — for , see [RS01].)
At some values of the parameter , has some remarkable properties. For instance, has a locality property [LSW01a] that makes it possible to relate its outer boundary with that of planar Brownian motion. This has led to the proof of conjectures concerning planar Brownian motion and simple random walks [LSW01a, LSW01b, LSW00b].
Smirnov [Smi01a, Smi01b] recently proved the existence and conformal invariance of the scaling limit of critical site percolation on the two-dimensional triangular lattice: he managed to prove Cardy’s formula [Car92] which is a formula for the limit of the probability of a percolation crossing between two arcs on the boundary of the domain. Combining this information with independence properties of percolation, Smirnov then showed that the scaling limit of the percolation interface is . This has led to the rigorous determination of critical exponents for this percolation model [LSW02a, SW01].
### 1.3 LERW and UST defined
The uniform spanning tree (UST), which can be interpreted as the critical random cluster model [Häg95], is a dependent model that has many remarkable features. In particular, it is very closely related to the loop-erased random walk, whose definition [Law80] we now briefly recall.
Consider any finite or recurrent connected graph , a vertex and a set of vertices . Loop-erased random walk (LERW) from to is a random simple curve joining to obtained by erasing the loops in chronological order from a simple random walk started at and stopped upon hitting . In other words, if is a simple random walk on started from and stopped at its first hitting time of , the loop-erasure is defined inductively as follows: ; if , then ; and otherwise where .
A spanning tree of a connected graph is a subgraph of such that for every pair of vertices in there is a unique simple path (that is, self-avoiding) in with these vertices as endpoints. A uniform spanning tree (UST) in a finite, connected graph is a sample from the uniform probability measure on spanning trees of . It has been shown [Pem91] that the law of the self-avoiding path with endpoints and in the UST is the same as that of LERW from to . See Figure 1.1.
David Wilson [Wil96] established an even stronger connection between LERW and UST by giving an algorithm to generate USTs using LERW. Wilson’s algorithm runs as follows. Pick an arbitrary ordering for the vertices in . Let . Inductively, for define to be the union of and a (conditionally independent) LERW path from to . (If , then .) Then, regardless of the chosen order of the vertices, is a UST on .
Wilson’s algorithm gives a natural extension of the definition of UST to infinite recurrent graphs. In fact, for transient graphs, there are two natural definitions which often coincide, but this interesting theory is somewhat removed from the topic of this paper. Many striking properties of UST and LERW have been discovered. See [Lyo98] for a survey of UST’s and [Law99] for a survey of properties of LERW in , .
Exploiting a link with domino tilings and deriving discrete analogs of Cauchy-Riemann equations, Richard Kenyon [Ken00a, Ken00b] rigorously established the values of various critical exponents predicted for the LERW [GB90, Maj92, Dup92] in two dimensions. In particular, he showed that the expected number of steps of a LERW joining two corners of the square in the square grid is of the order of magnitude of . He also showed conformal invariance for the leading term in the asymptotics of the probability that the LERW contains a given edge. This was the first mathematical evidence for full conformal invariance of the LERW scaling limit.
In [AB99, ABNW99] subsequential scaling limits of the UST measures in were shown to exist, using a compactness argument. Moreover, these papers prove that all the paths in the scaling limit that intersect a fixed bounded region are uniformly Hölder continuous. In [Sch00] the topology of subsequential scaling limits of the UST on was determined. In particular, it was shown that every subsequential scaling limit of LERW is a simple path.
### 1.4 A short description of SLE
We now briefly describe ; precise definitions are deferred to Section 2.1. Chordal is a random growing family of compact sets , in the closure of the upper half plane . The evolution of is given by the Loewner differential equation with “driving function” Brownian motion. It is known [RS01] that when the process is described by a random curve , in the sense that for every , is the unbounded component of . A corollary of our results is that this holds for as well. The curve satisfies and . If , then is a simple curve and .
There is another version of called radial . Radial also satisfies the description above, except that the upper half plane is replaced by the unit disk , is on the unit circle and .
Both radial and chordal versions of may be defined in an arbitrary simply connected domain by mapping over to using a fixed conformal map from or to .
### 1.5 The main results of the paper
Let be a simply connected domain with . For , let be the law of the loop-erasure of simple random walk on the grid , started at and stopped when it hits . See Figure 1.2. Let be the law of the image of the radial path under a conformal map from the unit disk to fixing . When the boundary of is very rough, the conformal map from to might not extend continuously to the boundary, but the proof of the following theorem in fact shows that even in this case the image of the path has a unique endpoint on .
On the space of unparameterized paths in , consider the metric , where the infimum is over all choices of parameterizations and in of and .
###### Theorem 1.1 (LERW scaling limit).
The measures converge weakly to as with respect to the metric on the space of curves.
Since is conformally invariant by definition, this theorem implies conformal invariance of the LERW. The theorem and proof apply also to some other walks on lattices in the plane where the scaling limit of the walk is isotropic Brownian motion. It even applies in the non-reversible setting. See Section 6 for further details.
There are two distinct definitions for the UST corresponding to a domain , as follows. Let denote the subgraph of consisting of all the edges and vertices which are contained in . If is connected, then we refer to the UST on as the UST on with free boundary conditions. Let denote the graph obtained from by contracting all the vertices outside of to a single vertex (and removing edges which become loops). Then the UST on is the UST on with wired boundary conditions.
Since the UST is built from LERW via Wilson’s Algorithm, it is not surprising that conformal invariance of the UST scaling limit should follow from that of the LERW scaling limit. In fact, [Sch00, Thm. 11.3] says just that.
###### Corollary 1.2 (UST scaling limit).
The wired and free UST scaling limits (as defined in [Sch00]) in a simply connected domain whose boundary is a smooth simple closed curve exist, and are conformally invariant. ∎
One can easily show, using [Sch00, Thm. 11.1.(i)], that the wired tree depends continuously on the domain, and hence for that case may be an arbitrary simply connected domain. However, some regularity assumption is needed for the free UST scaling limit: conformal invariance fails for the domain whose boundary contains the topologist’s sine curve (the closure of ).
The UST Peano curve is an entirely different curve derived from the UST in two dimensions. The curve is rather remarkable, as it is a natural random path visiting every vertex in an appropriate graph or lattice. We now roughly describe two natural definitions of this curve; further details appear in Section 4.
Let be a finite planar graph, with a particular embedding in the plane, and denote its planar dual, again with a particular embedding. Then there is a bijection between the edges of and those of , such that for every edge in , is a single point, and does not intersect any other edge of . Given a spanning tree of , let denote the graph whose vertices are the vertices of and whose edges are those edges such that . It is then easy to verify that is a spanning tree for . Therefore, if is a UST on , then is a UST on .
The UST Peano curve is a curve that winds between and and separates them. More precisely, consider the graph drawn in the plane by taking the union of and , where each edge or is subdivided into two edges by introducing a vertex at . The subgraph of the planar dual of containing all edges which do not intersect is a simple closed path — the UST Peano path. See Figure 1.3.
Some properties of the UST Peano path on have been studied in the physics literature; e.g., [Kas63, Dup87]. There, it has been called the Hamiltonian path on the Manhattan lattice. The reason for this name is as follows. On , say, orient each horizontal edge whose -coordinate is even to the right and each horizontal edge whose -coordinate is odd to the left. Similarly, orient down each vertical edge whose -coordinate is even, and orient up each vertical edge whose -coordinate is odd. Now rescale the resulting oriented graph by and translate it by . It is easy to check that a Hamiltonian path (a path visiting every vertex exactly once) respecting the orientation on the resulting oriented graph is the same as the UST Peano path of . It should be expected that the uniform measure on Hamiltonian paths in has the same scaling limit as that of the UST Peano path.
Given a domain , one can consider the UST Peano curve for the wired or for the free UST (which is essentially the same as the wired, by duality). However, the conjecture from [Sch00] regarding the convergence to chordal pertains to the UST Peano curve associated with the tree with mixed wired and free conditions.
Let be a domain whose boundary is a -smooth simple closed curve, and let be distinct boundary points. Let and denote the two complementary arcs of whose endpoints are and . For all , consider an approximation of the domain in the grid . (A precise statement of what it means for to be an approximation of will be given in Section 4.) Let denote the Peano curve associated to the UST on with wired boundary near and free boundary near . Then may be considered as a path in from a point near to a point near .
###### Theorem 1.3 (UST Peano path scaling limit).
The UST Peano curve scaling limit in with wired boundary on and free boundary on exists, and is equal to the image of the chordal path under any conformal map from to mapping to and to .
Again, the convergence is weak convergence of measures with respect to the metric . Figure 1.4 shows a sample of the UST Peano path on a fine grid.
As explained above, it was proved in [RS01] that each is generated by a path, except for . In Section 4.4, the remaining case is proved, using the convergence of the Peano curve.
Corollary 1.2 and Theorem 1.3 (and their proofs) apply to other reversible walks on planar lattices (the self-duality of does not play an important role); see Section 6.
To add perspective, we note that the convergence to SLE of the LERW and the UST Peano curve are two boundary cases of the conjectured convergence [RS01] of the critical FK random cluster measures with parameter . For these parameter values, the scaling limit of the interface of a critical cluster with mixed boundary values is conjectured to converge to chordal , where . The boundary case corresponds to the convergence of the UST Peano path to .
The outer boundary of the scaling limit of a macroscopic critical cluster is not the same as the scaling limit of a critical cluster outer boundary, because of “fjords” which are pinched off in the limit. The former is conjectured to “look like” , but a precise form of this conjecture is not yet known. In the case , however, such a correspondence is easy to explain. In , an arc of the Peano curve is surrounded on one side by a simple path in the tree, and on the other side by a simple path in the dual tree. Both these paths are LERW’s. Similar correspondences exist for the UST in a subdomain of , but one has to set appropriate boundary conditions. Thus, the convergence of LERW to also corresponds to the case , as .
Suppose that , and , as before. Consider the simple random walk on which is reflected off and stopped when it hits . Using an analogous method to the one of the present paper, one could handle the scaling limit of the loop-erasure of this walk. It is described by a variant of where the driving term is Brownian motion with time scaled by , but having an additional drift. The drift is not constant, but can be explicitly computed.
The identification of the scaling limit as one of the ’s should facilitate the derivation of critical exponents and also the asymptotic probabilities of various events, including some results which have not been predicted by arguments from physics. This was the case for critical site percolation on the triangular grid [Smi01a, Sch01, LSW02a, SW01].
Since loop-erased random walk is obtained in a deterministic way from simple random walk (by erasing its loops) and since simple random walk converges to Brownian motion in the scaling limit, it is natural to think that the scaling limit of LERW should simply be the process obtained by erasing the loops from a planar Brownian motion. The problem with this approach is that planar Brownian motion has loops at every scale, so that there is no simple algorithm to erase loops. In particular, there is no “first” loop. Our proof does use the relation between LERW and simple random walks, combined with the fact that quantities related to simple random walks, such as hitting probabilities, converge to their continuous conformally invariant counterparts.
The proof of each of our main theorems is naturally divided into two parts. The first part establishes the convergence to with respect to a weaker topology than the topology induced by the metric of paths, namely, we show that the Loewner driving process for the discrete random path converges to a Brownian motion. This part of the proof, which we consider to be the more important one, is essentially self-contained. The second part uses some regularity properties of the discrete processes from [Sch00] to prove convergence with respect to the stronger topology.
The method for the first part can be considered as a rather general method for identifying the scaling limit of a dependent system that is conjectured to be conformally invariant. It requires having some “observable” quantity that can be estimated well and a mild Markovian property, which we now describe. Suppose that to every simply connected domain containing there is associated a random path from to (e.g., the orientation reversal of LERW). The required property is that if is an arc with one endpoint in and we condition on (assuming this has positive probability, say), then the conditioned distribution of is the same as the random path in the domain conditioned to start at the other endpoint of . (Thus, is the state of a Markov chain whose transitions correspond to adding edges from to and modifying appropriately.) Interestingly, among the discrete processes conjectured to converge to , the LERW is the only one where the verification of this property is not completely trivial. (For LERW it is not trivial, but not difficult; see part 3 of Lemma 3.2.) The statement of this property for the UST Peano curve is in Lemma 4.1. The fact that satisfies this property follows from the Markovian property of its driving Brownian motion.
The particular choice of observable is not so important. What is essential is that one can conveniently calculate the asymptotics of the observable for appropriate large-scale configurations. The particular observable that we have chosen for the LERW convergence is the expected number of visits to a vertex by the simple random walk generating the LERW. Conformal invariance is not assumed but comes out of the calculation — hitting probabilities for random walks are discrete harmonic functions, which converge to continuous harmonic functions. One technical issue is to establish this convergence without any boundary smoothness assumption. Once the observable has been approximated, the conditional expectation and variance of increments of the Loewner driving function for the discrete process can be estimated, and standard techniques (the Skorokhod embedding) can be used to show that this random function approaches the appropriate Brownian motion.
Although Theorem 1.3 can probably be derived with some work from Corollary 1.2, instead, to illustrate our method we prove it by applying again the same general strategy of the proof of Theorem 1.1, with the choice of a different observable.
Actually, it is easier to explain the main ideas behind the proof of Theorem 1.3. Fix some vertex in and a subarc . Let be the event that the UST path (not the Peano-path, but the path contained in the UST) from to hits . By Wilson’s algorithm the probability of is the same as the probability that simple random walk started at reflected off first hits in . The latter probability can be estimated directly. If denotes the restriction of the Peano path to its first steps, then , the probability of conditioned on , is clearly a martingale with respect to . But, by the Markovian property discussed above, the value of may be estimated in precisely the same way that is estimated. The estimate turns out to be a function of the conformal geometry of the configuration . Knowing that this is a martingale for two appropriately chosen vertices is sufficient to characterize the large scale behavior of .
As mentioned above, in the case of LERW, the observable we chose to look at is the expected number of visits to a fixed vertex by the simple random walk generating the LERW . The walk can be considered as the union of with a sequence of loops based at vertices of . We look at the conditioned expectation of the number of visits of to given an arc of adjacent to the boundary of the domain. This is clearly a martingale with respect to the filtration obtained by taking larger and larger arcs . This quantity falls into two parts: the visits to in the loops based at , and those that are not. Each of these two parts can be estimated well by random-walk calculations. Translating the fact that this is a martingale to information about the Loewner driving process for inevitably leads to the identification of this driving process as appropriately scaled Brownian motion.
Actually, we first had a longer proof of convergence of LERW to , based on the fact that it is possible to construct the hull of a Brownian motion by adding Brownian loops to . This can be viewed as a particular case of the restriction properties of with Brownian loops added, which we study in the subsequent paper [LSW02b]. Let us also mention the following related open question. Consider a sequence of simple random walks on a lattice with lattice spacing , from to , and let denote the corresponding loop-erased paths. Theorem 1.1 shows that one can find a subsequence such that the law of the pair converges to a coupling of with Brownian motion. (That is, a law for a pair , where has the same distribution as the path and has the same distribution as Brownian motion.) The question is whether in this coupling, the is a deterministic function of the Brownian motion. In other words, is it possible to show that this is not a deterministic procedure to erase loops from a Brownian motion?
## 2 Preliminaries
The reading of this paper requires some background knowledge in several different fields. Some background about Loewner’s equation and is reviewed in the next subsection. It is assumed that the reader is familiar with some of the basic properties of Brownian motion (definition, strong Markov property, etc.). Some of the basic properties of conformal maps (Riemann’s mapping theorem, compactness, Koebe distortion) are also needed for the proof. This material may be learned from the first two chapters of [Pom92], for example. In terms of the theory of conformal mappings, this suffices for understanding the argument showing that the driving process of the LERW converges to Brownian motion. For improving the topology of convergence, some familiarity with the notion of extremal length (a.k.a. extremal distance) is also required. A possible source for that is [Ahl73]. The reader would also need to know some of the very basic properties of harmonic measure.
### 2.1 Loewner’s equation and SLE
We now review some facts concerning Loewner’s equations and stochastic Loewner evolutions. For more details, see e.g., [Sch00, RS01, LSW01a, LSW01b].
Suppose that is a simply connected domain with . Then there is a unique conformal homeomorphism which is onto the unit disk such that and is a positive real. If , then , and is called the capacity of from .
Now suppose that is a continuous simple curve in the unit disk with , and . For each , set , and . Since is increasing (by the Schwarz Lemma, say), one can reparametrize the path in such a way that . If that is the case, we say that is parametrized by capacity from . By standard properties of conformal maps ([Pom92, Proposition 2.5]), for each the limit
W(t):=limz→η(t)gt(z),
where tends to from within exists. One can also verify that
W:[0,∞)→∂U
is continuous. Assuming the parameterization by capacity, Loewner’s theorem states that satisfies the differential equation
∂tgt(z)=−gt(z)gt(z)+W(t)gt(z)−W(t). (2.1)
It is also clear that
∀z∈Ug0(z)=z. (2.2)
We call the driving function of the curve .
The driving function is sufficient to recover the two-dimensional path , because the procedure may be reversed, as follows. Suppose that is continuous. Then for every there is a solution of the ODE (2.1) with initial value up to some time , beyond which the solution does not exist. In fact, if and , then we have , since this is the only possible reason why the ODE cannot be solved beyond time . Then one defines and is the domain of definition of . The set is called the hull at time . If arises from a simple path as described in the previous paragraph, then we can recover from by using . However, if is an arbitrary continuous driving function, then in general need not be a path, and even if it is a path, it does not have to be a simple path.
Radial is the process , where the driving function is set to be , where is Brownian motion. Often, one takes the starting point to be random uniform in . It has been shown [RS01] that the hull is a.s. a simple curve for every if and that a.s. for every is not a simple curve if . For every , there is a.s. some random continuous path such that for all , is the component of containing . When , this was proved in [RS01], while for this will be proven in the current paper. This path is called the radial path.
Suppose that is a simply connected domain containing . If is a continuous simple curve joining to with only an endpoint in , one can reparametrize the path according to capacity and find its driving function , as before. The conformal map
^gt=ψD∖γ[0,t]:D∖γ[0,t]→U
still satisfies (2.1), but this time, . (Here, the parameterization chosen for is according to the capacity of .) Radial in is then simply the image under of radial in the unit disk.
Similarly, one can encode continuous simple curves from to in the closed upper half-plane via a variant of Loewner’s equation. For each time , there is a unique conformal map from onto satisfying the so-called hydrodynamic normalization
limz→∞gt(z)−z=0, (2.3)
where in . If we write near , it turns out that is monotone. Consequently, one can reparametrize in such a way that , that is when . This parameterization of is called the parameterization by capacity from infinity. (This notion of capacity is analogous to the notion of capacity in the radial setting, however, these are two distinct notions and should not be confused.) If is the conformal homeomorphism satisfying the hydrodynamic normalization, then is called the capacity of from . Assuming that is parameterized by capacity, the following analogue of Loewner’s equation holds:
∀t>0 ∀z∈Ht∂tgt(z)=2gt(z)−W(t), (2.4)
where the driving function is again defined by . As above, is determined by .
Conversely, suppose that is a real-valued continuous function. For , one can solve the differential equation (2.4) starting with , up to the first time where and collide (possibly, ). Let the hull be defined by . Then is a conformal map onto , and . In general, is not necessarily a simple curve. If , then is called chordal .
It turns out [LSW01b, §4.1] that the local properties of chordal and of radial are essentially the same. (That is the reason why the normalization was chosen over the seemingly more natural .) In particular, for every chordal is generated by a random continuous path, called the chordal path.
At some points in our proofs, we will need the following simple observation:
###### Lemma 2.1 (Diameter bounds on Kt).
There is a constant such that the following always holds. Let be continuous and let be the corresponding hull for Loewner’s chordal equation (2.4) with driving function . Set
k(t):=√t+max{|W(s)−W(0)|:s∈[0,t]}.
Then
∀t≥0C−1k(t)≤diamKt≤Ck(t).
Similarly, when is the radial hull for a continuous driving function , then
∀t≥0C−1min{k(t),1}≤diamKt≤Ck(t).
Proof. This lemma can be derived by various means. We will only give a detailed argument in the radial case. The chordal case is actually easier and can be derived using the same methods. It can also be seen as a consequence of the result in the radial setting (because chordal Loewner equations can be interpreted as scaling limits of radial Loewner equations).
We start by proving the upper bound on . Let . Then, as long as , we have . Hence, if , then for all , , and therefore . Hence, .
In order to derive the lower bound, we will compare capacity with harmonic measure. It is sufficient to consider the case where . Let denote the harmonic measure on from . Because is contained in the disk of radius with center , there is a universal constant such that . Hence, it suffices to give a lower bound for .
Since is analytic and nonzero in a neighborhood of , the function is harmonic in . Note that . Because as tends to the boundary of , the mean value property of implies the following relation between harmonic measure and capacity: . Since contains points in and , we have for all . Therefore,
It now remains to compare and . We still assume that . Let . If and then (2.1) shows that at . This implies that is non-decreasing. Hence, for all , we have and so that is bounded by the length of , which is equal to . This completes the proof of the lemma. ∎
### 2.2 A discrete harmonic measure estimate
In this section we introduce some notation and state an estimate relating discrete harmonic measure and continuous harmonic measure in domains in the plane. In order to get more quickly to the core of our method in Section 3.2, we postpone the proof of the harmonic measure estimate to Section 5.
A grid domain is a domain whose boundary consists of edges of the grid . For an arbitrary domain , and define the inner radius of with respect to ,
Let denote the set of all simply connected grid domains such that (i.e., and ).
Points in with integer coordinates will be called vertices, or lattice points. Let denote the lattice points in .
Let and a vertex in . If contains more than one edge incident with , then it may happen that the intersection of with a small disk centered at will not be connected. Hence, as viewed from , appears as more than one vertex. In particular, does not extend continuously to . This is a standard issue in conformal mapping theory, which is often resolved by introducing the notion of prime ends. But in the present case, there is a simpler solution which suffices for our purposes. Suppose , and is an edge incident with that intersects . The set of such pairs will be denoted . If is conformal, then will be shorthand for the limit of as along (which always exists, by [Pom92, Proposition 2.14]). Similarly, if a random walk first exists at , we say that it exited at if the edge was used when first hitting . A reader of this paper who chooses to be sloppy and not distinguish between and will not loose anything in the way of substance. We will not always be so careful to make this distinction.
If and , define as the probability that simple random walk started from and stopped at its first exit time of visits .
For any and , we define
λ=λ(w,u;D):=1−|ψ(w)|2|ψ(w)−ψ(u)|2=Re(ψ(u)+ψ(w)ψ(u)−ψ(w)). (2.5)
Note that is also equal to the imaginary part of the image of by the conformal map from onto the upper half-plane that maps onto and to . It is also the limit when of the ratio between the harmonic measure in of the neighborhood of in , taken respectively at and at (that is, it corresponds to the Poisson kernel). Therefore, can be viewed as the continuous analog of . Note that the function is discrete harmonic, on , which means that is equal to the average of on the neighbors of when .
###### Proposition 2.2 (Hitting probability).
For every there is some such that the following holds. Let satisfy , let and . Suppose and . Then
∣∣∣H(w,u)H(0,u)−λ(w,u;D)∣∣∣<ϵ. (2.6)
The proof is given in Section 5.
## 3 Conformal invariance of LERW
### 3.1 Loop-erased random walk background
We now recall some well-known facts concerning loop-erased random walks.
###### Lemma 3.1 (LERW reversal).
Let and let be simple random walk from stopped when it hits . Let be the loop-erasure of , and let be the loop-erasure of the time reversal of . Then has the same distribution as the time-reversal of .
See [Law91]. A simpler proof follows immediately from the symmetry of equation (12.2.3) in [Law99]. This result (and the proofs) also holds if we condition to exit at a prescribed , which correspond to the event (assuming this has positive probability).
Throughout our proof we will use the simple random walk and the loop-erasure of its time-reversal (so that and ). We use to denote the grid domains . Define for ,
nj:=min{n≥0 : Γ(n)=γj},
and note that for , by the definition of . Also set
Γj+1:=Γ[nj+1,nj].
More precisely, consider as the grid-path given by
Γj(m):=Γ(m+nj),m=0,1,…,nj−1−nj.
###### Lemma 3.2 (Markovian property).
Let and let . Suppose that the probability of the event is positive. Conditioned on this event, the following holds.
1. The paths and are conditionally independent.
2. For , the conditional law of is that of a simple random walk in started from and conditioned to leave through the edge .
3. The conditional law of is that of a simple random walk started from conditioned to leave at , and is the loop erasure of the time reversal of . ∎
Proof. Since is the loop-erasure of the reversal of , the event is equivalent to the statement that for each the first hit of to is through the edge . Let , . The strong Markov property of with the stopping times now implies the lemma. ∎
The following simple lemma will also be needed.
###### Lemma 3.3 (Expected visits).
Suppose that and that and are two vertices satisfying . Conditioned on and , the expected number of visits to by is .
Here, denotes the discrete Green’s function; that is, the expected number of visits to by a simple random walk started at , which is stopped on exiting .
Proof. Let be simple random walk from stopped on exiting and let be the last time such that . Then conditioned on and has the same distribution as conditioned on . But the path is independent from . Consequently, the expected number of visits of to conditioned on is equal to the expected number of visits to of . The lemma follows. ∎
### 3.2 The core argument
We keep the previous notation and also use the conformal maps satisfying and . Set and . Note that can also be viewed as a continuously growing simple curve from to , and therefore can be represented by Loewner’s equation. Let denote the (unique) continuous function such that solving the radial Loewner equation with driving function gives the path . Note that , where is the continuous capacity of from in (that is, the capacity of from in ). We denote by the continuous real-valued function with such that . We also define , so that .
###### Proposition 3.4 (The key estimate).
There exists a positive constant such that for all small positive , there exists such that the following holds. Let satisfy . For every with , let denote the random path from to obtained by loop-erasure of the time reversal of a simple random walk from to conditioned to hit in . Let
m:=min{j≥1:tj≥δ2 or |Δj|≥δ},
where and are as described above. Then
∣∣E[Δm]∣∣≤Cδ3, (3.1)
and
(3.2)
Recall that Lemma 3.1 says that has the same distribution as the chronological loop-erasure of random walk from to conditioned to hit at .
Here is a rough sketch of the proof. Let satisfy
Let denote the number of visits to by . (This is the quantity which we referred to in the introduction as the “observable”.) The proof is based on estimating the two sides of the equality
E[h+0]=E[E[h+0∣∣γ[0,m]]]. (3.4)
The estimate for the right-hand side will involve the distribution of and . We get the two relations (3.1) and (3.2) by considering two different choices for such a .
The estimates for the two sides of (3.4) are rather straightforward. Basically, each side is translated into expressions involving the Green’s functions and the hitting probabilities . These are then translated into analytic quantities using (2.6). Earlier versions of the proof required other estimates, somewhat more delicate, in addition to (2.6). Fortunately, it turned out that (2.6) is sufficient. Since we came across several different variants for the proof, based on choosing different observables, it may be said that the proof is inevitable, rather than accidental (and this also applies to Theorem 1.3). Basically, the reason the proof works is that the expected number of visits to in given can be estimated rather well given the rough geometry of in a scale much coarser than the scale of the grid. Similarly, it is important that can be estimated given the rough-geometry of , but this fact is not surprising.
In the following, we abbreviate the Green’s function and hitting probabilities in by and . The following lemma will be needed.
###### Lemma 3.5 (Green’s function bounds).
There is a constant such that for every and satisfying (3.3)
1/C≤GD(0,v)≤C (3.5)
holds. Also, given there is an such that if , then with the notations of Proposition 3.4
G0(0,v)−Gm(0,v)≤Cδ2. (3.6)
Proof of Proposition 3.4. Since it follows from the Koebe 1/4 theorem that if is small. (Apply [Pom92, Cor. 1.4] with to and .) Moreover, the continuous harmonic measure in at of any edge with a vertex on can be made arbitrarily small by requiring to be large. (A Brownian motion started at has probability going to to surround the disk before hitting , as .) By conformal invariance of harmonic measure, this implies that the diameter of can be made arbitrarily small. Applying this to the domains and using Lemma 2.1, we see that we may take large enough so that for all , for all , and . In particular, and . We also require to be larger than the of Lemma 3.5.
Suppose satisfies (3.3). Set and . For each , let denote the number of visits to by . Also let
h+j:=ℓ∑k=j+1hk,
which is the number of visits of by . Let , where is as in (2.5). Since, conditionally on , is a random walk in conditioned to leave at ,
E[h+j∣∣γ[0,j]]=Gj(0,v)Hj(v,γj)Hj(0,γj)
and Proposition 2.2 (together with (3.5)) implies that if is sufficiently large, for every
E[h+j∣∣γ[0,j]]=Gj(0,v)λj+O(δ3),
(This notation is shorthand for the statement that there is an absolute constant such that . We freely use this shorthand below.) In particular
E[m∑j=1hj]=E[h+0−h+m]=E[G0(0,v)λ0−Gm(0,v)λm]+O(δ3). (3.7)
We will now get a different approximation for the left-hand side. Applying Lemma 3.3 to the domain gives
E[hj∣∣γ[0,j]]=Gj−1(γj,v)Hj−1(v,γj).
Proposition 2.2 implies that for large enough,
E[hj∣∣γ[0,j]]=(λj−1+O(δ))Gj−1(γj | {"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": 0.9838330149650574, "perplexity": 311.64170294579503}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057329.74/warc/CC-MAIN-20210922041825-20210922071825-00124.warc.gz"} |
https://betadecay.wordpress.com/2009/10/ | ## Chet Raymo on the Sun
Posted in Science with tags , on October 8, 2009 by Grad Student
What follows is a beautiful Chet Raymo (of “Science Musings”) essay:
Let me say this as simply as I can.
The Sun, like the universe, is mostly hydrogen. An atom of hydrogen is a single proton and a single electron, bound together.
The center of the Sun is so hot — half-a-million miles of overbearing matter — that the electrons and protons can’t hold together (try holding hands in a surging, tumultuous crowd), so what you have is a hot soup — a plasma — of protons and electrons, flailing about independently.
The electrical force between like charges is repulsive, so every time two protons approach each other they swerve away. But protons are also subject to the strong nuclear force, which is attractive, and stronger than the electrical force at very close range. Normally, if two protons approach each other, they are repelled before they get close enough for the strong nuclear force to kick in.
But now heat up the soup. The protons whiz about faster. If they are moving fast enough, they can approach closely enough for the strong nuclear force to bring them crashing together. Fusion. How hot? About 10 million degrees, which is the sort of temperature you’d find at the center of the Sun.
When two protons combine, one flings off its positive charge and becomes a neutron. Then the proton-neutron pair combines with another proton to form a helium-three nucleus — two protons and a neutron — which quickly unites with another of the same and throws off two protons to become a helium-four nucleus — two protons and two neutrons. Got that? Hydrogen is fused into helium.
The ejected positive charges go off as positrons (antielectrons), which meet up with electrons in the soup and annihilate. (There are neutrinos involved too, but let’s ignore them.)
Now for the bookkeeping.
Add up the mass of the original particles in each interaction — six protons and two electrons — and add up the mass of the final particles — a helium-four nucleus and two protons. After the orgy of combination, some mass is missing! For each individual interaction as just described the amount of missing mass is miniscule, but in the seething caldron that is the Sun’s core it amounts to four billion kilograms of vanished mass every second. Hardly missed by our star — like a thimbleful of water dipped from the ocean — but for the Earth it is the difference between day and night. Hydrogen has been turned into helium and the vanished mass appears as energy. A lot of energy. The famous Einstein equation: Energy equals mass times the speed of light squared.
The star shines!
The universe blazes with light and life.
And, knowing this — and just think what a thing it is that we know it — how is it that we whine and carp and glower? How is it that we snipe and cavil and rue our fates and that of the world? Wallace Stevens answers ironically in his poem “Gubbinal,” smothers us in irony actually:
That strange flower, the sun,
Is just what you say.
The world is ugly,
That tuft of jungle feathers,
That animal eye,
Is just what you say.
That savage of fire,
That seed,
The world is ugly,
I love it! It perfectly encapsulates why I think science is beautiful. Though I have to say I don’t quite understand how the poem fits with the rest of the essay.
## Life, the Universe, and Everything: Entropy
Posted in cosmology with tags , on October 5, 2009 by Grad Student
Here’s the abstract of a curious paper by Egan and Lineweaver I saw on astro-ph a while back:
Using recent measurements of the supermassive black hole mass function we find that supermassive black holes are the largest contributor to the entropy of the observable Universe, contributing at least an order of magnitude more entropy than previously estimated. The total entropy of the observable Universe is correspondingly higher, and is $S_{obs} = 3.1^{+3.0}_{-1.7}\xt{104} k$. We calculate the entropy of the current cosmic event horizon to be $S_{CEH} = 2.6 \pm 0.3 \xt{122} k$, dwarfing the entropy of its interior, $S_{CEH int} = 1.2^{+1.1}_{-0.7}\xt{103} k$. We make the first tentative estimate of the entropy of dark matter within the observable Universe, $S_{dm} = 10^{88\pm1} k$. We highlight several caveats pertaining to these estimates and make recommendations for future work.
It is cool to think that black holes are the dominate contributors of entropy in the universe, but why is this important for understanding the universe? Here’s the opening line of the paper:
The entropy budget of the Universe is important because
its increase is associated with all irreversible processes, on
all scales, across all facets of nature: gravitational clustering,
accretion disks, supernovae, stellar fusion, terrestrial
weather, chemical, geological and biological processes
The entropy budget of the Universe is important because its increase is associated with all irreversible processes, on all scales, across all facets of nature: gravitational clustering, accretion disks, supernovae, stellar fusion, terrestrial weather, chemical, geological and biological processes.
Okay, we know this, any other reasons?
That the increase of entropy has not yet been capped by some limiting value, such as the holographic bound (’t Hooft 1993; Susskind 1995) at Smax 10123k (Frampton et al. 2008), is the reason dissipative processes are ongoing and that life can exist.
Mmmm, it appears that that’s it. I can’t quite grasp why this estimate of the universe’s entropy helps us understand the universe better. Perhaps it is important for understanding why the universe started out in such a low entropy state (cosmologists like Sean Carroll and Roger Penrose like to think about these things).
## Research: Numerical Problems
Posted in Science with tags , on October 2, 2009 by Grad Student
The coding I do in my research is relatively simple. I just solve some integrals that can’t be done analytically. Simple, right? You’d think, so, but I’m running into a classic problem, dividing by zero:
There’s bizarre stuff going on in the left side of the graph and there’s two divergent features at x=-0.2 and 0.4. It’s very simple why that’s happening. The denominator is going to zero at those parts. Here’s a graph of the denominator
The problem is, I don’t know how to circumvent this problem. That’s what the equation does, so… what do I do? Clearly I need to do some clever mathematical trick, perhaps express the quantity in another form without the singularity, I don’t know. It’s frustrating though.
UPDATE:
As I look at what the numerator is doing I’m encouraged. Take a look
Notice that the numerator is zero at all the places (and more) where the denominator is zero: x= -1 to -0.8, x=-0.4 and x=0.2. This suggests that I’m not making some obvious mistake in my physics. If I had a fully analytic solution to these integrals then I imagine the divergences would go away, I think. While this observation makes me feel better, I’m still stuck. My best guess is that since both the numerator and denominator go to zero at those points, the overall function should probably go to zero, much like the function:
$f(x)=\frac{(x-1)^2}{(x-1)}$
Both the numerator and denominator go to zero, but if you do you algebra correctly (or just take the limit as x approaches 1) the function goes to zero at the point where the denominator blows up (x = 1)
UPDATE II:
[Embarrassed clearing of throat] I found the solution, it was a missing minus sign.
## The Physics of Scaling Laws and Dimensional Analysis
Posted in Science with tags , , on October 2, 2009 by Grad Student
I’ve been puzzling over dimensional analysis and scaling laws lately. Believe it or not, professional physicists sometimes use dimensional analysis to do a quick back of the envelope calculation and usually get the correct answer to within an order of magnitude or so. In this post I’m going to give three examples of how the “magic” of dimensional analysis can be used to guess the answer. I’ll start with a simple example from introductory physics and move to more advanced examples.
1) If a baseball is dropped from a height ‘h’ above the ground, how long until it hits the ground neglecting air resistance? We know that since air resistance is neglected the size of the baseball is irrelevant, and Galileo tells us the mass of the baseball is also unimportant. That leaves the height from which the ball is dropped, h, which is in units of length [L], and the acceleration due to gravity, g, which is in units of length per time squared [L/T^2]. So, arranging these two quantities to give the dimension of time is simple:
$T=L^a\left(\frac{L}{T^2}\right)^b$
Which gives,
$T=L^{a+b} T^{-2b}$
We need time so the power for length must be zero, so a = -b, and b=-1/2. Thus a = 1/2 and b = -1/2 and voila:
$\Delta t = \sqrt{\frac{h}{g}}$
And this answer is only off by dimensionless factor of ~1.4 (root 2)!
2) As it turns out you can do precisely the same thing to figure out how large craters will be when you drop rocks into the sand (this was the topic of my previous post). The critical assumption you have to make is in deciding which variables are important. Mainly, you have to decide if density is the only important quantity for the sand. (It turns out that for other materials there are other dimensional quantities you must include.) Then you assume the only other important quantity is the kinetic energy of the ball. Finally, you do dimensional analysis exactly as I did above and you will find the same scaling law I’ve found in the last post for relating “meteor” kinetic energy (KE) to crater diameter (D):
$D \propto KE^{1/4}$
Notice how you can get the same answer as I did in the last post without thinking about all the physics and kinematics equations.
3) Now here’s a truly bizarre application of dimensional analysis to the Schrodinger equation that I saw on the first day of a quantum mechanics class. Let’s say that you’re tired of solving partial differential equations in three dimensions and you just want to guess what the ground state energy of an electron is in a hydrogen atom. You bypass the Schrodinger equation in all its glory and simply write the (classical) energy equation for an electron:
$E= \frac{p^2}{2m}-\frac{e^2}{4 \pi \epsilon_0 r}$
Now you think, in quantum mechanics you always have to throw in Planck’s constant, so how do you do that? Well we have momentum and position…mmm…momentum and position, then BAM!, you remember that planck’s contant (h-bar) has units of position times momentum because you remember the Heisenberg uncertainty principle. So let’s just postulate through dimensional analysisthat:
$p r = \hbar$
and substitute that into the energy equation, eliminating position:
$E= \frac{p^2}{2m}-\frac{e^2 p}{4 \pi \epsilon_0 \hbar}$
Now we want to find the lowest energy level, so we do the usual calculus to minimize this function, E(p), and find:
$p_{min}=\frac{e^2 m}{4 \pi \epsilon_0 \hbar}$
Which tells us that:
$E_{min}= -\frac{e^4m}{2(4 \pi \epsilon_0)^2 \hbar^2} \sim -13.6eV$
And out of shear dumb luck, that is not just the approximate answer, it’s the exact answer to the full solution of the Schrodinger equation! Before I celebrate too much, I should admit that I could have used h instead of h-bar (which is just h/2pi). In that case I would have gotten -0.34 eV, which is definitely in the right ball park, but not nearly as impressive as getting the exact answer.
Of course you could just use dimensional on all the relevant constants in the equation. You know that, being quantum mechanics you have include Planck’s constant. Obviously the electric force is involved so you have to include the fundamental unit of charge and the permittivity of free space in the following way:
$\frac{e^2}{4 \pi \epsilon_0}$
And finally you include the mass of the electron. If you have some experience with these types of problems you will know that you don’t have to include the proton mass because it’s so much heavier than the electron. Then you have to solve the following equation for a, b and c.
$ML^2 T^{-2}=\left[\frac{e^2}{4 \pi \epsilon_0}\right]^a \left[m_e\right]^b \left[\hbar\right]^c$
Where the brackets [] mean the units of the constant and the right hand side of the equation are the units of energy. When you do this you will find that a = 4, b = 1, and c = -2 which gives:
$\frac{m_e e^4}{(4 \pi \epsilon_0)^2\hbar^2} \sim 27.2 eV$
This result is only off by a factor of two! Also, you should know that a negative sign belongs there as we’re talking about bound energies. So, if the above were all you knew about the hydrogen atom, you would be able to rightly conclude that all the bound electronic energy levels are between 0 eV (which is to say the electron is free) and negative several tens of eV. And this is totally correct! Then when you have more time you can show that all the bound levels are between 0 eV and -13.6 eV.
I know it sounds crazy, just cobbling together relevant quantites to get important results, but it works. If you understand the physics of the problem well enough, and you can guess what the relevant parameters are, then you’re ready to pull out the envelope and start calculating. | {"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": 13, "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": 0.8233855962753296, "perplexity": 752.8069174815508}, "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/1502886109670.98/warc/CC-MAIN-20170821211752-20170821231752-00431.warc.gz"} |
http://mathhelpforum.com/trigonometry/43470-identities-print.html | # Identities
• Jul 11th 2008, 01:25 AM
gracey
Identities
Prove that identity tan2x - tanx = tanx sec2x
I managed to start i think but am not sure
(2tan x / 1-tan^2 x) - tan x
but i don't know what else
• Jul 11th 2008, 01:46 AM
kalagota
Quote:
Originally Posted by gracey
Prove that identity tan2x - tanx = tanx sec2x
I managed to start i think but am not sure
(2tan x / 1-tan^2 x) - tan x
but i don't know what else
$\tan x \sec 2x = \frac{\tan x}{\cos 2x} = \frac{\tan x}{2\cos^2 x - 1} = \frac{\tan x}{\frac{2}{\sec^2 x}-1}$
$= \frac{\tan x}{\frac{2 - \sec^2 x}{\sec^2 x}} = \frac{\tan x \sec^2 x}{2-\sec^2 x}$
this seems manageable..
EDIT: from what you got,
$\frac{2\tan x}{1-\tan^2 x} - \tan x = \frac{2\tan x - \tan x + \tan^3 x}{1-\tan^2 x}$
• Jul 11th 2008, 02:39 AM
nikhil
Here it is
Quote:
Originally Posted by gracey
Prove that identity tan2x - tanx = tanx sec2x
I managed to start i think but am not sure
(2tan x / 1-tan^2 x) - tan x
but i don't know what else
hi!gracey (sweet name)
(2tan x / 1-tan^2 x) - tan x
now what you should really do is take tanx common
tanx[(2/(1-tan^2x)-1]
now use
tan^2x=(1-cos2x)/1+cos2x
after substituting we get
tanx[2/(1-{(1-cos2x)/1+cos2x})-1]
=tanx[{(1+cos2x)/cos2x}-1]
=tanx[1/cos2x]
=tanxsec2x
hence proved
• Jul 11th 2008, 03:53 AM
Soroban
Hello, gracey!
A variation of nikhil's solution . . .
Quote:
Prove the identity: . $\tan2x - \tan x \:=\: \tan x\sec2x$
I managed to start i think, but am not sure
. . $\frac{2\tan x}{1-\tan^2\!x} - \tan x$
Factor: . $\tan x\left(\frac{2}{1-\tan^2\!x}-1\right) \;=\;\tan x\left(\frac{1+\tan^2\!x}{1-\tan^2\!x}\right)$
. . . . $= \;\tan x\left(\frac{\sec^2\!x}{1-\tan^2\!x}\right) \;=\;\tan x\left(\frac{\frac{1}{\cos^2\!x}}{1 - \frac{\sin^2\!x}{\cos^2\!x}}\right)$
. . . . $= \;\tan x\left(\frac{1}{\cos^2\!x - \sin^2\!x}\right) \;=\;\tan x\left(\frac{1}{\cos2x}\right)$
. . . . $= \;\tan x\sec2x$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9475296139717102, "perplexity": 8125.574489182541}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718034.77/warc/CC-MAIN-20161020183838-00213-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.maplesoft.com/support/help/MapleSim/view.aspx?path=MathExpressionComponent | Math Expression Component - Maple Help
Math Expression Component
Insert a math expression region in a Standard Maple worksheet or document.
Description
• The mathematical expression component displays 2-D math expressions passed from another embedded component. It also provides editing functionality for entering math expressions.
• The math expression component returns the value of its contents as a string of MathML elements. To convert this into a Maple expression, use the MathML[ImportModified] command.
• The mathematical expression component is customized by setting options in the Mathematical Expression context panel. To display the options in the context panel, select the embedded component. Actions are associated with components using routines in the DocumentTools package. For more information, see the Math Expression Component Example on this page.
• The Name property is used to reference the math expression component when using the routines from the DocumentTools package.
Component Palette Image
Mathematical Expression Properties
• The following table describes the control and use of the mathematical expression component options.
An x in the Get column indicates that the option can be read, that is, retrieved by using the DocumentTools[GetProperty] tool.
An x in the Set column indicates that the option can be written, that is, set by using the DocumentTools[SetProperty] tool.
Option Get Set Option Type autoFit x x true or false editable x x true or false expression x x expression fillcolor x x color minimumpixelheight x x positive integer minimumpixelwidth x x positive integer pixelHeight x x positive integer pixelWidth x x positive integer scrolldown x x true or false showBorders x x true or false type x string value x x MathML visible x x true or false
autoFit
Specifies whether to automatically resize the Math Expression component to fit its contents. By default, the value is false.
editable
Specifies whether the mathematical expression component is editable. By default, the value is true.
expression
The contents of the Math Expression component (MathContainer) as a Maple expression. The usual simplification rules are applied to the expression. For example, retrieving the expression from a MathContainer into which 3*4 has been entered will return 12. To obtain the contents exactly as they were entered, use InertForm[FromMathContainer] or the value property.
fillcolor
Specifies the background color of the component. The color can be given by RGB values (a list of three integers between 0 and 255) or a string representing a color name, for example, [255,0,0] or “Red”. The default value is [255,255,255] which corresponds to white.
minimumpixelheight
The minimum height of the mathematical expression region in pixels. Used with autoFit=true. By default, the value is 200.
minimumpixelwidth
The minimum width of the mathematical expression region in pixels. Used with autoFit=true. By default, the value is 300.
pixelHeight
The height of the mathematical expression region in pixels. By default, the value is 200.
pixelWidth
The width of the mathematical expression region in pixels. By default, the value is 300.
scrolldown
Specifies whether to allow automatic scrolling to the bottom of the Math Expression window to make new results visible, when the window size is restricted and the scroll bar is visible.
showBorders
Specifies if the border of the component is visible. By default, the value is true.
type
The type of the component. This returns "MathContainer".
value
The contents in the component in MathML. Note that since the content is in MathML, use of DocumentTools[Do] should either retrieve the value unparsed (i.e. %%ComponentName(value)) or use the expression property.
visible
Specifies if the component is visible. By default, the value is true.
• These properties can be accessed programmatically. You can access the contents of the MathContainer, for example, using expr:= DocumentTools:-GetProperty(ComponentName,expression); as demonstrated in the example.
Making the Expression Read-only
The mathematical expression component can be made read-only (non-editable).
To make the mathematical expression component read-only:
1 Click the component.
2 In the Mathematical Expression context panel, clear the check box Editable.
The expression is no longer editable.
Math Expression Component Example
Note: To interact with the example provided below, open this help page as a worksheet, and then execute the worksheet.
This example uses a List Box component to display some expressions. Selecting an expression from the List Box displays the integrated expression in the Math Expression component.
To insert the label, list box, and mathematical expression components, and then configure the label component:
1 Insert a Label, ListBox and MathExpression component using the Components palette. For information on displaying the Maple palettes, see the Show Palettes help page.
2 Click the Label component to display the context panel.
3 In the Caption field, enter Select a function to integrate.
To add the expressions to the list box:
1 Click the List Box component to display the context panel.
2 Click the Edit button of the Item List field.
3 Double-click the first row and replace ListBox with sin(x)*2.
4 Click Add, double-click the new field, and then replace the default entry with exp(x)^2.
5 Repeat step 4 and add the following items to the list: cos(x)/2 and sin(x)^2-cos(2*x).
6 Click OK to close the List Editor.
To display the integral of the expression selected from the list box:
1 Click the List Box component, and then select Edit Select Code.
This launches the Code Editor dialog.
2 Before the end use; statement in the dialog, enter the following command:
SetProperty("MathContainer0", value, (Int(parse(GetProperty("ListBox0", value)), x) = int(parse(GetProperty("ListBox0", value)), x)));
Note: Ensure the names of the components are correct (that is, that they match the Name fields for their components).
3 Click or from the File menu, select Save Code to accept all changes.
4 From the File menu, select Close Code Editor.
When you select an item from the list box, the math expression component displays the integral for the expression.
> with(DocumentTools):
You can verify the syntax of the embedded component action by executing the command in the worksheet.
> SetProperty("MathContainer0", value, (Int(parse(GetProperty("ListBox0", value)), x) = int(parse(GetProperty("ListBox0", value)), x)));
To display the contents of the Math Expression component, access the expression property or use the MathML[ImportModified] command on the value.
> GetProperty("MathContainer0",expression);
${∫}\frac{{1}}{{2}}{}{\mathrm{cos}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}\frac{{1}}{{2}}{}{\mathrm{sin}}{}\left({x}\right)$ (5.1)
> MathML[ImportModified](GetProperty("MathContainer0",value));
${∫}\left({{\mathrm{sin}}{}\left({x}\right)}^{{2}}{-}{\mathrm{cos}}{}\left({2}{}{x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{-}\frac{{1}}{{2}}{}{\mathrm{sin}}{}\left({x}\right){}{\mathrm{cos}}{}\left({x}\right){+}\frac{{1}}{{2}}{}{x}{-}\frac{{1}}{{2}}{}{\mathrm{sin}}{}\left({2}{}{x}\right)$ (5.2)
> | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "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": 0.8665291666984558, "perplexity": 2513.078465986157}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572021.17/warc/CC-MAIN-20220814083156-20220814113156-00484.warc.gz"} |
https://mathoverflow.net/questions/254325/stabilization-of-semistable-curves-in-a-concrete-case | # Stabilization of semistable curves in a concrete case
Some days ago I learned for the first time about the stabilization of a semistable curve (from Knudsen's article), but I am still quite confused. If $C/k$ is a semistable curve (i.e. we allow rational components with two nodes, but not less) we blow down all the unstable components (in whatever order) and get a stable curve $C'$. This operation may be performed in families: consider $\omega=\omega_{X/S}$ the canonical sheaf of $\pi:X\to S$ (a semistable curve over $S$), then $\pi^*\pi_*\omega$ surjects onto $\omega$ and the map $X\to \mathbb{P}(\pi_*\omega^n)$ where $n=3$ or greater (other references need $n=4$ for some reason).
I am having trouble understanding this map, apart from the case where $S=\operatorname{Spec} k$ in which case I just forget about the sheaves and see it in the intuitive way.
Let us consider the next simplest case: take $S=\operatorname{Spec}k[\epsilon]$ and $X/S$ such that its geometric fiber is $Y\cup Z$ where $Y$ is of genus $1$, $Z$ is of genus $0$ and $Y\cap Z=\{p_1,p_2\}$ such that $$\widehat{\mathcal{O}}_{p_i}\cong k[\epsilon][\![x,y]\!]/(xy-a_i\epsilon), a_i\in k$$
Concrete question: what is $\widehat{\mathcal{O}}_p$, where $p$ is the unique node of the stabilization of $X/S$ (whose inverse image is $Z$)?
My computations tell me that it should $k[\epsilon][\![x,y]\!]/(xy)$, but are not rigorous (I could elaborate on this, but I believe they do not make enough sense), independently from the $a_i$s. Is it right? If yes, I find it difficult to interpret it, can someone explain why every infinitesimal structure is collapsed?
I gave an example just to be specific, but if you have more general suggestions (behaviour at the nodes in more general families, for instance), of course they are very well welcome.
## 1 Answer
Consider the total space $S$ of your family over $k[\epsilon]$. Note that $S$ is a smooth surface, provided $a_i \ne 0$ (it could be singular at the singular points of $C$, but the first order conditions ensure that it is smooth). The normal bundle of $C$ in $S$ is trivial, hence $$O_Z(Z) = O_Z(-Y) = O_Z(-2).$$ Thus $Z$ is a smooth $(-2)$-curve on a smooth surface $S$. Contracting it you get a surface $S'$ with an ordinary double point. The curve $C'$ is a divisor on $S'$ passing through the singularity, hence it has an ordinary double point too. Thus, around this point it looks as $xy = \epsilon^2$. In particular, if you mod out $\epsilon^2$, you get $xy = 0$. | {"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": 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": 0.9344757199287415, "perplexity": 147.64522536054884}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487634616.65/warc/CC-MAIN-20210618013013-20210618043013-00143.warc.gz"} |
https://www.deepdyve.com/lp/ou_press/turbulence-closure-for-mixing-length-theories-GTXDAuD4Yo | # Turbulence closure for mixing length theories
Turbulence closure for mixing length theories Abstract We present an approach to turbulence closure based on mixing length theory with three-dimensional fluctuations against a two-dimensional background. This model is intended to be rapidly computable for implementation in stellar evolution software and to capture a wide range of relevant phenomena with just a single free parameter, namely the mixing length. We incorporate magnetic, rotational, baroclinic, and buoyancy effects exactly within the formalism of linear growth theories with non-linear decay. We treat differential rotation effects perturbatively in the corotating frame using a novel controlled approximation, which matches the time evolution of the reference frame to arbitrary order. We then implement this model in an efficient open source code and discuss the resulting turbulent stresses and transport coefficients. We demonstrate that this model exhibits convective, baroclinic, and shear instabilities as well as the magnetorotational instability. It also exhibits non-linear saturation behaviour, and we use this to extract the asymptotic scaling of various transport coefficients in physically interesting limits. convection, stars: evolution, stars: interiors, stars: rotation 1 INTRODUCTION An understanding of turbulent transport and stresses remains one of the major outstanding problems in the astrophysics of fluids. While many pieces of this puzzle are understood in broad strokes, the nature of this problem is such that the details are almost as important as the big picture. The magnetorotational instability (MRI), for instance, is understood conceptually but making predictions that match observed accretion discs is a persistent problem (Murphy & Pessah 2015). Similarly, the solar differential rotation is understood to arise from turbulent stresses but precisely how this works and in balance with what other forces remains uncertain (Schou et al. 1998). Significant progress has indeed been made with three-dimensional turbulence simulations (for examples, see Lee 2013; McKinney et al. 2014; Salvesen et al. 2016) but these are generally relevant only on short time-scales and in small volumes. Performing so-called global simulations over large times and distances requires a turbulence closure model to substitute for resolution at small scales (Launder & Spalding 1974; Canuto 1994). At the other extreme models of stellar evolution generally assume extremely simple analytical transport coefficients to overcome the tremendous gap between turbulent time-scales of minutes and nuclear time-scales of millions of years (Maeder 1995). A variety of such approaches have been developed. For instance, the mixing length theory of Böhm-Vitense (1958) provided a closure of convection. This was then put on firmer theoretical ground by Gough (1977, 2012) and extended to include additional phenomena (Smolec, Houdek & Gough 2011; Lesaffre et al. 2013). Kichatinov (1986) introduced an entirely different closure formalism, arriving at an expression for the so-called Λ-effect (Kichatinov 1987), and later incorporating it under the α–Λ formalism with Rudiger (Kichatinov & Rudiger 1993). What these formalisms have in common is a minimal set of free parameters: the mixing length formalism has just the mixing length, and the formalism of Kichatinov & Rudiger (1993) has just the anisotropy parameter. Another set of models has arisen, which aims to reproduce higher order moments of the turbulent fields. This increases the number of free parameters and a number of approaches have been developed to deal with this. For instance, Garaud, Gagnier & Verhoeven (2017) and Garaud et al. (2010) fit their free parameters against small-scale simulations, while Canuto (1997) fits his against experimental results. In addition, there are models, such as that of Canuto (1994), which fix at least some free parameters by introducing new assumptions, in this case regarding the various relevant time-scales. Regardless of the details of how they close the equations of turbulent moments, models of this sort generally take the form of physically motivated analytic expressions, which provide ready access to scaling laws. Their free parameters then serve to better their agreement with data, at the cost of being less straightforwardly interpreted and extended. The availability of growing computational resources in recent years has provided a new niche in this landscape in the form of computational closure models. These are models which do not seek analytic solutions but which are none the less distinct from attempts to simulate turbulence in all its detail. Some may introduce new dynamical fields, as in the k − ε model (Launder & Spalding 1974), while others invoke effective theories of small-scale motion (Canuto & Hartke 1986). The latter kind are essentially renormalized theories, which accept the cost of having to numerically accommodate complex behaviour in exchange for more precision over a wider variety of phenomena. Combined with perturbation theory, this approach represents a tunable middle-ground between expensive simulations and simple analytic models, allowing the computational cost to be traded off against fidelity to suit the problem at hand. The model we present here is in this spirit. We construct a mixing-length theory, which incorporates three-dimensional fluctuations against a two-dimensional axisymmetric background. This is done by treating each mode as growing with its linear growth rate before saturating at an amplitude set by the turbulent cascade (Lesaffre et al. 2013). Beyond this, the motion in each mode is taken to be uncorrelated. We treat the geometry of the flow in full generality, allowing for baroclinic effects as well as magnetism and rotational shears. To incorporate differential rotation, we use a time-dependent sheared coordinate system (Balbus & Schaan 2012). In this frame, there is a continual flow of modes across Fourier space, lending a time dependence to growth rates. Corrections to saturation amplitudes owing to this flow are incorporated perturbatively with the time derivatives of the growth rate. In Section 2, we describe our closure framework in more detail, paying particular attention to the choice of mixing length. We then develop a perturbative approach for correcting the saturation amplitude in Section 3. In section 4, we introduce the sheared coordinate system and the linearized equations of motion. Finally, in Section 6, we show results from our theory, including calculations for the solar convection zone and accretion discs. The software implementing our model is open source and available under a GPLv3 license. Details of the implementation are given in Appendix C. Tabulated transport coefficients produced by the code are also available under the same license and both may be found at github.com/adamjermyn/Mixer. 2 CLOSURE FORMALISM Turbulent phenomena generically exhibit a cascade of energy between large and small scales (Zhou, McComb & Vahala 1997; Lohse & Xia 2010). With some notable exceptions (Sukoriansky, Dikovskaya & Galperin 2007), this cascade begins at a large scale L0 set by the overall structure of the fluid flow and ends at an extremely small scale Lν related to the microscopic viscosity. Between these scales, yet far from each of them, lies the so-called inertial range where the fluid flow is scale-free (Kolmogorov 1941b). In this range, all correlations of the turbulent motion obey simple power laws. This statement was originally proved by Kolmogorov (1941b) for isotropic turbulence. It was later found to be a broader consequence of the renormalizability of the Navier–Stokes equation (Yakhot & Orszag 1986; Carati 1990) and consequently holds quite generally. This means that there is a single relevant scale L0 for a given turbulent flow, which fully characterises the turbulence as seen by measurements performed over length scales L ≫ L0. This is the modern interpretation and justification of the original mixing length hypothesis, which asserts that turbulent fluctuations on scales L ≪ L0 are not dynamically coupled to the large-scale (L ≫ L0) flow properties (Böhm-Vitense 1958). The scale-free nature of turbulence in the inertial range means that modes of significantly different wavevectors are uncorrelated. A natural extension of this is to assume that all modes of distinct wavevectors are at least approximately uncorrelated. That is, we assume that \begin{eqnarray} \langle \tilde{\boldsymbol {v}}_{\boldsymbol {k}} \otimes \tilde{\boldsymbol {v}}^*_{\boldsymbol {k}^{\prime }} \rangle &= (2{\pi} )^3 \delta ^3 (\boldsymbol {k}-\boldsymbol {k}^{\prime }) \mathsf {V}_{\boldsymbol {k}}, \end{eqnarray} (1)where v is the velocity, ⊗ denotes the outer product, 〈⋅⋅⋅〉 denotes the time-averaged expectation, $$\tilde{\boldsymbol {v}}_{\boldsymbol {k}}$$ is the amplitude of the Fourier mode with wavevector $$\boldsymbol {k}$$ and $$\mathsf {V}_{\boldsymbol {k}}$$ is the tensor specifying how different components of the same mode are correlated with one another. It is crucial to notice that the quantity $$\mathsf {V}_{\boldsymbol {k}}$$ is also the Reynolds stress of mode $$\boldsymbol {k}$$. This, and several other closely related quantities, are ultimately what we seek. These two-point correlation functions suffice to characterize not only the stresses but also all higher order correlations through Wick’s theorem and perturbation theory (Wick 1950; Isserlis 1918). To determine $$\mathsf {V}_{\boldsymbol {k}}$$, we begin by writing the linearized equations of motion as \begin{eqnarray} \partial _t \boldsymbol {v}(\boldsymbol {r}) = \mathcal {L}\left[\boldsymbol {v}(\boldsymbol {r}), \partial _i \boldsymbol {v}, \partial _i \partial _j \boldsymbol {v},\ldots , \boldsymbol {r}, t\right], \end{eqnarray} (2)where $$\mathcal {L}$$ is a linear operator of its first argument and $$\boldsymbol {v}$$ is the fluctuating part of the velocity field. In principle, we can work with this operator, though the derivatives of the velocity field make it highly inconvenient. Fortunately, at short length scales, the operator $$\mathcal {L}$$ may be treated as translation-invariant and so we may compute a Fourier transform in $$\boldsymbol {r}$$ without coupling different modes. This gives \begin{eqnarray} \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t} = \tilde{\mathcal {L}}\left[\tilde{\boldsymbol {v}}_{\boldsymbol {k}}, \boldsymbol {k}, t\right]. \end{eqnarray} (3)The modes are decoupled in this regime so $$\tilde{\mathcal {L}}$$ can be represented by a matrix $$\mathsf {L}$$, and we write \begin{eqnarray} \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t} = \mathsf {L}(\boldsymbol {k},t) \boldsymbol {v}_{\boldsymbol {k}}. \end{eqnarray} (4) When $$\mathsf {L}$$ is independent of t equation (4) is straightforward to solve and gives us \begin{eqnarray} \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t} = \sum _i v_{0,i} \hat{v}_{\boldsymbol {k},i} e^{\lambda _i t}, \end{eqnarray} (5)where v0, i are the initial mode amplitudes and $$\hat{v}_{\boldsymbol {k},i}$$ and λi are the normalized right eigenvectors and eigenvalues of $$\mathsf {L}$$, respectively. The vectors $$\hat{v}_{\boldsymbol {k},i}$$ then specify the modes of the system at a given wavevector. If the eigenvalues are not precisely degenerate then modes which begin in phase rapidly become uncorrelated and we may extend equation (1) to the modes at each wavevector and write \begin{eqnarray} \langle \tilde{\boldsymbol {v}}_{\boldsymbol {k},i} \otimes \tilde{\boldsymbol {v}}^*_{\boldsymbol {k}^{\prime },j} \rangle &= (2{\pi} )^3 \delta ^3 (\boldsymbol {k}-\boldsymbol {k}^{\prime }) \delta _{ij} \mathsf {V}_{\boldsymbol {k},i}. \end{eqnarray} (6)This result holds even when modes are degenerate. Because the time evolution of the Navier–Stokes equation is deterministic, the expectation 〈⋅⋅⋅〉 represents a sum over initial conditions. In this sum all relative phases between the modes are explored, so even degenerate modes become uncorrelated. Inserting equation (5) into equation (6) and summing over j and integrating over $$\boldsymbol {k}$$ gives us \begin{eqnarray} \mathsf {V}_{\boldsymbol {k},i} &= \hat{v}_{\boldsymbol {k},i} \otimes \hat{v}_{\boldsymbol {k},i} \langle |v_{0,i}|^2 \exp \left[2 t \Re \left[\lambda _i\right]\right] \rangle . \end{eqnarray} (7)Generally, some λi have positive real parts and so in a long-term expectation this exponential diverges. Indeed, it turns out that these growing modes are precisely those which matter! What happens of course is just that these modes eventually reach amplitudes where the linear approximation fails. By assumption, the system is stable over long times relative to the turbulent scale so this must result in these modes saturating. This has been variously described as mode crashing or the action of parasitic modes (Lesaffre, Balbus & Latter 2009; Pessah & Goodman 2009) but, regardless of the mechanism, it simply means that these modes exit the linear regime and find their growth impeded. To complete the closure, we must find the saturation amplitude. Relying again on the scale-free nature of turbulence, we note that this must be a power law in k. That is \begin{eqnarray} \langle \tilde{v}_{\boldsymbol {k}, i}^2 \rangle = \mathrm{Tr}\left[\mathsf {V}_{\boldsymbol {k}, i}\right] = \frac{A}{M} \left(\frac{k_0}{k}\right)^n, \end{eqnarray} (8)where A depends on the large-scale properties of the flow but is independent of $$\boldsymbol {k}$$, M is the number of modes per wavevector and n is the index of the turbulence. Following Kolmogorov (1941a), we choose n = 11/6 in our model. Appendix A contains a detailed discussion of this choice. The wavenumber k0 is just that of the characteristic scale, and is given by \begin{eqnarray} k_0 = \frac{2{\pi} }{L_0}. \end{eqnarray} (9)Replacing the divergent expression in equation (7) with this amplitude, we find \begin{eqnarray} \mathsf {V}_{\boldsymbol {k},i} = \frac{A}{M} \left(\frac{k_0}{k}\right)^n \hat{v}_{\boldsymbol {k},i} \otimes \hat{v}_{\boldsymbol {k},i}. \end{eqnarray} (10) It only remains to determine A. To do this, we note that there is one characteristic length scale L0 and one characteristic time scale, the growth rate ℜ[λi] of the mode. Because A has dimensions of velocity squared, we find \begin{eqnarray} \mathsf {V}_{\boldsymbol {k},i} = \frac{c}{M} L_0^2 \Re \left[\lambda _i\right]^2 \left(\frac{k_0}{k}\right)^n \hat{v}_{\boldsymbol {k},i} \otimes \hat{v}_{\boldsymbol {k},i}, \end{eqnarray} (11)where c is a dimensionless constant of order unity. This constant, known as the mixing length parameter, varies from theory to theory, so for clarity we set c = 1 in this work but this degree of freedom is important to note when comparing between models. In effect, what we have done is incorporate the non-linearity of turbulence by means of the spectrum while using linear growth rates to set the characteristic scale. In practice, the spectrum acts only to provide a convergent measure over modes (see Appendix A for further discussion), and it is the growth rate and the modes themselves that yield the anisotropies and other phenomena of interest. This is closely related to the approaches of Lesaffre et al. (2013) and Canuto & Hartke (1986). This prescription is easily extended in cases where there are additional dynamical fields, such as the turbulent displacement or a fluctuating magnetic field. The additional fields are simply incorporated into the vector describing the state and M is increased accordingly. We can continue to use equation (8) to fix the amplitude of the entire mode against that of the velocity as long as we know the turbulent index n. Up to this point this prescription is mathematically identical to that of Lesaffre et al. (2013), with the exception that we define the mixing wave vector as in equation (9), while they use π/L0 instead. In the next section, we introduce perturbative corrections to this model to capture a wider variety of phenomena. 3 PERTURBATIVE CORRECTIONS Now consider the case where the matrix $$\mathsf {L}$$ is time-dependent. Most of our reasoning about the behaviour of modes from the previous section still holds but, because the eigenvectors are time-dependent, we no longer have a well-defined notion of a mode as a long-running solution to the equations of motion. When the time dependence is periodic Floquet theory applies (Floquet 1883), but in the cases of interest the time dependence is aperiodic. To recover modes when the time evolution matrix itself evolves and does so aperiodically, we begin by expanding as \begin{eqnarray} \mathsf {L}(t) = \mathsf {L}(0) + t \frac{{\rm d}\mathsf {L}}{{\rm d}t} + \frac{1}{2}t^2 \frac{{\rm d}^2\mathsf {L}}{{\rm d}t^2} +\cdots \,. \end{eqnarray} (12)This series can be truncated to produce an approximation of $$\mathsf {L}$$, which is accurate in a certain window around t = 0. We may likewise write the velocity at a given wavevector as \begin{eqnarray} \tilde{\boldsymbol {v}}_{\boldsymbol {k}}(t) = \tilde{\boldsymbol {v}}_{\boldsymbol {k}}(0) + t\left. \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t}\right|_0 + \frac{1}{2}t^2\left. \frac{{\rm d}^2\tilde{\boldsymbol {v}}_{\boldsymbol {k}}(t)}{{\rm d}t^2}\right|_0 +\cdots \,. \end{eqnarray} (13)This suggests defining a new vector \begin{eqnarray} \Phi _{\boldsymbol {k}}(t) \equiv \left\lbrace \tilde{\boldsymbol {v}}_{\boldsymbol {k}}, \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t}, \frac{{\rm d}^2\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t^2},\ldots \right\rbrace , \end{eqnarray} (14)which, in principle, encodes the full time evolution of the velocity field. This vector evolves according to \begin{eqnarray} \frac{{\rm d}\Phi _{\boldsymbol {k}}}{{\rm d}t} = \mathsf {A} \Phi _{\boldsymbol {k}}, \end{eqnarray} (15)where $$\mathsf {A}$$ is formed of blocks given by \begin{eqnarray} \mathsf {A}_{ij} = {i \atopwithdelims ()j}\frac{{\rm d}^{i-j}}{{\rm d}t^{i-j}} \mathsf {L}. \end{eqnarray} (16)By definition though we also have \begin{eqnarray} \frac{{\rm d}\Phi _{\boldsymbol {k},i}}{{\rm d}t} = \Phi _{\boldsymbol {k},i+1}, \end{eqnarray} (17)where $$\Phi _{\boldsymbol {k},0} = \tilde{\boldsymbol {v}}_{\boldsymbol {k}}$$, $$\Phi _{\boldsymbol {k},1} = {\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}/{\rm d}t$$ and so on. Thus, we are searching for a simultaneous solution of equations (15) and (17). In order to close the system, we must truncate it at some finite order N. Doing so makes the assumption that the behaviour of the system at all greater N is known. Inspired by the solution for time-independent $$\mathsf {L}$$, we try an exponential behaviour. This truncates equation (17) such that it applies only to i < N − 1 and means that we are searching for vectors with \begin{eqnarray} \left(A \Phi _{\boldsymbol {k}}\right)_{N-1} = \lambda \Phi _{\boldsymbol {k},N-1} \end{eqnarray} (18)and \begin{eqnarray} \Phi _{\boldsymbol {k},i+1} = (\mathsf {A} \Phi _{\boldsymbol {k}})_i, i < N-1. \end{eqnarray} (19)These equations are most straightforwardly written as a general eigensystem and this has the advantage of restricting the dimension of the linear space to just those states obeying the constraint. This is possible because both $$\mathsf {A}$$ and the constraint are lower triangular in the same basis, and so each row may be substituted into the next, leading to an eigenproblem of the form \begin{eqnarray} \mathsf {Q} \Phi _{\boldsymbol {k,0}} = \lambda \mathsf {W} \Phi _{\boldsymbol {k,0}}, \end{eqnarray} (20)where $$\mathsf {Q}$$ and $$\mathsf {W}$$ are matrices acting only on the 0-block. For example, in the case where N = 2, our equations are \begin{eqnarray} \Phi _{\boldsymbol {k},1} &= \mathsf {M} \Phi _{\boldsymbol {k},0} \end{eqnarray} (21)and \begin{eqnarray} \mathsf {M} \Phi _{\boldsymbol {k},1} + \dot{\mathsf {M}} \Phi _{\boldsymbol {k},0} &= \lambda \Phi _{\boldsymbol {k},1}, \end{eqnarray} (22)which may be put in the form of equation (20) with \begin{eqnarray} \mathsf {Q} &= \mathsf {M}^2 + \dot{\mathsf {M}} \end{eqnarray} (23)and \begin{eqnarray} \mathsf {W} &= \mathsf {M}. \end{eqnarray} (24)The eigenvectors of this system are solutions of the original equation (4) because if $$\psi ^i_{\boldsymbol {k}}$$ is such an eigenvector then \begin{eqnarray} \tilde{\boldsymbol {v}}_{\boldsymbol {k},i}(t) \equiv \sum _{j=0}^{N} \frac{t^j}{j!}\psi ^i_{j} \end{eqnarray} (25)solves \begin{eqnarray} \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k},i}}{{\rm d}t} = \mathsf {L}(t) \tilde{\boldsymbol {v}}_{\boldsymbol {k},i}(t) \end{eqnarray} (26)over the time window for which $$\mathsf {L}$$ is well-approximated at N-th order. As a result, we say that ϕi(t) are the instantaneous modes of the system at N-th order and use them and in equation (11). In place of the eigenvalue, we use the instantaneous growth rate of the velocity, which is given by \begin{eqnarray} g \equiv \frac{1}{2}\frac{{\rm d} v^2}{{\rm d}t} = \frac{\Re \left(\Phi _{\boldsymbol {k},0} \cdot \Phi _{\boldsymbol {k},1}\right)}{|\Phi _{\boldsymbol {k},0}|^2}. \end{eqnarray} (27)This approximation is controlled in the sense that so long as $$\mathsf {L}(t)$$ converges as N grows, so does the inferred velocity history. In this work, we present results with N = 2 so that $$\mathsf {A}$$ involves both $$\mathsf {L}$$ and $$\dot{\mathsf {L}}$$. We leave the exploration of larger N to later work. 4 EQUATIONS OF MOTION We now specialise to the case of an ideal gas obeying the ideal MHD equations. This section largely follows the derivation of Balbus & Schaan (2012) so we present only the pieces necessary to understand later parts of this work as well as the few places where our derivation diverges from theirs. We take the background to be axisymmetric, the fluctuations to be adiabatic and we work in cylindrical coordinates. We neglect both the microscopic viscosity and the microscopic thermal diffusivity because these are both negligible in most circumstances in stellar physics.1 Because our closure model treats turbulent properties as local, we compute all background quantities at a reference point $$\boldsymbol {r}_0$$. Relative to this point, we define the Lagrangian separation $$\boldsymbol {\delta r}$$ and velocity $$\boldsymbol {\delta v}$$ equivalent to $$\boldsymbol {\xi }$$ and $${\rm D}\boldsymbol {\xi }/{\rm D}t$$ of Balbus & Schaan (2012). In addition, we take the Boussinesq approximation that density variations are ignored except in terms involving gravitational acceleration. With the above definitions, the continuity equation may be written as \begin{eqnarray} \nabla \cdot \boldsymbol {\delta r} = 0. \end{eqnarray} (28) In a fixed coordinate system differential rotation is difficult to analyse so we make two reference frame changes. First, we switch from an inertial frame to one rotating at \begin{eqnarray} \Omega _0 \equiv \Omega (\boldsymbol {r}_0). \end{eqnarray} (29)Secondly, we make a formal change of coordinates \begin{eqnarray} \phi \rightarrow \phi - t\delta \boldsymbol {r}\cdot \nabla \Omega , \end{eqnarray} (30)without altering the corresponding unit vectors. Under this last change the gradient transforms as \begin{eqnarray} \nabla &\rightarrow \nabla - t(\nabla \Omega )\partial _{\phi }. \end{eqnarray} (31)Because the operator $$\mathcal {L}$$ is most easily expressed in Fourier space, we define the transformed wavevector as \begin{eqnarray} \boldsymbol {q} \equiv \boldsymbol {k} - t k_\phi R \nabla \Omega . \end{eqnarray} (32)With this the transformed MHD and Navier–Stokes equations may be written as \begin{eqnarray} \boldsymbol {\delta \tilde{B}} = \boldsymbol {B} \cdot \boldsymbol {q} \boldsymbol {\delta \tilde{r}} \end{eqnarray} (33)and \begin{eqnarray} {\partial _t \delta \tilde{\boldsymbol {v}} + 2\boldsymbol {\Omega }\times \delta \tilde{\boldsymbol {v}}+ \hat{R} R \delta \tilde{\boldsymbol {r}}\cdot \nabla \Omega ^2}\nonumber\\ {\quad- \frac{1}{\gamma \rho }\left(\boldsymbol {\delta \tilde{r}}\cdot \nabla \sigma \right)\nabla \cdot \boldsymbol {\Pi } + \frac{i}{\rho }\boldsymbol {q}\cdot \boldsymbol {\delta \tilde{\Pi }}=0,} \end{eqnarray} (34)where σ is the specific entropy and \begin{eqnarray} \boldsymbol {\Pi } \equiv p \mathsf {I} - \frac{1}{\mu _0}\left(\boldsymbol {B}\otimes \boldsymbol {B} - \frac{1}{2}B^2 \mathsf {I}\right) \end{eqnarray} (35)is the pressure tensor with $$\mathsf {I}$$ the identity matrix. All quantities prefixed with δ are fluctuating, a tilde denotes the Fourier transformed function, and all other quantities are background fields evaluated at $$\boldsymbol {r}_0$$. It is straightforward to see that this is the same equation as that derived by Balbus & Schaan (2012) once the appropriate relations for the pressure and magnetic force are substituted. The fluctuation in the pressure tensor may be written as \begin{eqnarray} \boldsymbol {\delta \Pi } = \delta p \mathsf {I} - \frac{1}{\mu _0}\left(\boldsymbol {B}\otimes \boldsymbol {\delta B} + \boldsymbol {\delta B}\otimes \boldsymbol {B} - \mathsf {I} \boldsymbol {B}\cdot \boldsymbol {\delta B} \right), \end{eqnarray} (36)so in Fourier space \begin{eqnarray} \boldsymbol {\delta \tilde{\Pi }} = \delta \tilde{p} \mathsf {I} - \frac{1}{\mu _0}\left(\boldsymbol {B}\otimes \boldsymbol {\delta \tilde{B}} + \boldsymbol {\delta \tilde{B}}\otimes \boldsymbol {B} - \mathsf {I} \boldsymbol {B}\cdot \boldsymbol {\delta \tilde{B}} \right). \end{eqnarray} (37)Combining this with equation (33) and the Boussinesq approximation (see Appendix B), we find \begin{eqnarray} \boldsymbol {q}\cdot \boldsymbol {\delta \tilde{\Pi }} = \boldsymbol {q} \delta p - \frac{i}{\mu _0} (\boldsymbol {B}\cdot \boldsymbol {q})^2 \boldsymbol {\delta \tilde{r}}. \end{eqnarray} (38)Note that as did Balbus & Schaan (2012), we take $$\boldsymbol {B}\cdot \boldsymbol {q}$$ to be constant in time as implied by the Boussinesq and ideal-MHD conditions. We now depart from prior work and use this equation along with equation (34) taking the component perpendicular to $$\boldsymbol {q}$$ to eliminate δp and find \begin{eqnarray} 0 &=&\left(\partial _t \delta \tilde{\boldsymbol {v}} + 2\boldsymbol {\Omega }\times \delta \tilde{\boldsymbol {v}}+ \hat{R} R \delta \tilde{\boldsymbol {r}}\cdot \nabla \Omega ^2\right. \nonumber \\ &&- \left. \frac{1}{\gamma \rho }\left(\boldsymbol {\delta \tilde{r}}\cdot \nabla \sigma \right)\nabla \cdot \boldsymbol {\Pi } + \frac{1}{\mu _0 \rho } (\boldsymbol {B}\cdot \boldsymbol {q})^2 \boldsymbol {\delta \tilde{r}}\right)_{\perp \boldsymbol {q}}, \end{eqnarray} (39)where the notation $$\left(... \right)_{\perp \boldsymbol {q}}$$ denotes the component perpendicular to $$\boldsymbol {q}$$. To construct the matrix version $$\mathsf {L}$$ of these equations, we must choose a coordinate system. Both because of the constraint (28) and because equation (39) is written in the plane perpendicular to $$\boldsymbol {q}$$ we choose the unit vectors \begin{eqnarray} \hat{\boldsymbol {a}} \equiv \frac{\hat{\boldsymbol {q}}\times \hat{\boldsymbol {w}}}{\sqrt{1-(\hat{\boldsymbol {q}}\,\cdot\, \hat{\boldsymbol {w}})^2}} \end{eqnarray} (40)and \begin{eqnarray} \hat{\boldsymbol {b}} &\equiv \hat{\boldsymbol {q}}\times \hat{\boldsymbol {a}}, \end{eqnarray} (41)where $$\hat{w}$$ is any unit vector with $$\hat{\boldsymbol {w}} \cdot \hat{\boldsymbol {q}} \ne 1$$. This choice of basis ensures that our vectors are perpendicular to the wavevector. A choice of particular convenience for $$\hat{w}$$ is \begin{eqnarray} \hat{\boldsymbol {w}} = \frac{\nabla \Omega }{|\nabla \Omega |}. \end{eqnarray} (42)With this choice $$\hat{a}$$ is time-independent, because the component of $$\boldsymbol {q}$$ perpendicular to $$\boldsymbol {w}$$ is time-independent, and so we may write \begin{eqnarray} \boldsymbol {\delta \tilde{r}} = \alpha \hat{\boldsymbol {a}} + \beta \hat{\boldsymbol {b}} \end{eqnarray} (43)and \begin{eqnarray} \boldsymbol {\delta \tilde{v}} = \dot{\alpha } \hat{\boldsymbol {a}} + \dot{\beta } \hat{\boldsymbol {b}} + \beta \partial _t \hat{\boldsymbol {b}}. \end{eqnarray} (44)Note that there is a removable singularity when $$\hat{w} \parallel \hat{q}$$. The matrix $$\mathsf {L}$$ is then given by computing the relation between $$\partial _t \lbrace \alpha , \beta , \dot{\alpha }, \dot{\beta }\rbrace$$ and $$\lbrace \alpha , \beta , \dot{\alpha }, \dot{\beta }\rbrace$$. The result is quite unwieldy so we do not present it here but note that it is fully documented in the software in which we implement these equations. 5 STRESSES AND TRANSPORT The equations of motion contain the position and the velocity, so our expanded vector space is \begin{eqnarray} \Phi = \left\lbrace \delta \boldsymbol {r}, \delta \boldsymbol {v}, \partial _t \delta \boldsymbol {v},\ldots , \right\rbrace . \end{eqnarray} (45)Combining the linearized equations of motion with our closure scheme, we can compute the correlation function \begin{eqnarray} \langle \Phi \otimes \Phi \rangle = \int \frac{{\rm d}^3 \boldsymbol {k}}{(2{\pi} )^3} \sum _i \langle \Phi ^{i}_{\boldsymbol {k}}\otimes \Phi ^{i*}_{\boldsymbol {k}}\rangle , \end{eqnarray} (46)where the index i ranges over eigenvectors. This function contains all of the usual stresses and transport functions. For instance, the Reynolds stress is \begin{eqnarray} R \equiv \langle \delta \boldsymbol {v}\otimes \delta \boldsymbol {v}\rangle = \langle \Phi _1 \otimes \Phi _1 \rangle . \end{eqnarray} (47)Likewise up to a dimensionless constant of order unity the turbulent diffusivity is \begin{eqnarray} d \equiv \langle \delta \boldsymbol {v}\otimes \delta \boldsymbol {r}\rangle = \langle \Phi _1 \otimes \Phi _0 \rangle . \end{eqnarray} (48)and the turbulent viscosity is \begin{eqnarray} Q \equiv \langle \delta \boldsymbol {v}\otimes \delta \boldsymbol {r}\rangle +\langle \delta \boldsymbol {r}\otimes \delta \boldsymbol {v}\rangle = \langle \Phi _1 \otimes \Phi _0 \rangle +\langle \Phi _0 \otimes \Phi _1 \rangle . \end{eqnarray} (49)Similar expressions hold for the dynamo effect, the transport of magnetic fields, and material diffusion. 6 RESULTS In this section, we exhibit a number of results which come from applying our model to a wide variety of astronomically and physically relevant circumstances. We also compare with the results of Lesaffre et al. (2013) and Kichatinov & Rudiger (1993). We modify the former to use the convention in equation (9) to avoid spurious differences in scale. We likewise assume that our L0 is equal to three times the mixing length of Kichatinov & Rudiger (1993), as this is an inherent freedom in the formalism and resolves an otherwise-persistent scale difference between our model and theirs. These models have been well-tested against a variety of data, most notably helioseismic results, and so provide a useful reference for our work. We have also included more direct comparisons but, because direct experiments are extremely difficult to perform under most circumstances relevant to astrophysics, we have instead included comparisons with simulations and observations where available and applicable. Simulations are often the most useful comparison for stellar phenomena, because a variety of processes, including meridional circulation, can mask the effects of turbulent transport (Kitchatinov 2013). In accretion discs, however, there are several observable quantities, which are thought to correlate closely with the underlying turbulence and these provide very helpful constraints (King, Pringle & Livio 2007). These comparisons and calculations are not intended to be a complete collection of the results our model can produce, nor have we exhaustively explored the circumstances and dependencies of each result. Rather, it is our hope to demonstrate that there is a great deal of interesting physics in this model, that our perturbative corrections give rise to realistic results and reproduce many known results, and that there is much to warrant further exploration along these lines. 6.1 Rotating convection We begin with the effect of rotation on convection in the case of a rotating system with radial pressure and entropy gradients. It is useful to start by comparing our results with those from simulations. Fig. 1 shows the ratios $$\sqrt{\langle \delta v_r^2\rangle /\langle \delta v^2\rangle }$$, $$\sqrt{\langle \delta v_\theta ^2\rangle /\langle \delta v^2\rangle }$$ and $$\sqrt{\langle \delta v_\phi ^2\rangle /\langle \delta v^2\rangle }$$ for several rotation rates as a function of latitude. The positive latitudes come from table 2 of Chan (2001), while the negative are from table 2 of Käpylä, Korpi & Tuominen (2004). In order to match the units for the rotation rates, we put everything in terms of the coriolis number \begin{eqnarray} \mathrm{Co} \equiv \frac{\Omega h}{\langle \delta v^2\rangle ^{1/2}}, \end{eqnarray} (50)where, following the convention of Käpylä et al. (2004), 〈δv2〉1/2 was computed for a non-rotating system. Figure 1. View largeDownload slide The ratios $$\sqrt{\langle \delta v_r^2\rangle /\langle \delta v^2\rangle }$$ (blue), $$\sqrt{\langle \delta v_\theta ^2\rangle /\langle \delta v^2\rangle }$$ (red), and $$\sqrt{\langle \delta v_\phi ^2\rangle /\langle \delta v^2\rangle }$$ (purple) are shown for our model (solid) and for simulations by (Käpylä et al. 2004, dots, negative latitude) and (Chan 2001, dots, positive latitude) for a wide range of rotation rates as a function of latitude. The rotation rate is captured by the Coriolis number Co = Ωh/〈δv2〉1/2. Our model general overestimates the anisotropy but captures its variation well. Figure 1. View largeDownload slide The ratios $$\sqrt{\langle \delta v_r^2\rangle /\langle \delta v^2\rangle }$$ (blue), $$\sqrt{\langle \delta v_\theta ^2\rangle /\langle \delta v^2\rangle }$$ (red), and $$\sqrt{\langle \delta v_\phi ^2\rangle /\langle \delta v^2\rangle }$$ (purple) are shown for our model (solid) and for simulations by (Käpylä et al. 2004, dots, negative latitude) and (Chan 2001, dots, positive latitude) for a wide range of rotation rates as a function of latitude. The rotation rate is captured by the Coriolis number Co = Ωh/〈δv2〉1/2. Our model general overestimates the anisotropy but captures its variation well. Our model overestimates the anisotropy of the turbulence but captures its symmetries and trends well. For instance, we find that near the poles and in non-rotating systems the θ and ϕ components of the velocity fluctuations have identical magnitudes, in line with the simulations. We reproduce the trend of decreasing anisotropy towards the equator and decreasing anisotropy with increasing rotation, and, in cases where there are differences between the θ and ϕ velocities, we reproduce both their sign and magnitude. In particular, we find that $$\langle \delta v_r^2\rangle \ge \langle \delta v_\theta ^2 \rangle \ge \langle \delta v_\phi ^2\rangle$$, which is seen in these and other simulations (Rüdiger, Egorov & Ziegler 2005a). Likewise, we find that radial motion makes up a greater fraction of the total velocity near the poles than at the equator, and that as the Coriolis number increases $$\langle \delta v_r^2 - \delta v_\theta ^2 - \delta v_\phi ^2 \rangle \rightarrow 0$$, all of which is in agreement with the predictions of Rüdiger et al. (2005b). Our overestimate of the anisotropy may be due to our model incorporating the large-scale fields on all scales, as noted by Lesaffre et al. (2013). This suggests that a future refinement might be to use estimates of the large-scale modes to compute the environment of those at smaller scales, but we do not treat such complications for now. As a further comparison, we consider the off-diagonal Reynolds stresses of both Chan (2001) and Käpylä et al. (2004). These numbers were extracted from table 3 of the former and also table 3 of the latter and are shown along with our predictions in Fig. 2. In the former, they were straightforward to analyse but in the latter they do not provide a precise test because the simulations included a bulk shear. To correct for this, we used a linear expansion to subtract results across simulations, which were identical in all conditions other than the rotation and thereby determine the effect of the rotation alone. As we will see in Section 6.2 this procedure is problematic because the shear may interact non-linearly with the rotation. Furthermore, because these corrections are of the same order as the terms themselves some care must be taken in interpreting the results. Figure 2. View largeDownload slide The ratios $$\sqrt{\langle \delta v_r \delta v_\theta \rangle /\langle \delta v^2\rangle }$$ (red), $$\sqrt{\langle \delta v_\theta \delta v_\phi \rangle /\langle \delta v^2\rangle }$$ (purple), and $$\sqrt{\langle \delta v_r \delta v_\phi \rangle /\langle \delta v^2\rangle }$$ (blue) are shown from our model (solid) and from simulations by (Käpylä et al. 2004, dots, negative latitude) and (Chan 2001, dots, positive latitude) as a function of latitude. Note that Käpylä et al. (2004) cautions that the moderate rotation simulations had difficulty converging, and these results arise as the difference between two simulations, so it is not clear how significant this test is. Our model generally overestimates these stresses, and suggests a different symmetry for the variation (going as sin θ rather than sin (2θ)). Figure 2. View largeDownload slide The ratios $$\sqrt{\langle \delta v_r \delta v_\theta \rangle /\langle \delta v^2\rangle }$$ (red), $$\sqrt{\langle \delta v_\theta \delta v_\phi \rangle /\langle \delta v^2\rangle }$$ (purple), and $$\sqrt{\langle \delta v_r \delta v_\phi \rangle /\langle \delta v^2\rangle }$$ (blue) are shown from our model (solid) and from simulations by (Käpylä et al. 2004, dots, negative latitude) and (Chan 2001, dots, positive latitude) as a function of latitude. Note that Käpylä et al. (2004) cautions that the moderate rotation simulations had difficulty converging, and these results arise as the difference between two simulations, so it is not clear how significant this test is. Our model generally overestimates these stresses, and suggests a different symmetry for the variation (going as sin θ rather than sin (2θ)). Despite these difficulties some trends are clear and sustained between both sets of data. For instance, in the northern hemisphere (θ > 0), 〈δvrδvθ〉 < 0, while in both hemispheres 〈δvrδvϕ〉 < 0, in keeping with predictions and simulations by Rüdiger et al. (2005b). Likewise, we find that 〈vθvϕ〉 > 0 in the northern hemisphere, in agreement with the findings of Rüdiger et al. (2005a). Once more, however, our model overestimates these anisotropic terms by an amount, which is largely invariant as a function of rotation. This suggests that this overestimate is a systematic offset rather than an error in scaling. We also have some difficulty to reproduce the signs of some of the stresses, particularly in the results of Käpylä et al. (2004), though this could simply be a subtraction difficulty. This is supported by the fact that the simulations themselves do not agree on the signs of these terms and highlights the challenges of making comparisons of terms, which are small in magnitude relative to the scale of the turbulence. To better understand which trends are significant and which are artefacts, we have placed data from comparable rotation rates for the two sets of simulations side-by-side in Fig. 3. The top five panels show the same data as in Fig. 1, while the bottom three show the data from Fig. 2. In general, there is good agreement in the top five panels. The data of Käpylä et al. (2004) gives systematically larger anisotropies and the two sets of simulations occasionally differ on the relative magnitudes of the velocity components (i.e. their ordering), but otherwise the two are in good agreement. By contrast, the bottom three panels paint two very divergent pictures. Neither ordering, trends nor signs are consistent between the two sets of simulations. Only the magnitudes agree in these cases. Thus, the two sets of simulations agree that our model systematically overestimates anisotropies and that, beyond that, our model agrees with them to the extent that they agree with one another. Figure 3. View largeDownload slide The functions shown in Figs 1 and 2 are shown from our model (solid), simulations by (Käpylä et al. 2004, dots, negative latitude) and Chan (2001) (crosses, positive latitude) as a function of latitude. The most comparable pairs of rotation rates were placed side-by-side for each function. A solid black line is shown along the equator where the latitude is zero. There is reasonable agreement on the distribution of velocities in direction but not on the correlations between different velocity directions.. Figure 3. View largeDownload slide The functions shown in Figs 1 and 2 are shown from our model (solid), simulations by (Käpylä et al. 2004, dots, negative latitude) and Chan (2001) (crosses, positive latitude) as a function of latitude. The most comparable pairs of rotation rates were placed side-by-side for each function. A solid black line is shown along the equator where the latitude is zero. There is reasonable agreement on the distribution of velocities in direction but not on the correlations between different velocity directions.. Having compared in detail with these simulations, we now consider predictions which go beyond the domain where simulations are possible. In convection with radial gradients, the leading order effect is to transport heat and material radially. Fig. 4 shows 〈δvrδvr〉 and 〈δvrδrr〉, which are the correlation functions controlling this transport. Figure 4. View largeDownload slide The radial velocity correlation function 〈δvrδvr〉 (red) and the radial diffusivity 〈δvrδrr〉 (blue) are shown in linear scale for Ω < |N| (left-hand panel) and log-log scale for Ω > |N| (right-hand panel). These results are for uniform rotation at a latitude of π/4 with no magnetic field. On this and all subsequent figures $$v_r v_r/L_0^2|N|^2$$ should be read as $$\langle \delta v_r \delta v_r \rangle / L_0^2|N|^2$$ and similarly for other correlations. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, scale, and variation. The bumps in our results reflect parameter values where the numerical integration was more difficult. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 4. View largeDownload slide The radial velocity correlation function 〈δvrδvr〉 (red) and the radial diffusivity 〈δvrδrr〉 (blue) are shown in linear scale for Ω < |N| (left-hand panel) and log-log scale for Ω > |N| (right-hand panel). These results are for uniform rotation at a latitude of π/4 with no magnetic field. On this and all subsequent figures $$v_r v_r/L_0^2|N|^2$$ should be read as $$\langle \delta v_r \delta v_r \rangle / L_0^2|N|^2$$ and similarly for other correlations. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, scale, and variation. The bumps in our results reflect parameter values where the numerical integration was more difficult. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Both correlators vary at second order in Ω in the slow rotation limit as expected (Kitchatinov 2013; Lesaffre et al. 2013). In the rapid rotation limit on the other hand, they exhibit clear Ω−1 scaling, consistent with what is seen in other closure models and in simulations (Garaud et al. 2010). The quenching of turbulence in this limit arises because the Coriolis effect acts as a restoring force, stabilizing modes. The peak of each correlator is of order unity and occurs when Ω = 0. In fact, for the stress, the maximum is 0.254647 while for the diffusivity it is 0.28125, both of which are consistent to this precision with Lesaffre et al. (2013), noting that we used the definition in equation (9) for their mixing length. This is because our model is precisely the same as theirs in this limit. Based on this and the observed scalings a good approximation is \begin{eqnarray} \langle \delta v_r \delta r_r \rangle \approx \langle \delta v_r \delta v_r \rangle \approx \frac{1-(\Omega /|N|)^2}{1 - (\Omega /|N|)^3}. \end{eqnarray} (51) Next, we consider the effect of rotation on the r − θ correlation functions. These functions are responsible for latitudinal transport of heat, mass, and momentum and vanish as a result of spherical symmetry in the non-rotating limit. Fig. 5 shows 〈δvrδvθ〉 and 〈δvrδrθ〉 as a function of the rotation rate. Figure 5. View largeDownload slide The absolute value of the r − θ velocity correlation function 〈δvrδvθ〉 (red) and corresponding diffusivity 〈δvrδrr〉 (blue) are shown in log-log scale against rotation rate. These results are for uniform rotation at a latitude of π/4 with no magnetic field. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, scale, and variation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 5. View largeDownload slide The absolute value of the r − θ velocity correlation function 〈δvrδvθ〉 (red) and corresponding diffusivity 〈δvrδrr〉 (blue) are shown in log-log scale against rotation rate. These results are for uniform rotation at a latitude of π/4 with no magnetic field. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, scale, and variation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. In the slow-rotation regime both quantities scale as Ω2, while in the rapid rotation limit they scale as Ω−1. The peak is of order unity and occurs near Ω = |N|. This gives rise to the approximation \begin{eqnarray} \langle v_r r_\theta \rangle \approx \langle v_r v_\theta \rangle \approx \frac{(\Omega /|N|)^2}{1 + (\Omega /|N|)^3}. \end{eqnarray} (52)These scalings may be interpreted as a competition between symmetry breaking and quenching: the correlation function rises as rotation breaks symmetries but excessive rotation stabilises the system and quenches the turbulent motions. The symmetry is broken quadratically because, at first order, the Coriolis effect only couples radial and azimuthal motions. The properties of turbulence vary with latitude in a rotating system because the rotation axis picks out a preferred direction. Fig. 6 shows the r − r and r − θ stress and diffusivity correlations as a function of latitude. The r − r correlations vary similarly to one another, exhibiting a minimum at the equator and maxima on-axis. On-axis the rotation drops out of the equations and so the on-axis functions are just those for non-rotating convection. The effect of rotation is then largest at the equator, where the convective motion is predominantly perpendicular to the rotation axis. The correlation functions are smallest where the rotation has the largest effect because rotation primarily acts to stabilize modes. Figure 6. View largeDownload slide Various correlation functions are shown as a function of the angle θ from the rotation axis. The functions are the r − r (left-hand panel) and r − θ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for uniform rotation at Ω = 0.2|N| (top panel), Ω = |N| (middle panel), and Ω = 5|N| (bottom panel). Shown in purple (*, dashed) for comparison is the KR result, which agrees in sign and variation but not scale. For slow rotation the scale of the variation is generally smaller than we predict, while for fast rotation the variation is somewhat larger. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 6. View largeDownload slide Various correlation functions are shown as a function of the angle θ from the rotation axis. The functions are the r − r (left-hand panel) and r − θ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for uniform rotation at Ω = 0.2|N| (top panel), Ω = |N| (middle panel), and Ω = 5|N| (bottom panel). Shown in purple (*, dashed) for comparison is the KR result, which agrees in sign and variation but not scale. For slow rotation the scale of the variation is generally smaller than we predict, while for fast rotation the variation is somewhat larger. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. By contrast, the r − θ correlator is largest in magnitude at mid-latitudes, vanishing both on-axis and at the equator. On-axis this correlation function must vanish because the $$\hat{\theta }$$ unit vector is ill-defined. The sign change between the northern and southern hemispheres occurs because $$(\hat{\boldsymbol {r}} \times \boldsymbol {\Omega })_\phi$$ has the same sign everywhere while $$(\hat{\boldsymbol {\theta }} \times \boldsymbol {\Omega })_\phi$$ changes sign between the hemispheres. This also explains the vanishing correlation at the equator. The quantities of particular interest for studying the origins of differential rotation are the radial-azimuthal correlation functions 〈δvrδvϕ〉 and 〈δvrδrϕ〉. The former provides a stress coupling the angular momentum to radial motions known as the Λ-effect, while the latter provides a viscosity coupling radial shears to azimuthal motion and so acts as a proxy for the α-effect (Kichatinov & Rudiger 1993). Fig. 7 shows these quantities as a function of the rotation rate. In the slow-rotation limit, both scale as Ω before peaking near unity and falling off as Ω−2 in the rapid-rotation limit. The linear scaling at slow rotation rates is a consequence of the Coriolis effect directly coupling radial and azimuthal motions. These quantities fall off more rapidly than the others in the case of rapid rotation because it is preferentially the modes that couple strongly to the Coriolis effect, which are stabilized the most. The absolute scale of our Λ-effect is approximately what is seen in simulations, slightly overestimating relative to Käpylä et al. (2004) and similar to other theoretical predictions (Kitchatinov 2013; Gough 2012). Figure 7. View largeDownload slide The absolute value of the r − ϕ velocity correlation function 〈δvrδvϕ〉 (red) and corresponding diffusivity 〈δvrδrϕ〉 (blue) are shown in log-log scale versus rotation rate. These results are for uniform rotation at a latitude of π/4 with no magnetic field. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, variation, and scale up until Ω = |N|, at which point the behaviour differs significantly. Shown in grey (**, dotted) for comparison is 〈δvrδvϕ〉 from that of Lesaffre et al. (2013). This agrees precisely in the Ω → 0 limit and the agreement is good even near Ω ≈ 0.5|N|. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 7. View largeDownload slide The absolute value of the r − ϕ velocity correlation function 〈δvrδvϕ〉 (red) and corresponding diffusivity 〈δvrδrϕ〉 (blue) are shown in log-log scale versus rotation rate. These results are for uniform rotation at a latitude of π/4 with no magnetic field. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, variation, and scale up until Ω = |N|, at which point the behaviour differs significantly. Shown in grey (**, dotted) for comparison is 〈δvrδvϕ〉 from that of Lesaffre et al. (2013). This agrees precisely in the Ω → 0 limit and the agreement is good even near Ω ≈ 0.5|N|. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. 6.2 Differential rotation and convection We now turn to the dependence of convective transport coefficients on differential rotation. We expand our closure model to linear order in the shear and so restrict this analysis to cases where the dimensionless shear |R∇ln Ω| is at most of order unity. Fig. 8 shows the r − θ and r − ϕ velocity and diffusivity correlation functions as a function of differential rotation for a situation where ∇Ω is at an angle of π/4 relative to the pressure gradient. All four functions behave linearly near the origin, with intercept set by the stress and diffusivity in the uniform rotation limit. This is precisely as expected: the intercept is non-zero, giving rise to the Λ-effect, while the slope is non-zero, giving rise to the α-effect (Kichatinov & Rudiger 1993). Note that the favourable comparison of our results with those of Kitchatinov (2013) is helpful because their model was implemented in a two-dimensional solar model, which compared well with helioseismic observations. Figure 8. View largeDownload slide The r − θ (left-hand panel) and r − ϕ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions are shown in log-scale versus the differential rotation. These results are for a convecting region with differential rotation in the cylindrical radial direction, Ω = 0.1|N| and no magnetic field at a latitude of π/4. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2. This disagrees in sign and on the magnitude of the slope but agrees in the sign of the slope. Shown in grey (**, dotted) for comparison is 〈δvrδvϕ〉 of Lesaffre et al. (2013). This generally predicts smaller stresses though with the same sign and slope sign as our model. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 8. View largeDownload slide The r − θ (left-hand panel) and r − ϕ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions are shown in log-scale versus the differential rotation. These results are for a convecting region with differential rotation in the cylindrical radial direction, Ω = 0.1|N| and no magnetic field at a latitude of π/4. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2. This disagrees in sign and on the magnitude of the slope but agrees in the sign of the slope. Shown in grey (**, dotted) for comparison is 〈δvrδvϕ〉 of Lesaffre et al. (2013). This generally predicts smaller stresses though with the same sign and slope sign as our model. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. A key difference between our work and what we compare with in Fig. 8 is that, while we predict the same sign and comparable magnitude for the α-effect in the zero-shear limit, the effect changes sign near |R∇ln Ω| ≈ 0.5, indicating that, at least for this configuration, this is the point at which non-linear effects become important. This does not represent a particularly severe shear and highlights a key point that the correlation functions we find are generally non-linear in all of the small parameters in which one might wish to expand. Our model captures this non-linear behaviour despite being carried out to linear order in |R∇ln Ω|. This is because, in our expansion, the time evolution operator is what is expanded linearly. The resulting eigenvalues and eigenvectors are generally non-linear functions of this operator. This caution aside, there is a significant regime where the α–Λ expansion is valid and, in this regime, key quantities of interest are the derivatives of the various correlation functions with respect to the shear |R∇Ω|. Fig. 9 shows these derivatives as a function of Ω. The r − ϕ stress derivative is constant in Ω. This means that the stress scales as R∇Ω. This is as expected (see, e.g. equation 79 of Lesaffre et al. 2013) and indicates that there is a well-defined effective viscosity transporting angular momentum. This viscosity is given by \begin{eqnarray} \nu _{r\phi } \approx L_0^2 |N|. \end{eqnarray} (53) Figure 9. View largeDownload slide The derivatives of various correlation functions with respect to |R∇Ω| are shown as a function of Ω, with both axes log-scaled. The functions are the r − θ (left-hand panel) and r − ϕ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for a convecting region with differential rotation in the cylindrical radial direction and no magnetic field at a latitude of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 9. View largeDownload slide The derivatives of various correlation functions with respect to |R∇Ω| are shown as a function of Ω, with both axes log-scaled. The functions are the r − θ (left-hand panel) and r − ϕ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for a convecting region with differential rotation in the cylindrical radial direction and no magnetic field at a latitude of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. By contrast, the derivatives of the r − θ correlations as well as the r − ϕ diffusivity all diverge in the limit as Ω → 0. In particular, the r − θ correlations diverge as Ω−1, while the r − ϕ diffusivity diverges as Ω−2. These divergences are signatures of symmetry breaking. They indicate that the direction in which the R∇Ω → 0 limit is approached matters. That is, this limit can be approached by first letting Ω → 0 and then differentiating or by differentiating and then taking Ω → 0 and the divergence we find in the latter approach indicates that the order matters. When Ω = 0 and |R∇Ω| = 0, there is a symmetry between ±θ and between ±ϕ. As a result both the r − θ and r − ϕ terms vanish in this limit. When Ω ≠ 0 these symmetries are broken by the rotation and we know from Figs 5 and 7 that this occurs at first order for r − ϕ and second order for r − θ. In the opposing limit, the situation is different because in the time evolution described by equation (39) $$\mathsf {L}$$ is independent of |R∇Ω| when Ω = 0. There is, however, a dependence on |R∇Ω| through the time-dependence of $$\boldsymbol {q}$$. This breaks the ϕ symmetry because $$\partial _t \boldsymbol {q}$$ is proportional to qϕR∇Ω and hence is sensitive to ϕ. It does not, however, break the θ symmetry, because qϕR∇Ω is symmetric with respect to changing the signs of both θ and $$\boldsymbol {q}$$. It follows then that we should find divergences in the r − θ correlation derivatives owing to the path-dependence of the zero-rotation limit and that we should find the r − ϕ derivatives to be generally well-behaved. The curious divergence is then that in the r − ϕ diffusivity, because this correlation function does not suffer from a symmetry-derived path-dependence. This arises because the differential rotation means that $$\mathsf {L}$$ is time-dependent. This introduces polynomial corrections to the usual exponential growth, as discussed in Section 3. This formalism captures the fact that the differential rotation turns vertical displacement into ϕ displacements, which vary as polynomials in time. There are, therefore, modes with very small radial velocities, which nevertheless have large azimuthal displacements and these dominate the diffusivity derivative. These modes grow proportional to |R∇Ω| and their growth may proceed in the azimuthal direction until bounded by the Coriolis effect at a time Ω−1. As a result these modes contribute to the diffusivity as |R∇ln Ω| and hence lead to a diverging derivative in |R∇Ω| as Ω → 0. 6.3 Differential rotation and stable stratification Stably stratified regions are those with \begin{eqnarray} N^2 > 0, \end{eqnarray} (54)such that buoyancy acts to counter perturbations in the vertical direction. This tends to damp turbulence. In the presence of such damping, there can still be turbulence if there is also a shear. The classic example of this is the Kelvin–Helmholtz phenomenon, which can occur in such a system if the Richardson criterion \begin{eqnarray} \frac{|{\rm d}u/{\rm d}z|^2}{|N|^2} > \frac{1}{4} \end{eqnarray} (55)is satisfied (Zahn 1993). Here, u is the velocity and z is the coordinate parallel to the stratification. Even when this criterion is not satisfied, latitudinal shear can still generate turbulence (Canuto et al. 2008). These motions are suppressed in vertical extent by the stratification and hence are primarily confined to the plane perpendicular to the stratification direction. Fig. 10 shows the dependence on shear strength of all six stress components in a rotating stably stratified zone with latitudinal rotational shear. All six exhibit linear scaling with the shear strength. This is unusual in an otherwise-stable zone because it implies a viscosity which, to leading order, does not depend on the shear. That is, \begin{eqnarray} \nu _{ij} \approx L_0^2 N f_{ij}\left(\frac{\Omega }{|N|}\right), \end{eqnarray} (56)where fij is some function of the angular velocity. Fig. 11 shows the dependence of the stress components on Ω/|N| for fixed |R∇ln Ω| = 0.1. The r − θ and θ − ϕ stresses vary as Ω3 in the slow-rotation regime and as Ω2 for rapid rotation. The other components all scale as Ω2 in both regimes. Thus, for instance, frϕ = Ω/|N| because the viscosity is the derivative of the stress with respect to the shear, and hence \begin{eqnarray} \nu _{r\phi } \approx 10^{-5} L_0^2 \Omega . \end{eqnarray} (57) Figure 10. View largeDownload slide The absolute value of the r − r (red), r − θ (blue) and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data is computed for a point on the equator with radial differential rotation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 10. View largeDownload slide The absolute value of the r − r (red), r − θ (blue) and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data is computed for a point on the equator with radial differential rotation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 11. View largeDownload slide The absolute value of the r − r (red), r − θ (blue), and r − ϕ (purple) velocity correlation functions are shown as a function of Ω/|N| for fixed |R∇ln Ω| = 0.1, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction and no magnetic field. The data is computed for a point on the equator with radial differential rotation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 11. View largeDownload slide The absolute value of the r − r (red), r − θ (blue), and r − ϕ (purple) velocity correlation functions are shown as a function of Ω/|N| for fixed |R∇ln Ω| = 0.1, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction and no magnetic field. The data is computed for a point on the equator with radial differential rotation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. The scaling in equation (57) arises owing to the centrifugal term, which has a destabilising effect when Ω increases with $$\hat{R}$$. When |R∇Ω| = 0 this effect is not present so the system is stable but introducing a small differential rotation produces an acceleration proportional to $$\Omega R \boldsymbol {\delta r}\cdot \nabla \Omega$$ and hence \begin{eqnarray} \partial _t^2 \boldsymbol {\delta r} \approx g^2 \boldsymbol {\delta r} \propto \hat{R}\Omega R \boldsymbol {\delta r}\cdot \nabla \Omega , \end{eqnarray} (58)which means that the stress scales as Ω∇Ω and thence the viscosity scales as Ω. Because the magnitudes of the stresses are always ordered in the same way, the same terms are always the most significant. From largest to smallest, the stresses are r − r, r − ϕ, ϕ − ϕ, θ − θ, θ − ϕ, and r − θ. This group is nearly separated into diagonal stresses, which are larger, and off-diagonal stresses, which are smaller. The exception to this rule is the r − ϕ stress, which is special because it is the term which directly couples to the shear. The ordering of the remaining terms is not surprising because the off-diagonal stresses are typically mediated by a coupling between different directions, whereas the on-diagonal stresses require no such coupling. To better understand the effect of our perturbative corrections, we computed the same results without them. This produced stresses, which were zero to within numerical precision in all cases, indicating that the entire contribution in this case is coming from the perturbation. However with a different angle of differential rotation, we obtained non-zero results. It is instructive then to compare Fig. 12 with Fig. 13. These show the same correlation functions as each other in the same physical scenario, with differential rotation this time at an angle of π/4, but the former uses the first order perturbative expansion while the latter only expands to zeroth order. The difference between the two calculations is striking: many of the correlation functions have fundamentally different scalings when the perturbative corrections are taken into account. In particular, the r − θ and r − r stresses are both quadratic in the shear and the r − ϕ and θ − ϕ stresses both vary as the shear to the 3/2 power, whereas they are all linear in the shear in the expanded calculation. This difference relates in part to the centrifugal term, which couples the displacement to the acceleration. Without expanding the equations of motion, we would have δr ∝ δv, because the mode would need to be an eigenvector of $$\mathsf {M}$$. The modes which couple to the centrifugal term would still grow according to equation (58) but, for most modes, arranging for the displacement to couple to this term requires coupling to the stabilizing buoyant term too. To make this clearer, in Fig. 14, we have computed the growth rate as a function of wave-vector orientation without using the perturbative expansion. There are several rapidly growing regions, oriented at angles of ± π/4 relative to the vertical. These angles represent a compromise between maximizing the magnitude of the centrifugal acceleration and maximizing its projection on to the velocity, both subject to the Boussinesq condition that motion be in the plane perpendicular to $$\boldsymbol {q}$$. Figure 12. View largeDownload slide The absolute value of the r − r (red), r − θ (blue), and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. The correlation functions are evaluated at first order in the perturbative expansion rather than first order. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data are computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 12. View largeDownload slide The absolute value of the r − r (red), r − θ (blue), and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. The correlation functions are evaluated at first order in the perturbative expansion rather than first order. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data are computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 13. View largeDownload slide The absolute value of the r − r (red), r − θ (blue) and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. The correlation functions are evaluated at zeroth order in the perturbative expansion rather than first order. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data are computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 13. View largeDownload slide The absolute value of the r − r (red), r − θ (blue) and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. The correlation functions are evaluated at zeroth order in the perturbative expansion rather than first order. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data are computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 14. View largeDownload slide The square of the growth rate is shown as a function of wave-vector orientation on a logarithmic colour scale. The wave-vector is specified by a magnitude and two angles, θ(q) and ϕ(q), which are spherical angles relative to the $$\hat{z}$$ direction. These rates were computed with a zeroth-order expansion. Regions with squared growth rates below 10−16 are shown in white. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 14. View largeDownload slide The square of the growth rate is shown as a function of wave-vector orientation on a logarithmic colour scale. The wave-vector is specified by a magnitude and two angles, θ(q) and ϕ(q), which are spherical angles relative to the $$\hat{z}$$ direction. These rates were computed with a zeroth-order expansion. Regions with squared growth rates below 10−16 are shown in white. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. By contrast, the growth rates in the expanded system, shown in Fig. 15, are significant over a much wider swath of parameter space. This is because, in the expanded system, the displacement and velocity need not be parallel so the displacement can be chosen to maximize the centrifugal term while the velocity can be chosen to maximize the projection of the acceleration on to the velocity. Figure 15. View largeDownload slide The square of the growth rate is shown as a function of wave-vector orientation on a logarithmic colour scale. The wave-vector is specified by a magnitude and two angles, θ(q) and ϕ(q), which are spherical angles relative to the $$\hat{z}$$ direction. These rates were computed with a first-order expansion. Regions with squared growth rates below 10−16 are shown in white. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 15. View largeDownload slide The square of the growth rate is shown as a function of wave-vector orientation on a logarithmic colour scale. The wave-vector is specified by a magnitude and two angles, θ(q) and ϕ(q), which are spherical angles relative to the $$\hat{z}$$ direction. These rates were computed with a first-order expansion. Regions with squared growth rates below 10−16 are shown in white. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. 6.4 Baroclinic instability The baroclinic instability arises in otherwise stably stratified fluids when the entropy gradient is not parallel to the pressure gradient (Killworth 1980). In fact, this is part of a family of instabilities, which includes the convective instability (Lebovitz 1965). This family provides a continuous connection between the unstable convective and stably stratified limits. To explore, it consider Fig. 16 which shows the variation of r − r and r − θ correlation functions against the angle δ between the entropy gradient and the pressure gradient. The radial correlations peak when the two gradients are aligned. This is the convective limit. These correlations fall to zero in the opposing limit where the two gradients are anti-aligned, which is the stably stratified limit. In between these limits, the behaviour is approximately that of cos 2δ. Figure 16. View largeDownload slide Various correlation functions are shown as a function of the angle δ between the entropy and pressure gradients. The functions are the r − r (left-hand panel) and r − θ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for a non-rotating convective region on the equator and with no magnetic field. Shown in purple (*, dashed) for comparison is the KR result. This agrees in sign, and for r − r agrees in scale, but their r − θ prediction is considerably larger. Notably, this comparison is precisely as cos (δ) (left-hand panel) and sin (δ) (right-hand panel) and crosses zero at non-extremal angles. This is most likely because their theory is not designed for nearly stable regions with extreme baroclinicity. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 16. View largeDownload slide Various correlation functions are shown as a function of the angle δ between the entropy and pressure gradients. The functions are the r − r (left-hand panel) and r − θ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for a non-rotating convective region on the equator and with no magnetic field. Shown in purple (*, dashed) for comparison is the KR result. This agrees in sign, and for r − r agrees in scale, but their r − θ prediction is considerably larger. Notably, this comparison is precisely as cos (δ) (left-hand panel) and sin (δ) (right-hand panel) and crosses zero at non-extremal angles. This is most likely because their theory is not designed for nearly stable regions with extreme baroclinicity. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. By contrast, the r − θ correlations behave approximately as sin δ, and vanishes when δ = 0. This is because both the aligned and the anti-aligned limits are spherically symmetric and so must have this correlation function vanish. Deviations from the convective limit give rise to linear scaling so the convective baroclinic instability transports heat and momentum at first order in the baroclinicity. This is an entirely distinct phenomenon from the thermal wind balance, which is a large-scale effect while this results from integrating out the small-scale turbulent modes. In the stable limit perturbations arise quadratically, a deviation from the behaviour of sin δ. This is because there are no existing turbulent motions to perturb, and so each position and velocity component is linear in δ and gives rise to a quadratic two-point correlation function. 6.5 Stellar magnetism We now turn to the impact of the magnetic field on convective turbulence in stars. Fig. 17 shows 〈δvrδvr〉 in a mildly rotating (Ω = 0.1|N|) convection zone as a function of B for three polarisations; radial ($$\boldsymbol {B} \parallel \hat{\boldsymbol {r}}$$), latitudinal ($$\boldsymbol {B} \parallel \hat{\boldsymbol {\theta }}$$), and longitudinal ($$\boldsymbol {B} \parallel \hat{\boldsymbol {\phi }}$$). As the field increases, the stress falls off. This is because the field quenches the turbulence by providing a stabilizing restoring force, and is in general agreement with Canuto & Hartke (1986). Interestingly, the only significant differences are between the radial and angular field polarisations! The θ and ϕ polarisations show precisely the same behaviour out to very strong fields. This is a result of symmetry, because the radial stress is not sensitive to rotation about the radial direction. The deviation seen with strong fields is a numeric artefact and decreases with increasing integration time. Figure 17. View largeDownload slide The stress 〈δvrδvr〉 is shown as a function of magnetic field strength. The magnetic field is polarized radially (red), longitudinally (purple), and latitudinally (blue). The system is rigidly rotating at Ω = 0.1|N| at a latitude of π/4. All quantities are given in units of the mixing length and Brünt–Väisälä frequency. Figure 17. View largeDownload slide The stress 〈δvrδvr〉 is shown as a function of magnetic field strength. The magnetic field is polarized radially (red), longitudinally (purple), and latitudinally (blue). The system is rigidly rotating at Ω = 0.1|N| at a latitude of π/4. All quantities are given in units of the mixing length and Brünt–Väisälä frequency. By contrast, consider 〈δvrδvϕ〉, shown in Fig. 18. This component, along with the corresponding Maxwell stress, is responsible for transporting angular momentum. Interestingly, it shows differences amongst all polarisations, with the strongest difference between the θ polarization and the others. This is because the stress is mixed between different directions and so is sensitive to all variations in the magnetic field direction. The large difference of the θ polarization relative to the others reflects the fact that motion is damped perpendicular to the magnetic field so the θ polarization damps motion in both directions involved in this component of the stress, whereas the r and ϕ polarizations only damp motion in one of the two directions. Figure 18. View largeDownload slide The stress 〈δvrδvϕ〉 is shown as a function of magnetic field strength. The magnetic field is polarized radially (red), longitudinally (purple) and latitudinally (blue). The system is rigidly rotating at Ω = 0.1|N| at a latitude of π/4. All quantities are given in units of the mixing length and Brünt–Väisälä frequency. Figure 18. View largeDownload slide The stress 〈δvrδvϕ〉 is shown as a function of magnetic field strength. The magnetic field is polarized radially (red), longitudinally (purple) and latitudinally (blue). The system is rigidly rotating at Ω = 0.1|N| at a latitude of π/4. All quantities are given in units of the mixing length and Brünt–Väisälä frequency. 6.6 Magnetorotational instability As a final example, we consider the MRI (Chandrasekhar 1960). This instability arises in magnetized fluids undergoing Keplerian orbital motion. Fig. 19 shows the r − r, θ − ϕ, r − ϕ, and r − θ Reynolds and Maxwell stresses for an accretion disc with a vertical magnetic field. Contrary to predictions (Chandrasekhar 1960) none of the Reynolds stresses vanish in the zero-field limit. This is because the linear system supports short-term growing modes but, while they only grow in the short-time limit, our numerical methods are not sensitive to that effect at this order. In principle, at higher order, this phenomenon should become evident and so this may be interpreted as an artefact associated with our expanding to low order in |R∇Ω| > 1. Despite this, it is likely that other non-magnetic processes can destabilise these modes even in the long term and so we feel it is appropriate to at least consider them (cf. Luschgy & Pagès 2006). The Maxwell stresses by contrast do vanish as B → 0. This is to be expected because they are proportional to B2. Figure 19. View largeDownload slide From the top to bottom panel are the r − ϕ, r − r, r − θ, and θ − ϕ stresses. The Reynolds (velocity) stresses are in red and the Maxwell (magnetic) stresses are in blue. Note that it is the negative r − r Maxwell stress, which is shown to make the comparison with the Reynolds stresses clearer. In all cases, they are shown as a function of B for a Keplerian disc. The magnetic field is taken parallel to $$\hat{z}$$. The system is taken to be stably stratified in the vertical direction with |N| = Ω and hence L0 = h = R. All quantities are given in units of the mixing length and Ω. Figure 19. View largeDownload slide From the top to bottom panel are the r − ϕ, r − r, r − θ, and θ − ϕ stresses. The Reynolds (velocity) stresses are in red and the Maxwell (magnetic) stresses are in blue. Note that it is the negative r − r Maxwell stress, which is shown to make the comparison with the Reynolds stresses clearer. In all cases, they are shown as a function of B for a Keplerian disc. The magnetic field is taken parallel to $$\hat{z}$$. The system is taken to be stably stratified in the vertical direction with |N| = Ω and hence L0 = h = R. All quantities are given in units of the mixing length and Ω. As the magnetic field increases, the r − ϕ and θ − ϕ Reynolds stresses change sign. This indicates the onset of MRI modes, which have the opposite sign to the zero-field correlations. This effect saturates when vA ≈ Ωh, where h is the scale height of the disc. The total r − ϕ stress saturates at roughly 10−2(hΩ)2, which lies between those typically found in simulations and those inferred from observations (Starling et al. 2004; King et al. 2007). Note that at the saturation point the Maxwell and Reynolds stresses are comparable, and beyond this point the Maxwell stress increases while the Reynolds stress falls off. Above the saturation point the Reynolds stresses drop off as the magnetic field quenches the turbulence. This is precisely what is expected for the MRI (Balbus & Hawley 1991). The Maxwell stresses, however, continue to grow, again in line with expectations. Some care is required to interpret these results because they were computed for a fixed field and that field may or may not be stable under the action of the turbulence it generates (Pessah, Chan & Psaltis 2006). Furthermore, there are challenges with the α-disc prescription, which make the specific stress components more difficult to interpret (Pessah, Chan & Psaltis 2008). Nevertheless, it is encouraging that what we see matches well with both observations and simulations. 7 CONCLUSION We have derived a turbulent closure model, which incorporates shear, rotation, and magnetism as well as a full three-dimensional spectrum of fluctuations. We have also presented a new perturbative approach to incorporate time-dependence in the evolution equations. This model, which is implemented in an open source numerical software package, fully reproduces many known phenomena such as the MRI, baroclinic instability, rotational quenching, and more classic shear instabilities. Using this model, we have determined the asymptotic behaviour of a wide variety of correlation functions and transport coefficients under a wide range of circumstances, many of which do not appear in the literature. We have further explored the behaviour of turbulent transport coefficients in intermediate regimes where no single phenomenon dominates, such as in the critical MRI. In these cases, the behaviour is generally complex and does not separate easily into components associated with the different pieces of input physics. The closure formalism developed here fills a new niche in the landscape of solutions to turbulent transport, covering enough phenomena to be useful to understand those operating in stars, planets, and accretion discs, while being rapid enough to be incorporated into stellar evolution codes on nuclear time-scales. In the future, we hope to provide further refinements and comparisons with direct numerical simulations as well as experiments. In addition, it would be interesting to explore the results of this model to higher order in the shear and, even at this order, there are many results which deserve more analysis than we have given here. Acknowledgements ASJ acknowledges financial support from a Marshall Scholarship as well as support from the Institute of Astronomy, École Normale Supérieure (ENS), and Centre for Excellence in Basic Sciences (CEBS) to work at ENS Paris and CEBS in Mumbai. PL acknowledges travel support from the french PNPS (Programme National de Physique Stellaire) and from CEBS. CAT thanks Churchill College for his fellowship. SMC is grateful to the IOA for support and hopsitality and thanks the Cambridge-Hamied exchange program for financial support. The authors also thank Rob Izzard and Science and Technology Facilities Council Grant ST/L003910/1 for CPU cycles, which aided in this work. Footnotes 1 It would not be difficult, however, to incorporate them into this framework at a later date. REFERENCES Ashkenazi S., Steinberg V., 1999, Phys. Rev. Lett. , 83, 4760 CrossRef Search ADS Balbus S. A., Hawley J. F., 1991, ApJ , 376, 214 CrossRef Search ADS Balbus S. 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W., Vahala G., 1997, NASA Contractor Report 201718 APPENDIX A: TURBULENT INDEX The general question of which turbulent index to use and under what circumstances remains open though many specific cases are well understood. In the case of isotropic incompressible turbulence the Kolmogorov index is well-known to be n = 11/6 (Kolmogorov 1941a). There is more debate over the index to use for convection, with answers ranging from n = 5/2 (Benzi et al. 1994) to n = 21/10 (Procaccia & Zeitak 1989) and n = 2.4 ± 0.2 (Ashkenazi & Steinberg 1999). There has also been work attempting to determine the spectrum in a context-sensitive manner through energy balance arguments (Yakhot & Orszag 1986). In the magnetised case sources differ even more, with some suggesting that this range still applies (Dobrowolny, Mangeney & Veltri 1980), some arguing for a Kolmogorov-like spectrum (Goldreich & Sridhar 1995) and others giving a range of indices depending on geometry and the direction of the wavevector (Sridhar & Goldreich 1994). From numerical experiments with our closure model, we have found that the magnetic stress scales sufficiently rapidly with k that it is divergent for n = 11/6 and not for n = 8/3. This favours the scenario of Goldreich & Sridhar (1995), who argue that in the strongly magnetized limit the index ought to be n = 8/3. In order to consistently treat both the non-magnetic and the strongly magnetized limits, we choose a simple prescription in which n = 11/6 when one of |N|, or |R∇Ω| exceeds kvA and use n = 8/3 otherwise. This means that there is a critical wavenumber \begin{eqnarray} k_{\rm c} \equiv \frac{\max \left(|N|, |R\nabla \Omega |\right)}{v_A} \end{eqnarray} (A1)at which the spectrum changes. In the non-magnetic case, the evolution matrix is independent of the magnitude of the wavevector and so altering the index just alters the correlation coefficients by a multiplicative factor. In the magnetic case, the potential for error is larger because the magnitude of the wavevector is relevant but there appears to be no consensus on the best prescription and so we make do with what is available. APPENDIX B: BOUSSINESQ ODDITIES In this work, we have taken the Boussinesq approximation. In Fourier space this is \begin{eqnarray} \boldsymbol {q}\cdot \tilde{\delta \boldsymbol {r}}=0. \end{eqnarray} (B1)Taking the time derivative of both sides we see that \begin{eqnarray} \partial _t\left(\boldsymbol {q}\cdot \delta \tilde{\boldsymbol {r}}\right) = \boldsymbol {q}\cdot \delta \tilde{\boldsymbol {v}} + \delta \tilde{\boldsymbol {r}}\cdot \partial _t\boldsymbol {q} = 0. \end{eqnarray} (B2)As a result \begin{eqnarray} \delta \tilde{\boldsymbol {v}}\cdot \boldsymbol {q} = - \delta \tilde{\boldsymbol {r}}\cdot \partial _t\boldsymbol {q} \ne 0. \end{eqnarray} (B3)This is quite peculiar, but is just an artefact of our coordinate system. Because the wavevectors are time-dependent, maintaining the volume of a fluid parcel requires that the displacement be orthogonal to the wavevector, which actually means that the velocity is generally not orthogonal to the wavevector. APPENDIX C: SOFTWARE DETAILS The software used for this work is Mixer version 1, which we have released under a GPLv3 license at github.com/adamjermyn/Mixer. All data produced for this work are available at the same location as HDF5 tables with attributes documenting the physical inputs. Post-processing and visualization of the data was with the Python modules Numpy (van der Walt, Colbert & Varoquaux 2011) and Matplotlib (Hunter 2007) and the relevant scripts for this are included with Mixer. The core of Mixer is written in C++, for performance reasons, and the code is supplied with a Makefile, which supports compilation on both Linux and MacOS. Mixer makes use of the Eigen library (Guennebaud et al. 2010) for linear algebra. Mixer also uses the Cubature library for numerical integration. This library is an implementation of the algorithms by Genz & Malik (1980) and Berntsen, Espelid & Genz (1991). These integration routines are supplemented by a Python integration routine tailored for integrands with small support regions. The details will be explored in later work. In addition, many routines provide a Python interface. Currently Mixer supports only single-threaded operation, though it may be used inside parallelized scripts through the Python wrapper. The version of Mixer used to generate the data in this work was compiled against Cubature version 1.0.2 and Eigen version 3.3.3, though the code does not use any features which require recent versions, so many likely suffice. Mixer is optimized for convecting systems for which achieving accuracy better than 10−5 relative and absolute typically requires between 1 ms and 1 s on a single core of a 2016 Intel CPU. This is further improved when the differential rotation is minimal, in which case the perturbative expansion may be turned off to save a factor of several in runtime. In stably stratified zones and those with magnetic fields up to 103s may be required to achieve good convergence. In cases where the code has more difficulty, it is quite likely that Mixer becomes the bottleneck in simulations and so, under these circumstances, we recommend tabulating results in advance. This is still considerably more performant than direct numerical simulation, and the results can generally be guaranteed to converge at much higher precision, so that derivatives may be extracted as well. At various points in the software, we must divide by the magnitude of the velocity of an eigenmode. This may approach zero in some cases. To avoid dividing by zero in these cases, we place a lower bound on this magnitude, such that \begin{eqnarray} |\delta v|^2 \ge \epsilon , \end{eqnarray} (C1)where $$\epsilon = 10^{-20} L_0^2 |N|^2$$ in the calculations presented in this work. This corresponds to setting an upper bound on the length scale d of the displacements $$\delta \boldsymbol {r}$$, namely \begin{eqnarray} |\delta r|^2 \le L_0^3 |N| \epsilon ^{-1/2}, \end{eqnarray} (C2)which means that d = 1010L0 in this work. To verify that this numerical fix does not impact our results, we have examined the correlation functions in several scenarios as a function of this numerical cut off L. For example, Fig. C1 shows the r − θ and r − ϕ correlations as functions of d for a stably stratified differentially rotating system. The results are constant over many orders of magnitude so long as d > 104L0, which is easily satisfied by our default. Figure C1. View largeDownload slide The absolute values of 〈δvrδvθ〉 (left-hand panel) and 〈δvrδvϕ〉 (right-hand panel) are shown as functions of d, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction with |R∇ln Ω| = 10−3, Ω = 0.1|N| and no magnetic field. The data is computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure C1. View largeDownload slide The absolute values of 〈δvrδvθ〉 (left-hand panel) and 〈δvrδvϕ〉 (right-hand panel) are shown as functions of d, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction with |R∇ln Ω| = 10−3, Ω = 0.1|N| and no magnetic field. The data is computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monthly Notices of the Royal Astronomical Society Oxford University Press
# Turbulence closure for mixing length theories
, Volume 476 (1) – May 1, 2018
17 pages
/lp/ou_press/turbulence-closure-for-mixing-length-theories-GTXDAuD4Yo
Publisher
The Royal Astronomical Society
ISSN
0035-8711
eISSN
1365-2966
D.O.I.
10.1093/mnras/sty255
Publisher site
See Article on Publisher Site
### Abstract
Abstract We present an approach to turbulence closure based on mixing length theory with three-dimensional fluctuations against a two-dimensional background. This model is intended to be rapidly computable for implementation in stellar evolution software and to capture a wide range of relevant phenomena with just a single free parameter, namely the mixing length. We incorporate magnetic, rotational, baroclinic, and buoyancy effects exactly within the formalism of linear growth theories with non-linear decay. We treat differential rotation effects perturbatively in the corotating frame using a novel controlled approximation, which matches the time evolution of the reference frame to arbitrary order. We then implement this model in an efficient open source code and discuss the resulting turbulent stresses and transport coefficients. We demonstrate that this model exhibits convective, baroclinic, and shear instabilities as well as the magnetorotational instability. It also exhibits non-linear saturation behaviour, and we use this to extract the asymptotic scaling of various transport coefficients in physically interesting limits. convection, stars: evolution, stars: interiors, stars: rotation 1 INTRODUCTION An understanding of turbulent transport and stresses remains one of the major outstanding problems in the astrophysics of fluids. While many pieces of this puzzle are understood in broad strokes, the nature of this problem is such that the details are almost as important as the big picture. The magnetorotational instability (MRI), for instance, is understood conceptually but making predictions that match observed accretion discs is a persistent problem (Murphy & Pessah 2015). Similarly, the solar differential rotation is understood to arise from turbulent stresses but precisely how this works and in balance with what other forces remains uncertain (Schou et al. 1998). Significant progress has indeed been made with three-dimensional turbulence simulations (for examples, see Lee 2013; McKinney et al. 2014; Salvesen et al. 2016) but these are generally relevant only on short time-scales and in small volumes. Performing so-called global simulations over large times and distances requires a turbulence closure model to substitute for resolution at small scales (Launder & Spalding 1974; Canuto 1994). At the other extreme models of stellar evolution generally assume extremely simple analytical transport coefficients to overcome the tremendous gap between turbulent time-scales of minutes and nuclear time-scales of millions of years (Maeder 1995). A variety of such approaches have been developed. For instance, the mixing length theory of Böhm-Vitense (1958) provided a closure of convection. This was then put on firmer theoretical ground by Gough (1977, 2012) and extended to include additional phenomena (Smolec, Houdek & Gough 2011; Lesaffre et al. 2013). Kichatinov (1986) introduced an entirely different closure formalism, arriving at an expression for the so-called Λ-effect (Kichatinov 1987), and later incorporating it under the α–Λ formalism with Rudiger (Kichatinov & Rudiger 1993). What these formalisms have in common is a minimal set of free parameters: the mixing length formalism has just the mixing length, and the formalism of Kichatinov & Rudiger (1993) has just the anisotropy parameter. Another set of models has arisen, which aims to reproduce higher order moments of the turbulent fields. This increases the number of free parameters and a number of approaches have been developed to deal with this. For instance, Garaud, Gagnier & Verhoeven (2017) and Garaud et al. (2010) fit their free parameters against small-scale simulations, while Canuto (1997) fits his against experimental results. In addition, there are models, such as that of Canuto (1994), which fix at least some free parameters by introducing new assumptions, in this case regarding the various relevant time-scales. Regardless of the details of how they close the equations of turbulent moments, models of this sort generally take the form of physically motivated analytic expressions, which provide ready access to scaling laws. Their free parameters then serve to better their agreement with data, at the cost of being less straightforwardly interpreted and extended. The availability of growing computational resources in recent years has provided a new niche in this landscape in the form of computational closure models. These are models which do not seek analytic solutions but which are none the less distinct from attempts to simulate turbulence in all its detail. Some may introduce new dynamical fields, as in the k − ε model (Launder & Spalding 1974), while others invoke effective theories of small-scale motion (Canuto & Hartke 1986). The latter kind are essentially renormalized theories, which accept the cost of having to numerically accommodate complex behaviour in exchange for more precision over a wider variety of phenomena. Combined with perturbation theory, this approach represents a tunable middle-ground between expensive simulations and simple analytic models, allowing the computational cost to be traded off against fidelity to suit the problem at hand. The model we present here is in this spirit. We construct a mixing-length theory, which incorporates three-dimensional fluctuations against a two-dimensional axisymmetric background. This is done by treating each mode as growing with its linear growth rate before saturating at an amplitude set by the turbulent cascade (Lesaffre et al. 2013). Beyond this, the motion in each mode is taken to be uncorrelated. We treat the geometry of the flow in full generality, allowing for baroclinic effects as well as magnetism and rotational shears. To incorporate differential rotation, we use a time-dependent sheared coordinate system (Balbus & Schaan 2012). In this frame, there is a continual flow of modes across Fourier space, lending a time dependence to growth rates. Corrections to saturation amplitudes owing to this flow are incorporated perturbatively with the time derivatives of the growth rate. In Section 2, we describe our closure framework in more detail, paying particular attention to the choice of mixing length. We then develop a perturbative approach for correcting the saturation amplitude in Section 3. In section 4, we introduce the sheared coordinate system and the linearized equations of motion. Finally, in Section 6, we show results from our theory, including calculations for the solar convection zone and accretion discs. The software implementing our model is open source and available under a GPLv3 license. Details of the implementation are given in Appendix C. Tabulated transport coefficients produced by the code are also available under the same license and both may be found at github.com/adamjermyn/Mixer. 2 CLOSURE FORMALISM Turbulent phenomena generically exhibit a cascade of energy between large and small scales (Zhou, McComb & Vahala 1997; Lohse & Xia 2010). With some notable exceptions (Sukoriansky, Dikovskaya & Galperin 2007), this cascade begins at a large scale L0 set by the overall structure of the fluid flow and ends at an extremely small scale Lν related to the microscopic viscosity. Between these scales, yet far from each of them, lies the so-called inertial range where the fluid flow is scale-free (Kolmogorov 1941b). In this range, all correlations of the turbulent motion obey simple power laws. This statement was originally proved by Kolmogorov (1941b) for isotropic turbulence. It was later found to be a broader consequence of the renormalizability of the Navier–Stokes equation (Yakhot & Orszag 1986; Carati 1990) and consequently holds quite generally. This means that there is a single relevant scale L0 for a given turbulent flow, which fully characterises the turbulence as seen by measurements performed over length scales L ≫ L0. This is the modern interpretation and justification of the original mixing length hypothesis, which asserts that turbulent fluctuations on scales L ≪ L0 are not dynamically coupled to the large-scale (L ≫ L0) flow properties (Böhm-Vitense 1958). The scale-free nature of turbulence in the inertial range means that modes of significantly different wavevectors are uncorrelated. A natural extension of this is to assume that all modes of distinct wavevectors are at least approximately uncorrelated. That is, we assume that \begin{eqnarray} \langle \tilde{\boldsymbol {v}}_{\boldsymbol {k}} \otimes \tilde{\boldsymbol {v}}^*_{\boldsymbol {k}^{\prime }} \rangle &= (2{\pi} )^3 \delta ^3 (\boldsymbol {k}-\boldsymbol {k}^{\prime }) \mathsf {V}_{\boldsymbol {k}}, \end{eqnarray} (1)where v is the velocity, ⊗ denotes the outer product, 〈⋅⋅⋅〉 denotes the time-averaged expectation, $$\tilde{\boldsymbol {v}}_{\boldsymbol {k}}$$ is the amplitude of the Fourier mode with wavevector $$\boldsymbol {k}$$ and $$\mathsf {V}_{\boldsymbol {k}}$$ is the tensor specifying how different components of the same mode are correlated with one another. It is crucial to notice that the quantity $$\mathsf {V}_{\boldsymbol {k}}$$ is also the Reynolds stress of mode $$\boldsymbol {k}$$. This, and several other closely related quantities, are ultimately what we seek. These two-point correlation functions suffice to characterize not only the stresses but also all higher order correlations through Wick’s theorem and perturbation theory (Wick 1950; Isserlis 1918). To determine $$\mathsf {V}_{\boldsymbol {k}}$$, we begin by writing the linearized equations of motion as \begin{eqnarray} \partial _t \boldsymbol {v}(\boldsymbol {r}) = \mathcal {L}\left[\boldsymbol {v}(\boldsymbol {r}), \partial _i \boldsymbol {v}, \partial _i \partial _j \boldsymbol {v},\ldots , \boldsymbol {r}, t\right], \end{eqnarray} (2)where $$\mathcal {L}$$ is a linear operator of its first argument and $$\boldsymbol {v}$$ is the fluctuating part of the velocity field. In principle, we can work with this operator, though the derivatives of the velocity field make it highly inconvenient. Fortunately, at short length scales, the operator $$\mathcal {L}$$ may be treated as translation-invariant and so we may compute a Fourier transform in $$\boldsymbol {r}$$ without coupling different modes. This gives \begin{eqnarray} \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t} = \tilde{\mathcal {L}}\left[\tilde{\boldsymbol {v}}_{\boldsymbol {k}}, \boldsymbol {k}, t\right]. \end{eqnarray} (3)The modes are decoupled in this regime so $$\tilde{\mathcal {L}}$$ can be represented by a matrix $$\mathsf {L}$$, and we write \begin{eqnarray} \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t} = \mathsf {L}(\boldsymbol {k},t) \boldsymbol {v}_{\boldsymbol {k}}. \end{eqnarray} (4) When $$\mathsf {L}$$ is independent of t equation (4) is straightforward to solve and gives us \begin{eqnarray} \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t} = \sum _i v_{0,i} \hat{v}_{\boldsymbol {k},i} e^{\lambda _i t}, \end{eqnarray} (5)where v0, i are the initial mode amplitudes and $$\hat{v}_{\boldsymbol {k},i}$$ and λi are the normalized right eigenvectors and eigenvalues of $$\mathsf {L}$$, respectively. The vectors $$\hat{v}_{\boldsymbol {k},i}$$ then specify the modes of the system at a given wavevector. If the eigenvalues are not precisely degenerate then modes which begin in phase rapidly become uncorrelated and we may extend equation (1) to the modes at each wavevector and write \begin{eqnarray} \langle \tilde{\boldsymbol {v}}_{\boldsymbol {k},i} \otimes \tilde{\boldsymbol {v}}^*_{\boldsymbol {k}^{\prime },j} \rangle &= (2{\pi} )^3 \delta ^3 (\boldsymbol {k}-\boldsymbol {k}^{\prime }) \delta _{ij} \mathsf {V}_{\boldsymbol {k},i}. \end{eqnarray} (6)This result holds even when modes are degenerate. Because the time evolution of the Navier–Stokes equation is deterministic, the expectation 〈⋅⋅⋅〉 represents a sum over initial conditions. In this sum all relative phases between the modes are explored, so even degenerate modes become uncorrelated. Inserting equation (5) into equation (6) and summing over j and integrating over $$\boldsymbol {k}$$ gives us \begin{eqnarray} \mathsf {V}_{\boldsymbol {k},i} &= \hat{v}_{\boldsymbol {k},i} \otimes \hat{v}_{\boldsymbol {k},i} \langle |v_{0,i}|^2 \exp \left[2 t \Re \left[\lambda _i\right]\right] \rangle . \end{eqnarray} (7)Generally, some λi have positive real parts and so in a long-term expectation this exponential diverges. Indeed, it turns out that these growing modes are precisely those which matter! What happens of course is just that these modes eventually reach amplitudes where the linear approximation fails. By assumption, the system is stable over long times relative to the turbulent scale so this must result in these modes saturating. This has been variously described as mode crashing or the action of parasitic modes (Lesaffre, Balbus & Latter 2009; Pessah & Goodman 2009) but, regardless of the mechanism, it simply means that these modes exit the linear regime and find their growth impeded. To complete the closure, we must find the saturation amplitude. Relying again on the scale-free nature of turbulence, we note that this must be a power law in k. That is \begin{eqnarray} \langle \tilde{v}_{\boldsymbol {k}, i}^2 \rangle = \mathrm{Tr}\left[\mathsf {V}_{\boldsymbol {k}, i}\right] = \frac{A}{M} \left(\frac{k_0}{k}\right)^n, \end{eqnarray} (8)where A depends on the large-scale properties of the flow but is independent of $$\boldsymbol {k}$$, M is the number of modes per wavevector and n is the index of the turbulence. Following Kolmogorov (1941a), we choose n = 11/6 in our model. Appendix A contains a detailed discussion of this choice. The wavenumber k0 is just that of the characteristic scale, and is given by \begin{eqnarray} k_0 = \frac{2{\pi} }{L_0}. \end{eqnarray} (9)Replacing the divergent expression in equation (7) with this amplitude, we find \begin{eqnarray} \mathsf {V}_{\boldsymbol {k},i} = \frac{A}{M} \left(\frac{k_0}{k}\right)^n \hat{v}_{\boldsymbol {k},i} \otimes \hat{v}_{\boldsymbol {k},i}. \end{eqnarray} (10) It only remains to determine A. To do this, we note that there is one characteristic length scale L0 and one characteristic time scale, the growth rate ℜ[λi] of the mode. Because A has dimensions of velocity squared, we find \begin{eqnarray} \mathsf {V}_{\boldsymbol {k},i} = \frac{c}{M} L_0^2 \Re \left[\lambda _i\right]^2 \left(\frac{k_0}{k}\right)^n \hat{v}_{\boldsymbol {k},i} \otimes \hat{v}_{\boldsymbol {k},i}, \end{eqnarray} (11)where c is a dimensionless constant of order unity. This constant, known as the mixing length parameter, varies from theory to theory, so for clarity we set c = 1 in this work but this degree of freedom is important to note when comparing between models. In effect, what we have done is incorporate the non-linearity of turbulence by means of the spectrum while using linear growth rates to set the characteristic scale. In practice, the spectrum acts only to provide a convergent measure over modes (see Appendix A for further discussion), and it is the growth rate and the modes themselves that yield the anisotropies and other phenomena of interest. This is closely related to the approaches of Lesaffre et al. (2013) and Canuto & Hartke (1986). This prescription is easily extended in cases where there are additional dynamical fields, such as the turbulent displacement or a fluctuating magnetic field. The additional fields are simply incorporated into the vector describing the state and M is increased accordingly. We can continue to use equation (8) to fix the amplitude of the entire mode against that of the velocity as long as we know the turbulent index n. Up to this point this prescription is mathematically identical to that of Lesaffre et al. (2013), with the exception that we define the mixing wave vector as in equation (9), while they use π/L0 instead. In the next section, we introduce perturbative corrections to this model to capture a wider variety of phenomena. 3 PERTURBATIVE CORRECTIONS Now consider the case where the matrix $$\mathsf {L}$$ is time-dependent. Most of our reasoning about the behaviour of modes from the previous section still holds but, because the eigenvectors are time-dependent, we no longer have a well-defined notion of a mode as a long-running solution to the equations of motion. When the time dependence is periodic Floquet theory applies (Floquet 1883), but in the cases of interest the time dependence is aperiodic. To recover modes when the time evolution matrix itself evolves and does so aperiodically, we begin by expanding as \begin{eqnarray} \mathsf {L}(t) = \mathsf {L}(0) + t \frac{{\rm d}\mathsf {L}}{{\rm d}t} + \frac{1}{2}t^2 \frac{{\rm d}^2\mathsf {L}}{{\rm d}t^2} +\cdots \,. \end{eqnarray} (12)This series can be truncated to produce an approximation of $$\mathsf {L}$$, which is accurate in a certain window around t = 0. We may likewise write the velocity at a given wavevector as \begin{eqnarray} \tilde{\boldsymbol {v}}_{\boldsymbol {k}}(t) = \tilde{\boldsymbol {v}}_{\boldsymbol {k}}(0) + t\left. \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t}\right|_0 + \frac{1}{2}t^2\left. \frac{{\rm d}^2\tilde{\boldsymbol {v}}_{\boldsymbol {k}}(t)}{{\rm d}t^2}\right|_0 +\cdots \,. \end{eqnarray} (13)This suggests defining a new vector \begin{eqnarray} \Phi _{\boldsymbol {k}}(t) \equiv \left\lbrace \tilde{\boldsymbol {v}}_{\boldsymbol {k}}, \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t}, \frac{{\rm d}^2\tilde{\boldsymbol {v}}_{\boldsymbol {k}}}{{\rm d}t^2},\ldots \right\rbrace , \end{eqnarray} (14)which, in principle, encodes the full time evolution of the velocity field. This vector evolves according to \begin{eqnarray} \frac{{\rm d}\Phi _{\boldsymbol {k}}}{{\rm d}t} = \mathsf {A} \Phi _{\boldsymbol {k}}, \end{eqnarray} (15)where $$\mathsf {A}$$ is formed of blocks given by \begin{eqnarray} \mathsf {A}_{ij} = {i \atopwithdelims ()j}\frac{{\rm d}^{i-j}}{{\rm d}t^{i-j}} \mathsf {L}. \end{eqnarray} (16)By definition though we also have \begin{eqnarray} \frac{{\rm d}\Phi _{\boldsymbol {k},i}}{{\rm d}t} = \Phi _{\boldsymbol {k},i+1}, \end{eqnarray} (17)where $$\Phi _{\boldsymbol {k},0} = \tilde{\boldsymbol {v}}_{\boldsymbol {k}}$$, $$\Phi _{\boldsymbol {k},1} = {\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k}}/{\rm d}t$$ and so on. Thus, we are searching for a simultaneous solution of equations (15) and (17). In order to close the system, we must truncate it at some finite order N. Doing so makes the assumption that the behaviour of the system at all greater N is known. Inspired by the solution for time-independent $$\mathsf {L}$$, we try an exponential behaviour. This truncates equation (17) such that it applies only to i < N − 1 and means that we are searching for vectors with \begin{eqnarray} \left(A \Phi _{\boldsymbol {k}}\right)_{N-1} = \lambda \Phi _{\boldsymbol {k},N-1} \end{eqnarray} (18)and \begin{eqnarray} \Phi _{\boldsymbol {k},i+1} = (\mathsf {A} \Phi _{\boldsymbol {k}})_i, i < N-1. \end{eqnarray} (19)These equations are most straightforwardly written as a general eigensystem and this has the advantage of restricting the dimension of the linear space to just those states obeying the constraint. This is possible because both $$\mathsf {A}$$ and the constraint are lower triangular in the same basis, and so each row may be substituted into the next, leading to an eigenproblem of the form \begin{eqnarray} \mathsf {Q} \Phi _{\boldsymbol {k,0}} = \lambda \mathsf {W} \Phi _{\boldsymbol {k,0}}, \end{eqnarray} (20)where $$\mathsf {Q}$$ and $$\mathsf {W}$$ are matrices acting only on the 0-block. For example, in the case where N = 2, our equations are \begin{eqnarray} \Phi _{\boldsymbol {k},1} &= \mathsf {M} \Phi _{\boldsymbol {k},0} \end{eqnarray} (21)and \begin{eqnarray} \mathsf {M} \Phi _{\boldsymbol {k},1} + \dot{\mathsf {M}} \Phi _{\boldsymbol {k},0} &= \lambda \Phi _{\boldsymbol {k},1}, \end{eqnarray} (22)which may be put in the form of equation (20) with \begin{eqnarray} \mathsf {Q} &= \mathsf {M}^2 + \dot{\mathsf {M}} \end{eqnarray} (23)and \begin{eqnarray} \mathsf {W} &= \mathsf {M}. \end{eqnarray} (24)The eigenvectors of this system are solutions of the original equation (4) because if $$\psi ^i_{\boldsymbol {k}}$$ is such an eigenvector then \begin{eqnarray} \tilde{\boldsymbol {v}}_{\boldsymbol {k},i}(t) \equiv \sum _{j=0}^{N} \frac{t^j}{j!}\psi ^i_{j} \end{eqnarray} (25)solves \begin{eqnarray} \frac{{\rm d}\tilde{\boldsymbol {v}}_{\boldsymbol {k},i}}{{\rm d}t} = \mathsf {L}(t) \tilde{\boldsymbol {v}}_{\boldsymbol {k},i}(t) \end{eqnarray} (26)over the time window for which $$\mathsf {L}$$ is well-approximated at N-th order. As a result, we say that ϕi(t) are the instantaneous modes of the system at N-th order and use them and in equation (11). In place of the eigenvalue, we use the instantaneous growth rate of the velocity, which is given by \begin{eqnarray} g \equiv \frac{1}{2}\frac{{\rm d} v^2}{{\rm d}t} = \frac{\Re \left(\Phi _{\boldsymbol {k},0} \cdot \Phi _{\boldsymbol {k},1}\right)}{|\Phi _{\boldsymbol {k},0}|^2}. \end{eqnarray} (27)This approximation is controlled in the sense that so long as $$\mathsf {L}(t)$$ converges as N grows, so does the inferred velocity history. In this work, we present results with N = 2 so that $$\mathsf {A}$$ involves both $$\mathsf {L}$$ and $$\dot{\mathsf {L}}$$. We leave the exploration of larger N to later work. 4 EQUATIONS OF MOTION We now specialise to the case of an ideal gas obeying the ideal MHD equations. This section largely follows the derivation of Balbus & Schaan (2012) so we present only the pieces necessary to understand later parts of this work as well as the few places where our derivation diverges from theirs. We take the background to be axisymmetric, the fluctuations to be adiabatic and we work in cylindrical coordinates. We neglect both the microscopic viscosity and the microscopic thermal diffusivity because these are both negligible in most circumstances in stellar physics.1 Because our closure model treats turbulent properties as local, we compute all background quantities at a reference point $$\boldsymbol {r}_0$$. Relative to this point, we define the Lagrangian separation $$\boldsymbol {\delta r}$$ and velocity $$\boldsymbol {\delta v}$$ equivalent to $$\boldsymbol {\xi }$$ and $${\rm D}\boldsymbol {\xi }/{\rm D}t$$ of Balbus & Schaan (2012). In addition, we take the Boussinesq approximation that density variations are ignored except in terms involving gravitational acceleration. With the above definitions, the continuity equation may be written as \begin{eqnarray} \nabla \cdot \boldsymbol {\delta r} = 0. \end{eqnarray} (28) In a fixed coordinate system differential rotation is difficult to analyse so we make two reference frame changes. First, we switch from an inertial frame to one rotating at \begin{eqnarray} \Omega _0 \equiv \Omega (\boldsymbol {r}_0). \end{eqnarray} (29)Secondly, we make a formal change of coordinates \begin{eqnarray} \phi \rightarrow \phi - t\delta \boldsymbol {r}\cdot \nabla \Omega , \end{eqnarray} (30)without altering the corresponding unit vectors. Under this last change the gradient transforms as \begin{eqnarray} \nabla &\rightarrow \nabla - t(\nabla \Omega )\partial _{\phi }. \end{eqnarray} (31)Because the operator $$\mathcal {L}$$ is most easily expressed in Fourier space, we define the transformed wavevector as \begin{eqnarray} \boldsymbol {q} \equiv \boldsymbol {k} - t k_\phi R \nabla \Omega . \end{eqnarray} (32)With this the transformed MHD and Navier–Stokes equations may be written as \begin{eqnarray} \boldsymbol {\delta \tilde{B}} = \boldsymbol {B} \cdot \boldsymbol {q} \boldsymbol {\delta \tilde{r}} \end{eqnarray} (33)and \begin{eqnarray} {\partial _t \delta \tilde{\boldsymbol {v}} + 2\boldsymbol {\Omega }\times \delta \tilde{\boldsymbol {v}}+ \hat{R} R \delta \tilde{\boldsymbol {r}}\cdot \nabla \Omega ^2}\nonumber\\ {\quad- \frac{1}{\gamma \rho }\left(\boldsymbol {\delta \tilde{r}}\cdot \nabla \sigma \right)\nabla \cdot \boldsymbol {\Pi } + \frac{i}{\rho }\boldsymbol {q}\cdot \boldsymbol {\delta \tilde{\Pi }}=0,} \end{eqnarray} (34)where σ is the specific entropy and \begin{eqnarray} \boldsymbol {\Pi } \equiv p \mathsf {I} - \frac{1}{\mu _0}\left(\boldsymbol {B}\otimes \boldsymbol {B} - \frac{1}{2}B^2 \mathsf {I}\right) \end{eqnarray} (35)is the pressure tensor with $$\mathsf {I}$$ the identity matrix. All quantities prefixed with δ are fluctuating, a tilde denotes the Fourier transformed function, and all other quantities are background fields evaluated at $$\boldsymbol {r}_0$$. It is straightforward to see that this is the same equation as that derived by Balbus & Schaan (2012) once the appropriate relations for the pressure and magnetic force are substituted. The fluctuation in the pressure tensor may be written as \begin{eqnarray} \boldsymbol {\delta \Pi } = \delta p \mathsf {I} - \frac{1}{\mu _0}\left(\boldsymbol {B}\otimes \boldsymbol {\delta B} + \boldsymbol {\delta B}\otimes \boldsymbol {B} - \mathsf {I} \boldsymbol {B}\cdot \boldsymbol {\delta B} \right), \end{eqnarray} (36)so in Fourier space \begin{eqnarray} \boldsymbol {\delta \tilde{\Pi }} = \delta \tilde{p} \mathsf {I} - \frac{1}{\mu _0}\left(\boldsymbol {B}\otimes \boldsymbol {\delta \tilde{B}} + \boldsymbol {\delta \tilde{B}}\otimes \boldsymbol {B} - \mathsf {I} \boldsymbol {B}\cdot \boldsymbol {\delta \tilde{B}} \right). \end{eqnarray} (37)Combining this with equation (33) and the Boussinesq approximation (see Appendix B), we find \begin{eqnarray} \boldsymbol {q}\cdot \boldsymbol {\delta \tilde{\Pi }} = \boldsymbol {q} \delta p - \frac{i}{\mu _0} (\boldsymbol {B}\cdot \boldsymbol {q})^2 \boldsymbol {\delta \tilde{r}}. \end{eqnarray} (38)Note that as did Balbus & Schaan (2012), we take $$\boldsymbol {B}\cdot \boldsymbol {q}$$ to be constant in time as implied by the Boussinesq and ideal-MHD conditions. We now depart from prior work and use this equation along with equation (34) taking the component perpendicular to $$\boldsymbol {q}$$ to eliminate δp and find \begin{eqnarray} 0 &=&\left(\partial _t \delta \tilde{\boldsymbol {v}} + 2\boldsymbol {\Omega }\times \delta \tilde{\boldsymbol {v}}+ \hat{R} R \delta \tilde{\boldsymbol {r}}\cdot \nabla \Omega ^2\right. \nonumber \\ &&- \left. \frac{1}{\gamma \rho }\left(\boldsymbol {\delta \tilde{r}}\cdot \nabla \sigma \right)\nabla \cdot \boldsymbol {\Pi } + \frac{1}{\mu _0 \rho } (\boldsymbol {B}\cdot \boldsymbol {q})^2 \boldsymbol {\delta \tilde{r}}\right)_{\perp \boldsymbol {q}}, \end{eqnarray} (39)where the notation $$\left(... \right)_{\perp \boldsymbol {q}}$$ denotes the component perpendicular to $$\boldsymbol {q}$$. To construct the matrix version $$\mathsf {L}$$ of these equations, we must choose a coordinate system. Both because of the constraint (28) and because equation (39) is written in the plane perpendicular to $$\boldsymbol {q}$$ we choose the unit vectors \begin{eqnarray} \hat{\boldsymbol {a}} \equiv \frac{\hat{\boldsymbol {q}}\times \hat{\boldsymbol {w}}}{\sqrt{1-(\hat{\boldsymbol {q}}\,\cdot\, \hat{\boldsymbol {w}})^2}} \end{eqnarray} (40)and \begin{eqnarray} \hat{\boldsymbol {b}} &\equiv \hat{\boldsymbol {q}}\times \hat{\boldsymbol {a}}, \end{eqnarray} (41)where $$\hat{w}$$ is any unit vector with $$\hat{\boldsymbol {w}} \cdot \hat{\boldsymbol {q}} \ne 1$$. This choice of basis ensures that our vectors are perpendicular to the wavevector. A choice of particular convenience for $$\hat{w}$$ is \begin{eqnarray} \hat{\boldsymbol {w}} = \frac{\nabla \Omega }{|\nabla \Omega |}. \end{eqnarray} (42)With this choice $$\hat{a}$$ is time-independent, because the component of $$\boldsymbol {q}$$ perpendicular to $$\boldsymbol {w}$$ is time-independent, and so we may write \begin{eqnarray} \boldsymbol {\delta \tilde{r}} = \alpha \hat{\boldsymbol {a}} + \beta \hat{\boldsymbol {b}} \end{eqnarray} (43)and \begin{eqnarray} \boldsymbol {\delta \tilde{v}} = \dot{\alpha } \hat{\boldsymbol {a}} + \dot{\beta } \hat{\boldsymbol {b}} + \beta \partial _t \hat{\boldsymbol {b}}. \end{eqnarray} (44)Note that there is a removable singularity when $$\hat{w} \parallel \hat{q}$$. The matrix $$\mathsf {L}$$ is then given by computing the relation between $$\partial _t \lbrace \alpha , \beta , \dot{\alpha }, \dot{\beta }\rbrace$$ and $$\lbrace \alpha , \beta , \dot{\alpha }, \dot{\beta }\rbrace$$. The result is quite unwieldy so we do not present it here but note that it is fully documented in the software in which we implement these equations. 5 STRESSES AND TRANSPORT The equations of motion contain the position and the velocity, so our expanded vector space is \begin{eqnarray} \Phi = \left\lbrace \delta \boldsymbol {r}, \delta \boldsymbol {v}, \partial _t \delta \boldsymbol {v},\ldots , \right\rbrace . \end{eqnarray} (45)Combining the linearized equations of motion with our closure scheme, we can compute the correlation function \begin{eqnarray} \langle \Phi \otimes \Phi \rangle = \int \frac{{\rm d}^3 \boldsymbol {k}}{(2{\pi} )^3} \sum _i \langle \Phi ^{i}_{\boldsymbol {k}}\otimes \Phi ^{i*}_{\boldsymbol {k}}\rangle , \end{eqnarray} (46)where the index i ranges over eigenvectors. This function contains all of the usual stresses and transport functions. For instance, the Reynolds stress is \begin{eqnarray} R \equiv \langle \delta \boldsymbol {v}\otimes \delta \boldsymbol {v}\rangle = \langle \Phi _1 \otimes \Phi _1 \rangle . \end{eqnarray} (47)Likewise up to a dimensionless constant of order unity the turbulent diffusivity is \begin{eqnarray} d \equiv \langle \delta \boldsymbol {v}\otimes \delta \boldsymbol {r}\rangle = \langle \Phi _1 \otimes \Phi _0 \rangle . \end{eqnarray} (48)and the turbulent viscosity is \begin{eqnarray} Q \equiv \langle \delta \boldsymbol {v}\otimes \delta \boldsymbol {r}\rangle +\langle \delta \boldsymbol {r}\otimes \delta \boldsymbol {v}\rangle = \langle \Phi _1 \otimes \Phi _0 \rangle +\langle \Phi _0 \otimes \Phi _1 \rangle . \end{eqnarray} (49)Similar expressions hold for the dynamo effect, the transport of magnetic fields, and material diffusion. 6 RESULTS In this section, we exhibit a number of results which come from applying our model to a wide variety of astronomically and physically relevant circumstances. We also compare with the results of Lesaffre et al. (2013) and Kichatinov & Rudiger (1993). We modify the former to use the convention in equation (9) to avoid spurious differences in scale. We likewise assume that our L0 is equal to three times the mixing length of Kichatinov & Rudiger (1993), as this is an inherent freedom in the formalism and resolves an otherwise-persistent scale difference between our model and theirs. These models have been well-tested against a variety of data, most notably helioseismic results, and so provide a useful reference for our work. We have also included more direct comparisons but, because direct experiments are extremely difficult to perform under most circumstances relevant to astrophysics, we have instead included comparisons with simulations and observations where available and applicable. Simulations are often the most useful comparison for stellar phenomena, because a variety of processes, including meridional circulation, can mask the effects of turbulent transport (Kitchatinov 2013). In accretion discs, however, there are several observable quantities, which are thought to correlate closely with the underlying turbulence and these provide very helpful constraints (King, Pringle & Livio 2007). These comparisons and calculations are not intended to be a complete collection of the results our model can produce, nor have we exhaustively explored the circumstances and dependencies of each result. Rather, it is our hope to demonstrate that there is a great deal of interesting physics in this model, that our perturbative corrections give rise to realistic results and reproduce many known results, and that there is much to warrant further exploration along these lines. 6.1 Rotating convection We begin with the effect of rotation on convection in the case of a rotating system with radial pressure and entropy gradients. It is useful to start by comparing our results with those from simulations. Fig. 1 shows the ratios $$\sqrt{\langle \delta v_r^2\rangle /\langle \delta v^2\rangle }$$, $$\sqrt{\langle \delta v_\theta ^2\rangle /\langle \delta v^2\rangle }$$ and $$\sqrt{\langle \delta v_\phi ^2\rangle /\langle \delta v^2\rangle }$$ for several rotation rates as a function of latitude. The positive latitudes come from table 2 of Chan (2001), while the negative are from table 2 of Käpylä, Korpi & Tuominen (2004). In order to match the units for the rotation rates, we put everything in terms of the coriolis number \begin{eqnarray} \mathrm{Co} \equiv \frac{\Omega h}{\langle \delta v^2\rangle ^{1/2}}, \end{eqnarray} (50)where, following the convention of Käpylä et al. (2004), 〈δv2〉1/2 was computed for a non-rotating system. Figure 1. View largeDownload slide The ratios $$\sqrt{\langle \delta v_r^2\rangle /\langle \delta v^2\rangle }$$ (blue), $$\sqrt{\langle \delta v_\theta ^2\rangle /\langle \delta v^2\rangle }$$ (red), and $$\sqrt{\langle \delta v_\phi ^2\rangle /\langle \delta v^2\rangle }$$ (purple) are shown for our model (solid) and for simulations by (Käpylä et al. 2004, dots, negative latitude) and (Chan 2001, dots, positive latitude) for a wide range of rotation rates as a function of latitude. The rotation rate is captured by the Coriolis number Co = Ωh/〈δv2〉1/2. Our model general overestimates the anisotropy but captures its variation well. Figure 1. View largeDownload slide The ratios $$\sqrt{\langle \delta v_r^2\rangle /\langle \delta v^2\rangle }$$ (blue), $$\sqrt{\langle \delta v_\theta ^2\rangle /\langle \delta v^2\rangle }$$ (red), and $$\sqrt{\langle \delta v_\phi ^2\rangle /\langle \delta v^2\rangle }$$ (purple) are shown for our model (solid) and for simulations by (Käpylä et al. 2004, dots, negative latitude) and (Chan 2001, dots, positive latitude) for a wide range of rotation rates as a function of latitude. The rotation rate is captured by the Coriolis number Co = Ωh/〈δv2〉1/2. Our model general overestimates the anisotropy but captures its variation well. Our model overestimates the anisotropy of the turbulence but captures its symmetries and trends well. For instance, we find that near the poles and in non-rotating systems the θ and ϕ components of the velocity fluctuations have identical magnitudes, in line with the simulations. We reproduce the trend of decreasing anisotropy towards the equator and decreasing anisotropy with increasing rotation, and, in cases where there are differences between the θ and ϕ velocities, we reproduce both their sign and magnitude. In particular, we find that $$\langle \delta v_r^2\rangle \ge \langle \delta v_\theta ^2 \rangle \ge \langle \delta v_\phi ^2\rangle$$, which is seen in these and other simulations (Rüdiger, Egorov & Ziegler 2005a). Likewise, we find that radial motion makes up a greater fraction of the total velocity near the poles than at the equator, and that as the Coriolis number increases $$\langle \delta v_r^2 - \delta v_\theta ^2 - \delta v_\phi ^2 \rangle \rightarrow 0$$, all of which is in agreement with the predictions of Rüdiger et al. (2005b). Our overestimate of the anisotropy may be due to our model incorporating the large-scale fields on all scales, as noted by Lesaffre et al. (2013). This suggests that a future refinement might be to use estimates of the large-scale modes to compute the environment of those at smaller scales, but we do not treat such complications for now. As a further comparison, we consider the off-diagonal Reynolds stresses of both Chan (2001) and Käpylä et al. (2004). These numbers were extracted from table 3 of the former and also table 3 of the latter and are shown along with our predictions in Fig. 2. In the former, they were straightforward to analyse but in the latter they do not provide a precise test because the simulations included a bulk shear. To correct for this, we used a linear expansion to subtract results across simulations, which were identical in all conditions other than the rotation and thereby determine the effect of the rotation alone. As we will see in Section 6.2 this procedure is problematic because the shear may interact non-linearly with the rotation. Furthermore, because these corrections are of the same order as the terms themselves some care must be taken in interpreting the results. Figure 2. View largeDownload slide The ratios $$\sqrt{\langle \delta v_r \delta v_\theta \rangle /\langle \delta v^2\rangle }$$ (red), $$\sqrt{\langle \delta v_\theta \delta v_\phi \rangle /\langle \delta v^2\rangle }$$ (purple), and $$\sqrt{\langle \delta v_r \delta v_\phi \rangle /\langle \delta v^2\rangle }$$ (blue) are shown from our model (solid) and from simulations by (Käpylä et al. 2004, dots, negative latitude) and (Chan 2001, dots, positive latitude) as a function of latitude. Note that Käpylä et al. (2004) cautions that the moderate rotation simulations had difficulty converging, and these results arise as the difference between two simulations, so it is not clear how significant this test is. Our model generally overestimates these stresses, and suggests a different symmetry for the variation (going as sin θ rather than sin (2θ)). Figure 2. View largeDownload slide The ratios $$\sqrt{\langle \delta v_r \delta v_\theta \rangle /\langle \delta v^2\rangle }$$ (red), $$\sqrt{\langle \delta v_\theta \delta v_\phi \rangle /\langle \delta v^2\rangle }$$ (purple), and $$\sqrt{\langle \delta v_r \delta v_\phi \rangle /\langle \delta v^2\rangle }$$ (blue) are shown from our model (solid) and from simulations by (Käpylä et al. 2004, dots, negative latitude) and (Chan 2001, dots, positive latitude) as a function of latitude. Note that Käpylä et al. (2004) cautions that the moderate rotation simulations had difficulty converging, and these results arise as the difference between two simulations, so it is not clear how significant this test is. Our model generally overestimates these stresses, and suggests a different symmetry for the variation (going as sin θ rather than sin (2θ)). Despite these difficulties some trends are clear and sustained between both sets of data. For instance, in the northern hemisphere (θ > 0), 〈δvrδvθ〉 < 0, while in both hemispheres 〈δvrδvϕ〉 < 0, in keeping with predictions and simulations by Rüdiger et al. (2005b). Likewise, we find that 〈vθvϕ〉 > 0 in the northern hemisphere, in agreement with the findings of Rüdiger et al. (2005a). Once more, however, our model overestimates these anisotropic terms by an amount, which is largely invariant as a function of rotation. This suggests that this overestimate is a systematic offset rather than an error in scaling. We also have some difficulty to reproduce the signs of some of the stresses, particularly in the results of Käpylä et al. (2004), though this could simply be a subtraction difficulty. This is supported by the fact that the simulations themselves do not agree on the signs of these terms and highlights the challenges of making comparisons of terms, which are small in magnitude relative to the scale of the turbulence. To better understand which trends are significant and which are artefacts, we have placed data from comparable rotation rates for the two sets of simulations side-by-side in Fig. 3. The top five panels show the same data as in Fig. 1, while the bottom three show the data from Fig. 2. In general, there is good agreement in the top five panels. The data of Käpylä et al. (2004) gives systematically larger anisotropies and the two sets of simulations occasionally differ on the relative magnitudes of the velocity components (i.e. their ordering), but otherwise the two are in good agreement. By contrast, the bottom three panels paint two very divergent pictures. Neither ordering, trends nor signs are consistent between the two sets of simulations. Only the magnitudes agree in these cases. Thus, the two sets of simulations agree that our model systematically overestimates anisotropies and that, beyond that, our model agrees with them to the extent that they agree with one another. Figure 3. View largeDownload slide The functions shown in Figs 1 and 2 are shown from our model (solid), simulations by (Käpylä et al. 2004, dots, negative latitude) and Chan (2001) (crosses, positive latitude) as a function of latitude. The most comparable pairs of rotation rates were placed side-by-side for each function. A solid black line is shown along the equator where the latitude is zero. There is reasonable agreement on the distribution of velocities in direction but not on the correlations between different velocity directions.. Figure 3. View largeDownload slide The functions shown in Figs 1 and 2 are shown from our model (solid), simulations by (Käpylä et al. 2004, dots, negative latitude) and Chan (2001) (crosses, positive latitude) as a function of latitude. The most comparable pairs of rotation rates were placed side-by-side for each function. A solid black line is shown along the equator where the latitude is zero. There is reasonable agreement on the distribution of velocities in direction but not on the correlations between different velocity directions.. Having compared in detail with these simulations, we now consider predictions which go beyond the domain where simulations are possible. In convection with radial gradients, the leading order effect is to transport heat and material radially. Fig. 4 shows 〈δvrδvr〉 and 〈δvrδrr〉, which are the correlation functions controlling this transport. Figure 4. View largeDownload slide The radial velocity correlation function 〈δvrδvr〉 (red) and the radial diffusivity 〈δvrδrr〉 (blue) are shown in linear scale for Ω < |N| (left-hand panel) and log-log scale for Ω > |N| (right-hand panel). These results are for uniform rotation at a latitude of π/4 with no magnetic field. On this and all subsequent figures $$v_r v_r/L_0^2|N|^2$$ should be read as $$\langle \delta v_r \delta v_r \rangle / L_0^2|N|^2$$ and similarly for other correlations. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, scale, and variation. The bumps in our results reflect parameter values where the numerical integration was more difficult. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 4. View largeDownload slide The radial velocity correlation function 〈δvrδvr〉 (red) and the radial diffusivity 〈δvrδrr〉 (blue) are shown in linear scale for Ω < |N| (left-hand panel) and log-log scale for Ω > |N| (right-hand panel). These results are for uniform rotation at a latitude of π/4 with no magnetic field. On this and all subsequent figures $$v_r v_r/L_0^2|N|^2$$ should be read as $$\langle \delta v_r \delta v_r \rangle / L_0^2|N|^2$$ and similarly for other correlations. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, scale, and variation. The bumps in our results reflect parameter values where the numerical integration was more difficult. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Both correlators vary at second order in Ω in the slow rotation limit as expected (Kitchatinov 2013; Lesaffre et al. 2013). In the rapid rotation limit on the other hand, they exhibit clear Ω−1 scaling, consistent with what is seen in other closure models and in simulations (Garaud et al. 2010). The quenching of turbulence in this limit arises because the Coriolis effect acts as a restoring force, stabilizing modes. The peak of each correlator is of order unity and occurs when Ω = 0. In fact, for the stress, the maximum is 0.254647 while for the diffusivity it is 0.28125, both of which are consistent to this precision with Lesaffre et al. (2013), noting that we used the definition in equation (9) for their mixing length. This is because our model is precisely the same as theirs in this limit. Based on this and the observed scalings a good approximation is \begin{eqnarray} \langle \delta v_r \delta r_r \rangle \approx \langle \delta v_r \delta v_r \rangle \approx \frac{1-(\Omega /|N|)^2}{1 - (\Omega /|N|)^3}. \end{eqnarray} (51) Next, we consider the effect of rotation on the r − θ correlation functions. These functions are responsible for latitudinal transport of heat, mass, and momentum and vanish as a result of spherical symmetry in the non-rotating limit. Fig. 5 shows 〈δvrδvθ〉 and 〈δvrδrθ〉 as a function of the rotation rate. Figure 5. View largeDownload slide The absolute value of the r − θ velocity correlation function 〈δvrδvθ〉 (red) and corresponding diffusivity 〈δvrδrr〉 (blue) are shown in log-log scale against rotation rate. These results are for uniform rotation at a latitude of π/4 with no magnetic field. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, scale, and variation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 5. View largeDownload slide The absolute value of the r − θ velocity correlation function 〈δvrδvθ〉 (red) and corresponding diffusivity 〈δvrδrr〉 (blue) are shown in log-log scale against rotation rate. These results are for uniform rotation at a latitude of π/4 with no magnetic field. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, scale, and variation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. In the slow-rotation regime both quantities scale as Ω2, while in the rapid rotation limit they scale as Ω−1. The peak is of order unity and occurs near Ω = |N|. This gives rise to the approximation \begin{eqnarray} \langle v_r r_\theta \rangle \approx \langle v_r v_\theta \rangle \approx \frac{(\Omega /|N|)^2}{1 + (\Omega /|N|)^3}. \end{eqnarray} (52)These scalings may be interpreted as a competition between symmetry breaking and quenching: the correlation function rises as rotation breaks symmetries but excessive rotation stabilises the system and quenches the turbulent motions. The symmetry is broken quadratically because, at first order, the Coriolis effect only couples radial and azimuthal motions. The properties of turbulence vary with latitude in a rotating system because the rotation axis picks out a preferred direction. Fig. 6 shows the r − r and r − θ stress and diffusivity correlations as a function of latitude. The r − r correlations vary similarly to one another, exhibiting a minimum at the equator and maxima on-axis. On-axis the rotation drops out of the equations and so the on-axis functions are just those for non-rotating convection. The effect of rotation is then largest at the equator, where the convective motion is predominantly perpendicular to the rotation axis. The correlation functions are smallest where the rotation has the largest effect because rotation primarily acts to stabilize modes. Figure 6. View largeDownload slide Various correlation functions are shown as a function of the angle θ from the rotation axis. The functions are the r − r (left-hand panel) and r − θ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for uniform rotation at Ω = 0.2|N| (top panel), Ω = |N| (middle panel), and Ω = 5|N| (bottom panel). Shown in purple (*, dashed) for comparison is the KR result, which agrees in sign and variation but not scale. For slow rotation the scale of the variation is generally smaller than we predict, while for fast rotation the variation is somewhat larger. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 6. View largeDownload slide Various correlation functions are shown as a function of the angle θ from the rotation axis. The functions are the r − r (left-hand panel) and r − θ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for uniform rotation at Ω = 0.2|N| (top panel), Ω = |N| (middle panel), and Ω = 5|N| (bottom panel). Shown in purple (*, dashed) for comparison is the KR result, which agrees in sign and variation but not scale. For slow rotation the scale of the variation is generally smaller than we predict, while for fast rotation the variation is somewhat larger. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. By contrast, the r − θ correlator is largest in magnitude at mid-latitudes, vanishing both on-axis and at the equator. On-axis this correlation function must vanish because the $$\hat{\theta }$$ unit vector is ill-defined. The sign change between the northern and southern hemispheres occurs because $$(\hat{\boldsymbol {r}} \times \boldsymbol {\Omega })_\phi$$ has the same sign everywhere while $$(\hat{\boldsymbol {\theta }} \times \boldsymbol {\Omega })_\phi$$ changes sign between the hemispheres. This also explains the vanishing correlation at the equator. The quantities of particular interest for studying the origins of differential rotation are the radial-azimuthal correlation functions 〈δvrδvϕ〉 and 〈δvrδrϕ〉. The former provides a stress coupling the angular momentum to radial motions known as the Λ-effect, while the latter provides a viscosity coupling radial shears to azimuthal motion and so acts as a proxy for the α-effect (Kichatinov & Rudiger 1993). Fig. 7 shows these quantities as a function of the rotation rate. In the slow-rotation limit, both scale as Ω before peaking near unity and falling off as Ω−2 in the rapid-rotation limit. The linear scaling at slow rotation rates is a consequence of the Coriolis effect directly coupling radial and azimuthal motions. These quantities fall off more rapidly than the others in the case of rapid rotation because it is preferentially the modes that couple strongly to the Coriolis effect, which are stabilized the most. The absolute scale of our Λ-effect is approximately what is seen in simulations, slightly overestimating relative to Käpylä et al. (2004) and similar to other theoretical predictions (Kitchatinov 2013; Gough 2012). Figure 7. View largeDownload slide The absolute value of the r − ϕ velocity correlation function 〈δvrδvϕ〉 (red) and corresponding diffusivity 〈δvrδrϕ〉 (blue) are shown in log-log scale versus rotation rate. These results are for uniform rotation at a latitude of π/4 with no magnetic field. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, variation, and scale up until Ω = |N|, at which point the behaviour differs significantly. Shown in grey (**, dotted) for comparison is 〈δvrδvϕ〉 from that of Lesaffre et al. (2013). This agrees precisely in the Ω → 0 limit and the agreement is good even near Ω ≈ 0.5|N|. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 7. View largeDownload slide The absolute value of the r − ϕ velocity correlation function 〈δvrδvϕ〉 (red) and corresponding diffusivity 〈δvrδrϕ〉 (blue) are shown in log-log scale versus rotation rate. These results are for uniform rotation at a latitude of π/4 with no magnetic field. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2, which agrees in sign, variation, and scale up until Ω = |N|, at which point the behaviour differs significantly. Shown in grey (**, dotted) for comparison is 〈δvrδvϕ〉 from that of Lesaffre et al. (2013). This agrees precisely in the Ω → 0 limit and the agreement is good even near Ω ≈ 0.5|N|. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. 6.2 Differential rotation and convection We now turn to the dependence of convective transport coefficients on differential rotation. We expand our closure model to linear order in the shear and so restrict this analysis to cases where the dimensionless shear |R∇ln Ω| is at most of order unity. Fig. 8 shows the r − θ and r − ϕ velocity and diffusivity correlation functions as a function of differential rotation for a situation where ∇Ω is at an angle of π/4 relative to the pressure gradient. All four functions behave linearly near the origin, with intercept set by the stress and diffusivity in the uniform rotation limit. This is precisely as expected: the intercept is non-zero, giving rise to the Λ-effect, while the slope is non-zero, giving rise to the α-effect (Kichatinov & Rudiger 1993). Note that the favourable comparison of our results with those of Kitchatinov (2013) is helpful because their model was implemented in a two-dimensional solar model, which compared well with helioseismic observations. Figure 8. View largeDownload slide The r − θ (left-hand panel) and r − ϕ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions are shown in log-scale versus the differential rotation. These results are for a convecting region with differential rotation in the cylindrical radial direction, Ω = 0.1|N| and no magnetic field at a latitude of π/4. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2. This disagrees in sign and on the magnitude of the slope but agrees in the sign of the slope. Shown in grey (**, dotted) for comparison is 〈δvrδvϕ〉 of Lesaffre et al. (2013). This generally predicts smaller stresses though with the same sign and slope sign as our model. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 8. View largeDownload slide The r − θ (left-hand panel) and r − ϕ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions are shown in log-scale versus the differential rotation. These results are for a convecting region with differential rotation in the cylindrical radial direction, Ω = 0.1|N| and no magnetic field at a latitude of π/4. Shown in purple (*, dashed) for comparison is the result of Kichatinov & Rudiger (1993) with an anisotropy factor of 2. This disagrees in sign and on the magnitude of the slope but agrees in the sign of the slope. Shown in grey (**, dotted) for comparison is 〈δvrδvϕ〉 of Lesaffre et al. (2013). This generally predicts smaller stresses though with the same sign and slope sign as our model. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. A key difference between our work and what we compare with in Fig. 8 is that, while we predict the same sign and comparable magnitude for the α-effect in the zero-shear limit, the effect changes sign near |R∇ln Ω| ≈ 0.5, indicating that, at least for this configuration, this is the point at which non-linear effects become important. This does not represent a particularly severe shear and highlights a key point that the correlation functions we find are generally non-linear in all of the small parameters in which one might wish to expand. Our model captures this non-linear behaviour despite being carried out to linear order in |R∇ln Ω|. This is because, in our expansion, the time evolution operator is what is expanded linearly. The resulting eigenvalues and eigenvectors are generally non-linear functions of this operator. This caution aside, there is a significant regime where the α–Λ expansion is valid and, in this regime, key quantities of interest are the derivatives of the various correlation functions with respect to the shear |R∇Ω|. Fig. 9 shows these derivatives as a function of Ω. The r − ϕ stress derivative is constant in Ω. This means that the stress scales as R∇Ω. This is as expected (see, e.g. equation 79 of Lesaffre et al. 2013) and indicates that there is a well-defined effective viscosity transporting angular momentum. This viscosity is given by \begin{eqnarray} \nu _{r\phi } \approx L_0^2 |N|. \end{eqnarray} (53) Figure 9. View largeDownload slide The derivatives of various correlation functions with respect to |R∇Ω| are shown as a function of Ω, with both axes log-scaled. The functions are the r − θ (left-hand panel) and r − ϕ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for a convecting region with differential rotation in the cylindrical radial direction and no magnetic field at a latitude of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 9. View largeDownload slide The derivatives of various correlation functions with respect to |R∇Ω| are shown as a function of Ω, with both axes log-scaled. The functions are the r − θ (left-hand panel) and r − ϕ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for a convecting region with differential rotation in the cylindrical radial direction and no magnetic field at a latitude of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. By contrast, the derivatives of the r − θ correlations as well as the r − ϕ diffusivity all diverge in the limit as Ω → 0. In particular, the r − θ correlations diverge as Ω−1, while the r − ϕ diffusivity diverges as Ω−2. These divergences are signatures of symmetry breaking. They indicate that the direction in which the R∇Ω → 0 limit is approached matters. That is, this limit can be approached by first letting Ω → 0 and then differentiating or by differentiating and then taking Ω → 0 and the divergence we find in the latter approach indicates that the order matters. When Ω = 0 and |R∇Ω| = 0, there is a symmetry between ±θ and between ±ϕ. As a result both the r − θ and r − ϕ terms vanish in this limit. When Ω ≠ 0 these symmetries are broken by the rotation and we know from Figs 5 and 7 that this occurs at first order for r − ϕ and second order for r − θ. In the opposing limit, the situation is different because in the time evolution described by equation (39) $$\mathsf {L}$$ is independent of |R∇Ω| when Ω = 0. There is, however, a dependence on |R∇Ω| through the time-dependence of $$\boldsymbol {q}$$. This breaks the ϕ symmetry because $$\partial _t \boldsymbol {q}$$ is proportional to qϕR∇Ω and hence is sensitive to ϕ. It does not, however, break the θ symmetry, because qϕR∇Ω is symmetric with respect to changing the signs of both θ and $$\boldsymbol {q}$$. It follows then that we should find divergences in the r − θ correlation derivatives owing to the path-dependence of the zero-rotation limit and that we should find the r − ϕ derivatives to be generally well-behaved. The curious divergence is then that in the r − ϕ diffusivity, because this correlation function does not suffer from a symmetry-derived path-dependence. This arises because the differential rotation means that $$\mathsf {L}$$ is time-dependent. This introduces polynomial corrections to the usual exponential growth, as discussed in Section 3. This formalism captures the fact that the differential rotation turns vertical displacement into ϕ displacements, which vary as polynomials in time. There are, therefore, modes with very small radial velocities, which nevertheless have large azimuthal displacements and these dominate the diffusivity derivative. These modes grow proportional to |R∇Ω| and their growth may proceed in the azimuthal direction until bounded by the Coriolis effect at a time Ω−1. As a result these modes contribute to the diffusivity as |R∇ln Ω| and hence lead to a diverging derivative in |R∇Ω| as Ω → 0. 6.3 Differential rotation and stable stratification Stably stratified regions are those with \begin{eqnarray} N^2 > 0, \end{eqnarray} (54)such that buoyancy acts to counter perturbations in the vertical direction. This tends to damp turbulence. In the presence of such damping, there can still be turbulence if there is also a shear. The classic example of this is the Kelvin–Helmholtz phenomenon, which can occur in such a system if the Richardson criterion \begin{eqnarray} \frac{|{\rm d}u/{\rm d}z|^2}{|N|^2} > \frac{1}{4} \end{eqnarray} (55)is satisfied (Zahn 1993). Here, u is the velocity and z is the coordinate parallel to the stratification. Even when this criterion is not satisfied, latitudinal shear can still generate turbulence (Canuto et al. 2008). These motions are suppressed in vertical extent by the stratification and hence are primarily confined to the plane perpendicular to the stratification direction. Fig. 10 shows the dependence on shear strength of all six stress components in a rotating stably stratified zone with latitudinal rotational shear. All six exhibit linear scaling with the shear strength. This is unusual in an otherwise-stable zone because it implies a viscosity which, to leading order, does not depend on the shear. That is, \begin{eqnarray} \nu _{ij} \approx L_0^2 N f_{ij}\left(\frac{\Omega }{|N|}\right), \end{eqnarray} (56)where fij is some function of the angular velocity. Fig. 11 shows the dependence of the stress components on Ω/|N| for fixed |R∇ln Ω| = 0.1. The r − θ and θ − ϕ stresses vary as Ω3 in the slow-rotation regime and as Ω2 for rapid rotation. The other components all scale as Ω2 in both regimes. Thus, for instance, frϕ = Ω/|N| because the viscosity is the derivative of the stress with respect to the shear, and hence \begin{eqnarray} \nu _{r\phi } \approx 10^{-5} L_0^2 \Omega . \end{eqnarray} (57) Figure 10. View largeDownload slide The absolute value of the r − r (red), r − θ (blue) and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data is computed for a point on the equator with radial differential rotation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 10. View largeDownload slide The absolute value of the r − r (red), r − θ (blue) and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data is computed for a point on the equator with radial differential rotation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 11. View largeDownload slide The absolute value of the r − r (red), r − θ (blue), and r − ϕ (purple) velocity correlation functions are shown as a function of Ω/|N| for fixed |R∇ln Ω| = 0.1, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction and no magnetic field. The data is computed for a point on the equator with radial differential rotation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 11. View largeDownload slide The absolute value of the r − r (red), r − θ (blue), and r − ϕ (purple) velocity correlation functions are shown as a function of Ω/|N| for fixed |R∇ln Ω| = 0.1, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction and no magnetic field. The data is computed for a point on the equator with radial differential rotation. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. The scaling in equation (57) arises owing to the centrifugal term, which has a destabilising effect when Ω increases with $$\hat{R}$$. When |R∇Ω| = 0 this effect is not present so the system is stable but introducing a small differential rotation produces an acceleration proportional to $$\Omega R \boldsymbol {\delta r}\cdot \nabla \Omega$$ and hence \begin{eqnarray} \partial _t^2 \boldsymbol {\delta r} \approx g^2 \boldsymbol {\delta r} \propto \hat{R}\Omega R \boldsymbol {\delta r}\cdot \nabla \Omega , \end{eqnarray} (58)which means that the stress scales as Ω∇Ω and thence the viscosity scales as Ω. Because the magnitudes of the stresses are always ordered in the same way, the same terms are always the most significant. From largest to smallest, the stresses are r − r, r − ϕ, ϕ − ϕ, θ − θ, θ − ϕ, and r − θ. This group is nearly separated into diagonal stresses, which are larger, and off-diagonal stresses, which are smaller. The exception to this rule is the r − ϕ stress, which is special because it is the term which directly couples to the shear. The ordering of the remaining terms is not surprising because the off-diagonal stresses are typically mediated by a coupling between different directions, whereas the on-diagonal stresses require no such coupling. To better understand the effect of our perturbative corrections, we computed the same results without them. This produced stresses, which were zero to within numerical precision in all cases, indicating that the entire contribution in this case is coming from the perturbation. However with a different angle of differential rotation, we obtained non-zero results. It is instructive then to compare Fig. 12 with Fig. 13. These show the same correlation functions as each other in the same physical scenario, with differential rotation this time at an angle of π/4, but the former uses the first order perturbative expansion while the latter only expands to zeroth order. The difference between the two calculations is striking: many of the correlation functions have fundamentally different scalings when the perturbative corrections are taken into account. In particular, the r − θ and r − r stresses are both quadratic in the shear and the r − ϕ and θ − ϕ stresses both vary as the shear to the 3/2 power, whereas they are all linear in the shear in the expanded calculation. This difference relates in part to the centrifugal term, which couples the displacement to the acceleration. Without expanding the equations of motion, we would have δr ∝ δv, because the mode would need to be an eigenvector of $$\mathsf {M}$$. The modes which couple to the centrifugal term would still grow according to equation (58) but, for most modes, arranging for the displacement to couple to this term requires coupling to the stabilizing buoyant term too. To make this clearer, in Fig. 14, we have computed the growth rate as a function of wave-vector orientation without using the perturbative expansion. There are several rapidly growing regions, oriented at angles of ± π/4 relative to the vertical. These angles represent a compromise between maximizing the magnitude of the centrifugal acceleration and maximizing its projection on to the velocity, both subject to the Boussinesq condition that motion be in the plane perpendicular to $$\boldsymbol {q}$$. Figure 12. View largeDownload slide The absolute value of the r − r (red), r − θ (blue), and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. The correlation functions are evaluated at first order in the perturbative expansion rather than first order. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data are computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 12. View largeDownload slide The absolute value of the r − r (red), r − θ (blue), and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. The correlation functions are evaluated at first order in the perturbative expansion rather than first order. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data are computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 13. View largeDownload slide The absolute value of the r − r (red), r − θ (blue) and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. The correlation functions are evaluated at zeroth order in the perturbative expansion rather than first order. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data are computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 13. View largeDownload slide The absolute value of the r − r (red), r − θ (blue) and r − ϕ (purple) velocity correlation functions are shown as a function of |R∇ln Ω|, with both axes log-scaled. The correlation functions are evaluated at zeroth order in the perturbative expansion rather than first order. These results are for a stably stratified region with differential rotation in the radial direction, Ω = 0.1|N| and no magnetic field. The data are computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 14. View largeDownload slide The square of the growth rate is shown as a function of wave-vector orientation on a logarithmic colour scale. The wave-vector is specified by a magnitude and two angles, θ(q) and ϕ(q), which are spherical angles relative to the $$\hat{z}$$ direction. These rates were computed with a zeroth-order expansion. Regions with squared growth rates below 10−16 are shown in white. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 14. View largeDownload slide The square of the growth rate is shown as a function of wave-vector orientation on a logarithmic colour scale. The wave-vector is specified by a magnitude and two angles, θ(q) and ϕ(q), which are spherical angles relative to the $$\hat{z}$$ direction. These rates were computed with a zeroth-order expansion. Regions with squared growth rates below 10−16 are shown in white. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. By contrast, the growth rates in the expanded system, shown in Fig. 15, are significant over a much wider swath of parameter space. This is because, in the expanded system, the displacement and velocity need not be parallel so the displacement can be chosen to maximize the centrifugal term while the velocity can be chosen to maximize the projection of the acceleration on to the velocity. Figure 15. View largeDownload slide The square of the growth rate is shown as a function of wave-vector orientation on a logarithmic colour scale. The wave-vector is specified by a magnitude and two angles, θ(q) and ϕ(q), which are spherical angles relative to the $$\hat{z}$$ direction. These rates were computed with a first-order expansion. Regions with squared growth rates below 10−16 are shown in white. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 15. View largeDownload slide The square of the growth rate is shown as a function of wave-vector orientation on a logarithmic colour scale. The wave-vector is specified by a magnitude and two angles, θ(q) and ϕ(q), which are spherical angles relative to the $$\hat{z}$$ direction. These rates were computed with a first-order expansion. Regions with squared growth rates below 10−16 are shown in white. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. 6.4 Baroclinic instability The baroclinic instability arises in otherwise stably stratified fluids when the entropy gradient is not parallel to the pressure gradient (Killworth 1980). In fact, this is part of a family of instabilities, which includes the convective instability (Lebovitz 1965). This family provides a continuous connection between the unstable convective and stably stratified limits. To explore, it consider Fig. 16 which shows the variation of r − r and r − θ correlation functions against the angle δ between the entropy gradient and the pressure gradient. The radial correlations peak when the two gradients are aligned. This is the convective limit. These correlations fall to zero in the opposing limit where the two gradients are anti-aligned, which is the stably stratified limit. In between these limits, the behaviour is approximately that of cos 2δ. Figure 16. View largeDownload slide Various correlation functions are shown as a function of the angle δ between the entropy and pressure gradients. The functions are the r − r (left-hand panel) and r − θ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for a non-rotating convective region on the equator and with no magnetic field. Shown in purple (*, dashed) for comparison is the KR result. This agrees in sign, and for r − r agrees in scale, but their r − θ prediction is considerably larger. Notably, this comparison is precisely as cos (δ) (left-hand panel) and sin (δ) (right-hand panel) and crosses zero at non-extremal angles. This is most likely because their theory is not designed for nearly stable regions with extreme baroclinicity. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure 16. View largeDownload slide Various correlation functions are shown as a function of the angle δ between the entropy and pressure gradients. The functions are the r − r (left-hand panel) and r − θ (right-hand panel) velocity (red) and diffusivity (blue) correlation functions. These results are for a non-rotating convective region on the equator and with no magnetic field. Shown in purple (*, dashed) for comparison is the KR result. This agrees in sign, and for r − r agrees in scale, but their r − θ prediction is considerably larger. Notably, this comparison is precisely as cos (δ) (left-hand panel) and sin (δ) (right-hand panel) and crosses zero at non-extremal angles. This is most likely because their theory is not designed for nearly stable regions with extreme baroclinicity. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. By contrast, the r − θ correlations behave approximately as sin δ, and vanishes when δ = 0. This is because both the aligned and the anti-aligned limits are spherically symmetric and so must have this correlation function vanish. Deviations from the convective limit give rise to linear scaling so the convective baroclinic instability transports heat and momentum at first order in the baroclinicity. This is an entirely distinct phenomenon from the thermal wind balance, which is a large-scale effect while this results from integrating out the small-scale turbulent modes. In the stable limit perturbations arise quadratically, a deviation from the behaviour of sin δ. This is because there are no existing turbulent motions to perturb, and so each position and velocity component is linear in δ and gives rise to a quadratic two-point correlation function. 6.5 Stellar magnetism We now turn to the impact of the magnetic field on convective turbulence in stars. Fig. 17 shows 〈δvrδvr〉 in a mildly rotating (Ω = 0.1|N|) convection zone as a function of B for three polarisations; radial ($$\boldsymbol {B} \parallel \hat{\boldsymbol {r}}$$), latitudinal ($$\boldsymbol {B} \parallel \hat{\boldsymbol {\theta }}$$), and longitudinal ($$\boldsymbol {B} \parallel \hat{\boldsymbol {\phi }}$$). As the field increases, the stress falls off. This is because the field quenches the turbulence by providing a stabilizing restoring force, and is in general agreement with Canuto & Hartke (1986). Interestingly, the only significant differences are between the radial and angular field polarisations! The θ and ϕ polarisations show precisely the same behaviour out to very strong fields. This is a result of symmetry, because the radial stress is not sensitive to rotation about the radial direction. The deviation seen with strong fields is a numeric artefact and decreases with increasing integration time. Figure 17. View largeDownload slide The stress 〈δvrδvr〉 is shown as a function of magnetic field strength. The magnetic field is polarized radially (red), longitudinally (purple), and latitudinally (blue). The system is rigidly rotating at Ω = 0.1|N| at a latitude of π/4. All quantities are given in units of the mixing length and Brünt–Väisälä frequency. Figure 17. View largeDownload slide The stress 〈δvrδvr〉 is shown as a function of magnetic field strength. The magnetic field is polarized radially (red), longitudinally (purple), and latitudinally (blue). The system is rigidly rotating at Ω = 0.1|N| at a latitude of π/4. All quantities are given in units of the mixing length and Brünt–Väisälä frequency. By contrast, consider 〈δvrδvϕ〉, shown in Fig. 18. This component, along with the corresponding Maxwell stress, is responsible for transporting angular momentum. Interestingly, it shows differences amongst all polarisations, with the strongest difference between the θ polarization and the others. This is because the stress is mixed between different directions and so is sensitive to all variations in the magnetic field direction. The large difference of the θ polarization relative to the others reflects the fact that motion is damped perpendicular to the magnetic field so the θ polarization damps motion in both directions involved in this component of the stress, whereas the r and ϕ polarizations only damp motion in one of the two directions. Figure 18. View largeDownload slide The stress 〈δvrδvϕ〉 is shown as a function of magnetic field strength. The magnetic field is polarized radially (red), longitudinally (purple) and latitudinally (blue). The system is rigidly rotating at Ω = 0.1|N| at a latitude of π/4. All quantities are given in units of the mixing length and Brünt–Väisälä frequency. Figure 18. View largeDownload slide The stress 〈δvrδvϕ〉 is shown as a function of magnetic field strength. The magnetic field is polarized radially (red), longitudinally (purple) and latitudinally (blue). The system is rigidly rotating at Ω = 0.1|N| at a latitude of π/4. All quantities are given in units of the mixing length and Brünt–Väisälä frequency. 6.6 Magnetorotational instability As a final example, we consider the MRI (Chandrasekhar 1960). This instability arises in magnetized fluids undergoing Keplerian orbital motion. Fig. 19 shows the r − r, θ − ϕ, r − ϕ, and r − θ Reynolds and Maxwell stresses for an accretion disc with a vertical magnetic field. Contrary to predictions (Chandrasekhar 1960) none of the Reynolds stresses vanish in the zero-field limit. This is because the linear system supports short-term growing modes but, while they only grow in the short-time limit, our numerical methods are not sensitive to that effect at this order. In principle, at higher order, this phenomenon should become evident and so this may be interpreted as an artefact associated with our expanding to low order in |R∇Ω| > 1. Despite this, it is likely that other non-magnetic processes can destabilise these modes even in the long term and so we feel it is appropriate to at least consider them (cf. Luschgy & Pagès 2006). The Maxwell stresses by contrast do vanish as B → 0. This is to be expected because they are proportional to B2. Figure 19. View largeDownload slide From the top to bottom panel are the r − ϕ, r − r, r − θ, and θ − ϕ stresses. The Reynolds (velocity) stresses are in red and the Maxwell (magnetic) stresses are in blue. Note that it is the negative r − r Maxwell stress, which is shown to make the comparison with the Reynolds stresses clearer. In all cases, they are shown as a function of B for a Keplerian disc. The magnetic field is taken parallel to $$\hat{z}$$. The system is taken to be stably stratified in the vertical direction with |N| = Ω and hence L0 = h = R. All quantities are given in units of the mixing length and Ω. Figure 19. View largeDownload slide From the top to bottom panel are the r − ϕ, r − r, r − θ, and θ − ϕ stresses. The Reynolds (velocity) stresses are in red and the Maxwell (magnetic) stresses are in blue. Note that it is the negative r − r Maxwell stress, which is shown to make the comparison with the Reynolds stresses clearer. In all cases, they are shown as a function of B for a Keplerian disc. The magnetic field is taken parallel to $$\hat{z}$$. The system is taken to be stably stratified in the vertical direction with |N| = Ω and hence L0 = h = R. All quantities are given in units of the mixing length and Ω. As the magnetic field increases, the r − ϕ and θ − ϕ Reynolds stresses change sign. This indicates the onset of MRI modes, which have the opposite sign to the zero-field correlations. This effect saturates when vA ≈ Ωh, where h is the scale height of the disc. The total r − ϕ stress saturates at roughly 10−2(hΩ)2, which lies between those typically found in simulations and those inferred from observations (Starling et al. 2004; King et al. 2007). Note that at the saturation point the Maxwell and Reynolds stresses are comparable, and beyond this point the Maxwell stress increases while the Reynolds stress falls off. Above the saturation point the Reynolds stresses drop off as the magnetic field quenches the turbulence. This is precisely what is expected for the MRI (Balbus & Hawley 1991). The Maxwell stresses, however, continue to grow, again in line with expectations. Some care is required to interpret these results because they were computed for a fixed field and that field may or may not be stable under the action of the turbulence it generates (Pessah, Chan & Psaltis 2006). Furthermore, there are challenges with the α-disc prescription, which make the specific stress components more difficult to interpret (Pessah, Chan & Psaltis 2008). Nevertheless, it is encouraging that what we see matches well with both observations and simulations. 7 CONCLUSION We have derived a turbulent closure model, which incorporates shear, rotation, and magnetism as well as a full three-dimensional spectrum of fluctuations. We have also presented a new perturbative approach to incorporate time-dependence in the evolution equations. This model, which is implemented in an open source numerical software package, fully reproduces many known phenomena such as the MRI, baroclinic instability, rotational quenching, and more classic shear instabilities. Using this model, we have determined the asymptotic behaviour of a wide variety of correlation functions and transport coefficients under a wide range of circumstances, many of which do not appear in the literature. We have further explored the behaviour of turbulent transport coefficients in intermediate regimes where no single phenomenon dominates, such as in the critical MRI. In these cases, the behaviour is generally complex and does not separate easily into components associated with the different pieces of input physics. The closure formalism developed here fills a new niche in the landscape of solutions to turbulent transport, covering enough phenomena to be useful to understand those operating in stars, planets, and accretion discs, while being rapid enough to be incorporated into stellar evolution codes on nuclear time-scales. In the future, we hope to provide further refinements and comparisons with direct numerical simulations as well as experiments. In addition, it would be interesting to explore the results of this model to higher order in the shear and, even at this order, there are many results which deserve more analysis than we have given here. Acknowledgements ASJ acknowledges financial support from a Marshall Scholarship as well as support from the Institute of Astronomy, École Normale Supérieure (ENS), and Centre for Excellence in Basic Sciences (CEBS) to work at ENS Paris and CEBS in Mumbai. PL acknowledges travel support from the french PNPS (Programme National de Physique Stellaire) and from CEBS. CAT thanks Churchill College for his fellowship. SMC is grateful to the IOA for support and hopsitality and thanks the Cambridge-Hamied exchange program for financial support. The authors also thank Rob Izzard and Science and Technology Facilities Council Grant ST/L003910/1 for CPU cycles, which aided in this work. Footnotes 1 It would not be difficult, however, to incorporate them into this framework at a later date. REFERENCES Ashkenazi S., Steinberg V., 1999, Phys. Rev. Lett. , 83, 4760 CrossRef Search ADS Balbus S. A., Hawley J. F., 1991, ApJ , 376, 214 CrossRef Search ADS Balbus S. 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W., Vahala G., 1997, NASA Contractor Report 201718 APPENDIX A: TURBULENT INDEX The general question of which turbulent index to use and under what circumstances remains open though many specific cases are well understood. In the case of isotropic incompressible turbulence the Kolmogorov index is well-known to be n = 11/6 (Kolmogorov 1941a). There is more debate over the index to use for convection, with answers ranging from n = 5/2 (Benzi et al. 1994) to n = 21/10 (Procaccia & Zeitak 1989) and n = 2.4 ± 0.2 (Ashkenazi & Steinberg 1999). There has also been work attempting to determine the spectrum in a context-sensitive manner through energy balance arguments (Yakhot & Orszag 1986). In the magnetised case sources differ even more, with some suggesting that this range still applies (Dobrowolny, Mangeney & Veltri 1980), some arguing for a Kolmogorov-like spectrum (Goldreich & Sridhar 1995) and others giving a range of indices depending on geometry and the direction of the wavevector (Sridhar & Goldreich 1994). From numerical experiments with our closure model, we have found that the magnetic stress scales sufficiently rapidly with k that it is divergent for n = 11/6 and not for n = 8/3. This favours the scenario of Goldreich & Sridhar (1995), who argue that in the strongly magnetized limit the index ought to be n = 8/3. In order to consistently treat both the non-magnetic and the strongly magnetized limits, we choose a simple prescription in which n = 11/6 when one of |N|, or |R∇Ω| exceeds kvA and use n = 8/3 otherwise. This means that there is a critical wavenumber \begin{eqnarray} k_{\rm c} \equiv \frac{\max \left(|N|, |R\nabla \Omega |\right)}{v_A} \end{eqnarray} (A1)at which the spectrum changes. In the non-magnetic case, the evolution matrix is independent of the magnitude of the wavevector and so altering the index just alters the correlation coefficients by a multiplicative factor. In the magnetic case, the potential for error is larger because the magnitude of the wavevector is relevant but there appears to be no consensus on the best prescription and so we make do with what is available. APPENDIX B: BOUSSINESQ ODDITIES In this work, we have taken the Boussinesq approximation. In Fourier space this is \begin{eqnarray} \boldsymbol {q}\cdot \tilde{\delta \boldsymbol {r}}=0. \end{eqnarray} (B1)Taking the time derivative of both sides we see that \begin{eqnarray} \partial _t\left(\boldsymbol {q}\cdot \delta \tilde{\boldsymbol {r}}\right) = \boldsymbol {q}\cdot \delta \tilde{\boldsymbol {v}} + \delta \tilde{\boldsymbol {r}}\cdot \partial _t\boldsymbol {q} = 0. \end{eqnarray} (B2)As a result \begin{eqnarray} \delta \tilde{\boldsymbol {v}}\cdot \boldsymbol {q} = - \delta \tilde{\boldsymbol {r}}\cdot \partial _t\boldsymbol {q} \ne 0. \end{eqnarray} (B3)This is quite peculiar, but is just an artefact of our coordinate system. Because the wavevectors are time-dependent, maintaining the volume of a fluid parcel requires that the displacement be orthogonal to the wavevector, which actually means that the velocity is generally not orthogonal to the wavevector. APPENDIX C: SOFTWARE DETAILS The software used for this work is Mixer version 1, which we have released under a GPLv3 license at github.com/adamjermyn/Mixer. All data produced for this work are available at the same location as HDF5 tables with attributes documenting the physical inputs. Post-processing and visualization of the data was with the Python modules Numpy (van der Walt, Colbert & Varoquaux 2011) and Matplotlib (Hunter 2007) and the relevant scripts for this are included with Mixer. The core of Mixer is written in C++, for performance reasons, and the code is supplied with a Makefile, which supports compilation on both Linux and MacOS. Mixer makes use of the Eigen library (Guennebaud et al. 2010) for linear algebra. Mixer also uses the Cubature library for numerical integration. This library is an implementation of the algorithms by Genz & Malik (1980) and Berntsen, Espelid & Genz (1991). These integration routines are supplemented by a Python integration routine tailored for integrands with small support regions. The details will be explored in later work. In addition, many routines provide a Python interface. Currently Mixer supports only single-threaded operation, though it may be used inside parallelized scripts through the Python wrapper. The version of Mixer used to generate the data in this work was compiled against Cubature version 1.0.2 and Eigen version 3.3.3, though the code does not use any features which require recent versions, so many likely suffice. Mixer is optimized for convecting systems for which achieving accuracy better than 10−5 relative and absolute typically requires between 1 ms and 1 s on a single core of a 2016 Intel CPU. This is further improved when the differential rotation is minimal, in which case the perturbative expansion may be turned off to save a factor of several in runtime. In stably stratified zones and those with magnetic fields up to 103s may be required to achieve good convergence. In cases where the code has more difficulty, it is quite likely that Mixer becomes the bottleneck in simulations and so, under these circumstances, we recommend tabulating results in advance. This is still considerably more performant than direct numerical simulation, and the results can generally be guaranteed to converge at much higher precision, so that derivatives may be extracted as well. At various points in the software, we must divide by the magnitude of the velocity of an eigenmode. This may approach zero in some cases. To avoid dividing by zero in these cases, we place a lower bound on this magnitude, such that \begin{eqnarray} |\delta v|^2 \ge \epsilon , \end{eqnarray} (C1)where $$\epsilon = 10^{-20} L_0^2 |N|^2$$ in the calculations presented in this work. This corresponds to setting an upper bound on the length scale d of the displacements $$\delta \boldsymbol {r}$$, namely \begin{eqnarray} |\delta r|^2 \le L_0^3 |N| \epsilon ^{-1/2}, \end{eqnarray} (C2)which means that d = 1010L0 in this work. To verify that this numerical fix does not impact our results, we have examined the correlation functions in several scenarios as a function of this numerical cut off L. For example, Fig. C1 shows the r − θ and r − ϕ correlations as functions of d for a stably stratified differentially rotating system. The results are constant over many orders of magnitude so long as d > 104L0, which is easily satisfied by our default. Figure C1. View largeDownload slide The absolute values of 〈δvrδvθ〉 (left-hand panel) and 〈δvrδvϕ〉 (right-hand panel) are shown as functions of d, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction with |R∇ln Ω| = 10−3, Ω = 0.1|N| and no magnetic field. The data is computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. Figure C1. View largeDownload slide The absolute values of 〈δvrδvθ〉 (left-hand panel) and 〈δvrδvϕ〉 (right-hand panel) are shown as functions of d, with both axes log-scaled. These results are for a stably stratified region with differential rotation in the radial direction with |R∇ln Ω| = 10−3, Ω = 0.1|N| and no magnetic field. The data is computed for a point on the equator with differential rotation at an angle of π/4. All quantities are given in units of the mixing length and the Brünt–Väisälä frequency. © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society
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Monthly Notices of the Royal Astronomical SocietyOxford University Press
Published: May 1, 2018
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http://consigliogiovanifiuggi.it/tmfc/derivative-of-sigmoid-python.html | # Derivative Of Sigmoid Python
Deep Learning from first principles in Python, R and Octave – Part 3 The 3rd part implemented a multi-layer Deep Learning Network with sigmoid. We use the notation: θxi: = θ0 + θ1xi1 + ⋯ + θpxip. def __sigmoid(self, x): return 1 / (1 + exp(-x)) # The derivative of the Sigmoid function. If your output is 0, 1 value, if you're using binary classification, then the sigmoid activation function is a very natural choice for the output layer. Construction of sigmoid function based integral-derivative observer (SIDO) In this section, the specific formulation of proposed sigmoid function based integral-derivative observer (SIDO) is given and its stability is well-established using the concept of exponential stability and singular perturbation theory, as described in Theorem 2. As such, neural networks tend to employ a select few activation functions (identity, sigmoid, ReLU and their variants). Let's continue to code our Neural_Network class by adding a sigmoidPrime (derivative of sigmoid) function:. increase or decrease) and see if the performance of the ANN increased. Logistic regression - Sigmoid and Sigmoid derivative part 1 Python Lessons. Nesterov Momentum. If the sigmoid's output is a variable "out", then the derivative is simply out * (1-out). That's the essental difference. 5 (6,169 ratings) Course Ratings are calculated from individual students’ ratings and a variety of other signals, like age of rating and reliability, to ensure that they reflect course quality fairly and accurately. In this article, I will discuss about how to implement a neural network to classify Cats and Non-Cat images in python. We use the first derivative of the Sigmoid function to calculate the gradient since it is very convenient. tanh(number); Number: It can be a number or a valid numerical expression for which you want to find hyperbolic Tangent value. Convenience function griddata offering a simple interface to interpolation in N dimensions (N = 1, 2,. Let’s quickly recap the core concepts behind recurrent neural networks. In fact, The MathWorks just included it in their most recent update to the Neural Network toolbox. This is known as the partial derivative, with the symbol ∂. Derivative of sigmoid? I'm creating a neural network using the backpropagation technique for learning. def sigmoid_derivative (x): """ Compute the gradient (slope/derivative) of the sigmoid function with respect to its input x. The derivative of the logarithmic function is given by: f ' (x) = 1 / (x ln(b) ) x is the function argument. Lutfi Al-Sharif 14,856 views. Neural Network Back-Propagation Using Python You don't have to resort to writing C++ to work with popular machine learning libraries such as Microsoft's CNTK and Google's TensorFlow. We need the logistic function itself for calculating postactivation values, and the. How to properly derive the derivative of sigmoid function assuming the input is a matrix - i. In fact, The MathWorks just included it in their most recent update to the Neural Network toolbox. Sigmoid is used as the gating function for the 3 gates(in, out, forget) in LSTM, because it outputs a value between 0 and 1, there can be either no flow or complete flow of information throughout the gates. We then define the sigmoid_activation function on is the derivative of the sigmoid function? Thanks! Ben. Part 2 implemented the most elementary neural network with 1 hidden layer, but with any number of activation units in that layer, and a sigmoid activation at the output layer. You can store the output of the sigmoid function into variables and then use it to calculate the gradient. In mathematics, the softmax function, also known as softargmax or normalized exponential function,: 198 is a function that takes as input a vector of K real numbers, and normalizes it into a probability distribution consisting of K probabilities proportional to the exponentials of the input numbers. Congratulations on your pending nuptial. is_pow: return self. The default is -1 which indicates the last dimension. exp (logits), axis) logits: A non-empty Tensor. More formally, in integration theory it is a weak derivative, and in convex function theory the subdifferential of the absolute value at 0 is the interval. As the derivative of sigmoid(x) is sigmoid(x) * (1-sigmoid(x)) we pass the sigmoid(x) value as the state variable, which is then later fed into backward(). The Sigmoid Activation Function: Activation in Multilayer Perceptron Neural Networks The entire Python program is included as an image at the end of this article, and the file ("MLP_v1. Hello, for numerical differentiation, there are the following functions in octave-forge : deriv (f,x0[,h,O,N]) ## 1st, 2nd, 3rd or 4th order derivative of a function, computed ## using 2nd or 4th order method. We can store the output of the sigmoid function into variables and then use it to calculate the gradient. logsig is a transfer function. Here are plots of the sigmoid, \tanh and rectified linear functions: The \tanh(z) function is a rescaled version of the sigmoid, and its output range is [-1,1] instead of [0,1]. Part 2 implemented the most elementary neural network with 1 hidden layer, but with any number of activation units in that layer, and a sigmoid activation at the output layer. "Mastering Calculus for Deep learning / Machine learning / Data Science / Data Analysis / AI using Python " With this course, You start by learning the definition of function and move your way up for fitting the data to the function which is the core for any Machine learning, Deep Learning , Artificial intelligence, Data Science Application. Transfer functions calculate a layer’s output from its net input. In this post, I'm going to implement standard logistic regression from scratch. Once you feel comfortable with the concepts explained in those articles, you can come back and continue. It is well-known that the p-Sigmoid, or logistic function, can be used for piecewise approximations to other functions. Use the "Preview Post" button to make sure the code is presented as you expect before hitting the "Post Reply/Thread" button. I tested it out and it works, but if I run the code the way it is right now (using the derivative in the article), I get a super low loss and it's more or. We can think of this as probabilities. pyplot as plt import. random((3, 1)) - 1 # The Sigmoid function, which describes an S shaped curve. Next up in our top 3 activation functions list is the Softmax function. In this video, we’ll talk about how to compute derivatives for you to implement gradient descent for logistic regression. LSTMs belong to the family of recurrent neural networks which are very usefull for learning sequential data as texts, time series or video data. The outputs are then passed to the next layer. With softmax we have a somewhat harder life. 2k) SQL (822) Big Data Hadoop & Spark (852) Data Science (1. Derivative of Sigmoid Function Step 1-Applying Chain rule and writing in terms of partial derivatives. A derivative is just a fancy word for the slope or the tangent line to a given point. The focus of this article will be on the math behind simple neural networks and implementing the code in python from scratch. Sigmoid is a smoothed out perceptron. These neurons are called saturated neurons. using m Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. A derivative is just a fancy word for the slope or the tangent line to a given point. Sigmoid(double lo, double hi) Sigmoid function. So, if g of z is the sigmoid function, then the slope of the function is d, dz g of z, and so we know from calculus that it is the slope of g of x at z. Derivative ของ Tanh Function. # GRADED FUNCTION: sigmoid_derivative def sigmoid_derivative(x): """ Compute the gradient (also called the slope or derivative) of the sigmoid function with respect to its input x. def sigmoid_derivative (x): Compute the gradient (also called the slope or derivative) of the sigmoid function with respect to its input x. y) Example 23 Project: Wall-Following-Robot Author: seifullaah73 File: bpnn. This will lead to “Vanishing Gradients” problem. 88079708, 0. The course attempts to make the material as accessible as possible. Compute σ. What is the derivative of ReLU? ubuntu - black screen on ubuntu laptop after installing nvidia drivers; How to normalize vectors to unit norm in Python; How to Compute the Derivative of a Sigmoid Function (fully worked example) How to fix "Firefox is already running, but is not responding" How to use the Springer LNCS LaTeX template. Implement a neural network from scratch with Python/Numpy — Backpropagation. In this … Read more Derivative of Sigmoid Function. In the two-class logistic regression, the predicted probablies are as follows, using the sigmoid function:. With the help of Sigmoid activation function, we are able to reduce the loss during the time of training because it eliminates the gradient problem in machine learning model while training. First, calculate the derivative of sigmoid. 1 #Setting learning rate inputlayer_neurons = X. Sigmoid function is best used for modelling probability based networks. Williams On the Derivatives of the Sigmoid, Neural Networks, 6(1993), 845-853. The Sigmoid Activation Function: Activation in Multilayer Perceptron Neural Networks The entire Python program is included as an image at the end of this article, and the file ("MLP_v1. Today I learned about the Elliot Activation (or Sigmoid) Function. 5 >>> sigmoid(-6) 0. py") is provided as a download. Squashing functions limit the output to a range between 0 and 1, making these functions useful in the prediction of probabilities. Ask Question Asked 2 years, 11 months ago. You should be comfortable with variables and coefficients, linear equations. import numpy as np #Input array X=np. To create a logistic regression with Python from scratch we should import numpy and matplotlib libraries. exp(-x)) def sigmoid_derivative(x): return x * (1. Sigmoid has the property that for y=sigmoid(x), dy/dx= y(1-y) In python. The diagram below shows the architecture of a 2-layer Neural Network (note that the input layer is typically excluded when counting the number of layers in a Neural Network) Architecture of a 2-layer Neural Network. Unlike logistic regression, we will also need the derivative of the sigmoid function when using a neural net. It's very similar to linear regression, so if you are not familiar with it, I recommend you check out my last post, Linear Regression from Scratch in Python. An alternative to the sigmoid is the symmetrical sigmoid S(x) defined as S(x) = 2s(x. For values of in the domain of real numbers from − ∞ to + ∞, the S-curve shown on the right. It was first used in the work by L'Abbe Sauri (1774). For my implementation, values of 0. That is, the key equations you need in order to implement gradient descent for logistic regression. import numpy as np. Derivative ของ Tanh Function. The sigmoid function isn’t a step function however, the edge is “soft”, and the output doesn’t change instantaneously. The sigmoid and hyperbolic tangent activation functions cannot be used in networks with many layers due to the vanishing gradient problem. After completing this tutorial, you will know: How to forward-propagate an […]. Why not register and get more from Qiita? We will deliver articles that match you. We have to note that the numerical range of floating point numbers in numpy is limited. If you have a function that can be expressed as f (x) = 2x^2 + 3 then the derivative of that function, or the rate at which that function is changing, can be calculated with f' (x) = 4x. #' the activation function sigmoid <- function(x) { 1. , housing prices) as a linear function of input values (e. Free: Licensed under BSD, SymPy is free both as in speech and as in beer. It supports standard Python arithmetic and. The intuition behind using a sigmoid function to fit a binary decision is that it prevents extreme points from moving the zero crossing point (t. big o notation, euclidean, Java, modular inverse, multiplicative inverse, python, rsa, stranger things Sinc as a Neural Networks Activation Function Sinc function is a sinusoidal activation function in neural networks. is_var: return self. The derivative of the sigmoid, also known as sigmoid prime, will give us the rate of change, or slope, of the activation function at output sum. You must use the output of the sigmoid function for σ(x) not the gradient. Python was created out of the slime and mud left after the great flood. The network has three neurons in total — two in the first hidden layer and one in the output layer. First I plot sigmoid function, and derivative of all points from definition using python. # We pass the weighted sum of the inputs through this function to # normalise them between 0 and 1. One popular method was to perturb (adjust) the weights in a random, uninformed direction (ie. It can help in avoiding version conflicts and dependency issues among projects. But it doesn't gel when I think, precisely, of how to apply it to the results of i) linear combiner and ii) sigmoid activation function. The function to apply logistic function to any real valued input vector "X" is defined in python as """ function applies logistic function to a real valued input vector x""" def sigmoid(X): '''Compute the sigmoid function ''' den = 1. We saw our neural network gave a pretty good predictions of our test score based on how many hours we slept, and how many hours we studied the night before. GitHub Gist: instantly share code, notes, and snippets. With softmax we have a somewhat harder life. Sigmoid means ‘S’-shaped: the function maps (1;1) onto (0;1) — it is a “squashing function”. The sigmoid function is differentiable at every point and its derivative comes out to be. , “Observation of a new particle in the search for the Standard Model Higgsboson with the ATLAS detector at the LHC”, Phys. When the input data is transmitted into the neuron, it is processed, and an output is generated. Neural Networks for Machine Learning From Scratch 4. The course attempts to make the material as accessible as possible. Lines 2-5 import our required Python packages. The main goal of this reading is to understand enough statistical methodology to be able to leverage the machine learning algorithms in Python’s scikit-learn library and then apply this knowledge to solve a. This is a classification. simple sigmoid function with Python. Exercise: Now, implement the backward propagation step (derivative computation) of Figure 1. is non-decreasing, that is for all ; has horizontal asymptotes at both 0 and 1 (and as a consequence, , and ). # application of the chain rule to find derivative of the loss function with respect to weights2 and weights1 d_weights2 = np. Sigmoid is a smoothed out perceptron. It begins by looking sort of like the step function, except that the values between two points actually exist on a curve, which means that you can stack sigmoid functions to perform classification with multiple outputs. Tanh function is better than sigmoid function. Posted by Keng Surapong 2019-08-20 2020-01-31 Posted in Artificial Intelligence, Knowledge, Machine Learning, Python Tags: activation function, artificial intelligence, artificial neural network, converge, deep learning, deep neural networks, derivative, gradient, machine learning, multi-layer perceptron, neural networks, probability, sigmoid. After that, we can calculate the derivative of the predicted to the sop by calculating the derivative of the sigmoid function according to the figure below. exp ( - x )) Then, to take the derivative in the process of back propagation, we need to do differentiation of logistic function. Implementing gradient descent with Python. For vector inputs of length the gradient is , a vector of ones of length. The function was first introduced in 1993 by D. Python was created out of the slime and mud left after the great flood. Tanh function is better than sigmoid function. Sigmoid Function -- from Wolfram MathWorl. Tensorflow is an open-source machine learning library developed by Google. 2 Sigmoid gradient. One of my. I've gone over similar questions , but they seem to gloss over this part of the calculation. I tried different learning rates and found that 0. This may be somewhat abstract, so let's use another example. The link does not help very much with this. Sigmoid derivative. Learn how to handle data by normalizing inputs and reshaping images. The Logistic Sigmoid Activation Function. it fits the data which output variable either 0 or 1 and results a probability value for each data point. py GNU General Public License v3. The whole idea behind the other activation functions is to create non-linearity, to be able to model highly non-linear data that cannot be solved by a simple regression ! ReLU. sigmoid :: sigmoid_output_to_derivative (a) # a was created above using sigmoid() ## [1] 0. We will derive the Backpropagation algorithm for a 2-Layer Network and then will generalize for N-Layer Network. exp (-x)) plt. Python was created out of the slime and mud left after the great flood. During backpropagation through the network with sigmoid activation, the gradients in neurons whose output is near 0 or 1 are nearly 0. exp(-x)) # derivative of sigmoid # sigmoid(y) * (1. One such approximation is called softplus which is defined y = ln(1. Monotonic function: A function which is either entirely non-increasing or non-decreasing. Discover how to code ML algorithms from scratch including kNN, decision trees, neural nets, ensembles and much more in my new book, with full Python code and no fancy libraries. For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. Simple Neural Networks: Revised. Keras, on the other side, makes you focus on the big picture of what the LSTM does, and it’s great to quickly implement something that works. An activation function is said to saturate (without qualification) if it both left and right saturates. Sigmoid function go either from 0 to 1 or from -1 to 1 based on the convention. Let’s say we feed the value 5 into the first function, and let’s say further that computing the derivative of the first function at u = 5 gives us a value of 3—that is, d f 1 d. Those partial derivatives are going to be used during the training phase of your model, where a loss function states how much far your are from the correct result. To do so, we use the linspace method from the NumPy library. pyplot as plt. Whenever we want to use this function, we can supply the parameter True to get the derivative, We can omit this, or enter False to just get the output of the sigmoid. Python had been killed by the god Apollo at Delphi. If the sigmoid's output is a variable "out", then the derivative is simply out * (1-out). Hello again in the series of tutorials for implementing a generic gradient descent (GD) algorithm in Python for optimizing parameters of artificial neural network (ANN) in the backpropagation phase. The backpropagation algorithm was a major milestone in machine learning because, before it was discovered, optimization methods were extremely unsatisfactory. Here's the bottom line: I. Whilst I agree with the general consensus of responders that this is not the best way to solve the minimisation problem in the question, I have now resolved the challenge and can answer my own question to share the way one might overcome similar issues in using penalty methods to resolve optimisation problems in Python. Take a closer look at the sigmoid function's curve on the graph above. LogisticSigmoid [z] has no branch cut discontinuities. Memoization is a computer science term which simply means: don't recompute the same thing over and over. I have recently completed the Machine Learning course from. is_const elif self. The sigmoid function (logistic curve) is one of many curves use in neural networks. What is the role of this. Recommended for you. In this article, I will discuss about how to implement a neural network to classify Cats and Non-Cat images in python. The derivative of the logarithmic function is given by: f ' (x) = 1 / (x ln(b) ) x is the function argument. Python’s x % y returns a result with the sign of y instead, and may not be exactly computable for float arguments. io Find an R package R language docs Run R in your browser R Notebooks. import numpy as np. The sigmoid function squashes the outputs to a value between zero and one. Why is the de-facto standard sigmoid function, $\frac{1}{1+e^{-x}}$, so popular in (non-deep) neural-networks and logistic regression? Why don't we use many of the other derivable functions, with faster computation time or slower decay (so vanishing gradient occurs less). The weights indicate the importance of the input in the decision-making process. Minai and R. Backpropagation is an algorithm that calculate the partial derivative of every node on your model (ex: Convnet, Neural network). , the size of the house). derivative of cost function for Logistic Regression. You can vote up the examples you like or vote down the ones you don't like. I believe I'm doing something wrong, since the softmax function is commonly used as an activation function in deep learning (and thus cannot always have a derivative of $0$). ToTensor() to the raw data. 2 Sigmoid gradient. Discover how to code ML algorithms from scratch including kNN, decision trees, neural nets, ensembles and much more in my new book, with full Python code and no fancy libraries. Exercise: Now, implement the backward propagation step (derivative computation) of Figure 1. 3081 is the standard deviation relative to the values generated just by applying transforms. Recommended for you. The second function converts the sigmoid value of a number to its derivative. First, we have to talk about neurons, the basic unit of a neural network. According to Wikipedia, a sigmoid function is a mathematical function having a characteristic “S”-shaped curve or sigmoid curve. We need to increase the BIAS of the ACTIVATION function if this slope is -ve and we need to decrease our BIAS if the slope is +ve. How to build your own Neural Network from scratch in Python Python Nov 16, 2018 176 Inspiration: As a major aspect of my own adventure to pick up a superior comprehension of Deep Learning, I've chosen to assemble a Neural Network sans preparation without a profound learning library like TensorFlow. There are a number of nonlinear solvers in core MATLAB and different Toolboxes that can fit an ‘inverse sigmoid model’ to your data. import numpy as np def sigmoid_derivative(x): s = sigmoid(x) ds = s*(1-s) return ds Above we compute the gradient (also called the slope or derivative) of the sigmoid function with respect to its input x. The derivative of the sigmoid, also known as sigmoid prime, will give us the rate of change, or slope, of the activation function at output sum. I've gone over similar questions , but they seem to gloss over this part of the calculation. The Feedforward Backpropagation Neural Network Algorithm. Here's what a 2-input neuron looks like: 3 things are happening here. Monotonic function: A function which is either entirely non-increasing or non-decreasing. I thought the derivative of a sigmoid function output is just the slope of the sigmoid line at a specific point. It is the technique still used to train large deep learning networks. Larz60+ wrote Oct-18-2018, 05:31 PM: Please post all code, output and errors (it it's entirety) between their respective tags. 2 Sigmoid gradient. Quoting myself from this answer to a different question:. The derivative of the logarithmic function is given by: f ' (x) = 1 / (x ln(b) ) x is the function argument. This comment has been minimized. The output of a sigmoid function, superimposed on that of a threshold function, is shown in Figure 3. t theta of the cost function (Hessian’s matrix) and the gradient vector w. def sigmoid(z): return 1/(1+np. The backpropagation algorithm was a major milestone in machine learning because, before it was discovered, optimization methods were extremely unsatisfactory. Well, you calculated the sigmoid(Z2). Fitting a function to data with nonlinear least squares. It does also share its asymptotic properties with Sigmoid: although for very large values of $$x$$ the function approaches 1, it never actually equals it. We examined what the derivative of ReLU activation function is, and why it is this. Let this be a reminder to you to not rely on libraries too much for implementing your machine learning algorithms. The derivative of the sigmoid, also known as sigmoid prime, will give us the rate of change, or slope, of the activation function at output sum. Scientists tend to consume activation functions which have meaningful derivatives. Python basics, AI, machine learning and other tutorials Sigmoid and Sigmoid derivative functions. Memoization is a computer science term which simply means: don't recompute the same thing over and over. So, first we need to write out the function that calculates the derivative of our sigmoid, which gives us our gradient, or slope. Logistic regression is a classification algorithm used to assign observations to a discrete set of classes. Simple Neural Networks in Python. Important alternative hidden layer activation functions are logistic sigmoid and rectified linear units, and each has a different associated derivative term. I understand that passing True to the function to get the derivative can be convenient, but consider doing something less confusing. Hint: you may want to write your answer in terms of ˙(x). We can store the output of the sigmoid function into variables and then use it to calculate the gradient. Unlike logistic regression, we will also need the derivative of the sigmoid function when using a neural net. f'(x)=f(x)(1-f(x)) Sigmoid function is monotonic but its derivative is not monotonic. Let’s say we feed the value 5 into the first function, and let’s say further that computing the derivative of the first function at u = 5 gives us a value of 3—that is, d f 1 d. In this video, we’ll talk about how to compute derivatives for you to implement gradient descent for logistic regression. Process: $$f(z)=\\frac{1}{1+e^{-z}}$$ $$f'(z)=\\frac{e^{-z}}{{(1+e^{-z}})^{2}}$$ $$=\\frac{1+e^{-z}-1}{{(1+e^{-z}})^{2}}$$ =\\frac{1}{(1+e^{-z}}) – \\frac{1. Memoization is a computer science term which simply means: don't recompute the same thing over and over. It is of S shape with Zero centered curve. However, it wasn't until 1986, with the publishing of a paper by Rumelhart, Hinton, and Williams, titled "Learning Representations by Back-Propagating Errors," that the importance of the algorithm was. Logistic regression - Sigmoid and Sigmoid derivative part 1 Python Lessons. Summary: I learn best with toy code that I can play with. The network has three neurons in total — two in the first hidden layer and one in the output layer. Eli Bendersky has an awesome derivation of the softmax. Exercise: Implement the function sigmoid_grad() to compute the gradient of the sigmoid function with respect to its input x. GitHub Gist: instantly share code, notes, and snippets. That means, we can find the slope of the sigmoid curve at. Quoting myself from this answer to a different question:. Implementing a Neural Network from Scratch in Python – An Introduction. 0 - sigmoid(y)) # the way we use this y is already sigmoided def dsigmoid(): return y * (1. The logistic model uses the sigmoid function (denoted by sigma) to estimate the probability that a given sample y belongs to class 1 given inputs X and weights W, \begin{align} \ P(y=1 \mid x) = \sigma(W^TX) \end{align} where the sigmoid of our activation function for a given n is:. import numpy as np def sigmoid ( x ): return 1 / ( 1 + np. 5 >>> sigmoid(-6) 0. But Hornik (1993) shows that a sufficient condition for the universal approximation property without biases is that no derivative of the activation function vanishes at the origin, which implies that with the usual sigmoid activation functions, a fixed nonzero bias term can be used instead of a trainable bias. using m Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The rectified linear unit (ReLU) is defined as f(x)=max(0,x). The figure below illustrates how the parameters in layer affect the loss. In this section, we will take a very simple feedforward neural network and build it from scratch in python. It's a library called matplotlib which provides you a variety of functions to make a quick plot or figure, so you can examine your data sets just in a few minutes. As x goes to infinity, y goes to 1 (tends to fire): At x=0, y=1/2. array([[1],[1],[0]]) #Sigmoid Function def sigmoid (x): return 1/(1 + np. However, its background might confuse brains because of complex mathematical calculations. Some learning algorithms require the precise estimation of the sigmoid derivative - back-propagation included. And again, square bracket one superscript refers to this layer, and superscript square bracket two refers to the output layer. How to properly derive the derivative of sigmoid function assuming the input is a matrix - i. Some Deep Learning with Python, TensorFlow and Keras. Basically, instead of a sigmoid, let f(z·w) be the activation due to inputs z and weight w, then a(z) = z·w if z·w > 0 else 0. Any neural network has 1 input and 1 output layer. Sigmoid function is one commonly used activation function in this case. It is normally required to have a positive derivative at every real point. because the output is in [-1, 1], the mean of the output may be around 0, making it easy to learn the parameters of next layers. A sigmoid "function" and a sigmoid "curve" refer to the same object. If the number argument is a positive or Negative number, the tanh function. The Sigmoid activation function is used in our model. t theta of the cost function. io Find an R package R language docs Run R in your browser R Notebooks. If everything makes sense, then let's see our layer objects in the context of training. Since the output range of a sigmoid neuron is smooth, small changes in the inputs will result in small changes in the output. Scientists tend to consume activation functions which have meaningful derivatives. A simple neural network written in Python. Logistic Regression introduces the concept of the Log-Likelihood of the Bernoulli distribution, and covers a neat transformation called the sigmoid function. Often, sigmoid function refers to the special case of the logistic function shown in the figure above and defined by the formula. Learn Python programming. So, if g of z is the sigmoid function, then the slope of the function is d, dz g of z, and so we know from calculus that it is the slope of g of x at z. The derivative of , , is simply 1, in the case of 1D inputs. It was first used in the work by L'Abbe Sauri (1774). # GRADED FUNCTION: sigmoid_derivative def sigmoid_derivative(x): """ Compute the gradient (also called the slope or derivative) of the sigmoid function with respect to its input x. Lines 2-5 import our required Python packages. Sigmoid functions often arise as the integrals of bell-shaped functions having a single maximum. LSTMs are special kind of RNNs with capability of handling Long-Term dependencies. In mathematics, the softmax function, also known as softargmax or normalized exponential function,: 198 is a function that takes as input a vector of K real numbers, and normalizes it into a probability distribution consisting of K probabilities proportional to the exponentials of the input numbers. If we have the derivative, we can simply update the weights and biases by increasing/reducing with it. The second step uses the derivative we derived above. GitHub Gist: instantly share code, notes, and snippets. The sigmoid function looks like this (made with a bit of MATLAB code): Alright, now let's put on our calculus hats… First, let's rewrite the original equation to make it easier to work with. So why not use tanh?. By continuing to use Pastebin, you agree to our use of cookies as described in the Cookies Policy. Brian Thompson slides by Philipp Koehn tanh(x)sigmoid(x) = 1 1+e x relu(x) = max(0,x) – derivative of sigmoid: y0 Weight adjustment will be scaled by a. Let’s add the backpropagation function into our python code. But googling the actual derivative I find: s_deriv(x) = sigmoid(x) * (1-sigmoid(x)). import numpy as np. T he main reason behind deep learning is the idea that, artificial intelligence should draw inspiration from the brain. 1 and number of iterations = 300000 the algorithm classified all instances successfully. A neuron takes inputs, does some math with them, and produces one output. f'(x)=f(x)(1-f(x)) Sigmoid function is monotonic but its derivative is not monotonic. asked Jul 1, Python (1. array([[0,0,1,1]]). derivative of cost function for Logistic Regression. is_var: return self. Why does this simple neural network not learn? PythonIsGreat: 1: 304: Aug-30-2019, 05:49 PM Last Post: ThomasL : First neural network: Problem with the weight factos: 11112294: 0: 576: Jan-12-2019, 09:11 PM Last Post: 11112294 : neural network- undefined name with sigmoid function: kierie_001: 0: 999: Oct-18-2018, 04:08 PM Last Post: kierie_001. We will construct a very simple neural network in python with the following components: An input layer x; A fully connected hidden layer. using m Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Quotes "Neural computing is the study of cellular networks that have a natural property for storing experimental knowledge. In memoization we store previously computed results to avoid recalculating the same function. It is probably not difficult, for a feedforward model, there is just matrix multiplications and sigmoid functions, but it would be nice to have a routine that will do that directly on "net". Sigmoid has the property that for y=sigmoid(x), dy/dx= y(1-y) In python. So “self-gated” means that the gate is just the sigmoid of the activation itself. This is a classification. shape[1] #number of features. op in {'{', '(', '-', '!'}: return self [1]. Then, derivative would be in simpler form. sigmoid(x) = 1-e x, in the limit of x-> infinity. Discover how to code ML algorithms from scratch including kNN, decision trees, neural nets, ensembles and much more in my new book, with full Python code and no fancy libraries. Transfer functions calculate a layer’s output from its net input. Derivative of the sigmoid. Unlike logistic regression, we will also need the derivative of the sigmoid function when using a neural net. Finding the inflection point of a sigmoid function. tanh(number); Number: It can be a number or a valid numerical expression for which you want to find hyperbolic Tangent value. LSTMs also provide solution to Vanishing/Exploding Gradient problem. We use the first derivative of the Sigmoid function to calculate the gradient since it is very convenient. Not because gradient descent gets more complicated, it still ends up just being a matter of taking small steps downhill, it's that we need that pesky derivative in order to use gradient descent, and the derivative of a neural network cost function (with respect to its weights) is pretty intense. The Activation Function which is better than Sigmoid Function is Tanh function which is also called as Hyperbolic Tangent Activation Function. Hence, effectively making the local gradient to near 0. Among the various deep learning libraries I have used till date. A sigmoid "function" and a sigmoid "curve" refer to the same object. Differentiable means slope can find out in the sigmoid curve at any two points. The input layer of the network contains neurons encoding the values of the input pixels. reduce_sum (tf. Take a closer look at the sigmoid function's curve on the graph above. หรือเราจะคิดง่าย ๆ ว่า Tanh = ( Sigmoid x 2 ) - 1 คือ เอา Sigmoid มาคูณ 2 ให้ยืดจาก 0-1 เป็น 0-2 แล้ว ลบ 1 เพื่อเลื่อนลง จาก 0-2 เป็น -1-1. leaky_relu() Tensorflow is an open-source machine learning library developed by Google. Elliot under the title A Better Activation Function for Artificial Neural Networks. We may use chain rule: dG dθ = dG dhdh dzdz dθ and solve it one by one ( x and y are constants). Unlike logistic regression, we will also need the derivative of the sigmoid function when using a neural net. Similar to perceptrons, but modified so that small changes in their weights and bias cause only a small change in their output. numpy is the main package for scientific computing with Python. def sigmoid_derivative (x): """ Compute the gradient (slope/derivative) of the sigmoid function with respect to its input x. Part 2 implemented the most elementary neural network with 1 hidden layer, but with any number of activation units in that layer, and a sigmoid activation at the output layer. We are making this neural network, because we are trying to classify digits from 0 to 9, using a dataset called MNIST, that consists of 70000 images that are 28 by 28 pixels. 5 >>> sigmoid(-6) 0. This tutorial teaches gradient descent via a very simple toy example, a short python implementation. This should remind you of the LSTM, where we have “gates” in the form of sigmoids that control how much of a vector gets passed on to the next stage, by multiplying it between the output of the sigmoid, which is a number between 0 and 1. The output looks likes this:. ANNs, like people, learn by example. Neat! When I implement a deep NN from scratch, I usually use the arbitrary-value-when-x-equals-zero approach. What is the derivative of ReLU? ubuntu - black screen on ubuntu laptop after installing nvidia drivers; How to normalize vectors to unit norm in Python; How to Compute the Derivative of a Sigmoid Function (fully worked example) How to fix "Firefox is already running, but is not responding" How to use the Springer LNCS LaTeX template. In this section, we will take a very simple feedforward neural network and build it from scratch in python. is_const and self. Quotes "Neural computing is the study of cellular networks that have a natural property for storing experimental knowledge. We've used numpy's exponential function to create the sigmoid function and created an out variable to hold this in the derivative. It has five parameters: : the lower asymptote;: the upper asymptote when =. Here are plots of the sigmoid, \tanh and rectified linear functions: The \tanh(z) function is a rescaled version of the sigmoid, and its output range is [-1,1] instead of [0,1]. We use the first derivative of the Sigmoid function to calculate the gradient since it is very convenient. One popular method was to perturb (adjust) the weights in a random, uninformed direction (ie. Sigmoid function outputs in the range (0, 1), it makes it ideal for binary classification problems where we need to find the probability of the data belonging to a particular class. GitHub Gist: instantly share code, notes, and snippets. Hello again in the series of tutorials for implementing a generic gradient descent (GD) algorithm in Python for optimizing parameters of artificial neural network (ANN) in the backpropagation phase. However, without delving too much into brain analogies, I find it easier to simply describe neural networks as a mathematical function that maps a given input to the desired output. The data structures we use in numpy to represent the shapes ( vectors, matrices, etc) are called numpy arrays. py GNU General Public License v3. The high level idea is to express the derivation of dw^ { [l]} ( where l is the current layer) using the already calculated values ( dA^ { [l+1]} , dZ^ { [l+1]} etc ) of layer l+1. op in {'{', '(', '-', '!'}: return self [1]. The learning rate is 0. You can store the output of the sigmoid function into variables and then use it to calculate the gradient. imbalance_xgb. py GNU General Public License v3. For float64 the upper bound is. The function was first introduced in 1993 by D. If we look at the some mathematical functions we’ll realize that “sigmoid function” or “logistic function” below solves both of our problems i. import numpy as np def sigmoid ( x ): return 1 / ( 1 + np. If you came here to see some Python code, skip to the numerical solution. In this section, we will take a very simple feedforward neural network and build it from scratch in python. is_var: return self. Home / Artificial Intelligence / Machine Learning / MATLAB / Coursera: Machine Learning (Week 3) [Assignment Solution] - Andrew NG. A sigmoid function gives an output between zero to one for every input it gets. However, its background might confuse brains because of complex mathematical calculations. The ‘Deep Learning from first principles in Python, R and Octave’ series, so far included Part 1 , where I had implemented logistic regression as a simple Neural Network. Notice the pattern in the derivative equations below. Python had been killed by the god Apollo at Delphi. The signum function is the derivative of the absolute value function, up to the indeterminacy at zero. A derivative in which all but one of the variables is considered a constant. Brian Thompson slides by Philipp Koehn tanh(x)sigmoid(x) = 1 1+e x relu(x) = max(0,x) – derivative of sigmoid: y0 Weight adjustment will be scaled by a. Derivative of Sigmoid Function A virtual environment is an isolated copy of your environment that maintains its own version of the language, packages, and versions. The sigmoid function de fined as and represented in the following figure has small output changes in the range (0, 1) when the input varies in the range. Coursera’s machine learning course week three (logistic regression) 27 Jul 2015. Next, we will define the sigmoid function which will act as the activation function and the derivative of the sigmoid function which will help us in the backpropagation step: View the code on Gist. The sigmoid and hyperbolic tangent activation functions cannot be used in networks with many layers due to the vanishing gradient problem. Another application of the logistic function is in the Rasch model, used in item response theory. You can vote up the examples you like or vote down the ones you don't like. Sigmoid function is one commonly used activation function in this case. Gate: $$\sigma(x)$$. sigmoid function for activation. The code in pure Python takes you down to the mathematical details of LSTMs, as it programs the backpropagation explicitly. Squashing functions limit the output to a range between 0 and 1, making these functions useful in the prediction of probabilities. Not because gradient descent gets more complicated, it still ends up just being a matter of taking small steps downhill, it's that we need that pesky derivative in order to use gradient descent, and the derivative of a neural network cost function (with respect to its weights) is pretty intense. t theta of the cost function (Hessian’s matrix) and the gradient vector w. The Python implementation presented may be found in the Kite repository on Github. [note that and s'(x) are the same thing, just different notation. Constructor Summary; Sigmoid() Usual sigmoid function, where the lower asymptote is 0 and the higher asymptote is 1. Implement a neural network from scratch with Python/Numpy — Backpropagation. Instead, we'll use some Python and NumPy to tackle the task of training neural networks. You can think of the blue dots as male patients and the red dots as female patients, with the x- and y- axis being medical measurements. A generalisation of the logistic function to multiple inputs is the softmax activation function, used in multinomial logistic regression. Monotonic function: A function which is either entirely non-increasing or non-decreasing. The derivative of the loss with respect to the parameters and in the th layer. 0 - sigmoid(y)) # the way we use this y is already sigmoided def dsigmoid ( y ): return y * ( 1. The sigmoid derivative (greatest at zero) used in the backprop will help to push values away from zero. Williams On the Derivatives of the Sigmoid, Neural Networks, 6(1993), 845-853. Input layer : This layer consists of the neurons that do nothing than receiving the inputs and pass it on to the other layers. The activation function converts a layer’s inputs to outputs. Hello, for numerical differentiation, there are the following functions in octave-forge : deriv (f,x0[,h,O,N]) ## 1st, 2nd, 3rd or 4th order derivative of a function, computed ## using 2nd or 4th order method. Herein, softplus is a newer function than sigmoid and tanh. We will use Chain rule for calculating derivatives of loss function. Learn Python programming. 0 - sigmoid(y)) # the way we use this y is already sigmoided def dsigmoid(): return y * (1. This comment has been minimized. The simplest way to install the package is via pip: pip install SKompiler[full] Note that the [full] option includes the installations of sympy, sqlalchemy and astor, which are necessary if you plan to convert SKompiler's expressions to sympy. foo will dynamically request the Python runtime for a member with the specified name in this object. There are a number of nonlinear solvers in core MATLAB and different Toolboxes that can fit an ‘inverse sigmoid model’ to your data. The derivative of sigmoid function is plotted below. Tanh function is better than sigmoid function. First, the change in output accelerates close to $$x = 0$$, which is similar with the Sigmoid function. Sigmoid is defined as : Where:. # GRADED FUNCTION: sigmoid_derivative: def sigmoid_derivative (x): """ Compute the gradient (also called the slope or derivative) of the sigmoid function with respect to its input x. ReLu, Sigmoid, Tanh functions. This means that there is a derivative of the function and this is important for the training algorithm which is discussed more in Section 4. The first derivative of sigmoid function is: (1−σ(x))σ(x) Your formula for dz2 will become: dz2 = (1-h2)*h2 * dh2. The sigmoid activation function shapes the output at each layer. Nantomah On some prop erties and inequalities of the sigmoid function , RGMIA Res. The sigmoid is a squashing function whose output is in the range [0, 1]. LSTMs belong to the family of recurrent neural networks which are very usefull for learning sequential data as texts, time series or video data. As the value of n gets larger, the value of the sigmoid function gets closer and closer to 1 and as n gets smaller, the value of the sigmoid function is get closer and closer to 0. If the sigmoid's output is a variable "out", then the derivative is simply out * (1-out). Understanding and implementing Neural Network with SoftMax in Python from scratch Understanding multi-class classification using Feedforward Neural Network is the foundation for most of the other complex and domain specific architecture. Sigmoid函数的定义 2. To learn about Logistic Regression, at first we need to learn Logistic Regression basic properties, and only then we. GitHub Gist: instantly share code, notes, and snippets. I believe I'm doing something wrong, since the softmax function is commonly used as an activation function in deep learning (and thus cannot always have a derivative of 0). Sigmoid and Sigmoid derivative. exp (logits) / tf. The sigmoid function is differentiable at every point and its derivative comes out to be. In the script above, we first randomly generate 100 linearly-spaced points between -10 and 10. The output of a sigmoid function, superimposed on that of a threshold function, is shown in Figure 3. It can be calculated by applying the first derivative calculation twice in succession. More formally, in integration theory it is a weak derivative, and in convex function theory the subdifferential of the absolute value at 0 is the interval. The derivative of the sigmoid function is given here. The package has hard depedency on numpy, sklearn and xgboost. In fact, The MathWorks just included it in their most recent update to the Neural Network toolbox. We are building a basic deep neural network with 4 layers in total: 1 input layer, 2 hidden layers and 1 output layer. The straight forward answer is the bounds are $0$ and $1$ if we use the convention (and pretty much everywhere in ML) of [math]\text{sigmoid}(x. Unlike logistic regression, we will also need the derivative of the sigmoid function when using a neural net. You can store the output of the sigmoid function into variables and then use it to calculate the gradient. sigmoid(x) = 1-e x, in the limit of x-> infinity. Select an activation function from the menu below to plot it and its first derivative. The partial derivative of f with respect to x focuses only on how x is changing and ignores all other variables in. We will be explaining about it during this setup: From the above derivation we can infer that the derivative of a sigmoid function is the sigmoid function itself with the mathematical equation. is_num: return True elif self. Vlad is a versatile software engineer with experience in many fields. shape[1] #number of features. My title here refers to it as a "modern neural network" because while neural nets have been around since the 1950s, the use of backpropagation, a sigmoid function and the sigmoid's derivative in Andrew's script highlight the advances that have made neural nets so popular in machine learning today. The derivative of the sigmoid, also known as sigmoid prime, will give us the rate of change, or slope, of the activation function at output sum. The code in pure Python takes you down to the mathematical details of LSTMs, as it programs the backpropagation explicitly. The logistic function is a solution to the differential equation. import numpy as np def sigmoid_derivative(x): s = sigmoid(x) ds = s*(1-s) return ds Above we compute the gradient (also called the slope or derivative) of the sigmoid function with respect to its input x. 0 * X) d = 1. Picking a learning rate = 0. As the value of x gets larger, the value of the sigmoid function gets closer and closer to 1 and as x gets smaller, the value of the sigmoid function is approaching 0. Hence, even if the difference between actual output and desired output is very large, resulting in a large (z i − O. This function is easy to differentiate Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. If you could recall I already used the sigmoid function on my Logistic Regression from scratch blog post, go check it out. Sigmoid is continuous between exponential and linear. sigmoid(x) = 1-e x, in the limit of x-> infinity. Neural Networks Overview Neural Network Representation Computing a Neural Network's Output Vectorizing across multiple examples Explanation for Vectorized Implementation Activation functions Why do you need non-linear activation functions? Derivatives of activation functions g = sigmoid g = tanh g = ReLU / Leaky ReLU Gradient descent for Neural Networks Backpropagation intuition (optional. Sigmoid: A sigmoid function (A = 1 / 1 + e-x), which produces a curve shaped like the letter C or S, is nonlinear. def sigmoid(x): return 1 / (1 + np. If you are familiar with calculus and know how to take derivatives, if you take the derivative of the Sigmoid function, it is possible to show that it is equal to this formula. sigmoid(x) = x/4, in the limit of x-> 0 from either side. Properties. We are building a basic deep neural network with 4 layers in total: 1 input layer, 2 hidden layers and 1 output layer. Backpropagation is an algorithm that calculate the partial derivative of every node on your model (ex: Convnet, Neural network). The derivative of sigmoid function is sig(z) * (1 — sig(z)). The GD implementation will be generic and can work with any ANN architecture. The logistic model uses the sigmoid function (denoted by sigma) to estimate the probability that a given sample y belongs to class 1 given inputs X and weights W, \begin{align} \ P(y=1 \mid x) = \sigma(W^TX) \end{align} where the sigmoid of our activation function for a given n is:. How to properly derive the derivative of sigmoid function assuming the input is a matrix - i. 25], tanh is in the range of [0,1], and ReLU is in the range of {0,1}. Unlike logistic regression, we will also need the derivative of the sigmoid function when using a neural net. The Sigmoid Activation Function Using a mathematical definition, the sigmoid function [2] takes any range real number and returns the output value which falls in the range of 0 to 1. sigmoid(s)), takes the input s, runs it through the sigmoid function, gets the output and then uses that output as the input in the derivative. Deep Learning from first principles in Python, R and Octave – Part 3 The 3rd part implemented a multi-layer Deep Learning Network with sigmoid. Derivative of Sigmoid Function A virtual environment is an isolated copy of your environment that maintains its own version of the language, packages, and versions. We may use chain rule: dG dθ = dG dhdh dzdz dθ and solve it one by one ( x and y are constants). The vectorized python implementation of the sigmoid function is as follows: def sigmoid(x): return 1 / (1 + np. def RNNModel(vocab_size, max_len, rnnConfig, model_type): embedding_size = rnnConfig['embedding_size'] if model_type == 'inceptionv3': # InceptionV3. hNodes[j]) If h is a computed hidden node value using tanh, then the derivative is (1 - h)(1 + h). The derivatives of sigmoid are in the range of [0,0. I would like to know if there is a routine that will provide the derivatives of net (derivative of its outputs with respect to its inputs). op in {'+', '*', '*exp'}: return all (a. 8 seconds were needed. Neural Networks Overview Neural Network Representation Computing a Neural Network's Output Vectorizing across multiple examples Explanation for Vectorized Implementation Activation functions Why do you need non-linear activation functions? Derivatives of activation functions g = sigmoid g = tanh g = ReLU / Leaky ReLU Gradient descent for Neural Networks Backpropagation intuition (optional. Derivative of Sigmoid Function December 7, 2019 December 7, 2019 by yoursdata The sigmoid function is one of the most commonly used neural activations functions. And again, square bracket one superscript refers to this layer, and superscript square bracket two refers to the output layer. And the derivative of the sigmoid function can be written as: S′(x)=S(x)⋅(1−S(x)) How to get Derivative. With appropriate shifting and normalizing, there are a few reasonable (and time-tested) activation functions. Elliot under the title A Better Activation Function for Artificial Neural Networks. If you have no prior experience with neural networks, I would suggest you first read Part 1 and Part 2 of the series (linked above). If I were to use multiprocessing on my 2015 Macbook Air, it would at best make my web scraping task just less than 2x faster on my machine (two physical cores. def sigmoid(x): return 1 / (1 + np. Python | Tensorflow nn. Used in feedforward network. He was appointed by Gaia (Mother Earth) to guard the oracle of Delphi, known as Pytho. Whilst I agree with the general consensus of responders that this is not the best way to solve the minimisation problem in the question, I have now resolved the challenge and can answer my own question to share the way one might overcome similar issues in using penalty methods to resolve optimisation problems in Python. , Joshi et al. Since the expression involves the sigmoid function, its value can be. Oh, and those are called partial derivatives. And again, square bracket one superscript refers to this layer, and superscript square bracket two refers to the output layer. Rich Shepard was interested in plotting "S curves" and "Z curves", and a little bit of googling suggests that the S curve is a sigmoid and the Z curve is simply 1. Python | Tensorflow nn. The inverse hyperbolic functions are multiple-valued and as in the case of inverse trigonometric functions we restrict ourselves to principal values for which they can be considered as single-valued. Building a RNN-LSTM completely from scratch (no libraries!) In this post, we are going to build a RNN-LSTM completely from scratch only by using numpy (coding like it’s 1999). As the input diverges from 0 in either direction, the derivative approaches 0. The derivatives of sigmoid are in the range of [0,0. The link does not help very much with this. The simplest form of logistic regression is binary or binomial logistic regression in which the target or dependent variable can have only 2 possible types either 1 or 0. During backpropagation through the network with sigmoid activation, the gradients in neurons whose output is near 0 or 1 are nearly 0. f'(x)=f(x)(1-f(x)) Sigmoid function is monotonic but its derivative is not monotonic. For example: Is your favorite football team going to win the match today? — yes/no (0/1) Does a student pass in exam? — yes/no (0/1) The logistic function is. If you are familiar with calculus and know how to take derivatives, if you take the derivative of the Sigmoid function, it is possible to show that it is equal to this formula. The derivative of sigmoid function is plotted below. pyplot as plt. 5 (6,169 ratings) Course Ratings are calculated from individual students’ ratings and a variety of other signals, like age of rating and reliability, to ensure that they reflect course quality fairly and accurately. Natural Logarithm of Sigmoid. /end short summary. sigmoid(s)), takes the input s, runs it through the sigmoid function, gets the output and then uses that output as the input in the derivative. It was first used in the work by L'Abbe Sauri (1774). The following figure is taken from is the slide from the lecture on backpropagation. E is the final error Y – Z. I would like to know if there is a routine that will provide the derivatives of net (derivative of its outputs with respect to its inputs). vectorize def sigmoid_prime (z): return sigmoid (z)*(1-sigmoid (z)). Hot Network Questions sourdough bread crumb ripped in two. Next, initialize the parameters for our model including the number of epochs, learning rate, weights, biases, etc. Viewed 66k times. I tested it out and it works, but if I run the code the way it is right now (using the derivative in the article), I get a super low loss and it's more or. You perceive them as you are. def error_derivative(target, prediction): return - target + prediction The derivative of the output layer with respect to the sigmoid is. This is one of the 100+ free recipes of the IPython Cookbook, Second Edition, by Cyrille Rossant, a guide to numerical computing and data science in the Jupyter Notebook. Two common numpy functions used in deep learning are np. In this post, math behind the neural network learning algorithm and state of the art are mentioned. /end short summary. Sigmoid is continuous between exponential and linear. For example, in computer science, an image is represented by a 3D array of shape (length,height,depth=3). 2 of Pattern Recognition and Machine Learning (Springer 2006), Bishop shows that the logit arises naturally as the form of the posterior probability distribution in a Bayesian treatment of two-class classification. Hello again in the series of tutorials for implementing a generic gradient descent (GD) algorithm in Python for optimizing parameters of artificial neural network (ANN) in the backpropagation phase. 25], tanh is in the range of [0,1], and ReLU is in the range of {0,1}. Let’s now find that derivative as we did with linear regression. 8 seconds were needed. To learn about Logistic Regression, at first we need to learn Logistic Regression basic properties, and only then we. The link does not help very much with this. I'm using the standard sigmoid function f(x) = 1 / (1 + e^(-x)) and I've seen that its derivative is dy/dx = f(x)' = f(x) * (1 - f(x)) This may be a daft question. How backpropagation works, and how you can use Python to build a neural network. ” “Your interpretation of physical objects has everything to do with the historical trajectory of your brain – and little to do with the objects themselves. ANNs, like people, learn by example. More formally, in integration theory it is a weak derivative, and in convex function theory the subdifferential of the absolute value at 0 is the interval. We may use chain rule: dG dθ = dG dhdh dzdz dθ and solve it one by one ( x and y are constants). GitHub Gist: instantly share code, notes, and snippets. The sigmoid function squashes the outputs to a value between zero and one. (e x +1)) Notice that logarithm of the base is equal to 1. is non-decreasing, that is for all ; has horizontal asymptotes at both 0 and 1 (and as a consequence, , and ). Active 4 months ago. Backpropagation Algorithm - Outline The Backpropagation algorithm comprises a forward and backward pass through the network. First, we have to talk about neurons, the basic unit of a neural network. Also known as Logistic Regression. import numpy as np def sigmoid ( x ): return 1 / ( 1 + np. def sigmoid_derivative (x): """ Compute the gradient (slope/derivative) of the sigmoid function with respect to its input x. Next up in our top 3 activation functions list is the Softmax function. This means that there is a derivative of the function and this is important for the training algorithm which is discussed more in Section 4. We've produced generalized form for derivative of logarithm of sigmoid. # application of the chain rule to find derivative of the loss function with respect to weights2 and weights1 d_weights2 = np. An activation function is said to saturate (without qualification) if it both left and right saturates.
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https://physics.stackexchange.com/questions/122849/why-does-a-mirror-reverse-polarization-of-circularly-polarised-light | # Why does a mirror reverse polarization of circularly polarised light?
A glass mirror (with metal backing layer) will reverse the polarisation of circularly polarised light upon reflection.
A polished piece of metal will also reverse the polarisation of circularly polarised light upon reflection. (I have tested and confirmed this for myself).
wikipedia states the reason a mirror will reverse the polarisation of circularly polarised light is:
...[A]s a result of the interaction of the electromagnetic field with the conducting surface of the mirror, both orthogonal components are effectively shifted by one half of a wavelength.
However, my understanding of mirrors is that only a polished piece of metal will phase shift a wavelength by half a wavelength, whereas a glass mirror (with metal backing layer) will not produce a phase shift. For example wikipedia which states:
According to Fresnel equations there is only a phase shift if n2 > n1 (n = refractive index). This is the case in the transition of air to reflector, but not from glass to reflector
• What about from reflector to glass? Jul 3, 2014 at 0:21
• Yes, both a polished metal mirror and a glass mirror with metal backing ('reflector through glass') both reverse the polarisation of circularly polarised light. The simple test is to look through the circular polariser at either type of mirror; the reflection of the circular polariser appears completely opaque (ie. you cannot see your eye in reflection) Jul 3, 2014 at 2:50
While rob is correct about the quantum mechanical picture I think that this case is at least as easy to understand as in the classical description.
Classically circular polarization can be described in terms of a time-varying linear polarization, so let's just look at two points on a wave.
I'm going to chose a beam in the $$+z$$ direction to examine two points on the wave: one where the polarization currently points along $$+\hat{x} - \epsilon \hat{y}$$, and a very short time later where the polarization is in the $$+\hat{x} + \epsilon \hat{y}$$ direction. The wave has right-handed circular polarization.
Now we let the beam bounce off a mirror in the $$x\text{--}y$$ plane. This reverses the direction of propagation but leaves the time-order in which are two points of interest pass any given point unaffected. A little thought suffices to show that the reflected wave has left-handed circular polarization.
From a quantum-mechanical perspective, circularly-polarized light is made of photons with their spins parallel to their momentum. The mirror reverses the photons' momentum but does not affect their spins, so the dot product $\sigma\cdot p$ changes sign.
Both the quantum and classical approaches are examined in Beth's 1936 measurement of the angular momentum of light, one of my favorite underrated classic papers.
• Thanks Rob, I'll go to the library and get a copy of that paper later today. I think this explanation should be added to the Wikipedia entry. Jul 3, 2014 at 2:53
• I went to add it and someone had beaten me to it. :-)
– rob
Jul 3, 2014 at 15:43
Another way to understand how reflection of circularly polarized light causes reversal of the polarization direction is to hold a screw perpendicular to a mirror and look at its reflection, and then think about what you see. The field vectors in circularly polarized light trace out a corkscrew.
• Or, indeed, just do the right-hand rule in a mirror. Apr 10, 2019 at 16:19
To supplement the other answers, I'd like to add a note about how this emerges mathematically using notation and conventions from Jones calculus.
For a polarized (linear or circular) ray of light propagating along the $$+z$$ axis, the electric field can be written as the real part of oscillating complex phases: $$\begin{bmatrix}E_x(z,t)\\E_y(z,t)\\E_z(z,t)\end{bmatrix}=\Re\left(\,\begin{bmatrix}E_{0x}\mathrm{e}^{\mathrm{i}\phi_x}\\E_{0y}\mathrm{e}^{\mathrm{i}\phi_y}\\0\end{bmatrix}\mathrm{e}^{\mathrm{i}(kz-\omega t)}\right).$$ The Jones vector is $$\begin{bmatrix}E_{0x}\mathrm{e}^{\mathrm{i}\phi_x}\\E_{0y}\mathrm{e}^{\mathrm{i}\phi_y}\end{bmatrix},$$ which is essentially the electric field in the two nontrivial spatial dimensions, with the time-dependence and spatial oscillation stripped out. This vector is completely general for all polarized light states (it cannot describe unpolarized light). For instance, $$x$$- and $$y$$- linearly polarized light are expressed with Jones vectors $$\begin{bmatrix}1\\0\end{bmatrix}$$ and $$\begin{bmatrix}0\\1\end{bmatrix}$$ respectively. Right- and left-circularly polarized light are given (upto normalization and amplitude coefficients) by $$\begin{bmatrix}1\\-\mathrm{i}\end{bmatrix}$$ and $$\begin{bmatrix}1\\+\mathrm{i}\end{bmatrix}$$ respectively.
For the case of reflection, we will consider right-circularly polarized light as an example. The electric field of the beam incident upon the mirror is $$\Re\left(\,\begin{bmatrix}1\\-\mathrm{i}\\0\end{bmatrix}\mathrm{e}^{\mathrm{i}(kz-\omega t)}\right).$$ Upon reflection, the polarization state stays the same but the wavenumber is changed $$k\to -k$$. However, the Jones vector is defined in a right-handed coordinate system in which the ray propagates along $$+z$$. So, we'll need to transform the old Jones vector such that it's in a basis where $$+z$$ is reversed. The options are a 180° rotation around the $$y$$ or $$x$$ axes (remember mirror operations destroy handedness), which would transform the Jones vector from $$\begin{bmatrix}1\\-\mathrm{i}\end{bmatrix}$$ to either $$\begin{bmatrix}-1\\-\mathrm{i}\end{bmatrix}$$ or $$\begin{bmatrix}1\\+\mathrm{i}\end{bmatrix}$$ respectively. These are the same upto a minus sign; they are both the Jones vector for left-circularly polarized light. The same procedure can be used for the case of left-circularly polarized light incident on a mirror.
The reversal of circular polarization on reflection can be understood, using the law of conservation of angular momentum. Circularly polarized light carries angular momentum, which is parallel to the wave vector for right-hand circular polarization, and antiparallel for left-hand one. For normal incidence, reflection reverses the light wave vector, but not its angular momentum, because no angular momentum is transferred to the ideal mirror. Thus upon reflection the wave vector changes sign, whereas the angular momentum vector does not, hence circular polarization is reversed. However, circular polarization reversal occurs only for normal incidence. For example, for grazing incidence, by the same reasoning of angular momentum conservation, upon reflection the circular polarization maintained, i.e., no reversal of circular polarization occurs. | {"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": 22, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9134506583213806, "perplexity": 348.15586713087635}, "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/1659882571536.89/warc/CC-MAIN-20220811224716-20220812014716-00053.warc.gz"} |
https://people.maths.bris.ac.uk/~matyd/GroupNames/128i2/C2xD4%5E2.html | Copied to
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## G = C2×D42order 128 = 27
### Direct product of C2, D4 and D4
direct product, p-group, metabelian, nilpotent (class 2), monomial, rational
Series: Derived Chief Lower central Upper central Jennings
Derived series C1 — C22 — C2×D42
Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C2×D42
Lower central C1 — C22 — C2×D42
Upper central C1 — C23 — C2×D42
Jennings C1 — C22 — C2×D42
Generators and relations for C2×D42
G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 2428 in 1268 conjugacy classes, 484 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×24], C4 [×8], C4 [×10], C22, C22 [×22], C22 [×152], C2×C4 [×22], C2×C4 [×42], D4 [×32], D4 [×104], C23, C23 [×32], C23 [×184], C42 [×4], C22⋊C4 [×32], C4⋊C4 [×8], C22×C4, C22×C4 [×22], C22×C4 [×16], C2×D4 [×80], C2×D4 [×180], C24 [×24], C24 [×40], C2×C42, C2×C22⋊C4 [×8], C2×C4⋊C4 [×2], C4×D4 [×16], C22≀C2 [×32], C4⋊D4 [×32], C41D4 [×8], C23×C4 [×4], C22×D4 [×48], C22×D4 [×32], C25 [×4], C2×C4×D4 [×2], C2×C22≀C2 [×4], C2×C4⋊D4 [×4], C2×C41D4, D42 [×16], D4×C23 [×4], C2×D42
Quotients: C1, C2 [×31], C22 [×155], D4 [×16], C23 [×155], C2×D4 [×56], C24 [×31], C22×D4 [×28], 2+ 1+4 [×2], C25, D42 [×4], D4×C23 [×2], C2×2+ 1+4, C2×D42
Smallest permutation representation of C2×D42
On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 10)(6 11)(7 12)(8 9)(13 22)(14 23)(15 24)(16 21)(17 30)(18 31)(19 32)(20 29)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 9)(2 12)(3 11)(4 10)(5 26)(6 25)(7 28)(8 27)(13 30)(14 29)(15 32)(16 31)(17 22)(18 21)(19 24)(20 23)
(1 13 10 31)(2 14 11 32)(3 15 12 29)(4 16 9 30)(5 18 27 22)(6 19 28 23)(7 20 25 24)(8 17 26 21)
(1 29)(2 30)(3 31)(4 32)(5 24)(6 21)(7 22)(8 23)(9 14)(10 15)(11 16)(12 13)(17 28)(18 25)(19 26)(20 27)
G:=sub<Sym(32)| (1,27)(2,28)(3,25)(4,26)(5,10)(6,11)(7,12)(8,9)(13,22)(14,23)(15,24)(16,21)(17,30)(18,31)(19,32)(20,29), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,9)(2,12)(3,11)(4,10)(5,26)(6,25)(7,28)(8,27)(13,30)(14,29)(15,32)(16,31)(17,22)(18,21)(19,24)(20,23), (1,13,10,31)(2,14,11,32)(3,15,12,29)(4,16,9,30)(5,18,27,22)(6,19,28,23)(7,20,25,24)(8,17,26,21), (1,29)(2,30)(3,31)(4,32)(5,24)(6,21)(7,22)(8,23)(9,14)(10,15)(11,16)(12,13)(17,28)(18,25)(19,26)(20,27)>;
G:=Group( (1,27)(2,28)(3,25)(4,26)(5,10)(6,11)(7,12)(8,9)(13,22)(14,23)(15,24)(16,21)(17,30)(18,31)(19,32)(20,29), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,9)(2,12)(3,11)(4,10)(5,26)(6,25)(7,28)(8,27)(13,30)(14,29)(15,32)(16,31)(17,22)(18,21)(19,24)(20,23), (1,13,10,31)(2,14,11,32)(3,15,12,29)(4,16,9,30)(5,18,27,22)(6,19,28,23)(7,20,25,24)(8,17,26,21), (1,29)(2,30)(3,31)(4,32)(5,24)(6,21)(7,22)(8,23)(9,14)(10,15)(11,16)(12,13)(17,28)(18,25)(19,26)(20,27) );
G=PermutationGroup([(1,27),(2,28),(3,25),(4,26),(5,10),(6,11),(7,12),(8,9),(13,22),(14,23),(15,24),(16,21),(17,30),(18,31),(19,32),(20,29)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,9),(2,12),(3,11),(4,10),(5,26),(6,25),(7,28),(8,27),(13,30),(14,29),(15,32),(16,31),(17,22),(18,21),(19,24),(20,23)], [(1,13,10,31),(2,14,11,32),(3,15,12,29),(4,16,9,30),(5,18,27,22),(6,19,28,23),(7,20,25,24),(8,17,26,21)], [(1,29),(2,30),(3,31),(4,32),(5,24),(6,21),(7,22),(8,23),(9,14),(10,15),(11,16),(12,13),(17,28),(18,25),(19,26),(20,27)])
50 conjugacy classes
class 1 2A ··· 2G 2H ··· 2W 2X ··· 2AE 4A ··· 4H 4I ··· 4R order 1 2 ··· 2 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4
50 irreducible representations
dim 1 1 1 1 1 1 1 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 2+ 1+4 kernel C2×D42 C2×C4×D4 C2×C22≀C2 C2×C4⋊D4 C2×C4⋊1D4 D42 D4×C23 C2×D4 C22 # reps 1 2 4 4 1 16 4 16 2
Matrix representation of C2×D42 in GL5(ℤ)
-1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
1 0 0 0 0 0 1 -2 0 0 0 1 -1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
1 0 0 0 0 0 1 -2 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1
,
1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 -2 0 0 0 1 1
,
-1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 -2 0 0 0 0 1
G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,1,0,0,0,-2,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,-2,-1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,0,0,-2,1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,-2,1] >;
C2×D42 in GAP, Magma, Sage, TeX
C_2\times D_4^2
% in TeX
G:=Group("C2xD4^2");
// GroupNames label
G:=SmallGroup(128,2194);
// by ID
G=gap.SmallGroup(128,2194);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,570]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
×
𝔽 | {"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": 0.9827620387077332, "perplexity": 3474.6005238924636}, "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-2019-04/segments/1547583857913.57/warc/CC-MAIN-20190122140606-20190122162606-00485.warc.gz"} |
https://lingpipe-blog.com/2009/09/29/convexity-of-root-mean-square-error-or-why-committees-won-the-netflix-prize/ | ## Convexity of (Root) Mean Square Error, or Why Committees Won the Netflix Prize
As John Langord mentioned in his blog, the reason committees won the Netflix prize is because root mean square error is convex. Let’s see what that means with an example and with some math.
### A Regression Problem
The contest involves predicting ratings for $N$ movies. The actual ratings given the movies by users are known to Netflix to be $y = y_1,\ldots,y_N$, where the ratings are integers $y_n \in \{ 1, 2, 3, 4, 5 \}$ for $1 \leq n \leq N$. A submission consists of floating point approximations of real numbers $\hat{y} = \{ \hat{y}_1,\ldots,\hat{y}_N \}$ where $\hat{y}_n \in \mathbb{R}$ for $1 \leq n \leq N$.
### (Root Mean) Square Error
The error function used by Netflix was root mean square error (RMSE), which is defined as the square root of the mean of the square error:
$\mbox{RMSE}(\hat{y},y) = \sqrt{\mbox{MSE}(\hat{y},y)}$
$\mbox{MSE}(\hat{y},y) = \mbox{SE}(\hat{y},y)/N$
$\mbox{SE}(\hat{y},y) = \sum_{n=1}^N \mbox{SE}(\hat{y}_n,y_n)$
$\mbox{SE}(\hat{y}_n,y_n) = (y_n - \hat{y_n})^2$
The mean scales square error to a per-example error, and the square root puts it back on the original point scale. But note that the best RMSE is achieved at the point of best SE, so the winning entry is the one with the best SE, so we won’t need to worry about means or square roots, just square error. (Note that square error is just Euclidean distance between the entry vector and true answer vector.)
### Graphing Upward Curving Error
Suppose we have a true rating of $y_n = 4$ for movie $n$. We plot square error $\mbox{SE}(\hat{y}_n,y_n) = (y_n-\hat{y}_n)^2$ for $\hat{y}_n \in [1,5]$:
Note that the error curves upward. Note that the error is much higher for outlier guesses; a guess of 1 has an error of 9 in this case, whereas a guess of 2 has an error of 4 and a guess of 3 an error of only 1. This tends to drive estimates toward the center of the range to guard against huge error. By way of comparison, absolute (or taxicab distance) error would be $|y_n - \hat{y}_n|$ and would not penalize extreme guesses nearly as severely as squared error.
We have plotted as red circles the error for two guesses $3.0$ and $4.5$, with values
$\mbox{SE}(3.0,4)=(4-3.0)^2=1.0$
$\mbox{SE}(4.5,4) = (4-4.5)^2 = 0.25$.
Now consider the guess that results from averaging the guesses 3.0 and 4.5, namely 3.75. It has lower error than either 3.0 or 4.5,
$\mbox{SE}(3.75,4) = (4-3.75)^2 = 0.0625$
The error is plotted on the curve. Note that the error for the average of the guesses 0.0625 is not only less than the average of the errors, 1.25/2 = 0.625, it is less than the error of the individual guesses, 1.0 and 0.25.
$(\mbox{SE}(4.5,4) + \mbox{SE}(3.0,4))/2 \geq \mbox{SE}((4.5 + 3.0)/2,4)$
### Convexity
The reason committees fare so well with root mean square error is because the error function is convex. Convexity for functions means the same thing as the dictionary definition — curved upwards. When functions curve upwards, as square error does, the error for averaging two guesses is always lower than the average error from the two guesses.
Mathematically, a function $f$ is convex if for all $x$ and $x'$ and $\lambda \in [0,1]$,
$f(\lambda x + (1-\lambda) x') \leq \lambda f(x) + (1-\lambda) f(x')$
This says that the value of the function for a weigted average of inputs is less than or equal to the weighted average of the value of the function at those inputs. In the example above, $f$ is the square error function $f(x) = (y-x)^2$ and the examples are weighted equally with $\lambda = 1/2$.
We know square error is convex because its differentiable everywhere and its second derivative is positive:
$\frac{d^2}{d\hat{y}_n^2} \mbox{SE}(\hat{y}_n,y_n) = \frac{d^2}{d\hat{y}_n^2}(y_n - \hat{y}_n)^2 = 2 \hat{y}_n$
The result can also be easily established algebraically by expanding all of the terms, cancelling and rearranging.
Everything generalizes to multiple dimensions by additivity. Because the full square error is the sum of the dimensionwise square errors, all the partial second derivatives are positive and the entire error function is convex.
### Committees Average Their Members’ Predictions
All of the top-scoring Netflix Prize entries averaged the predictions of multiple base systems. Given the convexity of error, the error for the average of two equal scoring systems is less than or equal to their average error. So in general, two predictions with the same error can be averaged to produce a prediction with lower error. And as we saw in the example above, this can even happen when the predictions are not the same.
It can help to tune the weighting $\lambda$, typically by weighting the better scoring systems more heavily.
### Discrete Predictions and Posterior Variance
As we’ve been talking about in other blog posts, point estimates sweep posterior variance under the rug. The square error further pushes numerical scores toward non-comittal average ratings.
Suppose we build a discrete 5-way classifier and estimate a posterior $\mbox{Pr}(y_n=m)$ for $m \in \{ 1, 2, 3, 4, 5 \}$. The Bayesian estimator here would minimize expected square error with respect to the posterior, namely the expected rating, which weights each prediction by its estimated probability:
$\hat{y}_n = \sum_{m =1}^{5} m \cdot \mbox{Pr}(y_n=m)$
For instance, here are plots of two posterior estimates with the same mean prediction, 2.95, the first of which is for a film the system thinks the user will love or hate,
and one for which the system is pretty sure the user will be indifferent,
I find these posterior distributions much more useful than a single plot. This is how Amazon and NewEgg display their customer ratings, though neither site predicts how much you’ll like something numerically like Netflix does.
### 4 Responses to “Convexity of (Root) Mean Square Error, or Why Committees Won the Netflix Prize”
1. Max Gubin Says:
Actually, the situation with a real recommendation system is more complex because it can provide only a limited number of recommendations for a user, in other words, it provides only N top items with the best scores for a user. Any loss function that uses such a top window is extremely non-convex with many local minima (like loss functions in IR: P@N, NDCG). Popular Netflix methods like SVD and stochastic gradient descent won’t work well there. I won’t be surprised if people run into this problem when they try applying Netflix competition results in real systems.
2. lingpipe Says:
@Max Good point about convexity and ranking. Non-convex error functions make optimization-based estimation (training) difficult, too.
I don’t think the kind of non-convexity you’d get from top N or F-measure like scores would necessarily mean committee voting wouldn’t work for these kinds of tasks.
I think precision-at-N, where “relevant” means a customer rents it, is a better measure for Netflix’s task.
Furthermore, I think we need more results diversity — I don’t want it to recommend seven seasons of the Simpsons even if I will eventually rent them all and rank them highly. And I don’t want to keep being shown the same movie again and again as a recommendation, so there needs to be history.
I would also like Netflix and Amazon and PubMed to go deeper into more-like-this, because I find it useful for exploration.
Netflix’s own Cinematch system, which provided the baseline, was also based on regression. From what Netflix has said, they’re already incorporating some of the techniques from the competition. Certainly I’d think the features would port.
• Viviane Says:
Thank you Felipe. I have read your work. It is ineedd the first to my knowledge that uses the extended (also called Fisher’s non-central) hypergeometric distribution for IR. I found that sampling from extended hypergeometric becomes exponentially expensive as the sample size increases, and that there is ongoing research in statistics about .While researching on the use of the hypergeometric distribution for IR, I found a paper from Wilbur dating back to 1993! Wilbur models the vocabulary intersection between the query and a set of relevant documents using the central hypergeometric distribution. Little has been done since then, probably because the multinomial distribution is a good approximation to the hypergeometric for most IR scenarios, i.e., when the sample size (query) is cosiderably smaller than the population size (document). However, as we show in the paper, in the case of document-long queries, the multinomial approximation does not hold anymore, and the use of the “vanilla” hypergeometric distribution is required.
3. Search terms and the flu: prefering complex models | Ready-to-hand Says:
[…] One way of doing this is just taking a weighted average of the predictions of several simpler models. This works quite well when your measure of the value of your model is root mean squared error (RMSE)… […] | {"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": 35, "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": 0.8199058771133423, "perplexity": 822.1344809839725}, "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/1558232256724.28/warc/CC-MAIN-20190522022933-20190522044933-00452.warc.gz"} |
https://quantumcomputing.stackexchange.com/questions/18243/what-is-the-connection-between-bures-metric-and-finite-bures-distance | # What is the connection between Bures metric and (finite) Bures distance?
The Wikipedia page discussing the Bures metric introduces it as the Hermitian 1-form operator $$G$$ defined implicitly by $$\rho G+G\rho = \mathrm d\rho$$, and which induces the corresponding Bures distance, which reads $$[d(\rho,\rho+\mathrm d\rho)]^2 = \frac12 \operatorname{Tr}(\mathrm d\rho \,G).$$ Shortly thereafter, they give the corresponding finite version of this metric, which turns out to have the form $$[D(\rho_1,\rho_2)]^2 = 2\Big(1 - \operatorname{Tr}\left|\sqrt{\rho_1}\sqrt{\rho_2}\right|\Big).$$ These two expression do not look particularly similar. How does one go from one to the other? I suppose this should involve taking the geodesic curve connecting $$\rho_1$$ and $$\rho_2$$, and then integrating the infinitesimal form over it, but I'm not sure how to actually perform such a calculation.
• I just realised a possible derivation of this is (I think, I still have to work through it) in the second part of this great answer by @David Bar Moshe. I still think it might be worth having a more dedicated post here about this though, to make the information easier to retrieve.
– glS
Jul 2 at 9:16
The Bures metric is the limit of the Bures distance for two infinitesimally close density matrices $$\rho$$ and $$\rho+d\rho$$. The Bures distance however, is not unique. It depends on the space on which we integrate the Bures metric. The Bures distance for positive matrices differs from that of density matrices (with a unit trace), however they have the same limit when the separation becomes infinitesimal.
First, let me please remark that the calculation (by means of the Hamilton-Jacobi equation) in the previous answer for the case of the Fubini-Study distance of the space of pure states, is a much easier task than the general Bures distance evaluation for the space of density matrices. I haven't seen anywhere an evaluation of the Bures distance by integration of the Hamilton-Jacobi equation for the Bures metric. It should be a very nice exercise. It should be a case where the Hamilton- Jacobi equation has an exact solution which is not obtained by means of separation of variables.
The evaluation of the Bures distances in Bengtsson and Życzkowski (page 234 equations (9.28) to (9.32)), mentioned in the previous answer, was performed indirectly by minimizing the Euclidean or the spherical distance on the space of purifications, then showing that the infinitesimal version of which is the Bures metric by evaluating it on a horizontal vector (which by definition minimizes the distance).
Here, I'll perform the computation of the second part (the easier part), i.e., start from the Bures distance and obtain the Bures metric for the case of two infinitesimally separated density matrices by a direct computation: For the space of density matrices.
The Bures distance is given by: $$D_{\text{Bures}}(\rho_1, \rho_2) = \arccos(\sqrt{F}( \rho_1, \rho_2))$$ Where: $$\sqrt{F}( \rho_1, \rho _2) = \text{tr}\left(\sqrt{\sqrt{\rho _1}\rho _2\sqrt{\rho _1})}\right)$$
We need to evaluate this expression for
$$\rho_1 = \rho$$
and
$$\rho_2 = \rho+d\rho = \rho+ \frac{\partial\rho}{\partial \theta^a} d\theta^a := \rho + \partial_a \rho d\theta^a$$
($$\theta_a$$ are real coordinates locally parametrizing the space of density matrices). (Einstein's summation convention is implied for repeated indices).
When the density matrices are infinitesimally apart, the Uhlmnann's fidelity $$(\sqrt{F}$$ becomes close to unity and its arccosine can be approximated as:
$$D^2_{\text{Bures}}(\rho, \rho+d\rho) \approx 2- 2\sqrt{F}( \rho, \rho+d\rho)$$
Taking the Taylor series to the second order of the Uhlmann's fidelity, we obtain:
$$X \equiv\sqrt{\rho }(\rho+d\rho)\sqrt{\rho} = \rho^2 + \sqrt{\rho}\partial_a \rho \sqrt{\rho}d\theta^a + \frac{1}{2} \sqrt{\rho}\partial_a \partial_b \rho \sqrt{\rho}d\theta^a d\theta^b$$
We need to take the square root of the above expression. Suppose it has the form: $$\sqrt{X} = \sqrt{\sqrt{\rho }(\rho+d\rho)\sqrt{\rho}} = \rho + B_a d\theta^a + C_{ab} d\theta^a d\theta^b$$ To prove the uniqueness of the solution, let us assume that $$\rho$$ is invertible, thus none of its eigenvalues is equal to zero. Thus, for small enough $$|d\theta^a|$$ the square root will be positive definite. If we manage to find Hermitian solutions for $$B_a$$ and $$C_{ab}$$, then the result will be the unique Hermitian square root.
Substituting in $$\sqrt{X} \sqrt{X} = X$$, we obtain: $$\sqrt{\rho}\partial_a \rho \sqrt{\rho}= \rho B_a + B_a \rho \quad (1)$$ $$\frac{1}{2} \sqrt{\rho}\partial_a \partial_b\rho \sqrt{\rho}= \rho C_{ab} + C_{ab} \rho + \frac{1}{2} (B_a B_b + B_b B_a) \quad (2)$$ Using the definition of the symmetric logarithmic derivative: $$\partial_a \rho = \frac{1}{2}(\rho L_a + L_a \rho)$$ We can see that: $$B_a = \frac{1}{2} \sqrt{\rho} L_a \sqrt{\rho}$$ is a solution of (1) In the second equation, writing: $$\partial_a \partial_b\rho = \frac{1}{2} (\partial_a \partial_b + \partial_b \partial_a)\rho$$ and expressing all the derivatives of the density matrix by means of the symmetric logarithmic derivative and substituting in (2), we see that: $$C_{ab} =\frac{1}{16}\sqrt{\rho} (\partial_a L_b + \partial_b L_a + L_a L_b + L_b L_a) \sqrt{\rho}$$
is a solution of (2).
Substituting of the Taylor approximation of the Uhlmann fidelity into the Bures distance, we get the infinitesimal Bures distance: $$D^2_{\text{Bures}}(\rho, \rho+d\rho) = -2\mathrm{tr}(B_a)d\theta^a - \mathrm{tr}(C_{ab}) d\theta^ad\theta^b$$
The first term: $$\mathrm{tr}(B_a) = \mathrm{tr}(\rho L_a) = \frac{1}{2} \mathrm{tr}(\rho L_a+ L_a \rho) = \frac{1}{2} \mathrm{tr}(\partial_a \rho) = \partial_a \mathrm{tr}(\rho) = 0$$ The second term: $$\mathrm{tr}(C_{ab}) = \frac{1}{16}\mathrm{tr}\left(\rho(\partial_a L_b + \partial_b L_a + L_a L_b + L_b L_a) \right)$$ Using the identity $$\partial_a\partial_b\mathrm{tr}\rho = 0$$ We have: $$\mathrm{tr}\left(\rho(\partial_a L_b + \partial_b L_a)\right) = -2 \mathrm{tr}\left(\rho( L_a L_b + L_b L_a)\right)$$ Thus: $$\mathrm{tr}(C_{ab}) =- \frac{1}{16}\mathrm{tr}\left(\rho(L_a L_b + L_b L_a) \right)$$ Therefore: $$D^2_{\text{Bures}}(\rho, \rho+d\rho) = \frac{1}{8}\mathrm{tr}\left(\rho(L_a L_b + L_b L_a) \right) d\theta^ad\theta^b = \frac{1}{8}\mathrm{tr}\left(L_a (\rho L_b + L_b \rho) \right) d\theta^ad\theta^b = \frac{1}{4}\mathrm{tr}\left(L_a \partial_b\rho \right) d\theta^ad\theta^b$$
• very nice, thanks. The nontrivial part seems to be computing the Taylor approximation of $\sqrt{F}(\rho,\rho+d\rho)$. I wonder if reasoning in terms of its expression with the trace norm, $\sqrt{F}(\rho,\sigma)=\|\sqrt\rho\sqrt\sigma\|_1$ might make for a more "elegant" computation. The nontrivial part would probably then be to find an expansion for $\sqrt{\rho+d\rho}$. A few posts I found related to this are math.stackexchange.com/a/1320527/173147, mathoverflow.net/a/193921/84108, and physics.stackexchange.com/a/196720/58382
– glS
Jul 8 at 12:18
• Solving these seems to eventually boil down to solving a Sylvester equation, which makes sense as that is what computing the SLD also boils down to, I think.
– glS
Jul 8 at 12:18 | {"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": 35, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9834758043289185, "perplexity": 249.14764407467598}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363006.60/warc/CC-MAIN-20211204185021-20211204215021-00513.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition/chapter-2-graphs-and-functions-2-3-functions-2-3-exercises-page-215/31 | ## Precalculus (6th Edition)
The given relation defines a function. domain: $(-\infty, +\infty)$ range: $(-\infty, +\infty)$
Note that all vertical lines will pass through the graph in at most one point. This means that the graph pass the vertical line test. Thus, the given graph represents a function. The graph covers all real numbers from left to right so the function is defined for all values of $x$. This means the domain is $(-\infty, +\infty)$. The graph covers all real numbers from bottom to top so value of $y$ can be any real number. This means the range is $(-\infty, +\infty)$. | {"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": 0.5989094972610474, "perplexity": 114.57738238944407}, "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/1531676591140.45/warc/CC-MAIN-20180719144851-20180719164851-00230.warc.gz"} |
https://www.studypug.com/ca/grade5/introduction-to-fractions-fraction-of-a-number | # Fraction of a number
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0/4
##### Intros
###### Lessons
1. Introduction to Fraction of a Number:
2. Fraction of a number using shapes
3. Fraction of a number using fraction blocks
4. Fraction of a number using number lines
5. The 4 steps for fraction multiplication with a whole number
0/26
##### Examples
###### Lessons
1. Number Lines and Fraction of a Number
Use the number line to complete the multiplication sentence (fill in and/or analyze):
1. 5 × $\frac{1}{8}$ = ___
2. $\frac{2}{5}$ x 6 = ___
2. Fraction Blocks and Fraction of a Number
Use fraction blocks to solve for the fraction of each number.
1. 4 × $\frac{1}{6}$ = ?
2. $\frac{2}{3}$ × 6 = ?
3. Which model shows the answer to $\frac{1}{9}$ × 5?
4. Which model shows the answer to 3 x $\frac{3}{5}$ × 5?
3. Fraction Multiplication for Fraction of a Number
Use the steps for fraction multiplication to solve.
• Hint: (1) put whole number over 1, (2) cross cancel, (3) multiply numerators; multiply denominators, (4) write in lowest terms
1. 14 × $\frac{1}{7}$ = ?
2. $\frac{3}{11}$ × 11 = ?
3. 6 × $\frac{4}{5}$ = ?
4. $\frac{2}{35}$ × 70 = ?
5. 48 × $\frac{9}{16}$ = ?
6. Ben lives $\frac{53}{24}$ miles away from the mall. Trixie lives 6 times that distance from the mall. How many miles away from the mall does Trixie live?
4. Figuring out Fraction Multiplication Backwards
Fill in the blanks by working backwards.
1. ? × $\frac{1}{8}$ = 6
2. $\frac{5}{26}$ × ? = $\frac{5}{26}$
3. ? × $\frac{1}{9}$ = 0
4. ? × $\frac{3}{4}$ = 6
5. $\frac{6}{20}$ × ? = 7$\frac{1}{2}$
5. Fraction of a Number Picture Problems
Use the picture. Write the multiplication equation with a fraction and a whole number. Then, solve for the fraction of a number.
6. Fraction of a Number: Word Problem
Jerry Jepson Elementary School has a population of 380 students. Mrs. Kim's class has 14 girls and 10 boys.
1. If $\frac{1}{6}$ of the class is vegetarian, how many students are vegetarian?
2. If $\frac{3}{8}$ of the class has siblings, how many students have a brother or sister? How many students are only-children?
3. If $\frac{7}{38}$ of the school's students have pets how many students have pets? Out of these pet owners, $\frac{4}{7}$ have dogs as pets; how many students have dogs?
4. If $\frac{3}{5}$ of the school wears glasses and $\frac{1}{4}$ of Mrs. Kim's class wears glasses, what is the FRACTION of students wearing glasses in the class compared to the entire school?
###### Topic Notes
In this lesson, we will learn:
• How to understand fraction of a number using: models with shapes, models with fraction blocks, and fraction number lines
• The steps for multiplying a fraction with a whole number
Notes:
• When we are looking for a fraction of a number, we are using multiplication
• The word “of” in math usually signifies multiplication
• We can use models with shapes to find the fraction of a number:
• Ex. What is $\frac{1}{5}$ of 15?
• Use 15 circles to represent the whole number; divide into 5 equal parts; answer how many circles are in 1 of those parts:
Therefore, $\frac{1}{5}$ x 15 = 3
• We can also use models with fraction blocks to find the fraction of a number:
• Ex. What is $\frac{2}{3}$ × 5?
• Create a fraction block with 2 of 3 equal parts shaded in. Repeat 5 times.
• There are 10 parts shaded in, each one is worth $\frac{1}{3}$ of a whole.
• Therefore, $\frac{2}{3}$ x 5 = $\frac{10}{3}$ or 3 $\frac{1}{3}$
• We can also use number lines to find the fraction of a number:
• Ex. What is 4 × $\frac{1}{5}$ ?
• Split a line (from 0 to 1) into 5 equal parts, create 4 jumps of 1 part each ($\frac{1}{5}$)
• Generally, the fastest way to find the fraction of a number will be to do fraction multiplication with a whole number using these steps:
• Step 1: Put the whole number as a fraction over 1
Step 2: Cross cancel the numbers if possible
Step 3: Multiply the top numbers and then separately multiply the bottom numbers
• Ex. What is $\frac{6}{18}$ of 24? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 35, "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": 0.8543078303337097, "perplexity": 2053.530089614552}, "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/1674764499890.39/warc/CC-MAIN-20230131190543-20230131220543-00423.warc.gz"} |
http://mathoverflow.net/questions/25862/naive-questions-about-matrices-representing-endomorphisms-of-hilbert-spaces?answertab=active | # Naive questions about “matrices” representing endomorphisms of Hilbert spaces.
This is a very basic question and might be way too easy for MO. I am learning analysis in a very backwards way. This is a question about complex Hilbert spaces but here's how I came to it: I have in the past written a paper about (amongst other things) compact endomorphisms of $p$-adic Banach spaces (and indeed of Banach modules over a $p$-adic Banach algebra), and in this paper I continually used the notion of a "matrix" of an endomorphism as an essential crutch when doing calculations and proofs. I wondered at the time where more "conceptual" proofs existed, and probably they do, but I was too lazy to find them.
Now I find myself learning the basic theory of certain endomorphisms of complex separable Hilbert spaces (continuous, compact, Hilbert-Schmidt and trace class operators) and my instinct, probably wrong, is to learn the theory in precisely the same way. So this is the sort of question I find myself asking.
Say $H$ is a separable Hilbert space with orthonomal basis $(e_i)_{i\in\mathbf{Z}_{\geq1}}$. Say $T$ is a continuous linear map $H\to H$. Then $T$ is completely determined by its "matrix" $(a_{ij})$ with $Te_i=\sum_ja_{ji}e_j$. But are there completely "elementary" conditions which completely classify which collections of complex numbers $(a_{ij})$ arise as "matrices" of continuous operators?
I will ask a more precise question at the end, but let me, for the sake of exposition, tell you what the the answer is in the $p$-adic world.
In the $p$-adic world, $\sum_na_n$ converges iff $a_n\to 0$, and life is easy: the answer to the question in the $p$-adic world is that $(a_{ij})$ represents a continuous operator iff
(1) For all $i$, $\sum_j|a_{ji}|^2<\infty$ (equivalently, $a_{ji}\to 0$ as $j\to\infty$), and
(2) there's a universal bound $B$ such that $|a_{ij}|\leq B$ for all $i,j$.
[there is no inner product in the $p$-adic case, so no adjoint, and the conditions come out being asymmetric in $i$ and $j$]. See for example pages 8--9 of this paper of mine, although of course this isn't due to me---it's in Serre's paper on compact operators on $p$-adic Banach spaces from the 60s---see Proposition 3 of Serre's paper. In particular, in the $p$-adic world, one can identify the continuous maps $H\to H$ (here $H$ is a $p$-adic Banach space with countable ON basis $(e_i)$) with the collection of bounded sequences in $H$, the identification sending $T$ to $(Te_i)$.
In the real/complex world though, the analogue of this result fails: the sequence $(e_1,e_1,e_1,\ldots)$ is a perfectly good bounded sequence, but there is no continuous linear map $H\to H$ sending $e_i$ to $e_1$ for all $i$ (where would $\sum_n(1/n)e_n$ go?).
Let's consider the finite rank case, so $T$ is a continuous linear map $H\to H$ with image landing in $\mathbf{C}e_1$. Then by Riesz's theorem, $T$ is just "inner product with an element of $H$ and then multiply by $e_1$". Hence we have an additional condition on the $a_{ij}$, namely that $\sum_j|a_{ij}|^2<\infty$. Furthermore a continuous linear map is bounded, as is its adjoint.
This makes me wonder whether the following is true, or whether this is still too naive:
Q) Say $(a_{ij})$ $(i,j\in\mathbf{Z}_{\geq1})$ is a collection of complex numbers satisfying the following:
There is a real number $B$ such that
1) For all $i$, $\sum_j|a_{ij}|^2\leq B$
2) For all $j$, $\sum_j|a_{ij}|^2\leq B$
Then is there a unique continuous linear map $T:H\to H$ with $Te_i=\sum_ja_{ji}e_i$? My guess is that this is still too naive. Can someone give me an explicit counterexample? Or, even better, a correct "elementary" list of conditions characterising the continuous endomorphisms of a Hilbert space?
On the other hand, it clearly isn't a complete waste of time to think about matrix coefficients. For example there's a bijection between Hilbert-Schmidt operators $T:H\to H$ and collections $(a_{ij})$ of complexes with $\sum_{i,j}|a_{ij}|^2<\infty$, something which perhaps the experts don't use but which I find incredibly psychologically useful.
-
My "meta"-question ("what is a good characterisation of the (a_{ij})") has been answered by Laurent ("there is evidence to suggest there is none"). But my explicit question remains open: can someone give me a collection of (a_{ij}) with each row and column L^2-bounded by some universal B but such that the (a_{ij}) come from no matrix? – Kevin Buzzard May 25 '10 at 12:21
Kevin, have you tried looking in the classic book "Theorems and problems in functional analysis"? When I learned about Hilbert-Schmidt operators and the like, I remember finding its lists of problems to be a useful source of counterexamples, going deeper than "Counterexamples in analysis" (a largely "1-dimensional" book). – BCnrd May 25 '10 at 13:18
Chapter V of Halmos' "A Hilbert space problem book" is called "Infinite matrices".
It contains lots of nice results and problems, and also the statement that "there are no elegant and usable necessary and sufficient conditions [for a matrix to be the matrix of an operator]".
-
And to think my question is answered by a $p$-adic guy! Thanks Laurent. – Kevin Buzzard May 25 '10 at 11:05
[PS the reason I'm learning this stuff is that I'm giving some lectures on the trace formula.] – Kevin Buzzard May 25 '10 at 11:06
I looked in Halmos (Chapter IV of the 1967 edition, by the way, not Chapter V) and he says this and gives a couple of counterexamples to things but doesn't give a counterexample to the explicit question I asked. – Kevin Buzzard May 25 '10 at 12:20
It's chapter five in the 2nd edition, but there's still no counterexample... – Laurent Berger May 25 '10 at 12:35
So maybe the following is a counterexample to Kevin's original post. (It was created by computer scientist and friend Erik Vee as a "counterexample" to that exercise 3.14 in Zimmer (which says that a bounded operator is compact iff $a_{ij}$ goes to 0 as i and j go to $\infty$); and it was Robert Pollack who figured out that the reason it's not a counterexample is that it doesn't represent a continuous operator. So my role here is transcriber only.)
Define the matrix as follows. The first column is (1, 0, 0, ...), the second is (1/2, 1/2, 1/2, 1/2, 0, 0, ...), the nth has $\frac 1 n$ appearing $n^2$ times, followed by all zeros. Then $\ell_2$-norm of each column is exactly 1, and the $\ell_2$-norm of each row is bounded by $\sqrt{\frac{\pi^2}{6}}$. So this matrix has bounded $\ell_2$ rows and columns as necessary.
But this matrix cannot represent a continuous operator. If it did, then, since it satisfies Kevin's/Zimmer's criterion, this operator -- call it $A$ -- would be compact, and hence a uniform limit of the operators $A_n$ given by the first n rows of $A$. But the operator $A - A_n$ has, in its nth column, $\frac 1 n$ appearing $n^2 - n$ times, which means that that column's $\ell_2$ norm is $1 - \frac 1 n$, which is bounded away from zero.
It's still unclear to me if this matrix represents an unbounded linear map, or if it doesn't represent a well-defined map at all.
-
My (possibly naïve) understanding is that unbounded operators are not defined everywhere on a Hilbert space, and that they can be specified on at most a dense subspace. – S. Carnahan Jun 15 '10 at 5:18
@Anna M.: just to flag that gowers already gave an explicit counterexample. – Kevin Buzzard Jun 15 '10 at 10:30
@Scott: not quite. An unbounded operator by definition is a linear transformation $A$ defined on some subspace $D$ of $H$. $D=H$ is allowed, in which case $A$ is "everywhere defined". But we mostly care about closed operators, and the closed graph theorem says a closed everywhere defined operator is bounded. So as far as useful examples, you are right. Moreover, unbounded everywhere defined operators (which are not closed) typically (necessarily?) require the axiom of choice to define. For instance, it's easy to do if you have a Hamel basis for $H$. – Nate Eldredge Jun 15 '10 at 15:27
Not an answer, only a side question to Kevin Buzzard. Properly a comment.
Kevin, you mention in a comment to gowers' answer above that you have easy criteria for an infinite matrix representing a continuous operator on a Hilbert space to be be representing a compact or trace-class operator. What are these criteria?
-
If I didn't get it wrong, they were something like this: (a_ij) [assumed to represent a continuous endomorphism] is compact iff for all e>0 there's N such that |a_{ij}|<e for all i>N [do you believe this? It might be wrong.] and trace class iff sum_i(sum_j |a_{ij}|^2)^{1/2}<infty (this is almost the definition, IIRC). – Kevin Buzzard Jun 13 '10 at 7:17
Kevin, I'm not quite convinced by your criterion for trace class. Take the matrix which only has one nonzero column, and let that column be in $\ell^2$ but not in $\ell^1$. Then this is a rank one operator, hence trace class; but won't the quantity you give be infinite? – Yemon Choi Jun 13 '10 at 11:50
I really want to believe the compact criterion! (for one thing, I now think it matches Zimmer's exercise 3.14 in Essential Results of Functional Analysis.) I see that a compact operator satisfies the criterion (because it has to be the limit of its first n rows) but can't figure out the other direction... – Anna M. Jun 14 '10 at 20:52
Anna, I'm a bit worried that the criterion for compactness doesn't force the rows or columns to be $\ell^2$-vectors... – Yemon Choi Jun 15 '10 at 2:19
It doesn't matter, as the criterion for compactness only applies to matrices representing operators we know to be continuous. – Anna M. Jun 15 '10 at 2:45
Consider the n-by-n matrix that has $n^{-1/2}$ as every entry. That satisfies your condition with B=1. The image of the unit vector that has $n^{-1/2}$ as every coordinate is a vector that has 1 as every coordinate, and therefore norm $n^{1/2}$.
If you now put a whole lot of these as blocks down the diagonal, you can create an unbounded operator that satisfies your condition with B=1.
I'd say that the main general problem with the condition you suggested is that it is too tied to one particular basis. I'm not sure it's all that easy to come up with nice conditions of the kind you are looking for.
Additional remark: if you take spaces like ell_1 and ell_infinity, where the definition of the norm is much more closely tied to a particular basis, then it tends to be easier to find nice matrix conditions for boundedness.
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To make explicit what Tim was hinting at, note that a matrix defines a bounded linear operator on $\ell_1$ iff the columns form a bounded sequence in $\ell_1$, in which case the norm of the operator is the supremum of the $\ell_1$ norms of the columns. – Bill Johnson May 25 '10 at 16:28
Thanks gowers. You real guys don't know what you're missing---it's all infinitely easier in the p-adic world! Halmos says there's no easy criterion and I'm happy to believe this. On the other hand I think I have easy criteria for compactness, Hilbert-Schmidt and trace class (of the form "matrix must be continuous + ..."). – Kevin Buzzard May 25 '10 at 19:57
You are not the first to say that, Kevin. Many years ago Kurt Mahler told me much the same, using as one reason that every separable p-adic Banach space has a Schauder basis. (To be honest, I never checked out whether that was indeed the case, but who was I to question KM?) – Bill Johnson May 25 '10 at 21:58
One nice fact about a compact operator $T$ on a Hilbert space is that there is an ON basis $(e_n)$ s.t. $Te_n$ is orthogonal. Also, if for a bounded linear operator $T$ there is an ON basis $e_n$ s.t. $Te_n$ is orthogonal, then $T$ is compact iff $Te_n \to 0$; $T$ is HS iff $\sum \|Te_n\|^2<\infty$; $T$ is trace class iff $\sum \|Te_n\|<\infty$. So you can say quite a bit if you allow matrix representations with respect to two different ON basis. – Bill Johnson May 25 '10 at 22:09 | {"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": 0.9052361249923706, "perplexity": 360.90913330446745}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535924501.17/warc/CC-MAIN-20140909013105-00045-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://cre8math.com/2018/04/ | ## Calculus: Hyperbolic Trigonometry, II
Now on to some calculus involving hyperbolic trigonometry! Today, we’ll look at trigonometric substitutions involving hyperbolic functions.
$\displaystyle\int\sqrt{1+x^2}\,dx.$
The usual technique involving circular trigonometric functions is to put $x=\tan(\theta),$ so that $dx=\sec^2(\theta)\,d\theta,$ and the integral transforms to
$\displaystyle\int\sec^3(\theta)\,d\theta.$
In general, we note that when taking square roots, a negative sign is sometimes needed if the limits of the integral demand it.
This integral requires integration by parts, and ultimately evaluating the integral
$\displaystyle\int\sec(\theta)\,d\theta.$
And how is this done? I shudder when calculus textbooks write
$\displaystyle\int \sec(\theta)\cdot\dfrac{\sec(\theta)+\tan(\theta)}{\sec(\theta)+\tan(\theta)}\,d\theta=\ldots$
How does one motivate that “trick” to aspiring calculus students? Of course the textbooks never do.
Now let’s see how to approach the original integral using a hyperbolic substitution. We substitute $x=\sinh(u),$ so that $dx=\cosh(u)\,du$ and $\sqrt{1+x^2}=\cosh(u).$ Note well that taking the positive square root is always correct, since $\cosh(u)$ is always positive!
This results in the integral
$\displaystyle\int\cosh^2(u)\,du=\displaystyle\int\dfrac{1+\cosh(2u)}2\,du,$
which is quite simple to evaluate:
$\dfrac12u+\dfrac14\sinh(2u)+C.$
Now $u=\hbox{arcsinh}(x),$ and
$\sinh(2u)=2\sinh(u)\cosh(u)=2x\sqrt{1+x^2}.$
Recall from last week that we derived an explicit formula for $\hbox{arcsinh}(x),$ and so our integral finally becomes
$\dfrac12\left(\ln(x+\sqrt{1+x^2})+x\sqrt{1+x^2}\right)+C.$
You likely noticed that using a hyperbolic substitution is no more complicated than using the circular substitution $x=\sin(\theta).$ What this means is — no need to ever integrate
$\displaystyle\int\tan^m(\theta)\sec^n(\theta)\,d\theta$
again! Frankly, I no longer teach integrals involving $\tan(\theta)$ and $\sec(\theta)$ which involve integration by parts. Simply put, it is not a good use of time. I think it is far better to introduce students to hyperbolic trigonometric substitution.
Now let’s take a look at the integral
$\displaystyle\int\sqrt{x^2-1}\,dx.$
The usual technique? Substitute $x=\sec(\theta),$ and transform the integral into
$\displaystyle\int\tan^2(\theta)\sec(\theta)\,d\theta.$
Sigh. Those irksome tangents and secants. A messy integration by parts again.
But not so using $x=\cosh(u).$ We get $dx=\sinh(u)\,du$ and $\sqrt{x^2-1}=\sinh(u)$ (here, a negative square root may be necessary).
We rewrite as
$\displaystyle\int\sinh^2(u)\,du=\displaystyle\int\dfrac{\cosh(2u)-1}2\,du.$
This results in
$\dfrac14\sinh(2u)-\dfrac u2+C=\dfrac12(\sinh(u)\cosh(u)-u)+C.$
All we need now is a formula for $\hbox{arccosh}(x),$ which may be found using the same technique we used last week for $\hbox{arcsinh}(x):$
$\hbox{arccosh}(x)=\ln(x+\sqrt{x^2-1}).$
Thus, our integral evaluates to
$\dfrac12(x\sqrt{x^2-1}-\ln(x+\sqrt{x^2-1}))+C.$
We remark that the integral
$\displaystyle\int\sqrt{1-x^2}\,dx$
is easily evaluated using the substitution $x=\sin(\theta).$ Thus, integrals of the forms $\sqrt{1+x^2},$ $\sqrt{x^2-1},$ and $\sqrt{1-x^2}$ may be computed by using the substitutions $x=\sinh(u),$ $x=\cosh(u),$ and $x=\sin(\theta),$ respectively. It bears repeating: no more integrals involving powers of tangents and secants!
One of the neatest applications of hyperbolic trigonometric substitution is using it to find
$\displaystyle\int\sec(\theta)\,d\theta$
without resorting to a completely unmotivated trick. Yes, I saved the best for last….
So how do we proceed? Let’s think by analogy. Why did the substitution $x=\sinh(u)$ work above? For the same reason $x=\tan(\theta)$ works: we can simplify $\sqrt{1+x^2}$ using one of the following two identities:
$1+\tan^2(\theta)=\sec^2(\theta)\ \hbox{ or }\ 1+\sinh^2(u)=\cosh^2(u).$
So $\sinh(u)$ is playing the role of $\tan(\theta),$ and $\cosh(u)$ is playing the role of $\sec(\theta).$ What does that suggest? Try using the substitution $\sec(\theta)=\cosh(u)$!
No, it’s not the first think you’d think of, but it makes sense. Comparing the use of circular and hyperbolic trigonometric substitutions, the analogy is fairly straightforward, in my opinion. There’s much more motivation here than in calculus textbooks.
So with $\sec(\theta)=\cosh(u),$ we have
$\sec(\theta)\tan(\theta)\,d\theta=\sinh(u)\,du.$
But notice that $\tan(\theta)=\sinh(u)$ — just look at the above identities and compare. We remark that if $\theta$ is restricted to the interval $(-\pi/2,\pi/2),$ then as a result of the asymptotic behavior, the substitution $\sec(\theta)=\cosh(u)$ gives a bijection between the graphs of $\sec(\theta)$ and $\cosh(u),$ and between the graphs of $\tan(\theta)$ and $\sinh(u).$ In this case, the signs are always correct — $\tan(\theta)$ and $\sinh(u)$ always have the same sign.
So this means that
$\sec(\theta)\,d\theta=du.$
What could be simpler?
Thus, our integral becomes
$\displaystyle\int\,du=u+C.$
But
$u=\hbox{arccosh}(\sec(\theta))=\ln(\sec(\theta)+\tan(\theta)).$
Thus,
$\displaystyle\int \sec(\theta)\,d\theta=\ln(\sec(\theta)+\tan(\theta))+C.$
Voila!
We note that if $\theta$ is restricted to the interval $(-\pi/2,\pi/2)$ as discussed above, then we always have $\sec(\theta)+\tan(\theta)>0,$ so there is no need to put the argument of the logarithm in absolute values.
Well, I’ve done my best to convince you of the wonder of hyperbolic trigonometric substitutions! If integrating $\sec(\theta)$ didn’t do it, well, that’s the best I’ve got.
The next installment of hyperbolic trigonometry? The Gudermannian function! What’s that, you ask? You’ll have to wait until next time — or I suppose you can just google it….
## Calculus: Hyperbolic Trigonometry, I
love hyperbolic trigonometry. I always include it when I teach calculus, as I think it is important for students to see. Why?
1. Many applications in the sciences use hyperbolic trigonometry; for example, the use of Laplace transforms in solving differential equations, various applications in physics, modeling population growth (the logistic model is a hyperbolic tangent curve);
2. Hyperbolic trigonometric substitutions are, in many instances, easier than circular trigonometric substitutions, especially when a substitution involving $\tan(x)$ or $\sec(x)$ is involved;
3. Students get to see another form of trigonometry, and compare the new form with the old;
4. Hyperbolic trigonometry is fun.
OK, maybe that last reason is a bit of hyperbole (though not for me).
Not everyone thinks this way. I once had a colleague who told me she did not teach hyperbolic trigonometry because it wasn’t on the AP exam. What do you say to someone who says that? I dunno….
In any case, I want to introduce the subject here for you, and show you some interesting aspects of hyperbolic trigonometry. I’m going to stray from my habit of not discussing things you can find anywhere online, since in order to get to the better stuff, you need to know the basics. I’ll move fairly quickly through the introductory concepts, though.
The hyperbolic cosine and sine are defined by
$\cosh(x)=\dfrac{e^x+e^{-x}}2,\quad\sinh(x)=\dfrac{e^x-e^{-x}}2,\quad x\in\mathbb{R}.$
I will admit that when I introduce this definition, I don’t have an accessible, simple motivation for doing so. I usually say we’ll learn a lot more as we work with these definitions, so if anyone has a good idea in this regard, I’d be interested to hear it.
The graphs of these curves are shown below.
The graph of $\cosh(x)$ is shown in blue, and the graph of $\sinh(x)$ is shown in red. The dashed orange graph is $y=e^{x}/2,$ which is easily seen to be asymptotic to both graphs.
Parallels to the circular trigonometric functions are already apparent: $y=\cosh(x)$ is an even function, just like $y=\cos(x).$ Similarly, $\sinh(x)$ is odd, just like $\sin(x).$
Another parallel which is only slight less apparent is the fundamental relationship
$\cosh^2(x)-\sinh^2(x)=1.$
Thus, $(\cosh(x),\sinh(x))$ lies on a unit hyperbola, much like $(\cos(x),\sin(x))$ lies on a unit circle.
While there isn’t a simple parallel with circular trigonometry, there is an interesting way to characterize $\cosh(x)$ and $\sinh(x).$ Recall that given any function $f(x),$ we may define
$E(x)=\dfrac{f(x)+f(-x)}2,\quad O(x)=\dfrac{f(x)-f(-x)}2$
to be the even and odd parts of $f(x),$ respectively. So we might simply say that $\cosh(x)$ and $\sinh(x)$ are the even and odd parts of $e^x,$ respectively.
There are also many properties of the hyperbolic trigonometric functions which are reminiscent of their circular counterparts. For example, we have
$\sinh(2x)=2\sinh(x)\cosh(x)$
and
$\sinh(x+y)=\sinh(x)\cosh(y)+\sinh(y)\cosh(x).$
None of these are especially difficult to prove using the definitions. It turns out that while there are many similarities, there are subtle differences. For example,
$\cosh(x+y)=\cosh(x)\cosh(y)+\sinh(x)\sinh(y).$
That is, while some circular trigonometric formulas become hyperbolic just by changing $\cos(x)$ to $\cosh(x)$ and $\sin(x)$ to $\sinh(x),$ sometimes changes of sign are necessary.
These changes of sign from circular formulas are typical when working with hyperbolic trigonometry. One particularly interesting place the change of sign arises is when considering differential equations, although given that I’m bringing hyperbolic trigonometry into a calculus class, I don’t emphasize this relationship. But recall that $\cos(x)$ is the unique solution to the differential equation
$y''+y=0,\quad y(0)=1,\quad y'(0)=0.$
Similarly, we see that $\cosh(x)$ is the unique solution to the differential equation
$y''-y=0,\quad y(0)=1,\quad y'(0)=0.$
Again, the parallel is striking, and the difference subtle.
Of course it is straightforward to see from the definitions that $(\cosh(x))'=\sinh(x)$ and $(\sinh(x))'=\cosh(x).$ Gone are the days of remembering signs when differentiating and integrating trigonometric functions! This is one feature of hyperbolic trigonometric functions which students always appreciate….
Another nice feature is how well-behaved the hyperbolic tangent is (as opposed to needing to consider vertical asymptotes in the case of $\tan(x)$). Below is the graph of $y=\tanh(x)=\sinh(x)/\cosh(x).$
The horizontal asymptotes are easily calculated from the definitions. This looks suspiciously like the curves obtained when modeling logistic growth in populations; that is, finding solutions to
$\dfrac{dP}{dt}=kP(C-P).$
In fact, these logistic curves are hyperbolic tangents, which we will address in more detail in a later post.
One of the most interesting things about hyperbolic trigonometric functions is that their inverses have closed formulas — in striking contrast to their circular counterparts. I usually have students work this out, either in class or as homework; the derivation is quite nice, so I’ll outline it here.
So let’s consider solving the equation $x=\sinh(y)$ for $y.$ Begin with the definition:
$x=\dfrac{e^y-e^{-y}}2.$
The critical observation is that this is actually a quadratic in $e^y:$
$(e^y)^2-2xe^y-1=0.$
All that is necessary is to solve this quadratic equation to yield
$e^y=x\pm\sqrt{1+x^2},$
and note that $x-\sqrt{1+x^2}$ is always negative, so that we must choose the positive sign. Thus,
$y=\hbox{arcsinh}(x)=\ln(x+\sqrt{1+x^2}).$
And this is just the beginning! At this stage, I also offer more thought-provoking questions like, “Which is larger, $\cosh(\ln(42))$ or $\ln(\cosh(42))?$ These get students working with the definitions and thinking about asymptotic behavior.
Next week, I’ll go into more depth about the calculus of hyperbolic trigonometric functions. Stay tuned!
## Calculus: The Geometry of Polynomials, II
The original post on The Geometry of Polynomials generated rather more interest that usual. One reader, William Meisel, commented that he wondered if something similar worked for curves like the Folium of Descartes, given by the equation
$x^3+y^3=3xy,$
and whose graph looks like:
I replied that yes, I had success, and what I found out would make a nice follow-up post rather than just a reply to his comment. So let’s go!
Just a brief refresher: if, for example, we wanted to describe the behavior of $y=2(x-4)(x-1)^2$ where it crosses the x-axis at $x=1,$ we simply retain the $(x-1)^2$ term and substitute the root $x=1$ into the other terms, getting
$y=2(1-4)(x-1)^2=-6(x-1)^2$
as the best-fitting parabola at $x=1.$
$\displaystyle\lim_{x\to1}\dfrac y{(x-1)^2}=-6.$
For examples like the polynomial above, this limit is always trivial, and is essentially a simple substitution.
What happens when we try to evaluate a similar limit with the Folium of Descartes? It seems that a good approximation to this curve at $x=0$ (the U-shaped piece, since the sideways U-shaped piece involves writing $x$ as a function of $y$) is $y=x^2/3,$ as shown below.
To see this, we need to find
$\displaystyle\lim_{x\to0}\dfrac y{x^2}.$
After a little trial and error, I found it was simplest to use the substitution $z=y/x^2,$ and so rewrite the equation for the Folium of Descartes by using the substitution $y=x^2z,$ which results in
$1+x^3z^3=3z.$
Now it is easy to see that as $x\to0,$ we have $z\to1/3,$ giving us a good quadratic approximation at the origin.
Success! So I thought I’d try some more examples, and see how they worked out. I first just changed the exponent of $x,$ looking at the curve
$x^n+y^3=3xy,$
shown below when $n=6.$
What would be a best approximation near the origin? You can almost eyeball a fifth-degree approximation here, but let’s assume we don’t know the appropriate power and make the substitution $y=x^kz,$ with $k$ yet to be determined. This results in
$x^{3k-n}z^3+1=3zx^{k+1-n}.$
Now observe that when $k=n-1,$ we have
$x^{2n-3}z^3+1=3z,$
so that $\displaystyle\lim_{x\to0}z=1/3.$ Thus, in our case with $n=6,$ we see that $y=x^5/3$ is a good approximation to the curve near the origin. The graph below shows just how good an approximation it is.
OK, I thought to myself, maybe I just got lucky. Maybe introduce a change which will really alter the nature of the curve, such as
$x^3+y^3=3xy+1,$
whose graph is shown below.
Here, the curve passes through the x-axis at $x=1,$ with what appears to be a linear pass-through. This suggests, given our previous work, the substitution $y=(x-1)z,$ which results in
$x^3+(x-1)^3z^3=3x(x-1)z+1.$
We don’t have much luck with $\displaystyle\lim_{x\to1}z$ here. But if we move the $1$ to the other side and factor, we get
$(x-1)(x^2+x+1)+(x-1)^3z^3=3x(x-1)z.$
Nice! Just divide through by $x-1$ to obtain
$x^2+x+1+(x-1)^2z=3xz.$
Now a simple calculation reveals that $\displaystyle\lim_{x\to1}z=1.$ And sure enough, the line $y=x-1$ does the trick:
Then I decided to change the exponent again by considering
$x^n+y^3=3xy+1.$
Here is the graph of the curve when $n=6:$
It seems we have two roots this time, with linear pass-throughs. Let’s try the same idea again, making the substitution $y=(x-1)z,$ moving the $1$ over, factoring, and dividing through by $x-1.$ This results in
$x^{n-1}+x^{n-2}+\cdots+1+(x-1)^2z^3=3xz.$
It is not difficult to calculate that $\displaystyle\lim_{x\to1}z=n/3.$
Now things become a bit more interesting when $n$ is even, since there is always a root at $x=-1$ in this case. Here, we make the substitution $y=(x+1)z,$ move the $1$ over, and divide by $x+1,$ resulting in
$\dfrac{x^n-1}{x+1}+(x+1)^2z^3=3xz.$
But since $n$ is even, then $x^2-1$ is a factor of $x^n-1,$ so we have
$(x-1)(x^{n-2}+x^{n-4}+\cdots+x^2+1)+(x+1)^2z^3=3xz.$
Substituting $x=-1$ in this equation gives
$-2\left(\dfrac n2\right)=3(-1)z,$
which immediately gives $\displaystyle\lim_{x\to1}z=n/3$ as well! This is a curious coincidence, for which I have no nice geometrical explanation. The case when $n=6$ is illustrated below.
This is where I stopped — but I was truly surprised that everything I tried actually worked. I did a cursory online search for Taylor series of implicitly defined functions, but this seems to be much less popular than series for $y=f(x).$
Anyone more familiar with this topic care to chime in? I really enjoyed this brief exploration, and I’m grateful that William Meisel asked his question about the Folium of Descartes. These are certainly instances of a larger phenomenon, but I feel the statement and proof of any theorem will be somewhat more complicated than the analogous results for explicitly defined functions.
And if you find some neat examples, post a comment! I’d enjoy writing another follow-up post if there is continued interested in this topic.
## Bay Area Mathematical Artists, VII
We had yet another amazing meeting of the Bay Area Mathematical Artists yesterday! Just two speakers — but even so, we went a half-hour over our usual 5:00 ending time.
Our first presenter was Stan Isaacs. There was no real title to his presentation, but he brought another set of puzzles from his vast collection to share. He was highlighting puzzles created by Wayne Daniel.
Below you’ll see one of the puzzles disassembled. The craftsmanship is simply remarkable.
If you look carefully, you’ll see what’s going on. The outer pieces make an icosahedron, and when you take those off, a dodecahedron, then a cube…a wooden puzzle of nested Platonic solids! The pieces all fit together so perfectly. Stan is looking forward to an exhibition of Wayne’s work at the International Puzzle Party in San Diego later on this year. For more information, contact Stan at stan@isaacs.com.
Our second speaker was Scott Kim (www.scottkim.com), who’s presentation was entitled Motley Dissections. What is a motley dissection? The most famous example is the problem of the squared square — that is, dissecting a square with an integer side length into smaller squares with integer side lengths, but with all the squares different sizes.
One property of such a dissection is that no two edges of squares meet exactly corner to corner. In other words, edges always overlap in some way.
But there are of course many other motley dissections. For example, below you see a motley dissection of one rectangle into five, one pentagon into eleven, and finally, one hexagon into a triangle, square, pentagon and hexagon.
Look carefully, and you’ll see that no single edge in any of these dissections exactly matches any other. For these decompositions, Scott has proved they are minimal — so, for example, there is no motley dissection of one pentagon to ten or fewer. The proofs are not exactly elegant, but they serve their purpose. He also mentioned that he credits Donald Knuth with the term motley dissection, who used the term in a phone conversation not all that long ago.
Can you cube the cube? That is, can you take a cube and subdivide it into cubes which are all different? Scott showed us a simple proof that you can’t. But, it turns out, you can box the box. In other words, if the length, width, and height of the larger box and all the smaller boxes may be different, then it is possible to box the box.
Next week, Scott is off to the Gathering 4 Gardner in Atlanta, and will be giving his talk on Motley Dissections there. He planned an activity where participants actually build a boxed box — and we were his test audience!
He created some very elaborate transparencies with detailed instructions for cutting out and assembling. There were a very few suggestions for improvement, and Scott was happy to know about them — after all, it is rare that something works out perfectly the first time. So now, his success at G4G in Atlanta is assured….
We were so into creating these boxed boxes, that we happily stayed until 5:30 until we had two boxes completed.
I should mention that Scott also discussed something he terms pseudo-duals in two, three, and even four dimensions! There isn’t room to go into those now, but you can contact him through his website for more information.
As usual, we went out to dinner afterwards — and we gravitated towards our favorite Thai place again. The dinner conversation was truly exceptional this evening, revolving around an animated conversation between Scott Kim and magician Mark Mitton (www.markmitton.com).
The conversation was concerned with the way we perceive mathematics here in the U.S., and how that influences the educational system. Simply put, there is a lot to be desired.
One example Scott and Mark mentioned was the National Mathematics Festival (http://www.nationalmathfestival.org). Tens of thousands of kids and parents have fun doing mathematics. Then the next week, they go back to their schools and keep learning math the same — usually, unfortunately, boring — way it’s always been learned.
So why does the National Mathematics Festival have to be a one-off event? It doesn’t! Scott is actively engaged in a program he’s created where he goes into an elementary school at lunchtime one day a week and let’s kids play with math games and puzzles.
Why this model? Teachers need no extra prep time, kids don’t need to stay after school, and so everyone can participate with very little needed as far as additional resources are concerned. He’s hoping to create a package that he can export to any school anywhere where with minimal effort, so that children can be exposed to the joy of mathematics on a regular basis.
Mark was interested in Scott’s model: consider your Needs (improving the perception of mathematics), be aware of the Forces at play (unenlightened administrators, for example, and many other subtle forces at work, as Mark explained), and then decide upon Actions to take to move the Work (applied, pure, and recreational mathematics) forward.
The bottom line: we all know about this problem of attitudes toward mathematics and mathematics education, but no one really knows what to do about it. For Scott, it’s just another puzzle to solve. There are solutions. And he is going to find one.
We talked for over two hours about these ideas, and everyone chimed in at one time or another. Yes, my summary is very brief, I know, but I hope you get the idea of the type of conversation we had.
Stay tuned, since we are planning an upcoming meeting where we focus on Scott’s model and work towards a solution. Another theme throughout the conversation was that mathematics is not an activity done in isolation — it is a communal activity. So the Needs will not be addressed by a single individual, rather a group, and likely involving many members of many diverse communities.
A solution is out there. It will take a lot of grit to find it. But mathematicians have got grit in spades.
## Calculus: Linear Approximations, II
As I mentioned last week, I am a fan of emphasizing the idea of a derivative as a linear approximation. I ended that discussion by using this method to find the derivative of $\tan(x).$ Today, we’ll look at some more examples, and then derive the product, quotient and chain rules.
Differentiating $\sec(x)$ is particularly nice using this method. We first approximate
$\sec(x+h)=\dfrac1{\cos(x+h)}\approx\dfrac1{\cos(x)-h\sin(x)}.$
Then we factor out a $\cos(x)$ from the denominator, giving
$\sec(x+h)\approx\dfrac1{\cos(x)(1-h\tan(x))}.$
As we did at the end of last week’s post, we can make $h$ as small as we like, and so approximate by considering $1/(1-h\tan(x))$ as the sum of an infinite series:
$\dfrac1{1-h\tan(x)}\approx1+h\tan(x).$
Finally, we have
$\sec(x+h)\approx\dfrac{1+h\tan(x)}{\cos(x)}=\sec(x)+h\sec(x)\tan(x),$
which gives the derivative of $\sec(x)$ as $\sec(x)\tan(x).$
We’ll look at one more example involving approximating with geometric series before moving on to the product, quotient, and chain rules. Consider differentiating $x^{-n}.$ We first factor the denominator:
$\dfrac1{(x+h)^n}=\dfrac1{x^n(1+h/x)^n}.$
Now approximate
$\dfrac1{1+h/x}\approx1-\dfrac hx,$
so that, to first order,
$\dfrac1{(1+h/x)^n}\approx \left(1-\dfrac hx\right)^{\!\!n}\approx 1-\dfrac{nh}x.$
This finally results in
$\dfrac1{(x+h)^n}\approx \dfrac1{x^n}\left(1-\dfrac{nh}x\right)=\dfrac1{x^n}+h\dfrac{-n}{x^{n+1}},$
giving us the correct derivative.
Now let’s move on to the product rule:
$(fg)'(x)=f(x)g'(x)+f'(x)g(x).$
Here, and for the rest of this discussion, we assume that all functions have the necessary differentiability.
We want to approximate $f(x+h)g(x+h),$ so we replace each factor with its linear approximation:
$f(x+h)g(x+h)\approx (f(x)+hf'(x))(g(x)+hg'(x)).$
Now expand and keep only the first-order terms:
$f(x+h)g(x+h)\approx f(x)g(x)+h(f(x)g'(x)+f'(x)g(x)).$
And there’s the product rule — just read off the coefficient of $h.$
There is a compelling reason to use this method. The traditional proof begins by evaluating
$\displaystyle\lim_{h\to0}\dfrac{f(x+h)g(x+h)-f(x)g(x)}h.$
The next step? Just add and subtract $f(x)g(x+h)$ (or perhaps $f(x+h)g(x)$). I have found that there is just no way to convincingly motivate this step. Yes, those of us who have seen it crop up in various forms know to try such tricks, but the typical first-time student of calculus is mystified by that mysterious step. Using linear approximations, there is absolutely no mystery at all.
The quotient rule is next:
$\left(\dfrac fg\right)^{\!\!\!'}\!(x)=\dfrac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}.$
First approximate
$\dfrac{f(x+h)}{g(x+h)}\approx\dfrac{f(x)+hf'(x)}{g(x)+hg'(x)}.$
Now since $h$ is small, we approximate
$\dfrac1{g(x)+hg'(x)}\approx\dfrac1{g(x)}\left(1-h\dfrac{g'(x)}{g(x)}\right),$
so that
$\dfrac{f(x+h)}{g(x+h)}\approx(f(x)+hf'(x))\cdot\dfrac1{g(x)}\left(1-h\dfrac{g'(x)}{g(x)}\right).$
Multiplying out and keeping just the first-order terms results in
$\dfrac{f(x+h)}{g(x+h)}\approx f(x)g(x)+h\dfrac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}.$
Voila! The quotient rule. Now usual proofs involve (1) using the product rule with $f(x)$ and $1/g(x),$ but note that this involves using the chain rule to differentiate $1/g(x);$ or (2) the mysterious “adding and subtracting the same expression” in the numerator. Using linear approximations avoids both.
The chain rule is almost ridiculously easy to prove using linear approximations. Begin by approximating
$f(g(x+h))\approx f(g(x)+hg'(x)).$
Note that we’re replacing the argument to a function with its linear approximation, but since we assume that $f$ is differentiable, it is also continuous, so this poses no real problem. Yes, perhaps there is a little hand-waving here, but in my opinion, no rigor is really lost.
Since $g$ is differentiable, then $g'(x)$ exists, and so we can make $hg'(x)$ as small as we like, so the “$hg'(x)$” term acts like the “$h$” term in our linear approximation. Additionally, the “$g(x)$” term acts like the “$x$” term, resulting in
$f(g(x+h)\approx f(g(x))+hg'(x)f'(g(x)).$
Reading off the coefficient of $h$ gives the chain rule:
$(f\circ g)'(x)=f'(g(x))g'(x).$
So I’ve said my piece. By this time, you’re either convinced that using linear approximations is a good idea, or you’re not. But I think these methods reflect more accurately the intuition behind the calculations — and reflect what mathematicians do in practice.
In addition, using linear approximations involves more than just mechanically applying formulas. If all you ever do is apply the product, quotient, and chain rules, it’s just mechanics. Using linear approximations requires a bit more understanding of what’s really going on underneath the hood, as it were.
If you find more neat examples of differentiation using this method, please comment! I know I’d be interested, and I’m sure others would as well.
In my next installment (or two or three) in this calculus series, I’ll talk about one of my favorite topics — hyperbolic trigonometry. | {"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": 224, "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": 0.9089058041572571, "perplexity": 793.5339094386279}, "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-2019-09/segments/1550247504594.59/warc/CC-MAIN-20190221111943-20190221133943-00195.warc.gz"} |
https://socratic.org/questions/what-is-the-conjugate-pair-theorem | Precalculus
Topics
# What is the conjugate pair theorem?
Apr 17, 2015
In an acid-base neutralization, an acid and a base react to form water and salt.
In order for the reaction to carry out, there must be the transfer of protons between acids and bases. Proton acceptors and proton donors are the basis for these reactions, and are also referred to as conjugate bases and acids.
Apr 17, 2015
Although the Chemistry is interesting, the Conjugate Pair Theorem you want in algebra is:
For any polynomial with real coefficients, if $a + b i$ ($b \ne 0$) is a zero of the polynomial, then $a - b i$ is also a zero.
$a - b i$ and $a + b i$ are conjugates of each other, so they form a conjugate pair.
Another way of stating the Conjugate Pair Theorem is:
For any polynomial with real coefficients, imaginary zeros occur in conjugate pairs.
(By definition: a complex number $a + b i$ is imaginary if and only if $b \ne 0$.)
##### Impact of this question
3473 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 7, "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": 0.8992538452148438, "perplexity": 737.8168956011966}, "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/1642320300010.26/warc/CC-MAIN-20220116180715-20220116210715-00279.warc.gz"} |
https://www.physicsforums.com/threads/man-and-woman-on-boat-velocity-of-boat-after-dive.440335/ | # Man and Woman on Boat - Velocity of Boat after Dive
Gold Member
286
2
1. The problem statement, all variables and given/known data
A 180-lb man and a 120-lb woman stand side by side at the same end of a 300-lb boat, ready to dive, each with a 16-ft/s velocity relative to the boat. Determine the velocity of the boat after they have both dived, if (a) the woman dives first, (b) the man dives first.
(a) 9.20 ft/s (to the left)
(b) 9.37 ft/s (to the left)
2. Relevant equations
F=ma
3. The attempt at a solution
What does "...each with a velocity of 16-ft/s relative to the boat..." mean? If the man and woman are both standing on the boat, wouldn't their velocities relative to the boat be 0-ft/s?
Is 16-ft/s the velocities of each of them w.r.t. the boat afteir their respective dives?
Do they dive straight out or with x- and y- components (as in a ballistics eqn)?
Do we assume the motion (diving, boat's reaction) is all in the x-direction?
Aside from the question being highly vague, I've attempted (and failed) to solve the problem using impulse-momentum equations below.
____________________________________________________________
(Woman jumps first, creating an action-reaction pair of forces Fwand -Fw. So, she jumps and pushes off with a force equal to Fw. The boat feels -Fw.)
Impulse-momentum equation for Boat + Man (Eqn 1):
$$m_{(B+M)}v_{(B+M)}+F_{W}t=m_{(B+M)}v'_{(B+M)}$$
Impulse-momentum equation for Woman (Eqn 2):
$$m_Wv_W-F_Wt=m_Wv'_W$$
Adding Eqn 1 and Eqn 2:
$$0=m_{(B+M)}v'_{(B+M)}+m_Wv'_W$$
$$v'_{(B+M)}=\frac{-m_Wv'_W}{m_{(B+M)}}=-4ft/s$$
Then the man jumps...:
Impulse-momentum equation for Boat (Eqn 3):
$$\left ( m_B \right )\left ( \frac{-m_Wv'_W}{m_{(B+M)}} \right )+F_Mt=m_Bv'_B$$
Impulse-momentum equation for Man (Eqn 4):
$$\left ( m_M \right )\left ( \frac{-m_Wv'_W}{m_{(B+M)}} \right )-F_Mt=m_Mv'_M$$
Adding Eqn 3 and Eqn 4:
$$V'_B=\frac{\left ( m_B+m_M \right )(-4)-(m_M)(16)}{m_B}=-16ft/s$$
So, I found that the boat moves at -16 ft/s after both the man and woman jump. That's not the correct answer of -9.20 ft/s. Am I overcomplicating this whole matter? Is there some kind of m1v1+m2v1=m1v2+m2v2 way of solving it (conservation of linear momentum)?
Related Introductory Physics Homework News on Phys.org
#### tiny-tim
Homework Helper
25,789
242
A 180-lb man and a 120-lb woman stand side by side at the same end of a 300-lb boat, ready to dive, each with a 16-ft/s velocity relative to the boat. Determine the velocity of the boat after they have both dived, if (a) the woman dives first, (b) the man dives first.
What does "...each with a velocity of 16-ft/s relative to the boat..." mean? If the man and woman are both standing on the boat, wouldn't their velocities relative to the boat be 0-ft/s?
Is 16-ft/s the velocities of each of them w.r.t. the boat afteir their respective dives?
Do they dive straight out or with x- and y- components (as in a ballistics eqn)?
Do we assume the motion (diving, boat's reaction) is all in the x-direction?
Aside from the question being highly vague, I've attempted (and failed) to solve the problem using impulse-momentum equations below.
____________________________________________________________
Woman jumps first, creating an action-reaction pair of forces Fwand -Fw. So, she jumps and pushes off with a force equal to Fw. The boat feels -Fw.
Is there some kind of m1v1+m2v1=m1v2+m2v2 way of solving it (conservation of linear momentum)?
erm yes!
all the question means is that vperson - vboat = 16 ft/s (that's the x components … the y components won't affect the position of the boat, will they? )
so just put that into your conservation of momentum equations
Gold Member
286
2
erm yes!
all the question means is that vperson - vboat = 16 ft/s (that's the x components … the y components won't affect the position of the boat, will they? )
so just put that into your conservation of momentum equations
My conservation of momentum equations seem to get me to the same place:
Woman Jumps:
$$m_Bv_B+m_Mv_M+m_Wv_W=(m_B+m_M)v'_{(B+M)}+m_Wv_W$$
$$v'_{(B+M)}=\frac{-m_Wv'_W}{m_B+m_M}=-4ft/s$$
Then Man Jumps:
$$(m_B+m_M)v'_{(B+M)}=m_Bv''_B+m_Mv''_M$$
$$v''_B=\frac{(m_B+m_M)v'_{(B+M)}-m_Mv''_M}{m_B}=-16ft/s$$
#### tiny-tim
Homework Helper
25,789
242
I'm confused …
where does the given v1 - v2 = 16 come in that?
Gold Member
286
2
where does the given v1 - v2 = 16 come in
$$v_{person}-v_{boat}=16$$
$$v_{person}=v_{boat}+16$$
Woman Jumps:
$$(m_{boat}+m_{woman}+m_{man})v_{boat}=(m_{boat}+m_{man})v'_{boat}+m_{woman}(v_{boat}+16)$$
$$v'_{boat}=\frac{(m_{boat}+m_{woman}+m_{man})v_{boat}-m_{woman}(v_{boat}+16)}{m_{boat}+m_{man}}=-4ft/s$$
Then Man Jumps:
$$(m_{man}+m_{boat})v'_{boat}=m_{boat}v''_{boat}+m_{man}(v'_{boat}+16)$$
$$v''_{boat}=\frac{(m_{man}+m_{boat})v'_{boat}-m_{man}(v'_{boat}+16)}{m_{boat}}=-13.6ft/s$$
Still not getting the desired answer of -9.20 ft/s... Any other pointers?
#### tiny-tim
Homework Helper
25,789
242
It's very difficult to follow what you're doing without seeing any figures.
The initial speed of the boat is zero … have you used that?
Gold Member
286
2
Yes, I did use that. Then, all of the masses (m_m, m_w, and m_b) were all just their weights divided by 32.2.
$$v'_{boat}=\frac{\left [(300/32.2)+(120/32.2)+(180/32.2) \right ]0-(120/32.2)(0+16)}{(300/32.2)+(180/32.2)}$$
$$v''_{boat}=\frac{\left [(180/32.2)+(300/32.2) \right ](-4)-(180/32.2)(-4+16)}{(300/32.2)}$$
Doing the above calculations did not yield the correct answer of -9.20 ft/s for $v''_{boat}$
Last edited:
#### tiny-tim
Homework Helper
25,789
242
(just got up :zzz: …)
No, your (0 + 16) should be (v'boat + 16) …
v'woman = v'boat + 16 …
if the boat was fixed, v'woman would be 16, so since the boat isn't fixed, v'woman will obviously be less.
(and there's no need to divide everything by g … the ∑mv equation works just as well with weights instead of masses)
#### Xerxes1986
50
0
$$v_{person}-v_{boat}=16$$
$$v_{person}=v_{boat}+16$$
Woman Jumps:
$$(m_{boat}+m_{woman}+m_{man})v_{boat}=(m_{boat}+m_{man})v'_{boat}+m_{woman}(v_{boat}+16)$$
$$v'_{boat}=\frac{(m_{boat}+m_{woman}+m_{man})v_{boat}-m_{woman}(v_{boat}+16)}{m_{boat}+m_{man}}=-4ft/s$$
Then Man Jumps:
$$(m_{man}+m_{boat})v'_{boat}=m_{boat}v''_{boat}+m_{man}(v'_{boat}+16)$$
$$v''_{boat}=\frac{(m_{man}+m_{boat})v'_{boat}-m_{man}(v'_{boat}+16)}{m_{boat}}=-13.6ft/s$$
Still not getting the desired answer of -9.20 ft/s... Any other pointers?
You've almost got it with this equation:
$$(m_{boat}+m_{woman}+m_{man})v_{boat}=(m_{boat}+m_{man})v'_{boat}+m_{woman}(v_{boat}+16)$$
But you have made one mistake....the initial velocity of the boat ( v_{boat} ) is zero. So the whole left side of the eqn goes away. Also the velocity of the woman ISN'T $$(v_{boat}+16)$$, it is $$(v'_{boat}+16)$$. Fix that and your equation should yield -3.2ft/s for the velocity of the boat after the woman jumps. Then setup a similar equation for the man and you should get the correct answer of -9.2ft/s.
Gold Member
286
2
(just got up :zzz: …)
No, your (0 + 16) should be (v'boat + 16) …
v'woman = v'boat + 16 …
if the boat was fixed, v'woman would be 16, so since the boat isn't fixed, v'woman will obviously be less.
(and there's no need to divide everything by g … the ∑mv equation works just as well with weights instead of masses)
tiny-tim, I appreciate your help! I also learned something valuable, that if gravity (g) is going to cancel out in the end, don't bother converting all of the weights to masses. It's an unnecessary step. Awesome.
You've almost got it with this equation:
$$(m_{boat}+m_{woman}+m_{man})v_{boat}=(m_{boat}+m_{man})v'_{boat}+m_{woman}(v_{boat}+16)$$
...the velocity of the woman ISN'T $(v_{boat}+16)$, it is $(v'_{boat}+16)$. Fix that and your equation should yield -3.2ft/s for the velocity of the boat after the woman jumps. Then setup a similar equation for the man and you should get the correct answer of -9.2ft/s.
Xerxes, thanks for the help. That was where I went wrong... And I obtained the correct answers after working it out using $v'$ for the woman's jump and $v''$ for the man's jump.
Case closed on this ambiguously written textbook problem. Whew!
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• Solo and co-op problem solving | {"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": 0.7191537022590637, "perplexity": 2534.257695334916}, "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-2019-18/segments/1555578526966.26/warc/CC-MAIN-20190419021416-20190419043416-00197.warc.gz"} |
http://www.ck12.org/tebook/Human-Biology-Circulation-Teacher%2527s-Guide/r1/section/1.5/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 1.5: Unit Planning
Difficulty Level: At Grade Created by: CK-12
## Content Overview
Circulation: What is it? Why is it important?
In this activity driven unit, students explore these questions to understand their own cardiovascular systems, how they work, and how to keep them functioning effectively. Students make models of the heart and vessels, examine a mammalian heart, and listen to their own heartbeats to gain background for understanding basic structures of the cardiovascular system. They complete investigations of diffusion to connect the movement of molecules, with the passage of oxygen and nutrients out of capillary beds into the cells, and the passage of wastes and carbon dioxide from cells into the capillaries. By investigating how the body's control system regulates pressure and resistance in the vessels, students are introduced to negative feedback systems and their importance in regulating the cardiovascular system. With this background, students approach the concept of homeostasis, the internal body balance maintained by the circulatory system in the face of changing environmental conditions, and investigates how a healthy circulatory system maintains this balance. Students study risk factors for cardiovascular disease, based on their expanded knowledge of the circulatory system, and consider the evidence from the perspective of making good decisions about cardiovascular health and fitness.
Because the key ideas of the unit build in a spiral manner, it is advisable to complete sections 1-7 in sequence.
How is the unit structured?
Section 1 presents an overview of the circulatory system and an in-depth study of blood and its components.
Section 2 explores the heart's structure and how it functions.
Sections 3, 4, and 5 cover the blood vessels-arteries and arterioles, capillaries, and veins and venules.
Section 6 goes on to discuss pressure, flow and resistance in those blood vessels.
Section 7 covers cardiovascular health and the risk factors which affect it.
Why teach this unit? Connections to the Real World
According to the American Heart Association approximately 954,000\begin{align*}954,000\end{align*} people die each year from cardiovascular disease. America spent an estimated \$158.5\begin{align*}\158.5\end{align*} billion dollars on cardiovascular related health problems in 1997. Approximately, 57,490,000\begin{align*}57,490,000\end{align*} people in the US have one or more forms of cardiovascular disease. (Statistics obtained from the American Heart Association).
How is the unit structured?
PE Figure 1.1 Your circulatory system is like the streets of a city with lots of traffic flowing through the streets to and from different destinations.
PE Figure 2.20 The heart's two pumps work together side by side. Can you trace the flow of blood through the heart?
Rarely can you make it through the day without hearing the words “low-fat,” “healthy,” and “exercise.” Why? Because so much is known about cardiovascular health that our lives are filled with advice on how to keep our hearts healthy. An approach to teaching adolescents to make good decisions about their health is for them to explain the structure and function of their cardiovascular system. The more they know about how their bodies work, the better able they will be to make wise decisions that will lead to a healthy lifestyle.
A big part of choosing a healthy lifestyle involves the assessment and assumption of risk factors. This unit introduces the concept of risk. However, the discussion of risk could-and should-extend beyond food, exercise and the cardiovascular system. Many adolescents feel invulnerable-full of energy but lacking the experience of disease and death. At some point, a few will lose a friend or relative to disease or violence, and their understanding will grow. Risk is an inherent part of every youth's life. This unit provides a good opportunity to help them make decisions that could affect their lives for the long term.
This unit also stresses that cardiovascular health and fitness extend beyond personal choices to important public policy issues. The following summary questions provide a context for discussing cardiovascular health in a broader context:
Summary Questions to Consider throughout the Unit
Why don't more people donate blood or, in the event of premature death, healthy vital organs?
Should all packages of food contain information about possible health effects? For example, should a package of hamburger have a label warning consumers about the effects of eating fatty foods and red meat?
Is risk the same for everyone? How does risk vary from situation to situation?
Section 3: Arteries and Arterioles
Section 4: Capillaries
Section 5: Veins and Venules
PE Figure 5.1: The veins form a network of vessels that return blood to the heart.
Section 6: Pressure, Flow, and Resistance
PE Figure 6.8: This drawing shows how the cuff and stethoscope should be positioned to read the blood pressure.
Section 7: Cardiovascular Health
PE Figure 7.1: When arteries that feed the heart get narrow, the heart muscle doesn't get enough oxygen.
Unit Activities and Key Ideas
Section Key Ideas Activity
1. Circulation
Why is blood important to life?
• Complex animals like humans depend on the blood to transport food, water, oxygen, carbon dioxide and wastes to and from the body cells.
• Blood is made up of cells and plasma and blood diseases compromise the efficiency of the blood delivery system.
• The lymphatic system is a network of vessels and lymph nodes that function to recycle fluids which leak out of blood vessels back to the circulatory system.
Activity 1-1: Pathway of Blood through Your Body
Activity 1-2: Composition of Blood
Mini Activity: Blood Impressions
Mini Activity: Artificial Blood
2. The Heart
How does the heart pump blood?
• The heart is divided into four chambers: two “pumps” with two chambers each. One pump sends blood to the lungs for oxygen and back to the heart, the other sends the oxygenated blood throughout the body.
• The cardiac (heart) cycle is made up of a squeeze cycle and fill cycle: squeeze-fill; squeeze-fill; or systole-diastole, systole-diastole.
• The pacemaker is a specialized region of the heart which in combination with the nervous system and the endocrine system help to maintain homeostasis.
Mini Activity: Is Pumping Hard Work?
Activity 2-1: Exploring the Heart
Mini Activity: Heartbeats
Mini Activity: Word Origins
Activity 2-2: Siphon Pump
Mini Activity: What's Your Cardiac Output Today?
Enrichment 2-1: What Makes the Heart Beat Faster?
3. Arteries and Arterioles
How does blood get from the heart to all parts of the body?
• Arteries are thick, muscular vessels which carry blood away from the heart. Because arteries are elastic they expand during systole and their contraction during diastole propels the blood through the body.
• Arterioles are the smallest arteries, and have rings of muscle around them to serve as valves. By relaxing or contracting they control blood flow into the capillaries, and they control blood pressure in the arteries.
• Arteries and arterioles help maintain blood flow by maintaining blood pressure.
Activity 3-1: Blocked Arteries
4. Capillaries
How do oxygen and nutrients get from blood to cells?
• Capillaries are thin and permeable blood vessels that allow for the exchange of nutrients, gases and waste.
• This exchange of nutrients, gases and waste occurs through the process of diffusion.
• The capillary network is extensive; no cell is more than 2 cells away from a capillary.
Activity 4-1: Making a Capillary Bed Model
Mini Activity: Transport of Nutrients: Exploring Diffusion
Enrichment 4-1: Observing Goldfish Capillaries
Enrichment 4-2: Transport of Materials Exploring Diffusion
5. Veins and Venules
How does blood get back to the heart?
• Veins and venules are the counterparts of arteries and arterioles. They carry blood away from the capillaries Veins back to the heart.
• The walls of veins are thinner than those of arteries and can expand and collapse according to how much blood is in them.
• The pressure in veins is lower than the pressure in capillaries, so blood flows in one direction; from capillaries to venules to veins. Valves and the action of muscles help carry blood back one way to the heart.
Mini Activity: Observing Veins
Mini Activity: Percentages of Blood
Activity 5-1: The Direction of Blood Flow
6. Pressure, Flow, and Resistance
How is the right amount of blood directed to each part of the body?
• Blood pressure is an important indicator of how hard the heart is working to circulate blood through the body. High blood pressure can indicate a health problem.
• Friction builds up resistance in blood vessels contributing to an elevation of blood pressure.
• The endocrine and the nervous systems provide feedback that regulates basic functions such as breathing, heart rate and blood pressure.
Activity 6-1: Pressure, Resistance, and Flow
Mini Activity: The You in You
Mini Activity: Cold Toes
Activity 6-2: How a Controller Works
7. Cardiovascular Health
How can I keep my heart strong and my arteries clean and clear?
• Cardiovascular disease is the leading cause of death in this country.
• Cardiovascular risk factors can be genetic and/or environmental.
• Making healthy choices can reduce the risk of getting cardiovascular disease.
Mini Activity: Your Target Heart-Rate Zone
Activity 7-1: Pulse Rate
Mini Activity: Sources of Stress
Mini Activity: Risk profile
Enrichment 7-1: Cardiovascular Disease Risk Scoring
## Teacher's Guide Overview
This Circulation unit is built around a variety of student activities. Text material can be used to introduce, reinforce, and extend the concepts developed in the activities. The activities are the foundation of this unit, so the unit's success depends on students' involvement in the activities. Embedded activities are interrelated, since the concepts developed in one may be applied in another.
Section Planning
For each section, you'll find extensive advance planning for the student activities and the section topic. Key ideas, section objectives, background information, suggestions for introducing activities, and the materials needed for each activity are listed on the Section Planning page. Review this information ahead of time to ensure that materials for each activity are available when you need them.
Support for Embedded Activities
Embedded activities are those activities contained or “embedded” in the student edition. Procedures for each embedded activity are contained in the student edition. In the Teacher's Guide, you'll find activity planning information, activity assessment, and student reproducible pages for each embedded activity.
Enrichment Activities
Enrichment activities are activities found in the Teacher's Guide. These activities are designed to extend and enrich students' learning experiences. Complete Enrichment activities, including Teacher Activity Notes and the student procedures and reproducible pages, are located at the end of each appropriate section of the Teacher's Guide.
Group Work Activities
Learning science is a process that is both individual and social. Students in science classrooms often need to interact with their peers to develop a knowledge of scientific concepts and ideas, just as researchers, engineers, mathematicians, and physicians who are working in teams do to answer questions and to solve problems. The Group Work activities of the HumBio Curriculum for Middle Grades have been developed to foster a collaborative environment for groups of students. Students plan experiments, collect and review data, ask questions and offer solutions, use data to explain and justify their arguments, discuss ideas and negotiate conflicting interpretations , summarize and present findings, and explore the societal implications of the scientific enterprise. In short, Group Work activities provide an environment in which students are “doing science” as a team.
Projects
The research and action projects in HumBio are varied and provide students with time to explore a particular topic in depth. With Projects, students have the opportunity to take a position based on knowledge gained through research, debate an issue, and devise a plan of action. In this way, students can apply what they are learning to larger issues in the world around them.
Projects for this unit include
• Research Questions and Action Projects
• Be Heart Smart
• A Cafeteria Case Study
• Tasty Tidbits
• Past vs. Present
## Assessment Overview
Within each section of the unit there are suggestions for assessment that can be used individually or in combination to develop a complete assessment package. The list below describes the variety of assessment tools provided.
Apply Your Knowledge Questions appear throughout each section. They can be used as homework assignments and as ways to initiate a class discussion. These Questions are designed to assess
• communication skills
• depth of thought and preparation
• problem-solving skills
• ability to apply concepts to related or big ideas
• how well students relate their new knowledge to different problems
What Do You Think?
These Questions appear in each section. They provide students with opportunities to think and write about the concepts they are learning in a larger context. You can use these Questions to assess
• writing skills
• problem-solving abilities
• creativity and depth of thought
• the ability to analyze and summarize
Journal Writing prompts are suggested throughout the unit. These prompts provide opportunities for students to write critically and creatively about concepts and issues. The writing products can be used to assess
• writing skills
• depth of thought
• and the ability to explain and expand concepts
Review Questions
Review Questions are located at the end of each section. These Questions can be used for written responses or as the basis for class discussion. These Questions are designed to assess content knowledge and whether students can explain the concepts explored in the section.
Activity-Based Assessment
Inquiry-based student-centered activities are the foundation of the Human Biology Program. The unit is rich with relevant exciting activities that introduce, support, or reinforce concepts students are exploring. Within the Teacher's Guide, you'll find extensive teacher Information, including assessment strategies, for each type of activity:
• Embedded Activities
• Enrichment Activities
• MiniActivities
• GroupWork
• Projects
You can use students' products to assess their progress. These products include models, simultaneous, observations and reports of laboratory investigations, role plays, written responses to questions and written observations, student-designed explorations and procedures, poster presentations and classroom presentations.
PORTFOLIO ASSESSMENT
You may want to have your students develop a portfolio for the unit. A sample assessment portfolio for the unit might contain the following items:
• Written responses to three What Do You Think? questions
• An analysis of their two favorite Activities and how those activities helped then understand an important concept
• Two examples of written reports from library research from
Mini Activity: Cold Toes
Mini Activity: Artificial Blood
• An Activity Report from three investigations such as
Activities 2-1, 2-2, 4-1, 5-1
• An analysis or interpretation of graphs from Activity 3-1
• Example of constructing a model from
Activity 4-1: Making a Capillary Bed Model
• One example of an artistic creation from
Mini Activity: Blood Impressions
## Getting Started
The Human Biology Circulation unit emphasizes learning about the structure and function of the circulatory system and how to apply this knowledge to making good decisions about overall cardiovascular fitness.
As with all Human Biology units, the Circulation unit is built around inquiry. For each activity, teacher information is included on the section planning page and the activity page.
Plan the Unit to Fit Class Needs. For each activity, helpful hints, strategies and the materials needed are listed in the Teacher's Guide. Checking this information ahead of time will ensure that the materials will be available when needed. Teacher information for each activity is included in the Plan for each Section under the heading Advance Preparation. The important activity information is also listed in the Activity Planning page under the heading Advance Preparation.
Be aware of any special health problems your students may have which would make it uncomfortable for them to participate in certain activities such as taking measurements of pulse, blood pressure, and heart rates. Examples of such health problems include congenital or accidental cardiovascular conditions. For students unable to participate fully in these activities, you may wish to create an alternative assignment or to have them use data from another group. If the class is prepared appropriately, the affected students may want to share their special circumstances with the class to enhance the appreciation and understanding of all students.
Connect with Other Disciplines. The Circulation unit is particularly well suited for interdisciplinary use in combination with the units on Breathing and Digestion and Nutrition.
Although many schools include this unit through health or science classes, the interdisciplinary web provided shows you ways to expand the teaching of this material into other subject areas to reinforce learning for students. For interdisciplinary planning set meetings with your team early. The suggested interdisciplinary breakdown below provides some ideas for connections among subject areas. You are encouraged to tap the talents and interests of your team members as well as of your unique school and community resources in developing other suitable activities for this unit. Your school library/multimedia resource center and the science department at the high school may also be able to provide references that can be very helpful in teaching this unit.
As previously mentioned, continued emphasis upon healthy decision making leading to improved cardiovascular health involves all body systems affected by the circulatory system. Therefore, this unit is ideal for combining with the Human Biology units on Breathing and Digestion and Nutrition. The Circulation unit also links well with the Human Biology unit on the Nervous System because of the nervous system's role in controlling and coordinating homeostasis in the body.
Connections to other Human Biology units include Genetics with respect to hereditary predisposition to cardiovascular disease.
Use Current Events. Current events can be an important part of the circulation unit. Students can use current events to make group scrapbooks, bulletin boards, posters, or to give class presentations. Some examples of current event topics include articles explaining guidelines for healthy eating and exercise programs, medical advances in treating blocked arteries and heart disease, and information about new medications helpful in controlling high blood pressure.
Make Career Connections. Encourage student investigation of careers related to cardiovascular health such as in research, industrial production of medications, public education, and in health professions. Examples of health professions include a cardiologist, a surgical nurse specializing in cardiology, and a paramedic.
Use a Variety of Resources. For the duration of the unit, we encourage you and your students to use a wide variety of resources. The activities provide rich opportunities for students to explore many concepts, and the more they incorporate information from sources outside the classroom, the richer their experiences will be. Use your own creativity and the student activities in this unit to develop a series of lessons tailored to the needs of your students. Engage students through activities to help them learn about the function, and importance of the circulatory system. Use computer services for student and teacher information, networking (student pen pals, other schools, other teachers, other communities), and connecting with experts in the field.
Plan for Field Trips. Field trips to local hospitals, industrial sites, or universities need to be arranged in advance. Contact the public affairs offices of these institutions for assistance.
Possible guest speakers include specialists from the careers mentioned above.
If you select a guest speaker with a cardiovascular condition to address that condition, be sure to prepare your students appropriately so they will be sensitive and compassionate listeners. Prepare speakers by sharing with them the knowledge base of students.
Connect with the Home. Because lifestyle changes for improving cardiovascular health may involve changes in family eating and exercise patterns, students should be encouraged to take Apply Your Knowledge questions and Mini Activities home for further exploration. As a class, or as individuals, the new questions raised can become a part of ongoing research for everyone.
## Teaching Timelines
You can use these timelines as a place to start in designing your own timelines, or you can use them as they are laid out. If you're planning your own timeline, consider the inclusion of the Embedded activities first. The “Embedded activities” are included in the student edition. The Enrichment activities, Group Work activities, and Projects can then be included, depending on your time restrictions. The timelines are guides that can vary if some activities are done at home or in other classes in addition to science class.
Given your time constraints, it may not be possible to do all the activities on these timelines. If you need to remove activities, be careful not to remove any activities critical to sequential student understanding of the unit. You may want to divide the activities among interdisciplinary members of your teaching team.
Page references in these charts refer to the student edition, except when Enrichments are suggested. The page references for Enrichments refer to this Teacher's Guide.
Option 1: Three Week Timeline
Monday Tuesday Wednesday Thursday Friday
Week 1
Introduce Unit
Introduce Section 1
Activity 1-1: Pathway of Blood Through Your Body
Conclude Activity 1-1
Activity 2-1: Composition of Blood
Assign Mini Activities: Blood Impressions Artificial Blood
Introduce Section 2
Activity 1-2: Exploring the Heart
Continue Activity 2-1: Exploring the Heart
Activity 2-2: Siphon Pump
Week 2
Mini-Activities: Heart beats Word Origins What's Your Cardiac Output Today?
Activity 3-1:
Blocked Arteries
Review Sections 1, 2, 3
Introduce Section 4
Teaching Strategies
Enrichment 4-1:
Observing
Goldfish
Capillaries
Enrichment 4-2:
Transport of Materials Exploring diffusion
Introduce Section 5.
Activity 5-1: The Direction of Blood Flow
Week 3
Review Sections 3, 4, 5
Introduce Section 6.
Activity 6-1: Pressure, Resistance and Flow
Activity 6-2: How a Controller Works
Introduce Section 7
Activity 7-1: Pulse Rate
Assign Mini Activities: Your Target Heart Rate Zone
Risk Profile
Unit Review and Assessment
Option 2: Five Week Timeline
Monday Tuesday Wednesday Thursday Friday
Week 1
Introduce Unit
Introduce Section 1
Activity 1-1: Pathway of Blood Through Your Body
Conclude Activity 1-1
Activity 1-2: Composition of Blood
Conclude Activity 1-2
Assign Mini Activities: Blood Impressions Artificial Blood
Introduce Section 2
Introduce Activity 2-1: Exploring the Heart
Optional Mini Activity: Is Pumping Hard Work?
Activity 2-1: Exploring the Heart
Week 2
Optional Mini Activity: Heartbeats
Assign Mini Activity: Word Origins
Activity 2-2: Siphon Pump
Optional Mini Activity: What's Your Cardiac Output Today?
Enrichment 2-1: What Makes the Heart Beat Faster?
Introduce Section 3
Activity 3-1: Blocked Arteries
Review Sections 1, 2, 3
Introduce Section 4
Week 3
Enrichment 4-1:
Observing
Goldfish
Capillaries
Enrichment 4-2:
Transport of Material Exploring Diffusion
Introduce Section 5.
Activity 5-1: The Direction of Blood Flow
Mini Activities:
Observing Veins Percentages of Blood
Introduce Section 6
Activity 6-1: Pressure, Resistance, and Flow
Week 4
Mini Activities:
The You in You
Cold Toes
Activity 6-2: How a Controller Works
Enrichment 6-1:
Review Sections 4,5, 6
Including the Key Ideas
Introduce Section 7
Mini Activities:
Sources of Stress
Activity 7-1:
Pulse Rate
Week 5
Mini Activity:
Risk Profile
Enrichment 7-1:
Cardiovascular
Disease Risk
Scoring
Project Presentations, Culminating Activities
Unit Reviews
Including Unit Assessment
Project Presentations, Culminating Activities
Unit Reviews
Including Unit Assessment
Project Presentations, Culminating Activities
Unit Reviews
Including Unit Assessment
## Safety for Teachers
• Always perform an experiment or demonstration on your own before allowing students to perform the activity. Look for possible hazards. Alert students to possible dangers. Safety instructions should be given each time an experiment is begun.
• Wear glasses and not contact lenses. Make sure you and your students wear safety goggles in the lab when performing any experiments.
• Do not tolerate horseplay or practical jokes of any kind.
• Do not allow students to perform any unauthorized experiments.
• Never use mouth suction in filling pipettes with chemical reagents.
• Never “force” glass tubing into rubber stoppers.
• Use equipment that is heat resistant.
• Set good safety examples when conducting demonstrations and experiments.
• Turn off all hot plates and open burners when they are not in use and when leaving the lab.
• When students are working with open flames, remind them to tie back long hair and to be aware of loose clothing in order to avoid contact with flames.
• Make sure you and your students know the location of and how to use fire extinguishers, eyewash fountains, safety showers, fire blankets, and first-aid kits.
• Students and student aides should be fully aware of potential hazards and know how to deal with accidents. Establish and educate students on first-aid procedures.
• Teach students the safety precautions regarding the use of electricity in everyday situations. Make sure students understand that the human body is a conductor of electricity. Never handle electrical equipment with wet hands or when standing in damp areas. Never overload electrical circuits. Use 3\begin{align*}3-\end{align*}prong service outlets.
• Make sure that electrical equipment is properly grounded. A ground-fault circuit breaker is desirable for all laboratory AC circuits. A master switch to cut off electricity to all stations is desirable for all laboratory AC circuits.
• Make sure you and your students are familiar with how to leave the lab safely in an emergency. Be sure you know a safe exit route in the event of a fire or an explosion.
For Student Safety
Safety in the Classroom
• Wear safety goggles in the lab when performing any experiments. Tie back long hair and tuck in loose clothing while performing experiments, especially when working near or with an open flame.
• Never eat or drink anything while working in the science classroom. Only lab manuals, notebooks, and writing instruments should be in the work area.
• Do not taste any chemicals for any reason, including identification.
• Carefully dispose of waste materials as instructed by your teacher. Wash your hands thoroughly.
• Do not use cracked, chipped, or deeply scratched glassware, and never handle broken glass with your bare hands.
• Lubricate glass tubing and thermometers with water or glycerin before inserting them into a rubber stopper. Do not apply force when inserting or removing a stopper from glassware while using a twisting motion.
• Allow hot glass to cool before touching it. Hot glass shows no visible signs of its temperature and can cause painful burns. Do not allow the open end of a heated test tube to be pointed toward another person.
• Do not use reflected sunlight for illuminating microscopes. Reflected sunlight can damage your eyes.
• Tell your teacher if you have any medical problems that may affect your safety in doing lab work. These problems may include allergies, asthma, sensitivity to certain chemicals, epilepsy, or any heart condition.
• Report all accidents and problems to your teacher immediately.
HANDLING DISSECTING INSTRUMENTS and PRESERVED SPECIMENS
• Preserved specimens showing signs of decay should not be used for lab observation or dissection. Alert your teacher to any problem with the specimen.
• Dissecting instruments, such as scissors and scalpels, are sharp. Use a cutting motion directed away from yourself and your lab partner.
• Be sure the specimen is pinned down firmly in a dissecting tray before starting a dissection.
• In most cases very little force is necessary for making incisions. Excess force can damage delicate, preserved tissues.
• Do not touch your eyes while handling preserved specimens. First wash your hands thoroughly with warm water and soap. Also wash your hands thoroughly with warm water and soap when you are finished with the dissection.
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Feb 23, 2012 | {"extraction_info": {"found_math": true, "script_math_tex": 4, "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": 0.21783459186553955, "perplexity": 4891.855011095888}, "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/1464049281978.84/warc/CC-MAIN-20160524002121-00046-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/122416/why-does-the-speed-of-the-propellant-limit-the-speed-of-a-space-ship-in-open-spa | Why does the speed of the propellant limit the speed of a space ship in open space?
Isn't speed a relative thing in space? If so, why would the speed of a propellant matter? Why can't a space ship accelerate infinitely?
-
You're asking about the calculation for a finite amount of fuel, right? – David Z Jun 30 at 2:33
Imagine that the spaceship is 1/2 fuel, 1/2 useful cargo. Those pieces ultimately get divided. Then by momentum conservation, the final speed will be exactly like the average propellant speed, but in the opposite direction. If the fuel is a higher fraction of the mass, one may achieve a higher speed of the useful cargo than the propellant, but this increase is only logarithmic. – Luboš Motl Jun 30 at 4:48
This is why laser-pushed sails are the best option for an interstellar craft - You keep all the fuel at home, and you final velocity is limited only by the width of the Gaussian beam wavefront, the wavelength of that Gaussian laser beam, and the width, as well as the optical and thermal characteristics of the sail that you can afford. Also, compared with rockets, where your payload efficiency is around 1%, a laser-sail can give you payload efficiencies as high as 30% – lurscher Jun 30 at 5:10
@this: The same can be said of a rocket employing baking soda and vinegar, if the fuel tanks are big enough. – Beta Jun 30 at 9:10
This site can be a lot of help when understanding rocket science - projectrho.com/public_html/rocket/engines.php :) – Luaan Jun 30 at 16:30
Why can't a space ship accelerate infinitely?
Because a space ship needs to carry fuel, and because that fuel needs to be contained in a fuel tank.
That need to carry the fuel needed to make the spacecraft accelerate leads to the very nasty ideal rocket equation,
$$\Delta v = v_e \ln \left( \frac {m_{\text{initial}}} {m_{\text{final}}} \right)$$
The initial mass is the final mass plus the mass of the fuel and oxidizer that will eventually be expelled. The final mass includes the mass of the fuel tanks, the spacecraft structure, and whatever tiny payload that can be carried on top of that. Another way to look at that logarithmic growth in change in velocity is an exponential growth in fuel mass.
More fuel means larger fuel tanks and more structure to hold everything together. A delta V of 2.3 to 3 times the exhaust velocity is doable. That corresponds to a vehicle whose mass is initially 90% to 95% fuel. Beyond that, there's not much hope for a single stage rocket.
What about a multistage rocket? While multistage rockets appear to escape the tyranny of the rocket equation to some extent, they don't. They just change where the tradeoff occurs, and they add their own brand of exponential growth problems. The Saturn V + Apollo spacecraft that brought humans to the surface of the Moon and back was essentially a six stage rocket. The vehicle that took off from the surface of the Earth was arguably that largest, most powerful machine ever built. The vehicle that splashed down in the ocean when the mission was over was a tiny little thing that could barely hold three humans.
The only way to escape the tyranny of the rocket equation is the age-old answer to the age-old question "Doctor, it hurts when I do this: «bonk»" The answer: "Don't do that then."
Unfortunately, we don't know how to avoid the nastiness of the rocket equation. The only viable option currently that doesn't require carrying fuel is a solar sail. However, you can't accelerate forever with a solar sail because sunlight drops as the inverse square from distance to the Sun. Beyond solar sails? Now you're in the realm of science fiction.
The above assumes Newtonian mechanics. The relativistic rocket equation is far, far worse than the ideal (Newtonian) rocket equation. If the ideal rocket equation is evil, the relativistic rocket equation is evil incarnate. The relativistic rocket equation is one of the answers to the Fermi paradox.
-
Surely nuclear power or antimatter isn't science fiction. – this Jun 30 at 7:43
@this - nuclear power and antimatter don't change anything, an engine powered by them still needs to expel propellant in order to move forward, and the same rocket equation applies. The propellant needs to be stored in the rocket, it's finite, and the propellant speed (determined by the amount of energy that the nuclear or antimatter power source can provide) limits the spaceship speed. Currently used ion thrusters are examples of such engines, and they can be powered by nuclear reactors or antimatter once it's practical to do so. – Peteris Jun 30 at 8:44
@this If you created photons, I assure you you had a source of fuel that lost mass in the process. Also, for all intents and purposes, antimatter-powered anything is science fiction. – Chris White Jun 30 at 10:36
@this - The problem still exists. You have to carry (and consume) fuel to create those photons. In fact, photons are pretty much the very worst choice when it comes to rocket exhaust. The ratio of momentum squared to unit energy consumed to create that momentum is tiny with photons. – David Hammen Jun 30 at 11:01
Fusion is not science fiction. Look it up. It powers our planet. A rocket consisting of mostly fuel mass also complies with our physical laws. Your definition of science fiction must be very ( very ) broad. – this Jun 30 at 18:11
The maximum theoretical speed that a spaceship can reach isn't limited by anything (except the speed of light of course). However for a practical spaceship with a finite amount of fuel, the speed of the exhaust will set a practical maximum on the speed of the spaceship. This is because in order to accelerate to a higher speed, the spaceship would have to carry more fuel to begin with, but this additional fuel would increase the mass of the spaceship, making it even harder to accelerate. This relationship is exponential, which means for a reasonable rocket (one that you could actually build), the exhaust speed of the propellant sets a practical maximum on the final speed of the rocket.
If I recall correctly this practical limit is roughly twice the exhaust speed of the propellent. After this, the diminishing returns get too ridiculous.
-
Roughly three times the exhaust speed is perhaps a better number rather than twice, but it's the same concept. Three times exhaust speed means a vehicle that is initially 95% fuel. That's a tough but doable engineering problem. Twice exhaust speed means a vehicle that is only 86.5% fuel initially. That's a bit low. Modern rockets fall in the 90 to 95% range. That was being nitpicky. +1. – David Hammen Jun 30 at 11:28
As David said, in a rocket the final velocity is given by a function that grows logarithmically (i.e: really slowly) with the ratio between the mass of the rocket with and without fuel (the so-called dry mass). So if you somehow make your fuel carrying capacity a hundred times larger than the mass of the empty ship, you only increased your final velocity by something which is of the order of 2. If you increase that ratio to a whopping 1000 times, you only increase your final velocity in a factor which is of the order of 3. As those ratio get that large, your ship begins to have several structural and material problems as basically you are trying to move a whale worth of fuel on top of a paper craft. A realistic limit for chemical rockets is about a factor of 6-8, and probably as much as 15 could be achievable with fission fragment rockets.
This is why laser-pushed sails are the best option for an interstellar craft - You keep all the fuel at home, and your final velocity is limited only by the width of the Gaussian beam wavefront, the wavelength of that Gaussian laser beam, the width, as well as the optical and thermal characteristics of the sail that you can afford. Also, compared with rockets, where your payload efficiency is around 1%, a laser-sail can give you payload efficiencies as high as 30%
-
-1, for two reasons. The first and lesser reason is that your numbers are off. Your 100:1 spacecraft achieves 4.6 times exhaust velocity, and for your 1000:1 spacecraft the factor is 6.9. Do the math. The larger reason is your last paragraph (continued). – David Hammen Jun 30 at 11:44
As far as "laser-pushed sails" are concerned, that's pure science fiction for now. Beam propulsion is but one of many perpetually low TRL (technology readiness level) technologies. It's impossible to say which, if any, of those perpetually low TRL technologies is the "best". One of those technologies might eventually break through the low TRL barrier, but sans a magical crystal ball, there's no foretelling which one will do so. – David Hammen Jun 30 at 11:46
4.6 / 6.9 = 2 / 3, Which is why I said "of the order of", instead of "exactly", the other factor is constant in both cases – lurscher Jun 30 at 16:22
TRL is irrelevant as the OP is asking about how to achieve acceleration that is unbounded by the Tsiolkovsky equation. Regardless of what your feelings are toward laser-pushed sail readiness, we use what we know of physics as it currently stands as best as we can, and what current physics says is that laser-pushed sails are the most feasible solution for interstellar flight, like it or not – lurscher Jun 30 at 16:26
TRL most certainly is very important with regard to your claim that "laser-pushed sails are the best option for an interstellar craft". Proponents of alternate low TRL technologies would most certainly dispute this claim. The OP asked about being able to "accelerate infinitely", and beamed propulsion does not accomplish that end. – David Hammen Jun 30 at 16:40 | {"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": 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": 0.8139337301254272, "perplexity": 994.718684891271}, "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-2014-41/segments/1410657137108.99/warc/CC-MAIN-20140914011217-00035-ip-10-234-18-248.ec2.internal.warc.gz"} |
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The combined sedimentation and diffusion process of macromolecules in the centrifugal field was considered phenomenologically. For each dissolved particle a stochastic equation is estimated composed of two independent velocities: a diffusional and a sedimentational. With the help of the central limit theorem with Lindeberg conditions'' the formula for the distribution of …
### Centrifugal Pumps Fluid Flow Practical Calculations
The Volume formula is: V1 = V2 Then, A1.S1 = A2.S2 Or, A1 / A2 = S2 / S1 It is a simple lever machine since force is multiplied. The mechanical advantage is: MA = [S1 / S2 = A2 / A1]; can also be = [S1 / S2 = (π. r) / (π.R)]; or = [S1 /S2 = r / R] Where: A = Cross sectional area, in S = Piston distance moved, in
### Relative Centrifugal Field (RCF) - Beckman Coulter
There’s a simple formula for calculating the strength of a particular centrifugal field: where r stands for the radius, which is the distance in millimeters (mm) from the center of rotation to some point within the rotor, and RPM is the speed of rotation in revolutions per minute (rpm). Sometimes radial distances are given in centimeters.
### Molecular Weight Analysis in Centrifugal Fields
tive determinations in centrifugal fields up to 400,000 times the force of gravity, using a column of solution 12 mm high. If the height be reduced to 6 mm, it should be possible to reach 800,000 times gravity. An ultracentrifuge for measurements in this extreme field intensity is now being constructed in the writer's laboratory.
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https://repo.scoap3.org/record/30956 | # Fermion masses, mass-mixing and the almost commutative geometry of the Standard Model
Dąbrowski, Ludwik (0000 0004 1762 9868, Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, I-34136, Trieste, Italy) ; Sitarz, Andrzej (Instytut Fizyki Uniwersytetu Jagiellońskiego, Stanisława Łojasiewicza 11, 30-348, Kraków, Poland) (0000 0001 2286 5863, Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-656, Warszawa, Poland)
13 February 2019
Abstract: We investigate whether the Standard Model, within the accuracy of current experimental measurements, satisfies the regularity in the form of Hodge duality condition introduced and studied in [9]. We show that the neutrino and quark mass-mixing and the difference of fermion masses are necessary for this property. We demonstrate that the current data supports this new geometric feature of the Standard Model, Hodge duality, provided that all neutrinos are massive.
Published in: JHEP 1902 (2019) 068 | {"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": 0.8388080596923828, "perplexity": 2357.843105595645}, "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-2019-18/segments/1555578529472.24/warc/CC-MAIN-20190420080927-20190420102927-00343.warc.gz"} |
https://tex.stackexchange.com/questions/600821/error-that-occurred-during-compilation-pdflatex | # Error that occurred during compilation (PDFLatex)
Below is the error that occurred during the compilation.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
! LaTeX error: "xparse/unknown-argument-type"
!
! Unknown argument type '!' replaced by 'm'.
!
! See the LaTeX3 documentation for further information.
!
! For immediate help type H <return>.
!...............................................
l.105 }
In the .tex file, I used a template from the tikz&pgf documentation that includes the following code (which ends at line 105):
\documentclass[a4paper]{memoir}
\usepackage{tcolorbox}
\usepackage{tikz}
\usepackage{lipsum}
\tcbuselibrary{skins,hooks}
% Ex si solutii fara labeling
\NewTColorBox[auto counter,number within=section]{exercise}{+!O{}}{%
enhanced,colframe=green!20!black,colback=yellow!10!white,coltitle=green!40!black,
fonttitle=\bfseries,
underlay={\begin{tcbclipinterior}
(interior.north west) circle (2cm);
\draw[help lines,step=5mm,yellow!80!black,shift={(interior.northwest)}]
(interior.south west) grid (interior.north east);
\end{tcbclipinterior}},
title={Exercise~\thetcbcounter:},
label={exercise@\thetcbcounter},
after upper={\par\hfill\textcolor{green!40!black}%
{\itshape Solution on page~\pageref{solution@\thetcbcounter}}},
lowerbox=ignored,
savelowerto=solutions/exercise-\thetcbcounter.tex,
record={\string\solution{\thetcbcounter}{solutions/exercise-\thetcbcounter.tex}},
#1
}
\NewTotalTColorBox{\solution}{mm}{%
enhanced,colframe=red!20!black,colback=yellow!10!white,coltitle=red!40!black,
fonttitle=\bfseries,
underlay={\begin{tcbclipinterior}
(interior.north west) circle (2cm);
\draw[help lines,step=5mm,yellow!80!black,shift={(interior.north west)}]
(interior.south west) grid (interior.north east);
\end{tcbclipinterior}},
title={Solution of Exercise~\ref{exercise@#1} on page~\pageref{exercise@#1}:},
phantomlabel={solution@#1},
attach title to upper=\par,
}{\input{#2}}
\tcbset{no solution/.style={no recording,after upper=}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\chapter{Execices}%
%\vspace{-1cm}%
\tcbstartrecording\relax
\begin{exercise}
Compute the derivative of the following function:
\begin{equation*}
f(x)=\sin((\sin x)^2)
\end{equation*}
\tcblower
The derivative is:
\begin{align*}
f'(x) &= \left( \sin((\sin x)^2) \right)'=\cos((\sin x)^2) 2\sin x \cos x.
\end{align*}
\end{exercise}
\begin{exercise}[no solution]
It holds:
\begin{equation*}
\frac{d}{dx}\left(\ln|x|\right) = \frac{1}{x}.
\end{equation*}
\end{exercise}
\begin{exercise}
Compute the derivative of the following function:
\begin{equation*}
f(x)=(\sin(\sin x))^2
\end{equation*}
\tcblower
The derivative is:
\begin{align*}
f'(x) &= \left( (\sin(\sin x))^2 \right)'=2\sin(\sin x)\cos(\sin x)\cos x.
\end{align*}
\end{exercise}
\tcbstoprecording
\newpage
\tcbinputrecords
\end{document}
How can be this problem be solved?
• Welcome to TeX. SE! Please extend your code fragment to complete small but compilable document. – Zarko Jun 11 at 11:04
• I think we need to see your log file: I wonder what version of code you have. – Joseph Wright Jun 11 at 11:08
• Additionally is your latex up to date? This kind of problem can be seen by inproperly updated latex installations – daleif Jun 11 at 11:08
• Some info from log file: This is pdfTeX, Version 3.14159265-2.6-1.40.16 (MiKTeX 2.9 64-bit) (preloaded format=pdflatex 2019.3.9) 11 JUN 2021 14:00 entering extended mode.... . – demeter Jun 11 at 11:32
• Some updates after updating the Latex. A new error occurred, namely caused by the \tcblower. More precisely: LaTeX Warning: Reference 'solution@1.0.1' on page 1 undefined on input line 141 ! Undefined control sequence. \tcbverbatimwrite ...\immediate \openout \tcb@out #1 \tcb@verbatim@begin@hoo... l.141 \tcblower – demeter Jun 11 at 12:03 | {"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": 0.9506363272666931, "perplexity": 29612.52211094165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488257796.77/warc/CC-MAIN-20210620205203-20210620235203-00267.warc.gz"} |
https://cs.stackexchange.com/questions/18183/if-p-np-how-do-i-prove-i-can-find-the-maximum-clique-in-polynomial-time | # If P = NP, how do I prove I can find the maximum clique in polynomial time?
I want to prove that if P = NP, then there is a polynomial time algorithm for finding the largest clique in an undirected graph.
I understand how to use a verifier to find this but my issue is since P = NP it doesn't want me to use a verifier. I'm not sure how to approach this.
• basically its nearly by definition of NP completeness and that the problem is NP complete. – vzn Nov 21 '13 at 20:40
Assuming $P=NP$, then $CLIQUE \in P$, so you can test for a clique of size $k$ in polynomial time for all $k$. So you can just test for a clique of each size between $1$ and $n$ where $n$ is the number of nodes. | {"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": 0.722786545753479, "perplexity": 88.6372892488692}, "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/1627046154796.71/warc/CC-MAIN-20210804045226-20210804075226-00111.warc.gz"} |
http://physics.stackexchange.com/questions/46605/when-can-photon-field-amplitudes-be-written-as-field-operators | # When can photon field amplitudes be written as field operators?
Suppose I have some classical field equation for two photon fields with amplitudes $A_1(z),A_2(z)$ (plane waves) given as
${A}_1=\alpha f(A_1,A_2) \\ {{A}_2}=\beta g(A_1,A_2)$
Under what conditions can I make the replacement $A\rightarrow \hat{A}$ ? I am uncomfortable with the Bohr correspondence principle because the correspondence principle seems to be a very weak argument.
Edit: $z$ is position, $\alpha$,$\beta$ are real numbers. $f$ and $g$ are functions of the amplitudes.
-
If you're going to spend reputation on the bounty for this question, you might also want to spend some time cleaning up the question. What is $z$? What are $\alpha$ and $\beta$? What are $f$ and $g$? It's hard to tell what you're asking. – user1504 Dec 15 '12 at 15:25 | {"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": 0.6649801731109619, "perplexity": 202.82080985802108}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783396538.42/warc/CC-MAIN-20160624154956-00073-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://wrenstech.wordpress.com/2014/08/07/estimating-bb-muzzle-velocity-with-a-voice-recorder-and-a-curtain/comment-page-1/ | # Estimating BB Muzzle Velocity With A Voice Recorder And A Curtain
If you’re reading this blog, the chances are that at some point you’ve had a hankerin’ to do some science.
There’s a certain type of mind that looks at things and just wonders about them. Why does that transformer make that buzzing sound? What’s the muzzle energy of this BB gun lying on my desk here? Whether this is a useful activity or just intellectual navel-gazing is difficult to say.
The BB gun in question is pretty terrible – without a piece of electrical tape in the right place it tends to explode in every direction apart from the one in which it’s pointed. However, this does mean there’s a pretty hefty spring hiding in there, which is what got me wondering. How would I measure the muzzle velocity though?
One way of measuring projectile speed is to record it from the side with a high-speed camera against a backdrop of known feature size, à la Mythbusters:
I want to do this with the things in my desk and kitchen though (science right now), and my phone camera only goes up to 60fps, so that’s right out.
Walter Lewin has a great demo at the end of this lecture where he fires a rifle through two wires with a current flowing through them, and then measures the time between the wires breaking to calculate the speed. Unfortunately I don’t have this lying around in my kitchen either.
This is the method I came up with:
• Rest the BB gun on a chair, a known distance (4m) from a curtain (use a tape measure)
• Turn on my phone’s voice recorder
• Fire the BB gun at the curtain a bunch of times
This is all the data I need to find the average velocity. Primary school science: $speed = \frac{distance}{time}$
Opening up the recordings in Audacity:
We see two features:
• When the gun is fired
• When the BB hits the curtain.
I measured the times between these using the cursors in Audacity, and stuck it all into a spreadsheet. Subtracting the time for the sound to propagate back from the curtain (4m / 330 m/s = 19 ms) we get an average Time of Flight (ToF) of ~112ms.
Using Student’s t distribution it was straightforward to calculate a 95% confidence interval for the projectile’s average speed: between 33.6 and 38.3 metres per second. Ballin’.
After counting 100 of the BBs into a little jar and weighing with my kitchen scales, I also knew that the average projectile mass was 0.11g +- 0.05.
However. Substituting this value into $E_{K}=\frac{1}{2}mv^{2}$ does not give you the muzzle energy, because the BB’s speed is not constant, due to air drag. Neither does it give you the average KE. In fact, it’s pretty useless. We’re going to need to do some modelling.
I assumed that the drag force experienced by the ball was equal to its cross sectional area multiplied by the dynamic pressure:
$F = -\frac{1}{2}\rho v^{2} A$
Where $\rho$ is air density and $v$ is speed.
Newton’s second law then gives us
$m\ddot{x} + \frac{1}{2}\rho \dot{x}^{2}A = 0$
Where $m$ is the projectile’s mass and $x$ is its horizontal displacement from the muzzle.
Then:
$\textup{Let }R = \frac{\rho A}{2m}$
$\ddot{x} + R\dot{x}^{2} = 0$
At first this looked nasty, so I stuck everything into a spreadsheet and did Euler integration (difference equations).
Then I realised you can substitute $\dot{x}$ for $v$. Oops.
$\dot{v} + Rv^{2} = 0$
$\frac{dv}{dt} = -Rv^{2}$
Separate variables:
$\int{\frac{1}{v^2}dv} = -R\int{dt}$
$\frac{1}{2v} = Rt + c$
To find the constant, we’ll want to find the position in terms of time, so we rearrange and integrate:
$v = \frac{2}{Rt + c}$
$x =\frac{2}{R}ln(Rt + c) + d$
After measuring a line of BBs with a ruler, I found my constants and the specific function, shown in red above. Making the time step smaller shows that this is pretty much a perfect fit. Success! Using our model, we can now predict the BB’s velocity at every point in its 112 millisecond flight.
The answer to the original question? About 63m/s, with a KE of 0.22 Joules. Enough energy to lift an apple 20cm off of a table. Ouch. However, by the time the BB reaches the curtain 4 metres away, we can predict that it will have only 1/9th of its initial energy.
Using kitchen scales and a ruler to calculate the gun’s spring constant, it’s quick and easy to find the work of travel (about 1.4 Joules) and determine that the gun is approximately 15% efficient at turning spring energy into muzzle energy. | {"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": 18, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6822608709335327, "perplexity": 903.3981575133249}, "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/1516084886952.14/warc/CC-MAIN-20180117173312-20180117193312-00050.warc.gz"} |
http://www.ck12.org/book/CK-12-Middle-School-Math---Grade-7/r3/section/6.7/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 6.7: Simple Interest
Difficulty Level: At Grade Created by: CK-12
## Introduction
The Loan
Since Taylor has been working at the candy store, she has had her eye on a new bike. She has saved $56.00, but the bike is being sold for$156.00.
Taylor wouldn’t mind waiting, except for the fact that several of her friends are going to go on a big bike ride on the weekend and she wants to go with them. She has decided to ask her brother if she can borrow the money.
The one catch is that he wants to charge her interest. Her brother will loan her the $100.00, but he wants to charge her 15% interest per month. “I don’t think that is very fair,” Taylor tells him when he presents the deal. “It isn’t that much more that you have to pay back.” Taylor disagrees. How much is 15% interest on$100.00?
This lesson will teach you all about simple interest and about how to calculate simple interest. By the end of the lesson you will understand how to figure out the interest on Taylor's loan.
What You Will Learn
In this lesson, you will learn how to use the following skills.
• Use the simple interest equation to find an interest rate.
• Use the equation to find the time required to earn a given amount.
• Use the equation to find the new balance after a given time.
• Solve real-world problems involving simple interest.
Teaching Time
I. Use the Simple Interest Equation to find an Interest Rate
This lesson is all about borrowing money, paying money back and the fees associated with borrowing money. This is where you will learn all about interest.
What is interest?
Interest is a charge for money that is borrowed. When you borrow money, you pay the lender interest for borrowing the money. When you deposit money into a saving account at a bank, the bank pays you interest since the bank is borrowing money from you. So interest can be something that you have to pay or that is paid to you.
The amount of money that is invested or borrowed is called the principal. If you borrow $500.00 this is the principal. It is the initial amount before any interest is added on. The rate of interest is the percent charged or earned. We also have to consider the time in years that the money is borrowed or deposited when computing interest. How do we calculate interest? We can calculate interest by using an equation. Here is the equation. Take a few minutes to write this equation down in your notebook. If you know the amount of interest, the principal, and the time, you can find the interest rate. Example If you deposit$2,000 at USA Savings Bank, at the end of 2 years you will have received $240 in simple interest. What is the interest rate at USA Savings Bank? Change 0.06 to the percent 6%. The interest rate is 6% per year. Example If you borrow$3,600 from USA Savings Bank for 18 months, at the end of the 18 months you will repay $4,059 to the bank. What is the interest rate for this loan? The amount to be repaid includes the principal plus the interest. Subtract the principal from the amount to be repaid to find the amount of the interest. Since there are 12 months in a year, 18 months is or years. Change 0.085 to the percent 8.5%. The interest rate is 8.5% per year. 6S. Lesson Exercises Find the interest rate for each problem. 1. Jesse borrowed$500.00. At the end of the year he paid back $50.00 in interest. What was the interest rate? 2. Karen earned$200.00 in two years of simple interest on her initial investment of $400.00. What was the annual (yearly) interest rate? Take a few minutes to check your work with a friend. II. Use the Equation to Find the Time Required to Earn a Given Amount Now that you know how to find the interest rate, we can use the equation to calculate the amount of time it takes to earn a specific amount of interest. Example Ben deposited$1,200 in a certificate of deposit (CD) at a yearly interest rate of 5.5%. He earned $198 in simple interest. How long was the CD for? The CD was for 3 years. Example Joanna borrowed$500 at an annual interest rate of 8%. At the end of the loan period, she had to pay back $530. How long was the loan for? The amount to be repaid includes the principal plus the interest. Subtract the principal from the amount to be repaid to find the amount of the interest. of a year is of 12 months, or 9 months. The loan was for 9 months. 6T. Lesson Exercises Find the amount of time for each problem. 1. John earned$124.00 on a $1200 deposit at 2% annual interest rate. 2. Karen paid$25.00 on $600.00 at a 4% annual interest rate. Take a few minutes to check your work with a friend. III. Use the Equation to Find the New Balance After a Given Time We can also use the equation to figure out the amount of the interest. Let’s look at a few examples. Example Troy deposited$400 into his savings account. How much interest will he receive at the end of one year if the interest rate is 3% per year?
First we write the equation. Then we substitute the given values and solve.
Troy will receive $12 in interest at the end of one year. Example Courtney borrowed$7,500 for 4 years at an annual interest rate of 8%. How much interest will she pay on the loan?
Courtney will pay $2,400 interest on the loan. Once you know the interest, you can go back and add it to the Principal. Look at the first example. Example Troy deposited$400 into his savings account. We know that at the end of the year he received $12.00 in interest. We can add the amounts together to find the new balance. The new balance is$412.00.
In the second example, Courtney borrowed $7,500 and will pay$2400 interest on the loan. Here is the total she will pay to pay back the loan. We add the interest and the Principal.
The new balance is $9,900. IV. Solve Real-World Problems Involving Simple Interest Using simple interest is something that we do in real life all the time. This equation that you have learned is very important. Notice in Courtney’s example that she will pay back a lot of money in interest on the loan. This is always something that you have to consider when you borrow money. There is a lot of extra money that you end up paying because of the interest. That is why it is such a big deal when you hear people talk about a low interest rate. Example A car loan in the amount of$9,000 at an annual rate of 7% for 3 years is to be repaid in 36 monthly installments, including principal and interest. How much is each payment?
The amount to be repaid includes the principal plus the interest. First find the amount of interest.
Since the interest is $1,890, the amount to be repaid is , or$10,890.
Divide $10,890 by 36 to find the amount of each monthly payment. Each payment is$302.50.
## Real Life Example Completed
The Loan
Here is the original problem once again. Reread it and underline any important information. Then figure out the interest.
Since Taylor has been working at the candy store, she has had her eye on a new bike. She has saved $56.00, but the bike is being sold for$156.00.
Taylor wouldn’t mind waiting, except for the fact that several of her friends are going to go on a big bike ride on the weekend and she wants to go with them. She has decided to ask her brother if she can borrow the money.
The one catch is that he wants to charge her interest. Her brother will loan her the $100.00, but he wants to charge her 15% interest per month. “I don’t think that is very fair,” Taylor tells him when he presents the deal. “It isn’t that much more that you have to pay back.” Taylor disagrees. How much is 15% interest on$100.00?
To figure this out we multiply the 15% times $100.00. per month. If it takes her 3 months to pay him back, she will have to pay$45.00 in interest.
“No thanks,” Taylor says to her brother. “That isn’t a very good deal. I am going to ask Dad for a loan. I am sure that he won’t charge me interest.”
Taylor grinned at her brother and went off to find her Dad.
## Vocabulary
Interest
the amount of money added to a loan or to a deposit based on the amount of principal, an interest rate, and the amount of time for which the interest is paid.
Principal
the original amount of money borrowed or invested
Interest Rate
the percent that is being given for an investment or for a loan.
## Time to Practice
Directions: Find the simple interest on each amount (all interest rates are per year).
1. $500.00 at 4% for 2 years 2.$200.00 at 5% for 3 years
3. $5000.00 at 2% for 2 years 4.$600.00 at 10% for 1 year
5. $1200.00 at 4% for 2 years 6.$1500.00 at 3% for 1 year
7. $2300.00 at 2% for 2 years 8.$500.00 at 4% for 2 years
9. $2500.00 at 5% for 5 years 10.$1500.00 at 11% for 2 years
Directions: Find the interest rate for each loan.
11. principal: $2,500; time: 2 years; simple interest:$450
12. principal: $5,600; time: 9 months; simple interest:$357
Directions: Find the length of time for each loan.
13. principal: $1,250; interest rate: 6%; simple interest:$300
14. principal: $4,800; interest rate: 7.5%; simple interest:$900
Directions: Solve each problem.
15. Juan invested $5,000 in an account that pays 5% interest per year. If interest is split into four payments per year, how much is each interest payment? 16. Sophie put$330 in a savings account at a simple interest rate of 4% per year. Avi put $290 in a savings account at a simple interest rate of 5% per year. Who will have earned more interest after 2 years? How much more? 17. Madison invested in a certificate of deposit for 4 years at a 6% interest rate. At the end of the 4 years, the value of the certificate of deposit was$3,100. How much did Madison deposit originally?
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https://www.bartleby.com/solution-answer/chapter-96-problem-71e-calculus-10th-edition/9781285057095/3355e05d-a604-11e8-9bb5-0ece094302b6 | # : Which two series from the given below series are similar, a ) ∑ n = 0 ∞ n ( − 1 ) n ( 2 n + 1 ) ! . b ) ∑ n = 1 ∞ ( − 1 ) n − 1 ( 2 n − 1 ) ! . c ) ∑ n = 1 ∞ ( − 1 ) n − 1 ( 2 n + 1 ) !
### Calculus
10th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285057095
### Calculus
10th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285057095
#### Solutions
Chapter 9.6, Problem 71E
To determine
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1 - 3 / 31 1.Q-polynomial distance-regular graphs with a [sub] 1 [equal] 0 and a [sub] 2 [not equal] 0Štefko Miklavič, 2008, original scientific articleAbstract: Let ▫$\Gamma$▫ denote a ▫$Q$▫-polynomial distance-regular graph with diameter ▫$D \ge 3$▫ and intersection numbers ▫$a_1=0$▫, ▫$a_2 \ne 0$▫. Let ▫$X$▫ denote the vertex set of ▫$\Gamma$▫ and let ▫$A \in {\mathrm{Mat}}_X ({\mathbb{C}})$▫ denote the adjacency matrix of ▫$\Gamma$▫. Fix ▫$x \in X$▫ and let denote $A^\ast \in {\mathrm{Mat}}_X ({\mathbb{C}})$ the corresponding dual adjacency matrix. Let ▫$T$▫ denote the subalgebra of ▫$A{\mathrm{Mat}}_X ({\mathbb{C}})$▫ generated by ▫$A$▫, ▫$A^\ast$▫. We call ▫$T$▫ the Terwilliger algebra of ▫$\Gamma$▫ with respect to ▫$x$▫. We show that up to isomorphism there exists a unique irreducible ▫$T$▫-module ▫$W$▫ with endpoint 1. We show that ▫$W$▫ has dimension ▫$2D-2$▫. We display a basis for ▫$W$▫ which consists of eigenvectors for ▫$A^\ast$▫. We display the action of ▫$A$▫ on this basis. We show that ▫$W$▫ appears in the standard module of ▫$\Gamma$▫ with multiplicity ▫$k-1$▫, where ▫$k$▫ is the valency of ▫$\Gamma$▫.Found in: ključnih besedahSummary of found: ...let ▫$A \in {\mathrm{Mat}}_X ({\mathbb{C}})$▫ denote the adjacency matrix of ▫$\Gamma$▫. Fix ▫$x \in X$▫...Keywords: mathematics, graph theory, adjacency matrix, distance-regular graph, Terwilliger algebraPublished: 15.10.2013; Views: 1566; Downloads: 9 Full text (0,00 KB) 2.Adjacency preservers, symmetric matrices, and coresMarko Orel, 2012, original scientific articleAbstract: It is shown that the graph ▫$\Gamma_n$▫ that has the set of all ▫$n \times n$▫ symmetric matrices over a finite field as the vertex set, with two matrices being adjacent if and only if the rank of their difference equals one, is a core if ▫$n \ge 3$▫. Eigenvalues of the graph ▫$\Gamma_n$▫ are calculated as well.Found in: ključnih besedahSummary of found: ...adjacency preserver, symmetric matrix, finite field, eigenvalue of a graph, coloring,...Keywords: adjacency preserver, symmetric matrix, finite field, eigenvalue of a graph, coloring, quadratic formPublished: 15.10.2013; Views: 1515; Downloads: 81 Full text (0,00 KB) 3.O ekstremnih grafih z dano stopnjo in premerom/ožinoSlobodan Filipovski, 2018, doctoral dissertationFound in: ključnih besedahSummary of found: ... adjacency matrix, antipodal graphs, cages, excess, defect, Ramanujan...Keywords: adjacency matrix, antipodal graphs, cages, excess, defect, Ramanujan graphs, selfrepeats, degree/diameter problem, spectrum, Moore graphs, asymptotic density, distance matrices, Bermond and Bollobas problemPublished: 21.01.2019; Views: 536; Downloads: 0
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https://www.physicsforums.com/threads/why-do-couplings-run-with-log-of-energy.279326/ | # Why do couplings run with log of energy?
1. Dec 13, 2008
### franoisbelfor
Coupling constants run with the logarithm of energy.
This is a result of renormalization, its equations, and the
change of the effect of the virtual particle cloud with energy.
Is there a simple way to understand why this happens with the
log of energy, instead of with another function?
Logarithm is the integral of 1 over r; does this enter somehow?
François
2. Dec 13, 2008
### malawi_glenn
It comes from "the equations of renormalization", you calculate a lot of loop integrals which gives you Log's
And that is why in fact you "can do" renormalization, the log's are "slow functions"
3. Dec 13, 2008
### franoisbelfor
Well, I know that, as I wrote in my post. But why log? Why not something else?
Can one explain this in simple terms?
François
4. Dec 13, 2008
### malawi_glenn
why is not this "simple"??
The log just comes from math.
5. Dec 13, 2008
Staff Emeritus
Yes, but that by itself doesn't really tell you anything. You can always take an equation with a logarithm in it and express it as a differential equation with a 1/x in it instead. That DE may or may not provide additional insight.
As Malawi_Glenn points out, the key is that the coupling constant varies slowly. If the coupling constant varied quickly, we wouldn't call it a constant and wouldn't use the mathematical machinery that we do to describe it. We'd pick something else.
6. Dec 16, 2008
### njoshi3
In fact, it is the other way round:
You know for sure, that the theories we construct suffer from being well-behaved either at low or high energy limits. As a consequence, many of the integrals are divergent (either IR or UV).
To get a proper meaningful theory out of these divergent quantities (a theory which learns to live with the problems and still remains meaningful), one redefines the physical quantities, what is called re-normalization. And hence the coupling constants, for example, run (with energy).
Now, it is not that the quantities always run as log of energy... depending upon how you constructed your Lagrangian they can run more violently .... but, we select ONLY those theories which show a log dependance behavior: the logarithmic function is the slowest changing function and confirms the least sensitivity of your theory to the cut off energy (value of which you are not generally aware)
So, in short, it is not that the couplings run always logarithmically with energy, but it is your choice which decides the dependence and we (generally) chose theories which are renormalizable (and hence showing a log dependence)
7. Dec 21, 2008
### blechman
Expanding somewhat on what njoshi3 said:
The reason has to do with the phenomenon of "scaling." When you renormalize a coupling you are forced to introduce an energy scale (this is standard renormalization theory - see your favorite textbook), called the "subtraction scale" (M). Then by ordinary dimensional analysis, without any other energy scales in the problem (let's imagine that all the masses are zero for the moment) all dimensionless functions can only depend on the dimensionless scale E/M, where E is the energy, since this is the only dimensionless quantity left! Adding masses or dimensionful couplings means that there are more ratios you can construct, but that's fine.
It is an old trick that whenever you have a dimensionless ratio that can cover the entire positive real line, it is often very useful to take the log of that ratio, which maintains the dimensionless-ness of the ratio and is bijective, so it has an entire inverse (the exponential function). Find me another function that does this.
But it is NOT necessarily true that operators only run logarithmically with energy. They could run as a power law as well.
For example, QCD with a vanishing beta function runs power law, not logarithmically. This happens, for example, in supersymmetric QCD with Dirac gluinos (something I've been researching recently!). So the running can be more general, it's just that in the Standard Model, all the Lagrangian parameters have logarithmic running. But that's not true in the most general case.
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https://www.hepdata.net/search/?q=authors.affiliation%3A%22Fermilab%22&reactions=PBAR+P+--%3E+PBAR+P&page=1&size=50 | Showing 14 of 14 results
#### Measurement of the differential cross section $d\sigma/dt$ in elastic $p\bar{p}$ scattering at $\sqrt{s}=1.96$ TeV
The collaboration Abazov, Victor Mukhamedovich ; Abbott, Braden Keim ; Acharya, Bannanje Sripath ; et al.
Phys.Rev. D86 (2012) 012009, 2012.
Inspire Record 1117021
We present a measurement of the elastic differential cross section $d\sigma(p\bar{p}\rightarrow p\bar{p})/dt$ as a function of the four-momentum-transfer squared t. The data sample corresponds to an integrated luminosity of $\approx 31 nb^{-1}$ collected with the D0 detector using dedicated Tevatron $p\bar{p}$ Collider operating conditions at sqrt(s) = 1.96 TeV and covers the range $0.26 <|t|< 1.2 GeV^2$. For $|t|<0.6 GeV^2$, d\sigma/dt is described by an exponential function of the form $Ae^{-b|t|}$ with a slope parameter $b = 16.86 \pm 0.10(stat) \pm 0.20(syst) GeV^{-2}$. A change in slope is observed at $|t| \approx 0.6 GeV^2$, followed by a more gradual |t| dependence with increasing values of |t|.
0 data tables match query
#### $\bar{p}p$ elastic scattering at $\sqrt{s}$ = 1.8-TeV from |t| = $0.034-GeV/c^{2}$ to $0.65-GeV/c^{2}$
The collaboration Amos, Norman A. ; Avila, C. ; Baker, W.F. ; et al.
Phys.Lett. B247 (1990) 127-130, 1990.
Inspire Record 297541
0 data tables match query
#### The Real Part of the Forward Elastic Nuclear Amplitude for $p p$, $\bar{p} p$, $\pi^{+} p$, $\pi^{-} p$, $K^{+} p$, and $K^{-} p$ Scattering Between 70-GeV/c and 200-GeV/c
Fajardo, L.A. ; Majka, R. ; Marx, J.N. ; et al.
Phys.Rev. D24 (1981) 46, 1981.
Inspire Record 152596
0 data tables match query
#### pi+- p, K+- p, p p and anti-p p Elastic Scattering from 50-GeV/c to 175-GeV/c
The collaboration Ayres, D.S. ; Diebold, R. ; Maclay, G.J. ; et al.
Phys.Rev. D15 (1977) 3105, 1977.
Inspire Record 110409
The differential cross sections for the elastic scattering of π+, π−, K+, K−, p, and p¯ on protons have been measured in the t interval -0.04 to -0.75 GeV2 at five momenta: 50, 70, 100, 140, and 175 GeV/c. The t distributions have been parametrized by the quadratic exponential form dσdt=Aexp(B|t|+C|t|2) and the energy dependence has been described in terms of a single-pole Regge model. The pp and K+p diffraction peaks are found to shrink with α′∼0.20 and ∼0.15 GeV−2, respectively. The p¯p diffraction peak is antishrinking while π±p and K−p are relatively energy-independent. Total elastic cross sections are calculated by integrating the differential cross sections. The rapid decline in σel observed at low energies has stopped and all six reactions approach relatively constant values of σel. The ratio of σelσtot approaches a constant value for all six reactions by 100 GeV, consistent with the predictions of the geometric-scaling hypothesis. This ratio is ∼0.18 for pp and p¯p, and ∼0.12-0.14 for π±p and K±p. A crossover is observed between K+p and K−p scattering at |t|∼0.19 GeV2, and between pp and p¯p at |t|∼0.11 GeV2. Inversion of the cross sections into impact-parameter space shows that protons are quite transparent to mesons even in head-on collisions. The probability for a meson to pass through a proton head-on without interaction inelastically is ∼20% while it is only ∼6% for an incident proton or antiproton. Finally, the results are compared with various quark-model predictions.
0 data tables match query
#### Hadron-Proton Elastic Scattering at 50-GeV/c, 100-GeV/c and 200-GeV/c Momentum
Akerlof, C.W. ; Kotthaus, R. ; Loveless, R.L. ; et al.
Phys.Rev. D14 (1976) 2864, 1976.
Inspire Record 3655
0 data tables match query
#### Large Momentum Transfer Elastic Scattering of pi+-, K+-, and rho+- on Protons at 100-GeV/c and 200-GeV/c
Rubinstein, R. ; Baker, W.F. ; Eartly, David P. ; et al.
Phys.Rev. D30 (1984) 1413, 1984.
Inspire Record 202682
0 data tables match query
#### A Measurement of the proton-antiproton total cross-section at $\sqrt{s}$ = 1.8-TeV
The collaboration Avila, C. ; Baker, W.F. ; DeSalvo, R. ; et al.
Phys.Lett. B445 (1999) 419-422, 1999.
Inspire Record 478392
0 data tables match query
#### Charged-Particle Multiplicities in 100-GeV/c anti-p p Interactions
Ansorge, R.E. ; Bust, C.P. ; Carter, J.R. ; et al.
Phys.Lett. B59 (1975) 299-302, 1975.
Inspire Record 2603
0 data tables match query
#### A Luminosity Independent Measurement of the $\bar{p} p$ Total Cross-section at $\sqrt{s}=1$.8-tev
The collaboration Amos, Norman A. ; Avila, C. ; Baker, W.F. ; et al.
Phys.Lett. B243 (1990) 158-164, 1990.
Inspire Record 27474
0 data tables match query
#### $\bar{p}p$ elastic scattering at $\sqrt{s}$ = 1020-GeV
The collaboration Amos, Norman A. ; Avila, C. ; Baker, W.F. ; et al.
Nuovo Cim. A106 (1993) 123-132, 1993.
Inspire Record 338043
The antiproton-proton small-angle elastic-scattering distribution was measured at$\sqrt s$ GeV at the Fermilab Tevatron Collider. A fit to the nuclear-scattering distribution in the range 0.065≤|t|≤0.21 (GeV/c)2 givesb=(16.2±0.5±0.5) (GeV/c)−2 for the logarithmic slope parameter. Using the optical theorem and the luminosity from Collider parameters, we obtain σtoto(1+ρ2)1/2 =(61.7±3.7±4.4)mb.
0 data tables match query
#### GENERAL FEATURES OF CHARGED PARTICLE PRODUCTION IN ANTI-P P INTERACTIONS AT 100-GEV/C
Ward, C.P. ; Ward, D.R. ; Ansorge, R.E. ; et al.
Nucl.Phys. B153 (1979) 299-333, 1979.
Inspire Record 146577
0 data tables match query
#### Experimental Determination of Elastic and Topological Cross-sections in 48.9-GeV/c anti-p p Interactions
Zissa, D.E. ; Barnes, V.E. ; Carmony, D.D. ; et al.
Phys.Rev. D21 (1980) 3059, 1980.
Inspire Record 8434
The elastic and topological p¯p cross sections have been measured at 48.9 GeV/c in the Fermilab proportional-wire-chamber-30-in.-bubble-chamber hybrid spectrometer. The elastic cross section is 7.81±0.24 mb and the slope of the elastic differential cross section at t=0 is 13.4±0.8 GeV−2. Further, the moments of the inelastic topological-cross-section distribution are 〈nc〉=5.69±0.03, 〈nc〉D=2.10±0.02, and f2cc=1.67±0.12.
0 data tables match query
#### Elastic Scattering of Hadrons at 50-GeV to 200-GeV
Akerlof, C.W. ; Kotthaus, R. ; Loveless, R.L. ; et al.
Phys.Rev.Lett. 35 (1975) 1406, 1975.
Inspire Record 2687
0 data tables match query
#### Anti-Proton - Proton and Proton Proton Elastic Scattering at 100-GeV/c and 200-GeV/c
Kaplan, D.H. ; Karchin, P. ; Orear, J. ; et al.
Phys.Rev. D26 (1982) 723, 1982.
Inspire Record 178011
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https://www.arxiv-vanity.com/papers/1605.01403/ | # Gravitational waves from bubble collisions: analytic derivation
Ryusuke Jinno and Masahiro Takimoto Theory Center, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
###### Abstract
We consider gravitational wave production by bubble collisions during a cosmological first-order phase transition. In the literature, such spectra have been estimated by simulating the bubble dynamics, under so-called thin-wall and envelope approximations in a flat background metric. However, we show that, within these assumptions, the gravitational wave spectrum can be estimated in an analytic way. Our estimation is based on the observation that the two-point correlator of the energy-momentum tensor can be expressed analytically under these assumptions. Though the final expressions for the spectrum contain a few integrations that cannot be calculated explicitly, we can easily estimate it numerically. As a result, it is found that the most of the contributions to the spectrum come from single-bubble contribution to the correlator, and in addition the fall-off of the spectrum at high frequencies is found to be proportional to . We also provide fitting formulae for the spectrum.
preprint: KEK-TH-1900
## I Introduction
Gravitational waves (GWs) are one of the promising tools to probe the early universe. They provide a unique way to search for inflationary quantum fluctuations Starobinsky:1979ty , preheating Khlebnikov:1997di , topological defects Vilenkin:2000jqa ; Gleiser:1998na , and cosmic phase transitions (PTs) Witten:1984rs ; Hogan:1986qda . Especially, first-order PTs in the early universe have been attracted us because of their relation to high-energy physics beyond the standard model (SM), and in fact various extensions of the SM have been shown to predict first-order PTs with a large amount of GWs Espinosa:2008kw ; Ashoorioon:2009nf ; Das:2009ue ; Sagunski:2012pzo ; Kakizaki:2015wua ; Jinno:2015doa ; Apreda:2001tj ; Apreda:2001us ; Jaeckel:2016jlh ; Huber:2015znp ; Leitao:2015fmj ; Huang:2016odd ; Dev:2016feu ; Hashino:2016rvx ; Jinno:2016knw . On the observational side, ground-based GW experiments like KAGRA Somiya:2011np , VIRGO TheVirgo:2014hva and Advanced LIGO Harry:2010zz are now in operation, and space interferometers such as eLISA Seoane:2013qna , BBO Harry:2006fi and DECIGO Seto:2001qf have been proposed. Given that there is a growing possibility of their detecting GWs from cosmological sources in the near future, it would be worth reconsidering the theoretical predictions of GWs from first-order PTs.
First-order PTs proceed via the nucleation of bubbles, their expansion, collision and thermalization into light particles, and GWs are produced during this process. In the transition process, some of the released energy goes into heating up the plasma, while the rest is carried by the scalar field configuration (bubble wall) and/or the bulk motion of the surrounding fluid. Gravitational wave production by such localized structure of energy around the walls has been calculated by numerical simulations in the literature with so-called thin-wall and envelope approximations Kosowsky:1991ua ; Kosowsky:1992rz ; Kosowsky:1992vn ; Kamionkowski:1993fg aaa It is important to go beyond these approximations, especially when the bulk motion of the fluid dominates the released energy. In fact, it has been pointed out that the bulk motion can be a long-lasting GW source as sound waves Hindmarsh:2013xza ; Giblin:2014qia ; Hindmarsh:2015qta . . It has been shown that these approximations are valid especially when the energy of bubbles is dominated by the scalar field configuration Kosowsky:1991ua ; Weir:2016tov , and the latest result along this approach is found in Ref. Huber:2008hg . Analytic approaches have also been taken with some ansatz for correlator functions Caprini:2007xq ; Caprini:2009fx .
In this paper, we take an approach based on the evaluation of the correlation function of the energy-momentum tensor Caprini:2007xq , which is the only ingredient to obtain the spectrum. We point out that, under thin-wall and envelope approximations and in a flat background, this two-point correlator has a rather simple analytic expression and, as a result, the GW spectrum can also be expressed analytically. Though the final expression for the spectrum contains two remaining integrations, they can easily be estimated numerically. Our approach is not only free from statistical errors inherent to numerical simulations, but also enables us to specify the most effective bubble-wall configuration to the GW spectrum. At the current stage, our results are most relevant to strong phase transitions like near-vacuum ones, since the neglected effects such as the finite width of the bubble walls and/or the localized structure of the energy-momentum tensor remaining after collisions can be important when the scalar field is strongly coupled to the thermal plasma Hindmarsh:2013xza ; Giblin:2014qia ; Hindmarsh:2015qta bbb In addition, turbulent effects can contribute sizably to the GW spectrum Kamionkowski:1993fg ; Caprini:2006jb ; Gogoberidze:2007an ; Caprini:2009yp . . However, our method is extendable to the calculations without the envelope approximation JT , and such studies would be important in understanding how the localized structure after bubble collisions sources GWs.
The organization of the paper is as follows. In Sec. II we first make clear our assumptions in estimating the GW spectrum, i.e. thin-wall and envelope approximations, and then introduce basic ingredients such as the evolution equation and power spectrum of GWs. In Sec. III we present analytic expressions for the GW spectrum. Since two integrations cannot be performed explicitly, we evaluate them numerically in Sec. IV. We generalize our result to finite velocity case in Sec. V, and finally summarize in Sec. VI.
## Ii Basic ingredients
In this section we summarize basic ingredients for the calculation of GW spectrum. We first make clear the assumption and approximations used in the paper. We also explain the GW power spectrum around the time of sourcing from bubble collisions, and then show how to obtain the present spectrum.
### ii.1 Assumptions and approximations
#### ii.1.1 Thin wall and envelope approximation
In this subsection, we introduce the key assumptions to characterize the energy momentum tensor around the bubble wall, namely thin-wall and envelope approximations.
First, we introduce the thin-wall approximation, where all the energy of the bubble is assumed to be concentrated on the bubble wall with an infinitesimal width. We introduce the infinitesimal wall width for computational simplicity. The energy momentum tensor of the uncollided wall of a single bubble nucleated at can be written as
TBij(x) =ρ(x)ˆ(x−xN)iˆ(x−xN)j, (1)
with
ρ(x) =⎧⎪⎨⎪⎩4π3rB(t)3κρ04πrB(t)2lBrB(t)<|→x−→xN|
and
rB(t) =v(t−tN),r′B(t)=rB(t)+lB. (3)
Here , the hat on the vector indicates the unit vector in the direction of , is the bubble wall velocity, and represents the energy density released by the transitionccc Though the corresponding quantity is latent heat and not energy density in thermal environment, we use the word “energy density” throughout the paper, since . Also, indicates the efficiency factor, which determines the fraction of the released energy density which is transformed into the energy density localized around the wallddd This corresponds to the energy density of the bulk fluid around the wall when the bubble wall reaches a terminal velocity, while it is regarded as the energy density of the wall itself when the scalar field carries most of the energy. In the former case with so-called Jouguet detonation, the efficiency factor is related to the parameter introduced later Steinhardt:1981ct . Kamionkowski:1993fg . In addition, the Latin indices run over throughout the paper. Second, we assume that the energy momentum tensor of the bubble walls vanishes once they collide with others. In the literature this is called envelope approximation, whose validity in bubble collisions is confirmed in e.g. Ref. Kosowsky:1991ua . See Fig. 1 for a rough sketch of this approximation. These two assumptions make the calculation of the GW spectrum rather simple, as we will see later. Also, we regard the model-dependent quantities , and as free parameters constant in time.
#### ii.1.2 Transition rate
We assume that the bubble nucleation rate per unit time and volume can be written in the following form:
Γ(t) =Γ∗eβ(t−t∗), (4)
where indicates some fixed time typically around the transition time, is the nucleation rate at , and is assumed to be a constant. This parameter is often calculated with the instanton method from underlying models Linde:1977mm ; Linde:1981zj , and the typical time span of the phase transition is given by . We also assume that the phase transition completes in a short period compared to the Hubble time, i.e. , which typically holds for thermal phase transitions Kamionkowski:1993fg .
### ii.2 GW power spectrum around the transition time
In the following we express the GW spectrum in terms of the correlator of the energy-momentum tensor, following Ref. Caprini:2007xq .
#### ii.2.1 Equation of motion and its solution
In this paper we consider GWs sourced by the first order phase transition completed in a short period compared to the Hubble time. In such cases the background metric is well approximated by the Minkowski one. Including tensor perturbations, we write the metric as
ds2 =−dt2+(δij+2hij)dxidxj. (5)
The tensor perturbations satisfy the transverse and traceless condition and obey the following evolution equation
¨hij(t,→k)+k2hij(t,→k) =8πGΠij(t,→k), (6)
where is the Newton constant and indicates a Fourier mode of the corresponding object with being the wave vector. We take the convention for Fourier transformation to be and . The source term during the phase transition is given by the transverse and traceless projection of the energy momentum tensor
Πij(t,→k) =Kij,kl(^k)Tkl(t,→k), (7)
with being the energy momentum tensor, and being the projection
Kij,kl(^k) =Pik(^k)Pjl(^k)−12Pij(^k)Pkl(^k), (8) Pij(^k) ≡δij−^ki^kj. (9)
We assume that the source term is effective from to , and we set at the end of calculationeee Since the transition completes in a short period , and GWs are emitted only during this period, this procedure is expected not to affect the result. .
The solution of Eq. (6) is formally written in terms of the Green function satisfying and as
hij(t,→k) =8πG∫ttstartdt′Gk(t,t′)Πij(t′,→k)t
where . For , matching condition at gives
hij(t,→k) =Aij(→k)sin(k(t−tend))+Bij(→k)cos(k(t−tend)), (11)
with coefficients
Aij(→k) =8πGk∫tendtstartdtcos(k(tend−t))Πij(t,→k), (12) Bij(→k) =8πGk∫tendtstartdtsin(k(tend−t′))Πij(t,→k). (13)
#### ii.2.2 Power spectrum
Next we express the GW spectrum using Eq. (11). We define the equal-time correlator of the GWs by
⟨˙hij(t,→k)˙h∗ij(t,→q)⟩ =(2π)3δ(3)(→k−→q)P˙h(t,k), (14)
and also define the unequal-time correlator of the source term by
⟨Πij(tx,→k)Π∗ij(ty,→q)⟩ =(2π)3δ(3)(→k−→q)Π(tx,ty,k). (15)
Here the angular bracket denotes taking an ensemble average. Note that the in Eq. (15) appears due to the spacial homogeneity of the system. In terms the original energy-momentum tensor, the correlator is written as
Π(tx,ty,k) =Kij,kl(^k)Kij,mn(^k)∫d3rei→k⋅→r⟨TklTmn⟩(tx,ty,→r), (16)
where
⟨TklTmn⟩(tx,ty,→r) ≡⟨Tkl(tx,→x)Tmn(ty,→y)⟩, (17)
with . The L.H.S. depends only on because of the spacial homogeneity. Now let us consider the time . Since the GWs and the source term are related with each other through Eq. (11), the power spectrum of is written in terms of the source as
P˙h(t,k) =32π2G2∫tendtstartdtx∫tendtstartdtycos(k(tx−ty))Π(tx,ty,k). (18)
Though we put the argument in the L.H.S., the R.H.S. does not depend on it because there is no source term for and because we neglect the cosmic expansion. Since the total energy density of GWs is given by
ρGW(t) =⟨˙hij(t,→x)˙hij(t,→x)⟩T8πG, (19)
with being the oscillation and ensemble average, GW energy density per logarithmic frequency becomes
ΩGW(t,k) ≡1ρtotdρGWdlnk =2Gk3πρtot∫tendtstartdtx∫tendtstartdtycos(k(tx−ty))Π(tx,ty,k). (20)
with being the total energy density of the universe. Now all we have to do is to estimate , or the two-point function of the energy momentum tensor . Once the setup is defined, we can estimate this quantity analytically in principle. In fact, as shown later, this correlator can be expressed in an an analytical way under the thin-wall and envelope approximations (see Eqs. (49) and (58)).
For later convenience, we rewrite the expression for the GW spectrum as follows. We define the parameter as
which characterizes the fraction of the released energy density to that of radiation. Here and are the total and radiation energy density, respectively. Using thus defined, we have
ΩGW(t,k) =κ2(H∗β)2(α1+α)2Δ(k/β,v), (22)
where is given by
Δ(k/β,v) =38πGβ2ρtotκ2ρ20ΩGW(t,k) =34π2β2k3κ2ρ20∫tendtstartdtx∫tendtstartdtycos(k(tx−ty))Π(tx,ty,k). (23)
In deriving Eq. (22) we have used the Friedmann equation with being the Hubble parameter at the transition time. Note that the function depends only on the combination and the wall velocity , because the definition (22) factors out , and dependence, and because is a dimensionless quantity.
### ii.3 GW power spectrum at present
After produced, GWs are redshifted during propagation towards the present time. The relation between the scale factor just after the phase transition and at present is given by
a0a∗ =8.0×10−16(g∗100)−1(T∗100 GeV)−1, (24)
where denotes the temperature just after the phase transition, and indicates the total number of the relativistic degrees of freedom in the thermal bath at temperature . The present frequency is obtained by redshifting as
f =f∗(a∗a0) =1.65×10−5Hz(f∗β)(βH∗)(T∗102GeV)(g∗100)16, (25)
and the present GW amplitude is obtained from the fact that GWs are non-interacting radiation as
ΩGWh2 =1.67×10−5(g∗100)−13ΩGWh2∣∣t=tend =1.67×10−5κ2Δ(βH∗)−2(α1+α)2(g∗100)−13. (26)
## Iii Analytic expression
The following sections are mainly devoted to the calculation of (see Eq. (23)). We first focus on the case where the wall velocity is luminal, i.e., , since the final explanations become relatively simple in this case. Generalization to is straightforward and done in Sec. V.
In the expression of the GW spectrum (23), the only nontrivial quantity is the two-point correlator given by Eq. (16). If we can calculate this quantity, or equivalently the average of the product of the energy-momentum tensor with given , and , then we obtain the GW spectrum. In the following we show that this is indeed possible. For the energy momentum tensor to be nonzero at and with , the following two conditions are necessary and sufficient:
• No bubbles are nucleated inside the past light cones of and .
• Bubble(s) are nucleated on the past light cones of and , so that bubble walls are passing through the spacial points at time and at time .
In order to understand the former condition, one needs to notice that any spacial point is passed by bubble walls only once in the envelope approximation (see Fig. 1). Then, if bubble(s) nucleate inside the past light cone of or , either of the spacial points or is already passed by bubble walls before the evaluation time or . This makes it impossible for the energy-momentum tensor to be nonvanishing both at and , and therefore we need the former condition. On the other hand, the latter condition is necessary for bubble walls to be just passing through and at the evaluation time and . There are two possibilities for this condition: the bubble walls passing through and belong to one single nucleation point, or to two different nucleation points. We refer to these two as “single-bubble” and “double-bubble” contributions, respectively. Fig. 2 shows a schematic picture of these two contributions to the correlator . Here one may wonder why we consider the single-bubble contribution, since it is well known that a spherical object do not radiate GWs. The answer is that the single-bubble contribution takes into account the breaking of the original spherical symmetry of a bubble by collisions with others: see Appendix B on this point.
In the following discussion, we first make our notation clear. Then, after discussing the condition for no bubble nucleation inside the past light cones, we consider single- and double-bubble contributions separately. The final expressions are Eqs. (54) and (63), and those who need only the final GW spectrum may skip to Sec. IV.
### iii.1 Notations
We first fix our notations and conventions used in the following argument. We denote the two spacetime points in the two-point correlator as (see Fig. 3 and 4)
x =(tx,→x),y=(ty,→y). (27)
We sometimes use the time variables defined as
T ≡tx+ty2,td≡tx−ty, (28)
instead of . Also, we write their spacial separation as
→r≡→x−→y,r ≡|→r|. (29)
We often consider past light cones of and , which are denoted by and . The regions inside and are called and , respectively, and we write their union as . Since we consider bubbles with wall width , we also define the spacetime points
x+δ ≡(tx+lB,→x),y+δ≡(ty+lB,→y), (30)
whose past light cones are denoted by and , respectively. We also define the following regions
δVx ≡Vx+δ−Vx,δVy≡Vy+δ−Vy, (31)
whose intersection is denoted by
δVxy ≡δVx∩δVy. (32)
δV(y)x ≡δVx−Vy+δ,δV(x)y≡δVy−Vx+δ, (33)
as shown in Fig. 3. Also, in Fig. 4, we show how Fig. 3 looks in dimensions.
On a constant-time hypersurface at time , the two past light cones and form spheres, as shown in Fig. 5. We call these two spheres and , whose centers are labelled by and , respectively. The radii of and are given by
rx(t) ≡tx−t,ry(t)≡ty−t. (34)
These spheres and have an intersection for time with
txy ≡tx+ty−r2. (35)
Let us consider arbitrary points on and on , and we denote unit vectors from and to and as and , respectively. We parameterize these two unit vectors by the azimuthal and polar angles around as
nx ≡(sxcϕx,sxsϕx,cx),ny≡(sycϕy,sysϕy,cy), (36)
where the label has been omitted for simplicity. Also, we use shorthand notations etc. in the following. We sometimes need to label an arbitrary point on the intersection of and . We denote such point by , and also denote the unit vectors from and to as and , respectively. These unit vectors are parameterized by the azimuthal and polar angles , , and around . Especially, the cosines of the polar angles are given by
cx×(t)=cosθx×(t) =−rx(t)2+r2−ry(t)22rx(t)r, (37) cy×(t)=cosθy×(t) =ry(t)2+r2−rx(t)22ry(t)r. (38)
### iii.2 False vacuum probability
#### iii.2.1 Probability for one point to remain in the false vacuum
For illustrative purpose, we first consider the probability that a spacetime point is in the false vacuum. This occurs if and only if no bubbles are nucleated in . Dividing into infinitesimal four-dimensional regions so that , the probability that no bubbles are nucleated in is given by . Thus is written as Turner:1992tz
P(x) =∏i(1−ΓdVix)=e−I(x), (39)
with
I(x) =∫Vxd4zΓ(z). (40)
#### iii.2.2 Probability for two points to remain in the false vacuum
Next let us consider the probability that given two points and both remain in the false vacuum. This probability is expressed in the same way as before
P(x,y) =e−I(x,y),I(x,y)=∫Vxyd4zΓ(z). (41)
Below we assume spacelike separation , since only such configuration is relevant for the calculation of GW spectrum, due to the envelope approximationfff In the envelope approximation, it is impossible for two spacetime points and with timelike separation to be on bubble wall(s). This is because the spacial point is caught up before by the bubble wall which passed through . . Then is written as
I(x,y)=I(y)x+I(x)y, (42) I(y)x=∫txy−∞dtπ3rx(t)3Γ(t)(2−cx×(t))(1+cx×(t))2 +∫txtxydt4π3rx(t)3Γ(t) (43) I(x)y=I(y)x|x↔y. (44)
Here we have different integrands for and otherwise, because for the former the integrated volume do not form complete spheres. The time integration can be performed to give
I(x,y) =8πΓ(T)I(td,r), (45) I(td,r) =etd/2+e−td/2+t2d−(r2+4r)4re−r/2, (46)
where we have changed the variables from to , and adopted unit without loss of generality.
### iii.3 Single-bubble spectrum
We now evaluate the single-bubble contribution to the correlator (16). With the envelope approximation, the following two conditions are required in order for a single bubble to give nonvanishing energy-momentum tensor at both and :
• No bubbles are nucleated in .
• At least one bubble is nucleated in .
Note that the last condition reduces to “Only one bubble is nucleated in ” in the thin-wall limit . Below, we briefly derive the GW spectrum via single-bubble contribution starting from these two conditions. The final expression is Eq. (54), and the details of the calculation are summarized in Appendix A.
From above considerations, single-bubble contribution to the energy-momentum tensor is factorized in the following way (“” denotes “single”)
⟨TijTkl⟩(s)(tx,ty,→r) =P(tx,ty,r)∫txy−∞dtnΓ(tn)T(s)ij,kl(t,tx,ty,→r), (47)
where is the value of by the wall of the bubble nucleated at time (see Fig. 3 and 4). This is calculated as
T(s)ij,kl =(4π3rx(tn)3⋅κρ0⋅14πrx(tn)2lB) ×(4π3ry(tn)3⋅κρ0⋅14πry(tn)2lB) ×∫Rxyd3z(N×(tn))ijkl, (48)
with . Here is the ring made by rotating the diamond-shape shown in Fig. 5 around the axis . The integration by the nucleation time in Eq. (47) can be performed explicitly, and after taking the projection in Eq. (16) into account, we have
Kij,kl(^k)Kij,mn(^k)⟨TklTmn⟩(s)(tx,ty,→r) =2π9κ2ρ20Γ(T)e−r/2r5P(tx,ty,r) ×[12F0+14(1−(^r⋅^k)2)F1+116(1−(^r⋅^k)2)2F2], (49)
with functions given by
F0 =2(r2−t2d)2(r2+6r+12), (50) F1 =2(r2−t2d)[−r2(r3+4r2+12r+24) +t2d(r3+12r2+60r+120)], (51) F2 =12[r4(r4+4r3+20r2+72r+144) −2t2dr2(r4+12r3+84r2+360r+720) +t4d(r4+20r3+180r2+840r+1680)]. (52)
Note that we have changed the time variables from to . Also note that the correlator has now been successfully expressed analytically. Performing the integration over the angle between and in Eq. (16), we find
Π(s)(tx,ty,k) =4π29κ2ρ20Γ(T)∫∞0dre−r/2r3P(tx,ty,r) ×[j0(kr)F0+j1(kr)krF1+j2(kr)k2r2F2]. (53)
Then the integration over in Eq. (23) is performed by using the equality , and we obtain
Δ(s) =k312π∫∞0dtd∫∞tddre−r/2cos(ktd)r3I(td,r) ×[j0(kr)F0+j1(kr)krF1+j2(kr)k2r2F2], (54)
where denote the spherical Bessel functions given in Appendix A.
### iii.4 Double-bubble spectrum
Next we evaluate the double-bubble contribution to the correlator (16). With the envelope approximation, the following two conditions are necessary and sufficient for two different bubbles to give nonvanishing energy-momentum tensor at and :
• No bubbles are nucleated in .
• At least one bubble is nucleated in , and at least another is nucleated in .
Note that the last condition reduces to “Only one bubble is nucleated in each of and ” in the thin-wall limit . Below we derive the GW spectrum via double-bubble contribution starting from these two conditions. The final result is given by Eq. (63).
From above considerations, the two-bubble contribution to the energy-momentum tensor is decomposed as (“” denotes “double”)
⟨TijTkl⟩(d)(tx,ty,→r) =P(tx,ty,r) ∫txy−∞dtxnΓ(txn)∫δV(y)x∩Σtxnd3xnT(d)x,ij(txn,→xn;tx,→r) ×∫txy−∞dtynΓ(tyn)∫δV(x)y∩Σtynd3ynT(d)y,kl(tyn,→yn;ty,→r), (55)
where and are the value of the energy-momentum tensor by the bubble wall nucleated in and evaluated at the spacetime points and , respectively. They are given by
T(d)x,ij(txn,→xn;tx,→r) =(4π3rx(txn)3⋅κρ0⋅14πrx(txn)2lB)(nx)i(nx)j, T(d)y,kl(tyn,→yn;ty,→r) =(4π3ry(tyn)3⋅κρ0⋅14πry(tyn)2lB)(ny)i(ny)j. (56)
Here the arguments and in and are omitted for simplicity. Note that the time integration is over in Eq. (55), because the integration region or gives spherically symmetric contribution and thus vanishes (see Fig. 35, and notice that the nucleation points and run over the whole sphere for these nucleation times). Also note that the contribution to and that to factorize in Eq. (55) because the two bubbles nucleate independently of each other (see Fig. 3 and 4). There are no special directions except for , and therefore () is decomposed as follows after integration over the nucleation time :
∫txy−∞dtzn∫d3znT(d)z,ij(tzn,→zn;tz,→r) =A(d)z(tx,ty,r)δij+B(d)z(tx,ty,r)^ri^rj. (57)
Here and depend on both and because the integration region for is affected by the other points. After the projection by , only component survives:
Kij,kl(^k)Kij,mn(^k)⟨TklTmn⟩(d)(tx,ty,→r) =12P(tx,ty,r)B(d)x(tx,ty,r)B(d)y(tx,ty,r)(1−(^r⋅^k)2)2. (58)
Taking unit without loss of generality, we can calculate as
B(d)x(tx,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": 0.9786070585250854, "perplexity": 935.3026971117098}, "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-2023-14/segments/1679296948632.20/warc/CC-MAIN-20230327123514-20230327153514-00134.warc.gz"} |
https://crypto.stackexchange.com/questions/55744/analysing-communication-complexity-of-a-protocol | # Analysing communication complexity of a protocol
My question is more about the practical side of designing crypto. protocols and it is related to complexity. So, if you think it's irrelevant to this forum please kindly let me know and I will remove the question.
In many crypto. protocols the communication cost is very important. In the following, I consider a client-server protocol where the server computes a single value and sends it to the client. Assume the protocol requires modular arithmetics at a server-side on a finite field of size 120 bits, i.e. $\mathbb{F}_p$, where $|p|=120-bit$. Assume we have improved the previous protocol and now we can work on a smaller filed, e.g. $|p|=80-bit$.
As we know we cannot say the communication cost has been reduced by $40$ bits, as the value can be $1$ or $119$ bits value, because it involves a modular operation.
Question: How do we analyze/argue the communication improvement of the above protocol?
• "I do not think the number of bits can vary between 1 and 119"; actually, you could use a variable length encoding technique. Of course, if you include the bits to indicate the length of the field (or otherwise make the it unambiguous), and if all $2^{120}$ values are equiprobable, then I believe it can be shown that such a variable length encoding increases the average size of the message – poncho Feb 18 '18 at 19:40 | {"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": 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": 0.9011422991752625, "perplexity": 286.23476151590467}, "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/1620243991829.45/warc/CC-MAIN-20210514214157-20210515004157-00551.warc.gz"} |
https://electricala2z.com/renewable-energy/magnetism-electromagnetic-devices/ | Home / Renewable Energy / Magnetism and Electromagnetic Devices
# Magnetism and Electromagnetic Devices
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Magnetism and Electromagnetic Devices
In this section, you will learn about basic magnetism and electromagnetism. Three important electromagnetic devices are introduced: the relay, the generator, and the motor. All of these devices are important in renewable energy systems. Relays are commonly used in many applications to switch current on and off. Generators and motors are rotating electromagnetic machines that generate electrical power (in the case of the generator) and use electrical power to produce mechanical motion (in the case of the motor). Generators are used in wind and hydroelectric systems, and motors are used in solar tracking systems.
The Magnetic Field
A permanent magnet, such as the bar magnet shown in Figure 1, has a magnetic field surrounding it. All magnetic fields have their origin in moving charge, which in solid materials is caused by moving electrons. In certain materials, such as iron, atoms can be aligned so that the electron motion is reinforced, creating an observable field that extends in three dimensions.
To explain and illustrate magnetic fields, Michael Faraday drew lines of force, or flux lines, to represent the unseen field. Flux lines are widely used as a description of a magnetic field, showing the strength and direction of the field. The flux lines never cross. When lines are close together, the field is more intense; when they are farther apart, the field is weaker. The flux lines always extend from the North Pole (N) to the South Pole (S) of a magnet. Although the number of lines for any given magnet can be extremely large, only a few representative lines are usually shown in drawings.
When unlike poles of two permanent magnets are placed close together, their magnetic fields produce an attractive force, as indicated in Figure 2(a). When two like poles are brought close together, they repel each other, as shown in Figure 2(b).
Magnetic Flux
The group of force lines going from the north pole to the south pole of a magnet is called the magnetic flux and is symbolized by φ Several factors determine the strength of a magnet, including the material and physical geometry as well as the distance from the magnet. Magnetic flux lines tend to be more concentrated at the poles.
The mks unit of magnetic flux is the weber (Wb), which is a very large unit. In most practical situations, the microweber (μW), which is equal to 100 flux lines, is more appropriate. The magnetic flux density (B) is the amount of flux, φ per unit area (A) perpendicular to the magnetic field. Flux density is defined mathematically as follows;
$B=\phi /A$
Figure 1: Magnetic Lines of Force around a Bar Magnet
Figure 2: Magnetic Attraction and Repulsion
The flux density is in Wb/m2 when the magnetic flux is in Weber and the area is in square meters. One Wb/m2 defines the Tesla (T), which is the mks unit. The Tesla represents a large unit; the strongest permanent magnets are above 5 T. The Gauss (G) is the much smaller cgs unit for flux density (104G = 1T). The meter used to measure flux density is named the gauss meter (rather than the teslameter).
Figure 3: Magnetic Field around a Current-Carrying Conductor. The red arrows show the direction of electron current.
Electromagnetism
Current produces a magnetic field around a conductor, as illustrated in Figure 3. The invisible lines of force of the magnetic field form a concentric circular pattern around the conductor and are continuous along its length. Although the magnetic field cannot be seen, it is capable of producing visible effects. For example, if a current-carrying wire is inserted through a sheet of paper in a perpendicular direction, iron filings placed on the surface of the paper arrange themselves along the magnetic lines of force in concentric rings, as illustrated in Figure 4(a)Figure 4(b) shows that the needle of a compass placed in the magnetic field points in the direction of the lines of force.
Electromagnet
Figure 5(a) illustrates a basic magnetic circuit with a coil of wire around a magnetic material. The current through the coil creates a magnetic field represented by flux lines along the magnetic path. An electromagnet works on the same principle except that an air gap exists in the magnetic material so that the magnetic field set up by the current in the coil of the wire extends from the North Pole to the South Pole. When the current through the coil reverses direction, the magnetic field also reverses direction, as shown in Figure 5(b). An electromagnet can have various configurations, but a U-shape magnetic core is shown.
Figure 4: Visible Effects of a Magnetic Field
Figure 5: Basic Electromagnetic Circuit and Electromagnet
Relay
Figure 6 shows the basic operation of an armature-type relay with a normally open (NO) contact and one normally closed (NC) contact. When there is no current through the coil, the armature is held against the upper contact by the spring, thus providing continuity from terminal 1 to terminal 2, as shown in Figure 6(a). When energized with coil current, the armature is pulled down by the attractive force of the magnetic field and makes a connection with the lower contact to provide continuity from terminal 1 to terminal 3, as shown in Figure 6(b). A typical armature relay and its schematic symbol are shown in Figure 7.
Another widely used type of relay is the reed relay. Like the armature relay, the reed relay uses an electromagnetic coil. The contacts are thin reeds of magnetic material and are usually located inside the coil. Reed relays are faster, are more reliable, and produce less contact arcing than armature relays. However, they have a less current-handling capability and are more susceptible to mechanical shock.
Figure 6: Basic Structure of a Single-Pole, Double-Throw Armature Relay
Figure 7: Typical Armature Relay
AC Generators
The AC generator is an electromagnetic machine that produces a sinusoidal voltage. The basic principle of an ac generator can be understood using a simplified single-loop model, as shown in Figure 8. The loop is mechanically driven by a rotating force from a motor shaft, wind turbine blades, or water-driven turbine blades. As the loop rotates through the magnetic field, a voltage is induced across the slip rings. When a load is connected via the brushes, a current is produced and power is delivered to the load.
Each revolution of the loop produces one cycle of a sine wave. The positive and negative peaks occur when the loop cuts through the maximum number of flux lines. The rate at which the loop spins determines the time for one complete cycle and the frequency. If it takes 1/60 of a second to make a revolution between a single set of poles, the period of the sine wave is 1/60 second and its frequency is 60 Hz.
The single-loop generator in Figure 8 produces only a very tiny voltage. Instead of using permanent magnets, a practical generator usually has hundreds of loops that are wound on a magnetic core forming an electromagnet for the rotor. Two basic types of generator are the rotating-armature and the rotating-field. In a motor or generator, the armature is the power producing component.
Figure 8: Simplified AC Generator
Rotating-Armature Generator
In a rotating-armature generator that has multiple loops and many pole-pairs, the magnetic field is stationary and is supplied by permanent magnets or electromagnets operated from dc. With electromagnets, field windings are used instead of permanent magnets. These windings provide a fixed magnetic field that interacts with the rotor coils. The output power is taken from the rotor through the slip rings and brushes.
Rotating-Field Generator
Figure 9 shows how a rotating-field generator can produce three-phase sine waves with an electromagnetic rotor. A permanent magnet rotor is shown for simplicity. AC is generated in each set of windings as the North Pole and the south pole of the rotor alternately sweep by a stator winding. The stator winding is separated by 120° causing the sine wave outputs also to have 120° between them. Three-phase (abbreviated 3φ) is generated by power companies, which is then converted to single-phase (1φ) for residential users.
Figure 9: Rotating-Field Generator
Motors
The rotor in dc motors contains the armature winding, which sets up a magnetic field. The rotor moves because of the attractive force between opposite poles and the repulsive force between like poles, as illustrated in the simplified diagram of Figure 10. A force of attraction exists between the south pole of the rotor and the north pole of the stator (and vice versa). As the two poles near each other, the polarity of the rotor current is suddenly switched by the commutator, reversing the magnetic poles of the rotor. The commutator serves as a mechanical switch to reverse the current in the armature just as the unlike poles are near each other, thus continuing the rotation of the rotor. A shaft is connected to the rotor; as the rotor moves, the shaft turns to provide mechanical torque.
Figure 10: Simplified DC Motor
Some dc motors do not use a commutator to reverse the polarity of the current. Instead of supplying current to a moving rotor, the magnetic field is rotated in the stator windings using an electronic controller.
Review Questions
1. Discuss the repulsive and attractive forces in magnets.
2. How is an electromagnetic field produced?
3. Describe the difference between a generator and a motor.
4. Name two basic types of AC generator.
1. All magnets have two poles: north and south. Like poles repel; unlike poles attract.
2. Moving charge (current) in a conductor creates a magnetic field.
3. A motor converts electrical energy to mechanical (rotational) energy. A generator does the opposite.
4. Rotating armature and rotating field.
Did you find apk for android? You can find new Free Android Games and apps. | {"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": 0.8163760304450989, "perplexity": 664.9921193462535}, "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/1558232257553.63/warc/CC-MAIN-20190524064354-20190524090354-00377.warc.gz"} |
http://mathoverflow.net/questions/85880?sort=newest | ## Is there a general method of determining the line of best fit for any given set of data? [closed]
Is there a general method of determining the line of best fit (using the principle of least squares or any other principle) for any given set of data points$? If there is no general method, what is/are the next best options? This problem is motivated by the difficulty in deciding which curve will best fit a given data set. If the data is roughly linear, I can use linear regression. If the data shows a quadratic behavior, I can guess that a quadratic curve will best fit the data and according I will try to find the best quadratic fit. But if the data show no particular trend of if it has a trend which I am not able to determine by simple observation, it is difficult to guess which model will best fit the data. For example, using linear or quadratic regression on a data that has the hidden pattern$y=x^{2.5}\ln x$(which is difficult to guess) is not effective. Hence I am looking for general a method of regression using least squares or any other principle that will work for all kinds of data. - Would that really give a line of best fit?$\;\;\$ – Ricky Demer Jan 17 2012 at 4:43
This question seems a little too broad, considering there is a whole field devoted to it. You might want to read en.wikipedia.org/wiki/Curve_fitting if you haven't already. – William DeMeo Jan 17 2012 at 5:09
Is the purpose of such methods to find an inherent trend in the data? What if...the data is just inherently trendless?(!) – Timothy Foo Jan 17 2012 at 9:30 | {"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": 0.2960270643234253, "perplexity": 550.3918140502133}, "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/1368704933573/warc/CC-MAIN-20130516114853-00050-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/280305/direct-product-of-group-order-2 | # Direct product of group order 2
If $P$ is group of order 2, how many subgroups (trivial and proper) has the group $P \times P \times P$? Labelling the elements of $P$ to be $e$ and $a$, list the proper subgroups.
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What have you tried? – Tobias Kildetoft Jan 16 '13 at 20:49
oeis.org/A006116 – Hagen von Eitzen Jan 16 '13 at 20:56
I tried to do $P\times P$ first and the elements are $(e,e),(e,a),(a,e),(a,a)$, right? – bbr4in Jan 16 '13 at 21:02
That's a good start. Your list of the elements is correct, but what are the subgroups of $P \times P$? – Hew Wolff Jan 16 '13 at 21:30
It's good to keep in mind that there is only one group of order two, so you may always think of that one! – andybenji Jan 16 '13 at 21:35
$P = \{e, a\} \cong \{0, 1\} = \mathbb{Z}_2$. There is one and only one group of order $2$ and it is cyclic. So $e$ is the identity, $a^2 = aa = e$.
Yes, your computation of $P\times P = \{(e,e),(e,a),(a,e),(a,a)\}$ is correct. $P\times P$ has 3 subgroups of order 2, the trivial subgroup $\{e, e\},$ and $P\times P$ itself: $5$ in all, 4 not including the improper subset $P\times P$.
Can you try and compute $(P\times P) \times P = P\times P \times P$? The order of $P\times P \times P$ is $8$, and it consists of 3-tuples, e.g., $(e, e, e)$ is the identity.
Hint: append each of the elements you computed for $P \times P$ first with $e$, then append the same element from $P\times P$ with $a$: e.g. $(e, e) \to (e, e, e), (e, e, a); (e, a) \to (e, a, e), (e, a, a)$...
Your task is to determine the subgroups (proper and trivial) in this group of eight 3-tuples:
Hint: there are 15 subgroups, excluding $P\times P \times P$, (so 16 in all). Can you find the proper subgroups? One of order $1$: the identity, the rest are of order $2$ or $4$: use the definition of direct product to compute which elements form subgroups of order 2, and which comprise subgroups of order 4, and confirm that there are no other subgroups than those of order $1$ and $2$ and $4$.
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Start first by computing the subgroups of $P\times P$. One, of course, is $\{(e, e)\}$. By the definition of the direct product, if the operation on $P$ is $*$, then compute $(e, a)(e, a) = (e*e, a*a) = (e, e)$. (Recall, a is of order 2, so $a*a = e$). So what can you say about $\{(e, e), (e, a)\}$?...etc. – amWhy Jan 16 '13 at 22:08
There are certainly subgroups of order 4. – Gerry Myerson Jan 16 '13 at 23:13
I count 16 subgroups. – Gerry Myerson Jan 16 '13 at 23:31
@user52187: feel free to follow up with any questions you still have... – amWhy Jan 17 '13 at 22:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7504920363426208, "perplexity": 334.4766561143272}, "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/1440644064538.31/warc/CC-MAIN-20150827025424-00245-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/453342/is-there-any-use-of-a-quadratic-equation-of-state-in-flrw-cosmology | Is there any use of a quadratic equation of state in FLRW cosmology?
Consider standard FLRW cosmology. Usually, the relation between energy density $$\rho$$ and pressure $$p$$ of a cosmological fluid component is linear: $$$$\tag{1} p = w \, \rho,$$$$ where $$w$$ is just a constant ($$w = 0$$ for the dust model, $$w = \frac{1}{3}$$ for ultra-relativistic radiation, $$w = - 1$$ for the cosmological constant, etc). I'm interested in applications of the following non-linear equation of state (quadratic in this case): $$$$\tag{2} p = -\, \kappa \, \rho^2,$$$$ where $$\kappa$$ is a positive constant. I never saw any textbook or paper discussing such an equation of state. Is there a reason for that? I would like to find some references on it.
Notice that there is no sound speed from (2) and that the four main energy conditions may be violated, depending on the value of the energy density $$\rho$$ relative to the constant $$\kappa$$: \begin{align} &\textsf{NEC:} \qquad \rho + p \ge 0. \tag{3} \\[12pt] &\textsf{WEC:} \qquad \rho \ge 0, \quad \text{and} \quad \rho + p \ge 0. \tag{4} \\[12pt] &\textsf{SEC:} \qquad \rho + 3 p \ge 0, \quad \text{and} \quad \rho + p \ge 0. \tag{5} \\[12pt] &\textsf{DEC:} \qquad \rho \ge |\, p \,|. \tag{6} \end{align} These energy conditions are a subject of much controversy today, and nothing in physics imposes that they need to be satisfied in all cases.
The state equation (2) describes phantom energy when $$\rho > 1/\kappa$$ and may lead to a Big Rip scenario.
Local conservation of energy-momentum imposes the following constraint: $$$$\tag{7} \nabla_{\mu} \, T^{\mu \nu} = 0 \qquad \Rightarrow \qquad \dot{\rho} \, a + 3 \, (\rho + p) \, \dot{a} = 0.$$$$ Substituting (2) into (7) gives the energy density as a function of the scale factor $$a(t)$$: $$$$\tag{8} \rho = \frac{\mu}{\mu \kappa + a^3} = \frac{\rho_0}{\kappa \, \rho_0 + (1 - \kappa \, \rho_0)(a / a_0)^3},$$$$ where $$\mu$$ is an integration constant and $$\rho_0$$ is the energy density today (at time $$t_0$$ for which $$a(t_0) \equiv a_0$$). Notice that (8) gives $$\rho \approx 1 / \kappa = \text{cste}$$ (the fluid is behaving like a cosmological constant) when $$a / a_0 \ll 1$$, and $$\rho \propto a^{-3}$$ (like dust) when $$a / a_0 \gg 1$$ (but only if $$\kappa \, \rho_0 < 1$$, so that energy density stay positive: $$\rho > 0$$). So (2) may be a model for inflation before a matter (i.e. dust) era.
So was the equation of state (2) already discussed before (surely!)? Any reference on it?
EDIT: Here's a cool graph made with Mathematica, of the evolution of the scale factor $$a(t)/a_0$$ for a model containing a cosmological constant ($$\Lambda > 0$$) and a fluid described by equation of state (2) (with $$\kappa$$ very small). For the parameters selected, the universe is spatially closed ($$k = 1$$), and there's a Big Bounce without any singularity (no Big Bang and no Big Rip). There's a timid inflation era, a matter domination era and a Dark energy era (second inflation): | {"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": 30, "wp-katex-eq": 0, "align": 1, "equation": 4, "x-ck12": 0, "texerror": 0, "math_score": 0.9928779602050781, "perplexity": 495.2512070817028}, "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-2019-43/segments/1570986657949.34/warc/CC-MAIN-20191015082202-20191015105702-00297.warc.gz"} |
http://physics.stackexchange.com/questions/114109/conservation-of-energy-or-lack-thereof-in-doppler-cooling | # Conservation of energy (or lack thereof) in Doppler cooling [duplicate]
I did not find my question answered elsewhere, so here it is.
I'm doing a project in my first optics course, and we are reading a bit about Doppler cooling. I understand that a laser is tuned to a frequency where the photon energy $E_p = h \nu$ corresponds to an energy slightly below the energy $E_{abs}$ required to excite the gas atoms, and that an atom moving towards the laser at a high speed (relative to rest of the gas) will experience a blueshift that allows the atom to absorb the photon, thus slowing down the atom.
I also understand that the photon is re-emitted in a random direction, which of course means that the atom's velocity in the direction of the laser will be decreased. What I'm not sure I understand is how the re-emission of the photon doesn't just bring the atom back to the same kinetic energy (but with a different direction of motion).
I have not been able to find a source explaining this, but here's my own attempt at an explanation (which I am not sure is correct, which is why I am asking here): The absorbed photon has an energy (in the frame of the laboratory) of $E_1 = h \nu_1$ which is slightly less than the absorption energy $E_{abs}$. But after the absorption the atom's speed is reduced, thus the photon will be emitted with a different frequency $\nu_2$ (in the frame of the laboratory). Since the speed of the atom (relative to the frame of the laboratory) has been reduced by the absorption, the energy $E_2$ of the emitted photon will be closer to $E_{abs}$, thus $|E_{abs} - E_2| > |E_{abs} - E_1|$ (the last step of the explanation only works if the atom's speed perpendicular to the laser is not greater than the speed parallel to the laser for most atoms, but since the laser is tuned to only hit the fast moving atoms, this seems improbable enough to not invalidate the explanation).
Is this explanation correct or have I made mistake? I have taken a special relativity course but no quantum mechanics yet, so please keep the quantum mechanics to a minimum if possible.
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## marked as duplicate by Neuneck, DavePhD, Danu, Brandon Enright, Kyle KanosMay 24 '14 at 13:51
After writing my answer, I did some more googling and came across physics.stackexchange.com/q/5851 . It looks like a duplicate to me. – Floris May 23 '14 at 23:12
I'm not sure the questions are necessarily the same, or at least they are stated differently (although I don't know how similar is too similar). In any case, the answers on that other question definitely covered my question too. – Johnny Hansen May 23 '14 at 23:31
The questions are quite different but the answer applies. So even if this is not closed as a duplicate I think that the link between the two is useful to record. – Floris May 24 '14 at 1:07
Here are some intuitive arguments; I believe they are valid but it's been a very long time since I thought about these things - so I am open to comments / improvements.
The simple description of Doppler cooling:
Because of the detuning, the probability of a photon absorption is greatest for an atom moving towards the light source - and in the process of absorbing the photon, the atom loses momentum in the direction it was going.
When it re-emits the photon, it will be in a random direction; therefore there will be no net impact on the momentum of the atoms. This is sufficient to see that the atoms "slow down" in the beam direction, but not enough to see that they lose energy in total.
To see that the energy is lower, imagine that the atom had an initial momentum $2p$ where $p$ = momentum of the photon. After the absorption, the atom has momentum $p$. Now if we re-emit the photon at 90 degrees to the original direction, the net momentum is $\sqrt{2}p$ and thus the kinetic energy of the atom, which scales with momentum squared, is lower than it was. Only if the photon is re-emitted in the original direction do you see no net change in energy.
This does beg the question: what happened to the energy? It appears that we did no work on the system - the same amount of energy came in (one photon) as went out (same photon, barring tiny relativistic effects that are much smaller than the drop in kinetic energy of the atom). I don't think that the shift in the wavelength explains this.
I suspect this is where entropy comes in: since this process actually decreases the entropy of the system, we must have done work on the system: $\Delta U = -\Delta S\cdot T$. So energy needed to reduce the entropy was taken from the kinetic energy of the atom.
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Here's a stab at it:
I don't believe you need to consider the change in frequency of the scattered light to understand how Doppler Cooling works - indeed, a conservation of momentum argument is usually more insightful.
In a nutshell, you've got the core part of it down. The laser field that the atom interacts with is red-detuned in frequency to an excited state transition such that atoms with any velocity in the direction of the beam path sees the frequency of the laser Doppler shifted by some amount proportional to its speed closer to its transition frequency. This Doppler shift makes the photons appear to have a higher energy in the atomic frame. It's not as if the photon really has two separate energies, but rather this is really just a sort-of addition between the energy of the photon and a portion of the kinetic energy of the moving atom. The atom and the photon are meeting half-way, as it were.
In the momentum picture, the atom experiences a "collision" with the photon going in the opposite direction, leading to a reduction in the velocity of the atom along that direction. When the photon is remitted, it is emitted in a random direction. While the atom now moves with a new velocity in a direction opposite to the emitted photon, the photon is emitted with energy matching the transition energy - slightly higher than the frequency of the absorbed light.
While no net energy is lost in our system when the photon is absorbed (the collision is elastic), the way the energy is spread between the atom and the photon is not the same as when the photon is emitted. In this subsequent "inelastic" anticollision (emission), the photon leaves with more than it started. The number of scattering events is high enough that through the course of many emissions the net contribution to motion of the atom not along the beam path and atom trajectory averages to 0.
Basically, dumping a unit of momentum consistently from one specific direction into a bunch of random directions that cancel out. You're still conserving energy in the system, but a small bit of energy is being removed in the form of emitting photons in other directions.
You may also be interested in looking at the Doppler Cooling limit (when absorbing and remitting a photon does not lead to a further loss of kinetic energy due to the absorption and emission process both behaving elastically), or also how magnetic fields are used in conjunction to prepare atomic traps and optical molasses.
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I don't believe that the difference in energy between the photon arriving and leaving explains the difference in kinetic energy of the atom from before the absorption to after the emission. – Floris May 24 '14 at 1:22
I'm not sure how compelling I find it either, but if energy is conserved only in the same frame, the photon being absorbed at a Doppler shifted frequency will certainly be emitted at a different frequency relative to the atom. The atom is traveling at a different velocity when it emits compared to when it absorbs. If the frequency of both events is that of the transition in both cases with shifting taken into account, the photon leaving must have a different energy than that being absorbed. Entropy alone tells us state-wise what must be changing, but not the mechanism or change involved. – sakanojo May 24 '14 at 21:04
Aaaand I see this was marked as duplicate. Will check out that other answer ... – sakanojo May 24 '14 at 21:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8900010585784912, "perplexity": 262.5012494172169}, "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/1448398464386.98/warc/CC-MAIN-20151124205424-00012-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/167948/is-there-literature-on-approaching-semisimplicity-of-l-adic-cohomology-using-van | # Is there literature on approaching semisimplicity of l-adic cohomology using vanishing cycles
Let $K$ be a finitely generated field, and $X/K$ a smooth proper variety. Let $G$ denote the absolute Galois group $\mathrm{Gal}(\bar{K}/K)$. Let $\ell$ be a prime different from $\mathrm{char}(K)$. A longstanding famous conjecture (due to, I think, Tate, but also Grothendieck–Serre) is:
Conjecture. The representation of $G$ on $H_{\mathrm{ét}}^{i}(X_{\bar{K}}, \mathbb{Q}_{\ell})$ is semisimple.
The conjecture is known if $X$ is an abelian variety (hence if $X$ is a curve), and a few other cases.
One way of approaching this might be to mimic Deligne's proof of Weil I, and one readily reduces to the case where:
• $f \colon X \to \mathbb{P}^{1}$ is a Lefschetz pencil (finitely many singular fibres, and each singular fibre has a unique singularity, that is a quadratic singularity);
• with fibres of dimension $n$;
• by induction the conjecture is known for varieties of dimension $\le n$.
The groups $H_{\mathrm{ét}}^{i}(X_{\bar{K}}, \mathbb{Q}_{\ell})$, for $i \ne n+1$ are semisimple via an induction argument.
One can then study $H_{\mathrm{ét}}^{n+1}(X_{\bar{K}}, \mathbb{Q}_{\ell})$ via the Leray spectral sequence of $f$. Here one should use perverse sheaves, in order to obtain $E_{2}$-degeneration of the spectral sequence.
Lei Fu does some computations on this spectral sequence in “On the semisimplicity conjecture and Galois representations”. Using those computations, one remains with a factor $H_{\mathrm{ét}}^{1}(\mathbb{P}^{1}, R^{n}f_{*}\mathbb{Q}_{\ell})$ for which one has to prove semisimplicity. Lei Fu reduces this to his own conjecture about Galois representations related to function fields.
My question is whether there is any literature that has pursued this path along Deligne's Weil I. With this I mean the following: Consider the subsheaf $\mathcal{E} \subset R^{n}f_{*}\mathbb{Q}_{\ell}$ of vanishing cycles. Then we have a surjection $H_{\mathrm{ét}}^{1}(\mathbb{P}^{1}, \mathcal{E}) \to H_{\mathrm{ét}}^{1}(\mathbb{P}^{1}, R^{n}f_{*}\mathbb{Q}_{\ell})$.
Has anyone studied the semisimplicity of $H_{\mathrm{ét}}^{1}(\mathbb{P}^{1}, \mathcal{E})$? Is there literature about this? Are there any further/partial results?
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https://link.springer.com/article/10.1007%2Fs00022-019-0503-1 | Journal of Geometry
, 110:48
# Envelopes of one-parameter families of framed curves in the Euclidean space
• Donghe Pei
• Masatomo Takahashi
• Haiou Yu
Open Access
Article
## Abstract
As a one-parameter family of singular space curves, we consider a one-parameter family of framed curves in the Euclidean space. Then we define an envelope for a one-parameter family of framed curves and investigate properties of envelopes. Especially, we concentrate on one-parameter families of framed curves in the Euclidean 3-space. As applications, we give relations among envelopes of one-parameter families of framed space curves, one-parameter families of Legendre curves and one-parameter families of spherical Legendre curves, respectively.
## Keywords
Envelope one-parameter families of framed curves singularity
## Mathematics Subject Classification
58K05 53A40 57R45
## 1 Introduction
For regular plane curves, envelopes are classical objects in differential geometry [1, 2, 6, 7, 10, 12]. An envelope of a family of curves is a new curve which is tangent to curves of the given family. An envelope of a family of submanifolds in the Euclidean space were studied in [3, 15]. Using the notion of stability for germs of family, they investigate the singularities of the envelope of the submanifolds in the Euclidean space via Legendrian singularity theory. However, by using implicit functions, the definition and calculation of the envelope of space curves are complicated.
On the other hand, if we look at the classical concept of envelope we see that the envelope is the set of characteristic points and avoiding singularities of the elements of the family. Moreover, for singular space curves, the classical definitions of envelopes are vague. In [14], the second author clarified the definition of the envelope for families of singular plane curves. In [11], we clarified the definition of the envelope for families of singular spherical curves. In this paper we pursue second author’s method and treat the envelope of smooth space curves with singular points. We would like to define an envelope of parametric space curves with singular points in the Euclidean space. As a singular curve, we consider a framed curve in the Euclidean space. A framed curve in the Euclidean space is a smooth curve with a moving frame (cf. [8]). It is a generalization of a regular curve with linear independent condition, a Legendre curve in the unit tangent bundle over the Euclidean plane and a Legendre curve in the unit spherical bundle over the unit sphere.
In Sect. 2, we consider one-parameter families of framed curves in the Euclidean space. We give a moving frame and the curvature of a one-parameter family of framed curves. Then we give the existence and uniqueness theorems in terms of the curvatures of the one-parameter families of framed curves. In Sect. 3, we define an envelope of a one-parameter family of framed curves in the Euclidean space. We obtain that the envelope is also a framed curve. As a main result, we give a necessary and sufficient condition that a one-parameter family of framed curves has an envelope (Theorem 3.3). In Sect. 4, we focus on the framed curves in the Euclidean 3-space. The theory of one-parameter families of framed curves relate to the theory of surfaces with singular points (cf. [4, 5]). The envelope is independent of rotated frames and reflected frames of the framed curves. We also define a parallel curve of a one-parameter family of framed space curves. The parallel curve is also a one-parameter family of framed curves. Then the envelope of parallel curves and a parallel curve of the envelope of a one-parameter family of framed curves coincide (Proposition 4.7). In Sect. 5, as applications, we give relations among envelopes of one-parameter families of framed space curves, one-parameter families of Legendre curves in the unit tangent bundle over the Euclidean plane and one-parameter families of Legendre curves in the unit spherical bundle over the unit sphere, respectively. In Sect. 6, we give concrete examples of envelopes of one-parameter families of framed curves.
All maps and manifolds considered here are differentiable of class $$C^\infty$$.
## 2 Framed curves and one-parameter families of framed curves in the Euclidean space
We first recall some definitions and results of framed curves in the Euclidean space. For more details see [8].
Let $${\mathbb {R}}^n$$ be the n-dimensional Euclidean space equipped with the inner product $$\varvec{a} \cdot \varvec{b} = \sum _{i=1}^{n}a_{i}b_{i}$$, where $$\varvec{a} = (a_1, \ldots , a_n)$$ and $$\varvec{b} = (b_1, \ldots , b_n)$$. We denote the norm of $$\varvec{a}$$ by $$|\varvec{a}|=\sqrt{\varvec{a} \cdot \varvec{a}}$$. Let $$\varvec{a}_{1},\ldots ,\varvec{a}_{n-1}\in {\mathbb {R}}^n$$ be vectors $$\varvec{a}_{i}=(a_{i1},\ldots ,a_{in})$$ for $$i=1,\ldots ,n-1$$. The vector product is given by
\begin{aligned} \varvec{a}_{1} \times \cdots \times \varvec{a}_{n-1}=\mathrm{det} \left( \begin{array}{cccc} a_{11} &{} \cdots &{} a_{1n} \\ \vdots &{} \ddots &{} \vdots \\ a_{n-11} &{} \cdots &{} a_{n-1n}\\ \varvec{e}_1 &{} \cdots &{} \varvec{e}_n \end{array} \right) =\sum ^{n}_{i=1} \mathrm{det}(\varvec{a}_{1},\ldots ,\varvec{a}_{n-1},\varvec{e}_{i})\varvec{e}_{i}, \end{aligned}
where $$\{\varvec{e}_1, \ldots , \varvec{e}_n\}$$ is the canonical basis of $${\mathbb {R}}^n$$. Then we have $$(\varvec{a}_{1}\times \cdots \times \varvec{a}_{n-1})\cdot \varvec{a}_{i} = 0$$ for $$i = 1,\ldots ,n-1$$. Note that for the case of $$n = 3$$,
\begin{aligned} \varvec{a}_{1} \times \varvec{a}_{2}=\mathrm{det} \left( \begin{array}{ccc} a_{11} &{}\quad a_{12} &{}\quad a_{13} \\ a_{21} &{}\quad a_{22} &{}\quad a_{23}\\ \varvec{e}_1 &{}\quad \varvec{e}_2 &{}\quad \varvec{e}_3 \end{array} \right) =\mathrm{det} \left( \begin{array}{ccc} \varvec{e}_1 &{}\quad \varvec{e}_2 &{}\quad \varvec{e}_3\\ a_{11} &{}\quad a_{12} &{}\quad a_{13} \\ a_{21} &{}\quad a_{22} &{}\quad a_{23} \end{array} \right) . \end{aligned}
Let $$S^{n-1}=\{(x_{1},\ldots ,x_{n})\in \mathbb {R}^n~|~x_{1}^2+\cdots +x_{n}^2=1\}$$ be the unit sphere. We denote the set $$\Delta _{n-1}$$,
\begin{aligned} \Delta _{n-1}&= \{\varvec{\nu }=(\nu _{1},\ldots ,\nu _{n-1})\in {\mathbb {R}}^{n}\times \cdots \times {\mathbb {R}}^{n}|\nu _{i}\cdot \nu _{j}=\delta _{ij},i,j=1,\ldots ,n-1\}\\&= \{\varvec{\nu }=(\nu _{1},\ldots ,\nu _{n-1})\in S^{n-1}\times \cdots \times S^{n-1}|\nu _{i}\cdot \nu _{j}=0,i\ne j,i,j=1,\ldots ,n-1\}. \end{aligned}
Then $$\Delta _{n-1}$$ is a $$n(n-1)/2$$-dimensional smooth manifold.
Let I be an interval of $${\mathbb {R}}$$.
### Definition 2.1
Let $$(\gamma , \varvec{\nu }): I \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ be a smooth mapping. We say that $$(\gamma , \varvec{\nu })$$ is a framed curve if $$\dot{\gamma }(t) \cdot \nu _{i}(t)=0$$ for all $$t \in I$$ and $$i=1,\ldots ,n-1$$, where $$\varvec{\nu }=(\nu _{1},\ldots ,\nu _{n-1})$$.
We define $$\varvec{\mu }:I\rightarrow S^{n-1}$$ by $$\varvec{\mu }(t) = \nu _{1}(t)\times \cdots \times \nu _{n-1}(t)$$. By definition, $$(\varvec{\nu }(t),\varvec{\mu }(t)) \in \Delta _{n}$$ and $$\{\varvec{\nu }(t),\varvec{\mu }(t)\}$$ is a moving frame along $$\gamma (t)$$. Then we have the Frenet type formula.
\begin{aligned} \left( \begin{array}{cc} \dot{\varvec{\nu }}(t)\\ \dot{\varvec{\mu }}(t) \end{array}\right) = A(t)\left( \begin{array}{cc} \varvec{\nu }(t)\\ \varvec{\mu }(t) \end{array}\right) , \ \dot{\gamma }(t)=\alpha (t)\varvec{\mu }(t) \end{aligned}
where $$A(t) = (\alpha _{ij}(t)) \in \mathfrak {o}(n), i, j = 1, \ldots , n$$, $$\mathfrak {o}(n)$$ is the set of $$n \times n$$ alternative matrices and $$\alpha :I \rightarrow {\mathbb {R}}$$ is a smooth function. We say that the mapping $$(\alpha _{ij}, \alpha )$$ is the curvature of the framed curve $$(\gamma , \varvec{\nu })$$.
### Definition 2.2
Let $$(\gamma , \varvec{\nu })$$ and $$(\widetilde{\gamma }, \widetilde{\varvec{\nu }}) : I \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ be framed curves. We say that $$(\gamma , \varvec{\nu })$$ and $$(\widetilde{\gamma }, \widetilde{\varvec{\nu }})$$ are congruent as framed curves if there exist a special orthogonal matrix $$X\in SO(n)$$ and a constant vector $${\varvec{x}}\in {\mathbb {R}}^{n}$$ such that $$\widetilde{\gamma }(t)=X(\gamma (t))+{\varvec{x}}$$ and $$\widetilde{\varvec{\nu }}(t)=X(\varvec{\nu }(t))$$ for all $$t\in I$$.
Then we have the existence and uniqueness theorems in terms of the curvatures of the framed curves.
### Theorem 2.3
(Existence Theorem for framed curves [8]). Let $$(\alpha _{ij}, \alpha ) : I \rightarrow \mathfrak {o}(n)\times {\mathbb {R}}$$ be a smooth mapping. Then there exists a framed curve $$(\gamma ,\varvec{\nu }):I \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ whose associated curvature is given by $$(\alpha _{ij}, \alpha )$$.
### Theorem 2.4
(Uniqueness Theorem for framed curves [8]). Let $$(\gamma , \varvec{\nu })$$ and $$(\widetilde{\gamma }, \widetilde{\varvec{\nu }}) : I \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ be framed curves with the curvatures $$(\alpha _{ij},\alpha )$$ and $$(\widetilde{\alpha }_{ij},\widetilde{\alpha })$$ respectively. Then $$(\gamma , \varvec{\nu })$$ and $$(\widetilde{\gamma }, \widetilde{\varvec{\nu }})$$ are congruent as framed curves if and only if $$(\alpha _{ij},\alpha )$$ and $$(\widetilde{\alpha }_{ij},\widetilde{\alpha })$$ coincide.
Next, we consider one-parameter families of framed curves in the Euclidean space. Let $$\Lambda$$ be open interval of $$\mathbb {R}$$.
### Definition 2.5
Let $$(\gamma , \varvec{\nu }): I \times \Lambda \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ be a smooth mapping. We say that $$(\gamma , \varvec{\nu })$$ is a one-parameter family of framed curves if $$\gamma _t(t, \lambda ) \cdot \nu _{i}(t, \lambda )=0$$ for all $$(t, \lambda )\in I \times \Lambda$$ and $$i=1,\ldots ,n-1$$, where $$\varvec{\nu }=(\nu _{1},\ldots ,\nu _{n-1})$$.
By definition, $$(\gamma (\cdot ,\lambda ), \varvec{\nu }(\cdot ,\lambda )): I \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ is a framed curve for each fixed parameter $$\lambda \in \Lambda$$. It is easy to see that $$(-\gamma , \varvec{\nu })$$ and $$(\gamma , -\varvec{\nu })$$ are also one-parameter families of framed curves.
We define $$\varvec{\mu }(t,\lambda )=\nu _{1}(t,\lambda )\times \cdots \times \nu _{n-1}(t,\lambda )$$. Then $$\{\varvec{\nu }(t,\lambda ),\varvec{\mu }(t,\lambda )\}$$ is a moving frame along $$\gamma (t,\lambda )$$ and we have the Frenet type formula.
\begin{aligned} \left( \begin{array}{cc} \varvec{\nu }_{t}(t,\lambda )\\ \varvec{\mu }_{t}(t,\lambda ) \end{array}\right)= & {} A(t,\lambda )\left( \begin{array}{cc} \varvec{\nu }(t,\lambda )\\ \varvec{\mu }(t,\lambda ) \end{array}\right) , \quad \left( \begin{array}{cc} \varvec{\nu }_{\lambda }(t,\lambda )\\ \varvec{\mu }_{\lambda }(t,\lambda ) \end{array}\right) = B(t,\lambda )\left( \begin{array}{cc} \varvec{\nu }(t,\lambda )\\ \varvec{\mu }(t,\lambda ) \end{array}\right) ,\\ \gamma _{t}(t,\lambda )= & {} \alpha (t,\lambda )\varvec{\mu }(t,\lambda ),~ \gamma _{\lambda }(t,\lambda )=P(t,\lambda )\left( \begin{array}{cc} \varvec{\nu }(t,\lambda )\\ \varvec{\mu }(t,\lambda ) \end{array}\right) , \end{aligned}
where $$A(t,\lambda ) = (\alpha _{ij}(t,\lambda )), B(t,\lambda ) = (\beta _{ij}(t,\lambda ))\in \mathfrak {o}(n), i, j = 1, \ldots , n$$, $$\alpha :I \times \Lambda \rightarrow {\mathbb {R}}$$ is a smooth function and $$P:I\times \Lambda \rightarrow {\mathbb {R}}^{n},~P(t,\lambda )=(P_{1}(t,\lambda ), \ldots ,P_{n}(t,\lambda ))$$ is a smooth mapping.
By $$\gamma _{t\lambda }(t,\lambda )=\gamma _{\lambda t}(t,\lambda ),~\nu _{t\lambda }(t,\lambda )=\nu _{\lambda t}(t,\lambda )$$ and $$\varvec{\mu }_{t\lambda }(t,\lambda )=\varvec{\mu }_{\lambda t}(t,\lambda )$$, we have the integrability condition
\begin{aligned} \begin{aligned}&(\alpha _{ij}(t,\lambda ))_{\lambda }+(\alpha _{ij}(t,\lambda )) (\beta _{ij}(t,\lambda ))=(\beta _{ij}(t,\lambda ))_{t}+(\beta _{ij} (t,\lambda ))(\alpha _{ij}(t,\lambda )),\\&\alpha (t,\lambda )\beta _{ni}(t,\lambda )=(P_{i})_{t}(t,\lambda ) +\sum _{j=1}^{n}P_{j}(t,\lambda )\alpha _{ji}(t,\lambda ),~(i=1,\ldots ,n-1),\\&\alpha _{\lambda }(t,\lambda )+\alpha (t,\lambda )\beta _{nn}(t,\lambda ) =(P_{n})_{t}(t,\lambda )+\sum _{j=1}^{n}P_{j}(t,\lambda )\alpha _{jn}(t,\lambda ) \end{aligned} \end{aligned}
(2.1)
for all $$(t,\lambda )\in \mathrm {I}\times \mathrm {\Lambda }$$. We call the mapping $$(\alpha _{ij},\beta _{ij},\alpha ,P_{1},\ldots ,P_{n})$$ with the integrability condition (2.1) the curvature of the one-parameter family of framed curves $$(\gamma ,\varvec{\nu })$$.
### Definition 2.6
Let $$(\gamma ,\varvec{\nu })$$ and $$(\widetilde{\gamma }, \widetilde{\varvec{\nu }}) : I \times \Lambda \rightarrow {\mathbb {R}}^{n}\times \Delta _{n-1}$$ be one-parameter families of framed curves. We say that $$(\gamma ,\varvec{\nu })$$ and $$(\widetilde{\gamma }, \widetilde{\varvec{\nu }})$$ are congruent as one-parameter families of framed curves if there exist a special orthogonal matrix $$X \in SO(n)$$ and a constant vector $${\varvec{x}}\in {\mathbb {R}}^{n}$$ such that $$\widetilde{\gamma }(t, \lambda ) = X(\gamma (t, \lambda ))+{\varvec{x}}$$ and $$\widetilde{\varvec{\nu }}(t, \lambda ) = X(\varvec{\nu }(t, \lambda ))$$ for all $$(t, \lambda )\in I \times \Lambda$$.
We give the existence and uniqueness theorems in terms of the curvatures of the one-parameter families of framed curves.
### Theorem 2.7
(Existence Theorem for one-parameter families of framed curves). Let $$(\alpha _{ij},\beta _{ij},\alpha ,P_{1},\ldots ,P_{n}): I \times \Lambda \rightarrow \mathfrak {o}(n)\times \mathfrak {o}(n)\times {\mathbb {R}}^{n+1}$$ be a smooth mapping with the integrability condition. Then there exists a one-parameter family of framed curves $$(\gamma ,\varvec{\nu }) : I \times \Lambda \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ whose associated curvature is given by $$(\alpha _{ij},\beta _{ij},\alpha ,P_{1},\ldots ,P_{n})$$.
### Proof
We denote that M(n) is the set of $$n\times n$$ matrices and $$I_n$$ is the identity matrix. Choose any fixed value $$(t_0, \lambda _0) \in I \times \Lambda$$. We consider an initial value problem,
\begin{aligned} F_t(t,\lambda )=A(t,\lambda )F(t,\lambda ),~ F_\lambda (t,\lambda )=B(t,\lambda )F(t,\lambda ),~ F(t_0,\lambda _0)=I_n, \end{aligned}
where $$F(t,\lambda ) \in M(n)$$ and $$A(t,\lambda ) = (\alpha _{ij}(t,\lambda )), B(t,\lambda ) = (\beta _{ij}(t,\lambda ))\in \mathfrak {o}(n)$$ for $$i, j = 1, \ldots , n$$. By
\begin{aligned} \begin{aligned} F_{t\lambda }&=A_\lambda F+AF_\lambda =A_\lambda F+ABF=(A_\lambda +AB)F,\\ F_{\lambda t}&=B_t F+BF_t=B_t F+BAF=(B_t+BA)F. \end{aligned} \end{aligned}
and the integrability condition $$A_\lambda +AB=B_t+BA$$, we have $$F_{t\lambda }=F_{\lambda t}$$. Since $$I\times \Lambda$$ is simply connected, there exists a solution $$F(t,\lambda )$$. Since $$A(t,\lambda ),B(t,\lambda )\in \mathfrak {o}(n)$$,It follows that Open image in new window is constant. Therefore, we haveLet Open image in new window. Since
\begin{aligned} (\partial /\partial t)(\mathrm{det} F(t,\lambda ))=0,~(\partial /\partial \lambda )(\mathrm{det} F(t,\lambda ))=0, \end{aligned}
we have
\begin{aligned} \mathrm{det} F(t,\lambda )=\mathrm{det} F(t_0,\lambda _0)=\mathrm{det} I_n=1. \end{aligned}
Hence $$F(t,\lambda )$$ is a special orthogonal matrix. Then $$\varvec{\mu }(t,\lambda )=\nu _1(t,\lambda )\times \cdots \times \nu _{n-1}(t,\lambda )$$. Next we consider differential equations
\begin{aligned} \gamma _{t}=\alpha \varvec{\mu },~\gamma _{\lambda }=P_{1}\nu _{1}+\cdots +P_{n-1}\nu _{n-1}+P_{n}\varvec{\mu }. \end{aligned}
By the integrability condition, we have $$\gamma _{t\lambda }(t,\lambda )=\gamma _{\lambda t}(t,\lambda )$$ for all $$(t,\lambda )\in I\times \Lambda$$. Then there exists a solution $$\gamma (t,\lambda )$$. It follows that $$(\gamma ,\varvec{\nu }) : I \times \Lambda \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ is a one-parameter family of framed curves whose associated curvature is given by $$(\alpha _{ij},\beta _{ij},\alpha ,P_{1},\ldots ,P_{n})$$. $$\square$$
### Lemma 2.8
If $$(\alpha _{ij},\beta _{ij},\alpha ,P_{1},\ldots ,P_{n})(t,\lambda ) =(\widetilde{\alpha }_{ij},\widetilde{\beta }_{ij},\widetilde{\alpha }, \widetilde{P}_{1},\ldots ,\widetilde{P}_{n})(t,\lambda )$$ and $$(\gamma ,\varvec{\nu })(t_0,\lambda _0)=(\widetilde{\gamma },\widetilde{\varvec{\nu }})(t_0,\lambda _0)$$ for some point $$(t_0, \lambda _0) \in I \times \Lambda$$, then $$(\gamma ,\varvec{\nu })$$=$$(\widetilde{\gamma },\widetilde{\varvec{\nu }})$$.
### Proof
We put $$\nu _{n}(t,\lambda )=\varvec{\mu }(t,\lambda )$$ and $$\widetilde{\nu }_{n}(t,\lambda )=\widetilde{\varvec{\mu }}(t,\lambda )$$. We define a smooth function $$f:I\times \Lambda \rightarrow \mathbb {R}$$ by
\begin{aligned} f(t,\lambda )=\sum ^{n}_{i=1}\nu _{i}(t,\lambda )\cdot \widetilde{\nu }_{i}(t,\lambda ). \end{aligned}
By $$\varvec{\nu }(t_0,\lambda _0)=\widetilde{\varvec{\nu }}(t_0,\lambda _0)$$, we have $$\nu _{i}(t_0,\lambda _0)=\widetilde{\nu }_{i}(t_0,\lambda _0)~(i=1,\ldots ,n)$$ and hence $$f(t_0,\lambda _0)=n$$. Since $$\alpha _{ij}(t,\lambda )=\widetilde{\alpha }_{ij}(t,\lambda )$$ and $$\alpha _{ij}(t,\lambda )=-\alpha _{ji}(t,\lambda )$$, we have
\begin{aligned} f_{t}(t,\lambda )= & {} \sum _{i=1}^{n}\bigl (\nu _{i_{t}}(t,\lambda ) \cdot \widetilde{\nu }_{i}(t,\lambda )+\nu _{i}(t,\lambda ) \cdot (\widetilde{\nu }_{i})_{t}(t,\lambda )\bigr )\\= & {} \sum _{i=1}^{n}\left\{ \left( \sum _{j=1}^{n}\alpha _{ij} (t,\lambda )\nu _{j}(t,\lambda )\right) \cdot \widetilde{\nu }_{i}(t,\lambda )+ \nu _{i}(t,\lambda )\cdot \left( \sum _{j=1}^{n}\widetilde{\alpha }_{ij} (t,\lambda )\widetilde{\nu }_{j}\right) \right\} \\= & {} 2\sum _{i=1}^{n}\sum _{j=1}^{n}\bigl (\alpha _{ij} (t,\lambda )+\alpha _{ji}(t,\lambda )\bigr )\nu _{i}(t,\lambda ) \cdot \widetilde{\nu }_{j}(t,\lambda )=0. \end{aligned}
In the same way, we have $$f_\lambda (t,\lambda )=0$$. It follows that f is constant with value n. By the Cauchy-Schwarz inequality, we have
\begin{aligned} \nu _{i}(t,\lambda )\cdot \widetilde{\nu }_{i}(t,\lambda )&\leqslant |\nu _{i}(t,\lambda )||\widetilde{\nu }_{i}(t,\lambda )|=1 \end{aligned}
for each $$i=1,\ldots ,n$$. If one of these inequalities is strict, the value of $$f(t,\lambda )$$ would be less than n. Therefore, these inequalities are equalities, that is, $$\nu _{i}(t,\lambda )\cdot \widetilde{\nu }_{i}(t,\lambda )=1$$. Then we have $$|\nu _{i}(t,\lambda )-\widetilde{\nu }_{i}(t,\lambda )|^2=0$$ and hence $$\nu _{i}(t,\lambda )=\widetilde{\nu }_{i}(t,\lambda )$$ for all $$(t,\lambda )\in I\times \Lambda$$ and $$i=1,\ldots ,n$$. Since $$\gamma _{t}(t,\lambda )=\alpha (t,\lambda )\varvec{\mu }(t,\lambda ), \widetilde{\gamma }_{t}(t,\lambda )=\widetilde{\alpha }(t,\lambda ) \widetilde{\varvec{\mu }}(t,\lambda )$$, $$\gamma _{\lambda }(t,\lambda )=P_{1}(t,\lambda )\nu _{1}(t,\lambda ) +\cdots +P_{n-1}(t,\lambda )\nu _{n-1}(t,\lambda )+P_{n}(t,\lambda )\varvec{\mu }(t,\lambda )$$, $$\widetilde{\gamma }_{\lambda }(t,\lambda )=\widetilde{P}_{1}(t,\lambda ) \widetilde{\nu }_{1}(t,\lambda )+\cdots +\widetilde{P}_{n-1}(t,\lambda ) \widetilde{\nu }_{n-1}(t,\lambda )+\widetilde{P}_{n}(t,\lambda ) \widetilde{\varvec{\mu }}(t,\lambda )$$, and the assumptions $$\alpha (t,\lambda )=\widetilde{\alpha }(t,\lambda ), P_{i}(t,\lambda )=\widetilde{P}_{i}(t,\lambda ) \ (i=1,\dots ,n)$$, we have
\begin{aligned} (\gamma (t,\lambda )-\widetilde{\gamma }(t,\lambda ))_{t}=0,~ (\gamma (t,\lambda )-\widetilde{\gamma }(t,\lambda ))_{\lambda }=0. \end{aligned}
It follows that $$\gamma (t,\lambda )-\widetilde{\gamma }(t,\lambda )$$ is constant. By the condition $$\gamma (t_{0},\lambda _{0})= \widetilde{\gamma }(t_{0},\lambda _{0})$$, we have $$\gamma (t,\lambda )=\widetilde{\gamma }(t,\lambda )$$ for all $$(t,\lambda )\in I\times \Lambda .$$ $$\square$$
### Theorem 2.9
(Uniqueness Theorem for one-parameter families of framed curves). Let $$(\gamma ,\varvec{\nu })$$ and $$(\widetilde{\gamma },\widetilde{\varvec{\nu }}): I \times \Lambda \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ be one-parameter families of framed curves with the curvatures $$(\alpha _{ij},\beta _{ij},\alpha ,P_{1},\ldots ,P_{n})$$ and $$(\widetilde{\alpha }_{ij},\widetilde{\beta }_{ij},\widetilde{\alpha }, \widetilde{P}_{1},\ldots ,\widetilde{P}_{n})$$, respectively. Then $$(\gamma ,\varvec{\nu })$$ and $$(\widetilde{\gamma },\widetilde{\varvec{\nu }})$$ are congruent as one-parameter families of framed curves if and only if $$(\alpha _{ij},\beta _{ij},\alpha ,P_{1},\ldots ,P_{n})$$ and $$(\widetilde{\alpha }_{ij},\widetilde{\beta }_{ij},\widetilde{\alpha }, \widetilde{P}_{1},\ldots ,\widetilde{P}_{n})$$ coincide.
### Proof
Suppose that $$(\gamma , \varvec{\nu })$$ and $$(\widetilde{\gamma }, \widetilde{\varvec{\nu }})$$ are congruent as one-parameter families of framed curves, there exist a matrix $$X\in SO(n)$$ and a constant vector $${\varvec{x}}\in {\mathbb {R}}^{n}$$ such that
\begin{aligned} \widetilde{\gamma }(t,\lambda )=X(\gamma (t,\lambda ))+{\varvec{x}}, ~\widetilde{\varvec{\nu }}(t,\lambda )=X(\varvec{\nu }(t,\lambda )) \end{aligned}
for all $$(t,\lambda )\in I\times \Lambda$$. Since the definition of $$\varvec{\mu }$$, we have $$\widetilde{\varvec{\mu }}(t,\lambda )=X(\varvec{\mu }(t,\lambda ))$$ for all $$(t,\lambda )\in I\times \Lambda$$. By a direct calculation, we have
\begin{aligned} \widetilde{\alpha }_{ij}(t,\lambda )= & {} (\widetilde{\nu }_{i})_{t}(t,\lambda ) \cdot \widetilde{\nu }_{j}(t,\lambda )\\= & {} X((\nu _{i})_{t}(t,\lambda ))\cdot X(\nu _{j}(t,\lambda ))=(\nu _{i})_{t}(t,\lambda )\cdot \nu _{j}(t,\lambda ) =\alpha _{ij}(t,\lambda ),\\ \widetilde{\beta }_{ij}(t,\lambda )= & {} (\widetilde{\nu }_{i})_{\lambda }(t,\lambda ) \cdot \widetilde{\nu }_{j}(t,\lambda )\\= & {} X((\nu _{i})_{\lambda }(t,\lambda )) \cdot X(\nu _{j}(t,\lambda ))=(\nu _{i})_{\lambda }(t,\lambda )\cdot \nu _{j}(t,\lambda ) =\beta _{ij}(t,\lambda ),\\ \widetilde{\alpha }(t,\lambda )= & {} \widetilde{\gamma }_{t}(t,\lambda )\cdot \widetilde{\varvec{\mu }}(t,\lambda )\\= & {} X(\gamma _{t}(t,\lambda ))\cdot X(\varvec{\mu }(t,\lambda ))=\gamma _{t}(t,\lambda )\cdot \varvec{\mu }(t,\lambda ) =\alpha (t,\lambda ),\\ \widetilde{P}_{i}(t,\lambda )= & {} \widetilde{\gamma }_{\lambda }(t,\lambda ) \cdot \widetilde{\nu }_{i}(t,\lambda )=X(\gamma _{\lambda }(t,\lambda ))\cdot X(\nu _{i}(t,\lambda ))\\= & {} \gamma _{\lambda }(t,\lambda )\cdot \nu _{i}(t,\lambda ) =P_{i}(t,\lambda )~(i=1,\ldots ,n-1),\\ \widetilde{P}_{n}(t,\lambda )= & {} \widetilde{\gamma }_{\lambda }(t,\lambda ) \cdot \widetilde{\varvec{\mu }}(t,\lambda )\\= & {} X(\gamma _{\lambda }(t,\lambda ))\cdot X(\varvec{\mu }(t,\lambda ))=\gamma _{\lambda }(t,\lambda )\cdot \varvec{\mu }(t,\lambda ) =P_{n}(t,\lambda ). \end{aligned}
Therefore, $$(\alpha _{ij},\beta _{ij},\alpha ,P_{1},\ldots ,P_{n})$$ and $$(\widetilde{\alpha }_{ij},\widetilde{\beta }_{ij}, \widetilde{\alpha },\widetilde{P}_{1},\ldots ,\widetilde{P}_{n})$$ coincide.
Conversely, suppose that $$(\alpha _{ij},\beta _{ij},\alpha ,P_{1},\ldots ,P_{n})$$ and $$(\widetilde{\alpha }_{ij},\widetilde{\beta }_{ij}, \widetilde{\alpha },\widetilde{P}_{1},\ldots ,\widetilde{P}_{n})$$ coincide. Let $$(t_0, \lambda _0)$$ $$\in I \times \Lambda$$ be fixed. By using a congruence as one-parameter families of framed curves, we may assume $$\gamma (t_0, \lambda _0)= \widetilde{\gamma }(t_0, \lambda _0)$$ and $$\varvec{\nu }(t_0, \lambda _0) = \widetilde{\varvec{\nu }}(t_0, \lambda _0)$$. By Lemma 2.8, we have $$\gamma (t, \lambda ) = \widetilde{\gamma }(t, \lambda )$$ and $$\varvec{\nu }(t, \lambda ) = \widetilde{\varvec{\nu }}(t, \lambda )$$ for all $$(t,\lambda ) \in I \times \Lambda$$. $$\square$$
## 3 Envelopes of one-parameter families of framed curves in the Euclidean space
Let $$(\gamma ,\varvec{\nu }): I \times \Lambda \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ be a one-parameter family of framed curves with the curvature $$(\alpha _{ij},\beta _{ij},\alpha ,$$ $$P_{1},\ldots ,P_{n})$$ and let $$e:U \rightarrow I \times \Lambda$$, $$e(u) = (t(u), \lambda (u))$$ be a smooth curve, where U is an interval of $${\mathbb {R}}$$. We denote $$E_\gamma =\gamma \circ e:U \rightarrow {\mathbb {R}}^{n}$$, $$E_{\nu _{i}}=\nu _{i}\circ e:U \rightarrow S^{n-1}$$ and $$E_{\varvec{\nu }}=\varvec{\nu }\circ e : U \rightarrow \Delta _{{n-1}}$$.
### Definition 3.1
We call $$E_\gamma$$ an envelope (and e a pre-envelope) for the one-parameter family of framed curves $$(\gamma ,\varvec{\nu })$$, when the following conditions are satisfied.
1. (i)
The function $$\lambda$$ is non-constant on any non-trivial subinterval of U. (The Variability Condition.)
2. (ii)
For all $$u\in U$$, the curve $$E_\gamma$$ is tangent at u to the curve $$\gamma (t, \lambda )$$ at the parameter $$(t(u),\lambda (u))$$, meaning that the tangent vectors $$E_\gamma '(u) = (dE /du)(u)$$ and $$\varvec{\mu }(e(u))$$ are linearly dependent. (The Tangency Condition.)
This definition is a generalization of the definition of an envelope of a one-parameter family of Legendre curves in [11, 14]. Note that the tangency condition is equivalent to the condition $$E_\gamma '(u) \cdot \nu _{i}(e(u))=E_\gamma '(u) \cdot E_{\nu _{i}}(u) = 0$$ for all $$u\in U$$ and $$i=1,\ldots ,n-1.$$
### Proposition 3.2
Let $$(\gamma , \varvec{\nu }) : I\times \Lambda \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ be a one-parameter family of framed curves with the curvature $$(\alpha _{ij},\beta _{ij},\alpha ,P_{1},\ldots ,P_{n})$$. Suppose that $$e : U \rightarrow I\times \Lambda , e(u) = (t(u), \lambda (u))$$ is a pre-envelope and $$E_\gamma : U \rightarrow {\mathbb {R}}^n$$ is an envelope of $$(\gamma , \varvec{\nu })$$. Then $$(E_\gamma , E_{\varvec{\nu }}) : U \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ is a framed curve with the curvature
\begin{aligned} \alpha _{ijE_\gamma }(u)= & {} t'(u)\alpha _{ij}(e(u)) + \lambda '(u)\beta _{ij}(e(u)),\\ \alpha _{E_\gamma }(u)= & {} t'(u)\alpha (e(u)) + \lambda '(u)P_{n}(e(u)). \end{aligned}
### Proof
By definition, $$E_{\nu _{i}}(u) \cdot E_{\nu _{ j}}(u) = \nu _{i}(e(u))\cdot \nu _{j}(e(u))=\delta _{ij}$$ for all $$u \in U$$ and $$i,j=1,\ldots ,n-1$$. Since $$E_\gamma$$ is an envelope, $$E_\gamma '(u) \cdot E_{\nu _{ i}}(u) = 0$$ for all $$u \in U$$ and $$i=1,\ldots ,n-1$$. It follows that $$(E_\gamma , E_{\varvec{\nu }}) : U \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ is a framed curve. By a direct calculation, we have the curvature
\begin{aligned} \alpha _{ij E_\gamma }(u)= & {} E_{\nu _{i}}'(u) \cdot E_{\nu _{j}}(e(u)) = \bigl (t'(u)(\nu _{i})_{t}(e(u)) + \lambda '(u)(\nu _{i})_{\lambda } (e(u))\bigr ) \cdot \nu _{j}(e(u))\\= & {} t'(u)\alpha _{ij}(e(u)) + \lambda '(u)\beta _{ij}(e(u)),\\ \alpha _{E_\gamma }(u)= & {} E_\gamma '(u) \cdot \varvec{\mu }(e(u)) = \bigl (t'(u)\gamma _t(e(u)) + \lambda '(u)\gamma _{\lambda } (e(u))\bigr ) \cdot \varvec{\mu }(e(u))\\= & {} t'(u)\alpha (e(u)) + \lambda '(u)P_{n}(e(u)). \end{aligned}
$$\square$$
As a main result, we formulate a necessary and sufficient condition that a one-parameter family of framed curves has an envelope. The envelope theorem is as follows:
### Theorem 3.3
Let $$(\gamma ,\varvec{\nu }):I \times \Lambda \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ be a one-parameter family of framed curves and let $$e : U \rightarrow I \times \Lambda$$ be a smooth curve satisfying the variability condition. Then the following are equivalent.
1. (1)
$$e:U \rightarrow I \times \Lambda$$ is a pre-envelope of $$(\gamma , \varvec{\nu })$$ and $$E_\gamma$$ is an envelope of $$(\gamma , \varvec{\nu })$$.
2. (2)
$$\gamma _\lambda (e(u))\cdot \nu _{i}(e(u)) = 0$$ for all $$u \in U$$ and $$i=1,\ldots ,n-1$$.
3. (3)
$$P_i(e(u))=0$$ for all $$u \in U$$ and $$i=1,\dots ,n-1$$.
### Proof
Suppose that e is a pre-envelope of $$(\gamma ,\varvec{\nu })$$. By the tangency condition, there exists a smooth function $$c:U \rightarrow \mathbb {R}$$ such that $$E_\gamma '(u) = c(u)\varvec{\mu }(e(u))$$. By differentiate $$E_\gamma (u) = \gamma \circ e(u)$$, we have
\begin{aligned} E_\gamma '(u)= t'(u)\gamma _t(e(u)) + \lambda '(u)\gamma _{\lambda }(e(u)). \end{aligned}
It follows from $$\gamma _t(t, \lambda )=\alpha (t, \lambda )\varvec{\mu }(t, \lambda )$$ that $$\bigl (t'(u)\alpha (e(u))-c(u)\bigr )\varvec{\mu }(e(u))+\lambda '(u)\gamma _{\lambda }(e(u))=0.$$ Then we have $$\lambda '(u)\gamma _{\lambda }(e(u))\cdot \nu _{i}(e(u))=0.$$ By the variability condition, we have $$\gamma _{\lambda }(e(u))\cdot \nu _{i}(e(u))=0$$ for all $$u\in U$$ and $$i=1,\ldots ,n-1$$.
Conversely, suppose that $$\gamma _{\lambda }(e(u)) \cdot \nu _{i}(e(u))=0$$ for all $$u \in U$$ and $$i=1,\ldots ,n-1$$. Since
\begin{aligned} E_\gamma '(u)\cdot \nu _{i}(e(u))&=\bigl (t'(u)\gamma _t(e(u)) + \lambda '(u)\gamma _{\lambda }(e(u))\bigr ) \cdot \nu _{i}(e(u)) = 0, \end{aligned}
e is a pre-envelope of $$(\gamma , \varvec{\nu })$$. Therefore, (1) and (2) are equivalent.
By using the Frenet type formula of the one-parameter family of framed curves, $$P_i(t,\lambda )=\gamma _{\lambda }(t,\lambda ) \cdot \nu _i(t,\lambda )$$ for all $$(t,\lambda ) \in I \times \Lambda$$ and $$i=1,\dots ,n-1$$. Therefore, (2) and (3) are equivalent. $$\square$$
We say that the singular set of $$\gamma :I\times \Lambda \rightarrow {\mathbb {R}}^n$$ is the subset of the domain $$I\times \Lambda$$ defined by
\begin{aligned} \mathrm{rank}(\gamma _t(t,\lambda ),\gamma _\lambda (t,\lambda ))\le 1. \end{aligned}
Here we discuss the relation between the envelope $$E_\gamma$$ of $$(\gamma , \varvec{\nu })$$ and the singular set of $$\gamma$$. In order to consider the result, we need the following lemma.
### Lemma 3.4
Let $$\varvec{a},\varvec{b}:U\rightarrow {\mathbb {R}}^n$$ be smooth maps. Suppose that the set of non-zero points of smooth function $$k:U\rightarrow {\mathbb {R}}$$ is dense in U. If $$k(u)\varvec{a}(u)$$ and $$\varvec{b}(u)$$ are linearly dependent, then $$\varvec{a}(u)$$ and $$\varvec{b}(u)$$ are linearly dependent for all $$u\in U$$.
### Proof
Since rank$$(k(u)\varvec{a}(u),\varvec{b}(u))\le 1$$. By the condition and continuous property, we have rank$$(\varvec{a}(u),\varvec{b}(u))\le 1$$ for all $$u\in U$$. $$\square$$
### Proposition 3.5
Let $$(\gamma ,\varvec{\nu }):I\times \Lambda \rightarrow {\mathbb {R}}^n\times \Delta _{n-1}$$ be a one-parameter family of framed curves, and let $$e:U\rightarrow I\times \Lambda$$ be a smooth curve satisfying the variability condition. If the set of regular points of $$\gamma$$ on e(U) is dense in U and trace of e lies in the singular set of $$\gamma$$, then e is a pre-envelope of $$(\gamma ,\varvec{\nu })$$ (and $$E_\gamma$$ is an envelope).
### Proof
Since e belong to the singular set of $$\gamma$$, we have rank$$(\gamma _t(e(u)),\gamma _\lambda (e(u)))\le 1$$ for all $$u\in U$$. Therefore $$\gamma _t(e(u))=\beta (e(u))\varvec{\mu }(e(u))$$ and $$\gamma _\lambda (e(u))$$ are linearly dependent. By the assumption, the set of non-zero points of $$\beta \circ e(u)$$ is dense in U. It follows from Lemma 3.4 that $$\varvec{\mu }(e(u))$$ and $$\gamma _\lambda (e(u))$$ are linearly dependent. Therefore $$\gamma _\lambda (e(u))\cdot \nu (e(u))=0$$ for all $$u\in U$$. By Theorem 3.3, e is a pre-envelope of $$(\gamma ,\varvec{\nu })$$. $$\square$$
### Proposition 3.6
Let $$(\gamma , \varvec{\nu }) : I \times \Lambda \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ be a one-parameter family of framed curves. Suppose that $$e : U \rightarrow I\times \Lambda$$ is a pre-envelope and $$E_\gamma$$ is an envelope of $$(\gamma ,\varvec{\nu })$$. Then $$e : U \rightarrow I\times \Lambda$$ is also a pre-envelope of $$(-\gamma ,\varvec{\nu })$$, $$(\gamma ,-\varvec{\nu })$$. Moreover, $$-E_\gamma$$ is an envelope of $$(-\gamma ,\varvec{\nu })$$ and $$E_\gamma$$ is an envelope of $$(\gamma ,-\varvec{\nu })$$.
### Proof
Since $$e : U \rightarrow I\times \Lambda$$ is a pre-envelope, we have $$\gamma _{\lambda }(e(u))\cdot \nu _{i}(e(u))=0$$ for all $$u \in U$$ and $$i=1,\ldots ,n-1$$. It follows that $$-\gamma _{\lambda }(e(u))\cdot \nu _{i}(e(u))=0$$ and $$\gamma _{\lambda }(e(u))\cdot (-\nu _{i}(e(u)))=0$$ for all $$u \in U$$. Thus $$e : U \rightarrow I\times \Lambda$$ is also a pre-envelope of $$(-\gamma ,\varvec{\nu })$$, $$(\gamma ,-\varvec{\nu })$$. $$-E_\gamma$$ is an envelope of $$(-\gamma ,\varvec{\nu })$$ and $$E_\gamma$$ is an envelope of $$(\gamma ,-\varvec{\nu })$$. $$\square$$
Let $$\widetilde{I}$$ and $$\widetilde{\Lambda }$$ be intervals of $${\mathbb {R}}$$.
### Definition 3.7
We say that a map $$\Phi : \widetilde{I} \times \widetilde{\Lambda } \rightarrow I \times \Lambda$$ is a one-parameter family of parameter change if $$\Phi$$ is a diffeomorphism of the form $$\Phi (s, k) = (\phi (s, k), \varphi (k))$$.
### Proposition 3.8
Let $$(\gamma , \varvec{\nu }) : I \times \Lambda \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ be a one-parameter family of framed curves with the curvature $$(\alpha _{ij},\beta _{ij},\alpha ,P_{1},\ldots ,P_{n})$$. Suppose that $$\Phi : \widetilde{I}\times \widetilde{\Lambda } \rightarrow I\times \Lambda , \Phi (s,k)= (\phi (s, k), \varphi (k))$$ is a one-parameter family of parameter change. Then $$(\widetilde{\gamma }, \widetilde{\varvec{\nu }}) = (\gamma \circ \Phi , \varvec{\nu } \circ \Phi ) : \widetilde{I} \times \widetilde{\Lambda } \rightarrow {\mathbb {R}}^{n} \times \Delta _{n-1}$$ is also a one-parameter family of framed curves with the curvature
\begin{aligned} \widetilde{\alpha }_{ij}(s,k)= & {} \alpha _{ij}(\Phi (s,k))\phi _s(s,k),~ \widetilde{\beta }_{ij}(s,k)\\= & {} \alpha _{ij}(\Phi (s,k))\phi _k(s,k) +\beta _{ij}(\Phi (s,k))\varphi ^{\prime }(k),\\ \widetilde{\alpha }(s,k)= & {} \alpha (\Phi (s,k))\phi _s(s,k), ~\widetilde{P}_{i}(s,k)=P_{i}(\Phi (s,k))\varphi ^{\prime }(k)~(i=1,\ldots ,n-1),\\ \widetilde{P}_{n}(s,k)= & {} \alpha (\Phi (s,k))\phi _k(s,k)+P_{n}(\Phi (s,k)) \varphi ^{\prime }(k). \end{aligned}
Moreover, if $$e: U \rightarrow I\times \Lambda$$ is a pre-envelope and $$E_\gamma$$ is an envelope, then $$\Phi ^{-1}\circ e : U \rightarrow \widetilde{I} \times \widetilde{\Lambda }$$ is a pre-envelope and $$E_\gamma$$ is also an envelope of $$(\widetilde{\gamma }, \widetilde{\varvec{\nu }})$$.
### Proof
Since $$\widetilde{\gamma }_s(s, k) = \gamma _t(\Phi (s, k))\phi _s(s,k)$$ and $$\gamma _t(t, \lambda ) \cdot \nu _{i}(t, \lambda ) = 0$$, we have $$\widetilde{\gamma }_s(s, k) \cdot \widetilde{\nu }_{i}(s, k) = 0$$ for all $$(s, k) \in \widetilde{I} \times \widetilde{\Lambda }$$ and $$i=1,\ldots ,n-1$$. Therefore, $$(\widetilde{\gamma }, \widetilde{\nu })$$ is a one-parameter family of framed curves. By a direct calculation, we have the curvature
\begin{aligned} \widetilde{\alpha }_{ij}(s,k)= & {} (\widetilde{\nu }_{i})_{s}(s,k) \cdot \widetilde{\nu }_{j}(s,k)=(\nu _{i})_{t}(\Phi (s,k)) \phi _s(s,k) \cdot \nu _{j}(\Phi (s,k))\\= & {} \alpha _{ij}(\Phi (s,k))\phi _s(s,k),\\ \widetilde{\beta }_{ij}(s,k)= & {} (\widetilde{\nu }_{i})_{k}(s,k) \cdot \widetilde{\nu }_{j}(s,k)=((\nu _{i})_{t}(\Phi (s,k))\phi _k(s,k)\\&+(\nu _{i})_{\lambda }(\Phi (s,k))\varphi ^{\prime }(k))\cdot {\nu }_{j}(\Phi (s,k))\\= & {} \alpha _{ij}(\Phi (s,k))\phi _k(s,k)+\beta _{ij}(\Phi (s,k)) \varphi ^{\prime }(k),\\ \widetilde{\alpha }(s,k)= & {} \widetilde{\gamma }_s(s,k)\cdot \widetilde{\varvec{\mu }}(s,k)=\gamma _t(\Phi (s,k)) \phi _s(s,k) \cdot \varvec{\mu }(\Phi (s,k))\\= & {} \alpha (\Phi (s,k))\phi _s(s,k),\\ \widetilde{P}_{i}(s,k)= & {} \widetilde{\gamma }_k(s,k)\cdot \widetilde{\nu }_{i}(s,k)=(\gamma _{t}(\Phi (s,k))\phi _k(s,k)\\&+\gamma _{\lambda }(\Phi (s,k))\varphi ^{\prime }(k))\cdot \nu _{i}(\Phi (s,k))\\= & {} P_{i}(\Phi (s,k))\varphi ^{\prime }(k)~(i=1,\ldots ,n-1),\\ \widetilde{P}_{n}(s,k)= & {} \widetilde{\gamma }_k(s,k)\cdot \widetilde{\varvec{\mu }}(s,k)=(\gamma _{t}(\Phi (s,k))\phi _k(s,k)\\&+\gamma _{\lambda }(\Phi (s,k))\varphi ^{\prime }(k))\cdot \varvec{\mu }(\Phi (s,k))\\= & {} \alpha (\Phi (s,k))\phi _k(s,k)+P_{n}(\Phi (s,k))\varphi ^{\prime }(k). \end{aligned}
By the form of the diffeomorphism $$\Phi (s, k)=(\phi (s, k), \varphi (k)),~\Phi ^{-1}:I\times \Lambda \rightarrow \widetilde{I} \times \widetilde{\Lambda }$$ is given by the form $$\Phi ^{-1}(t, \lambda ) = (\psi (t,\lambda ), \varphi ^{-1}(\lambda ))$$, where $$\psi$$ is a smooth function. It follows that $$\Phi ^{-1} \circ e(u) = (\psi (t(u), \lambda (u)), \varphi ^{-1}(\lambda (u)))$$. Since $$(d/du)\varphi ^{-1}(\lambda (u)) = (\varphi ^{-1})_\lambda (\lambda (u))\lambda '(u),$$ the variability condition holds. Moreover, we have
\begin{aligned}&\widetilde{\gamma }_k(s, k) \cdot \widetilde{\nu }_{i}(s, k) = (\gamma _t(\Phi (s, k))\phi _k(s, k) + \gamma _{\lambda }(\Phi (s, k))\varphi '(k)) \cdot \nu _{i}(\Phi (s, k)) \\&= \varphi '(k)\gamma _{\lambda }(\Phi (s, k)) \cdot \nu _{i}(\Phi (s, k)). \end{aligned}
It follows that
\begin{aligned} \widetilde{\gamma }_k(\Phi ^{-1} \circ e(u)) \cdot \widetilde{\nu }_{i}(\Phi ^{-1} \circ e(u)) = \varphi '(\varphi ^{-1}(\lambda (u)))\gamma _{\lambda }(e(u)) \cdot \nu _{i}(e(u)) = 0 \end{aligned}
for all $$u\in U$$ and $$i=1,\ldots ,n-1$$. By Theorem 3.3, $$\Phi ^{-1} \circ e$$ is a pre-envelope of $$(\widetilde{\gamma }, \widetilde{\nu })$$. Therefore, $$\widetilde{\gamma } \circ \Phi ^{-1} \circ e = \gamma \circ \Phi \circ \Phi ^{-1} \circ e = \gamma \circ e = E_\gamma$$ is also an envelope of $$(\widetilde{\gamma }, \widetilde{\varvec{\nu }})$$. $$\square$$
## 4 Envelopes of one-parameter families of framed curves in $$\mathbb {R}^{3}\times \Delta _{2}$$
In this section, we concentrate on the case of $$n=3$$. We use the following notations. Let $$(\gamma ,\nu _{1},\nu _{2}):I\times \Lambda \rightarrow \mathbb {R}^{3}\times \Delta _{2}$$ be a one-parameter family of framed curves and $$\varvec{\mu }(t,\lambda )=\nu _{1}(t,\lambda )\times \nu _{2}(t,\lambda )$$. We have the Frenet type formula.s
\begin{aligned} \left( \begin{array}{cccc} \nu _{1t}(t,\lambda )\\ \nu _{2t}(t,\lambda )\\ \varvec{\mu }_t(t,\lambda ) \end{array}\right)= & {} \left( \begin{array}{ccc} 0 &{}\quad \ell (t,\lambda ) &{}\quad m(t,\lambda )\\ -\ell (t,\lambda ) &{}\quad 0 &{}\quad n(t,\lambda ) \\ -m(t,\lambda ) &{}\quad -n(t,\lambda ) &{}\quad 0 \end{array}\right) \left( \begin{array}{cccc} \nu _{1}(t,\lambda )\\ \nu _{2}(t,\lambda )\\ \varvec{\mu }(t,\lambda ) \end{array}\right) ,\\ \left( \begin{array}{cccc} \nu _{1\lambda }(t,\lambda )\\ \nu _{2\lambda }(t,\lambda )\\ \varvec{\mu }_\lambda (t,\lambda ) \end{array}\right)= & {} \left( \begin{array}{ccc} 0 &{}\quad L(t,\lambda ) &{}\quad M(t,\lambda )\\ -L(t,\lambda ) &{}\quad 0 &{}\quad N(t,\lambda ) \\ -M(t,\lambda ) &{}\quad -N(t,\lambda ) &{}\quad 0 \end{array}\right) \left( \begin{array}{cccc} \nu _{1}(t,\lambda )\\ \nu _{2}(t,\lambda )\\ \varvec{\mu }(t,\lambda ) \end{array}\right) ,\\ \gamma _{t}(t,\lambda )= & {} \alpha (t,\lambda )\varvec{\mu }(t,\lambda ),~ \gamma _{\lambda }(t,\lambda )=P(t,\lambda )\nu _{1}(t,\lambda )\\&+Q(t,\lambda )\nu _{2}(t,\lambda )+R(t,\lambda )\varvec{\mu }(t,\lambda ). \end{aligned}
Then we have the integrability condition
\begin{aligned} \begin{aligned} L_{t}(t,\lambda )&=M(t,\lambda )n(t,\lambda )-N(t,\lambda )m(t,\lambda ) +\ell _{\lambda }(t,\lambda ),\\ M_{t}(t,\lambda )&=N(t,\lambda )\ell (t,\lambda )-L(t,\lambda )n(t,\lambda ) +m_{\lambda }(t,\lambda ),\\ N_{t}(t,\lambda )&=L(t,\lambda )m(t,\lambda )-M(t,\lambda )\ell (t,\lambda ) +n_{\lambda }(t,\lambda ),\\ P_{t}(t,\lambda )&=Q(t,\lambda )\ell (t,\lambda )+R(t,\lambda )m(t,\lambda ) -\alpha (t,\lambda ) M(t,\lambda ),\\ Q_{t}(t,\lambda )&=-P(t,\lambda )\ell (t,\lambda )+R(t,\lambda )n(t,\lambda ) -\alpha (t,\lambda ) N(t,\lambda ),\\ R_{t}(t,\lambda )&=-P(t,\lambda )m(t,\lambda )-Q(t,\lambda )n(t,\lambda ) +\alpha _{\lambda }(t,\lambda ) \end{aligned} \end{aligned}
(4.1)
for all $$(t,\lambda )\in \mathrm {I}\times \mathrm {\Lambda }$$. We call the mapping $$(\ell ,m,n,\alpha ,L,M,N,P,Q,R)$$ with the integrability condition (4.1) the curvature of the one-parameter family of framed curves $$(\gamma ,\nu _{1},\nu _{2})$$.
By using the above notations and Theorem 3.3, we have the following Corollary.
### Corollary 4.1
Let $$(\gamma , \nu _1,\nu _2) : I\times \Lambda \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ be a one-parameter family of framed curves with the curvature $$(\ell ,m,n,\alpha ,L,M,N,P,Q,R)$$ and let $$e : U \rightarrow I \times \Lambda$$ be a smooth curve satisfying the variability condition. Then the following are equivalent.
1. (1)
$$e : U \rightarrow I \times \Lambda$$ is a pre-envelope and $$E_\gamma$$ is an envelope of $$(\gamma , \nu _1, \nu _2)$$.
2. (2)
$$\gamma _{\lambda }(e(u)) \cdot \nu _1(e(u))=0$$ and $$\gamma _{\lambda }(e(u)) \cdot \nu _2(e(u))=0$$ for all $$u \in U$$.
3. (3)
$$P(e(u))=0$$ and $$Q(e(u))=0$$ for all $$u \in U$$.
We give two examples. One is an example that a one-parameter family of framed curves has an envelope and the other is that a one-parameter family of framed curves does not have an envelope.
### Example 4.2
Let $$(\gamma ,\nu _1,\nu _2): {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}^3 \times \Delta _2$$ be
\begin{aligned} \gamma (t,\lambda )= & {} \left( \frac{1}{2}t^{2}+\lambda , \frac{1}{3}t^{3},\frac{1}{4}t^{4}\right) , \nu _{1}(t,\lambda )=\frac{(-t,1,0)}{1+t^{2}}, \nu _{2}(t,\lambda )\\= & {} \frac{(-t^{2},-t^{3},1+t^{2})}{\sqrt{(1+t^{2})(1+t^{2}+t^{4})}}. \end{aligned}
Then $$(\gamma ,\nu _1,\nu _2)$$ is a one-parameter family of framed curves. Since $$\gamma _\lambda (t,\lambda )=(1,0,0)$$, we have $$\gamma _\lambda (t,\lambda ) \cdot \nu _1(t,\lambda )=-t/(1+t^2)$$ and $$\gamma _\lambda (t,\lambda ) \cdot \nu _2(t,\lambda )=-t^2/\sqrt{(1+t^2)(1+t^2+t^4)}$$. By Corollary 4.1, $$e:{\mathbb {R}}\rightarrow {\mathbb {R}}\times {\mathbb {R}}, e(u)=(0,u)$$ is a pre-envelope of $$(\gamma ,\nu _1,\nu _2)$$. Hence $$E(u)=\gamma \circ e(u)=(u,0,0)$$ is an envelope of $$(\gamma ,\nu _1,\nu _2)$$. For more general cases see in Example 6.1.
### Example 4.3
Let $$(\gamma ,\nu _1,\nu _2): {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}^3 \times \Delta _2$$ be
\begin{aligned} \gamma (t,\lambda )= & {} \left( \frac{1}{2}t^{2},\frac{1}{3}t^{3} +\lambda ,\frac{1}{4}t^{4}\right) , \nu _{1}(t,\lambda )=\frac{(-t,1,0)}{1+t^{2}}, \nu _{2}(t,\lambda )\\= & {} \frac{(-t^{2},-t^{3},1+t^{2})}{\sqrt{(1+t^{2})(1+t^{2}+t^{4})}}. \end{aligned}
Then $$(\gamma ,\nu _1,\nu _2)$$ is also a one-parameter family of framed curves. However, since $$\gamma _\lambda (t,\lambda )=(0,1,0)$$, we have $$\gamma _\lambda (t,\lambda ) \cdot \nu _1(t,\lambda )=1/(1+t^2)\ne 0$$ for all $$(t,\lambda )\in I\times \Lambda$$ and $$\gamma _\lambda (t,\lambda ) \cdot \nu _2(t,\lambda )=-t^3/\sqrt{(1+t^2)(1+t^2+t^4)}$$. By Corollary 4.1, $$(\gamma ,\nu _1,\nu _2)$$ does not have an envelope.
### 4.1 Rotated frame and reflected frame of a one-parameter family of framed curves
Let $$(\gamma , \nu _{1},\nu _{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ be a one-parameter family of framed curves with the curvature $$(\ell ,m,n,\alpha ,L, M,N,P,Q,R)$$. For the normal plane of $$\gamma (t,\lambda )$$, spanned by $$\nu _{1}(t,\lambda )$$ and $$\nu _{2}(t,\lambda )$$, there are other frames by rotations and reflections. We define $$(\overline{\nu }_{1}(t,\lambda ),\overline{\nu }_{2}(t,\lambda ))\in \Delta _{2}$$ by
\begin{aligned} \left( \begin{array}{c}\overline{\nu }_1(t,\lambda ) \\ \overline{\nu }_2(t,\lambda )\end{array}\right) =\left( \begin{array}{cc}\cos \theta (t,\lambda ) &{} -\sin \theta (t,\lambda ) \\ \sin \theta (t,\lambda ) &{} \cos \theta (t,\lambda )\end{array}\right) \left( \begin{array}{c}\nu _{1}(t,\lambda ) \\ \nu _{2}(t,\lambda )\end{array}\right) , \end{aligned}
where $$\theta : I \times \Lambda \rightarrow {\mathbb {R}}$$ is a smooth function. Then $$(\gamma , \overline{\nu }_{1},\overline{\nu }_{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ is also a one-parameter family of framed curves and $$\overline{\varvec{\mu }}(t,\lambda )=\overline{\nu }_{1}(t,\lambda ) \times \overline{\nu }_{2}(t,\lambda )=\nu _{1}(t,\lambda )\times \nu _{2}(t,\lambda ) =\varvec{\mu }(t,\lambda )$$. By a direct calculation, the curvature $$(\overline{\ell },\overline{m},\overline{n},\overline{\alpha }, \overline{L},\overline{M},\overline{N}, \overline{P},\overline{Q},\overline{R})$$ of $$(\gamma ,\overline{\nu }_{1},\overline{\nu }_{2})$$ is given by
\begin{aligned}&(\ell -\theta _{t},m\cos \theta -n\sin \theta ,m\sin \theta +n\cos \theta , \alpha ,L-\theta _{\lambda },M\cos \theta -N\sin \theta ,\\&\quad M\sin \theta +N\cos \theta ,P\cos \theta -Q\sin \theta ,P\sin \theta +Q\cos \theta ,R). \end{aligned}
We call the moving frame $$\{\overline{\nu }_{1}(t,\lambda ),\overline{\nu }_{2}(t,\lambda ),\varvec{\mu }(t,\lambda )\}$$ a rotated frame along $$\gamma (t,\lambda )$$ by $$\theta (t,\lambda )$$.
On the other hand, we define $$(\widetilde{\nu }_{1}(t,\lambda ),\widetilde{\nu }_{2}(t,\lambda ))\in \Delta _{2}$$ by
\begin{aligned} \left( \begin{array}{c}\widetilde{\nu }_1(t,\lambda ) \\ \widetilde{\nu }_2(t,\lambda )\end{array}\right)&=\left( \begin{array}{cc}1 &{}\quad 0 \\ 0&{}\quad -1\end{array}\right) \left( \begin{array}{cc}\cos \theta (t,\lambda ) &{}\quad -\sin \theta (t,\lambda ) \\ \sin \theta (t,\lambda ) &{}\quad \cos \theta (t,\lambda )\end{array}\right) \left( \begin{array}{c}\nu _{1}(t,\lambda ) \\ \nu _{2}(t,\lambda )\end{array}\right) , \end{aligned}
where $$\theta :I \times \Lambda \rightarrow {\mathbb {R}}$$ is a smooth function. Then $$(\gamma , \widetilde{\nu }_{1},\widetilde{\nu }_{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ is also a one-parameter family of framed curves and $$\widetilde{\varvec{\mu }}(t,\lambda )=\widetilde{\nu }_{1}(t,\lambda )\times \widetilde{\nu }_{2}(t,\lambda )=\nu _{2}(t,\lambda )\times \nu _{1}(t,\lambda ) =-\varvec{\mu }(t,\lambda )$$. By a direct calculation, the curvature $$(\widetilde{\ell },\widetilde{m},\widetilde{n},\widetilde{\alpha },\widetilde{L},\widetilde{M},\widetilde{N}, \widetilde{P},\widetilde{Q}, \widetilde{R})$$ of $$(\gamma ,\widetilde{\nu }_{1},\widetilde{\nu }_{2})$$ is given by
\begin{aligned}&(-\ell +\theta _{t},-m\cos \theta +n\sin \theta ,m\sin \theta +n\cos \theta ,-\alpha ,-L+\theta _{\lambda },-M\cos \theta +N\sin \theta ,\\&\quad M\sin \theta +N\cos \theta ,-P\cos \theta +Q\sin \theta ,P\sin \theta +Q\cos \theta ,-R). \end{aligned}
We call the moving frame $$\{\widetilde{\nu }_{1}(t,\lambda ),\widetilde{\nu }_{2}(t,\lambda ),-\varvec{\mu }(t,\lambda )\}$$ a reflected frame along $$\gamma (t,\lambda )$$ by $$\theta (t,\lambda )$$. By using Corollary 4.1, we have the following result.
### Proposition 4.4
Under the above notations, if $$e : U \rightarrow I\times \Lambda$$ is a pre-envelope of $$(\gamma , \nu _{1},\nu _{2})$$, then $$e : U \rightarrow I\times \Lambda$$ is also a pre-envelope of $$(\gamma ,\overline{\nu }_{1},\overline{\nu }_{2})$$ and $$(\gamma , \widetilde{\nu }_{1},\widetilde{\nu }_{2})$$.
### Proof
By Corollary 4.1, we have $$P(e(u))=Q(e(u))=0$$ for all $$u \in U$$. It follows that $$\overline{P}(e(u))=\overline{Q}(e(u))=0$$ and $$\widetilde{P}(e(u))=\widetilde{Q}(e(u))=0$$ for all $$u \in U$$. Therefore, we have the result. $$\square$$
It follows that the envelope is independent of rotated frames and reflected frames of the framed curves.
### 4.2 Parallel curves of one-parameter families of framed curves
The parallel curve of a framed curve is defined in [9]. We define a parallel curve of a one-parameter family of framed curves.
### Definition 4.5
Let $$(\gamma , \nu _{1},\nu _{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ be a one-parameter family of framed curves and $$\theta : I\times \Lambda \rightarrow {\mathbb {R}}$$ be a smooth function which satisfy $$\theta _{t}(t,\lambda )=\ell (t,\lambda )$$ and $$\theta _{\lambda }(t,\lambda )=L(t,\lambda )$$ for all $$(t,\lambda ) \in I \times \Lambda$$. Then we define a parallel curve $$\gamma _{(a,b)}:I \times \Lambda \rightarrow {\mathbb {R}}^{3}$$ of the one-parameter family of framed curves $$(\gamma ,\nu _1,\nu _2)$$ by
\begin{aligned} \gamma _{(a,b)}(t,\lambda )= & {} \gamma (t,\lambda )+(a\cos \theta (t,\lambda ) +b\sin \theta (t,\lambda ))\nu _{1}(t,\lambda )\\&+(-a\sin \theta (t,\lambda ) +b\cos \theta (t,\lambda ))\nu _{2}(t,\lambda ) \end{aligned}
where $$a,b\in {\mathbb {R}}$$.
### Remark 4.6
Since $$\theta _{t}(t,\lambda )=\ell (t,\lambda )$$ and $$\theta _{\lambda }(t,\lambda )=L(t,\lambda )$$, $$\ell _{\lambda }(t,\lambda )=L_{t}(t,\lambda )$$ holds for all $$(t,\lambda )\in I\times \Lambda$$. By the integrability condition (4.1), $$\ell _{\lambda }(t,\lambda )=L_{t}(t,\lambda )$$ is equivalent to $$M(t,\lambda )n(t,\lambda )-N(t,\lambda )m(t,\lambda )=0$$.
### Proposition 4.7
Let $$(\gamma , \nu _{1},\nu _{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ be a one-parameter family of framed curves with the curvature $$(\ell ,m,n,\alpha ,L,M,N,P,Q,R)$$ and $$\theta : I\times \Lambda \rightarrow {\mathbb {R}}$$ be a smooth function which satisfy $$\theta _{t}(t,\lambda )=\ell (t,\lambda )$$ and $$\theta _{\lambda }(t,\lambda )=L(t,\lambda )$$ for all $$(t,\lambda ) \in I \times \Lambda$$. Then $$(\gamma _{(a,b)}, \nu _{1},\nu _{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ is also a one-parameter family of framed curves with the curvature $$(\ell ,m,n,\alpha +m(a\cos \theta +b\sin \theta )+n(a\sin \theta -b\cos \theta ),L,M,N,P,Q,R+M(a\cos \theta +b\sin \theta )+N(-a\sin \theta +b\cos \theta ))$$. Moreover, if $$e : U \rightarrow I\times \Lambda$$ is a pre-envelope of $$(\gamma , \nu _{1},\nu _{2})$$, then $$e : U \rightarrow I\times \Lambda$$ is also a pre-envelope of $$(\gamma _{(a,b)},\nu _{1},\nu _{2})$$. We have $$E_{\gamma _{(a,b)}}(u)=(E_\gamma )_{(a,b)}(u)$$ for all $$u\in U$$, where $$E_{\gamma _{(a,b)}}=\gamma _{(a,b)}\circ e$$.
### Proof
Since
\begin{aligned} (\gamma _{(a,b)})_{t}(t,\lambda )= & {} \bigl (\alpha \varvec{\mu } +(-a\theta _{t}\sin \theta +b\theta _{t}\cos \theta )\nu _{1}\\&+(a\cos \theta +b\sin \theta )(\ell \nu _{2}+m\varvec{\mu })\\&+(-a\theta _{t}\cos \theta -b\theta _{t}\sin \theta )\nu _{2}\\&+(-a\sin \theta +b\cos \theta )(-\ell \nu _{1}+n\varvec{\mu })\bigr )(t,\lambda )\\= & {} \bigl ((\alpha +m(a\cos \theta +b\sin \theta )+n(-a\sin \theta +b\cos \theta ))\varvec{\mu }\bigr )(t,\lambda ),\\ (\gamma _{(a,b)})_{\lambda }(t,\lambda )= & {} \bigl (P\nu _{1}+Q\nu _{2} +R\varvec{\mu }+(-a\theta _{\lambda }\sin \theta \\&+b\theta _{\lambda }\cos \theta )\nu _{1} +(a\cos \theta +b\sin \theta )(L\nu _{2}+M\varvec{\mu })\\&+(-a\theta _{\lambda }\cos \theta -b\theta _{\lambda }\sin \theta )\nu _{2}\\&+(-a\sin \theta +b\cos \theta )(-L\nu _{1}+N\varvec{\mu })\bigr )(t,\lambda )\\= & {} \bigl (P\nu _{1}+Q\nu _{2}+(R+M(a\cos \theta +b\sin \theta )\\&+N(-a\sin \theta +b\cos \theta ))\varvec{\mu }\bigr )(t,\lambda ), \end{aligned}
we have $$(\gamma _{(a,b)})_{t}(t,\lambda )\cdot \nu _{1}(t,\lambda )=0$$ and $$(\gamma _{(a,b)})_{t}(t,\lambda )\cdot \nu _{2}(t,\lambda )=0$$ for all $$(t,\lambda ) \in I \times \Lambda$$. Hence $$(\gamma _{(a,b)}, \nu _{1},\nu _{2})$$ is a one-parameter family of framed curves. It follows that $$\gamma (t,\lambda )$$ and $$\gamma _{(a,b)}(t,\lambda )$$ have the same moving frame $$\{\nu _{1}(t,\lambda ),\nu _{2}(t,\lambda ),\varvec{\mu }(t,\lambda )\}$$. By a direct calculation, we have the curvature $$\bigl (\ell ,m,n,\alpha +m(a\cos \theta +b\sin \theta ) +n(-a\sin \theta +b\cos \theta ),L,M,N,P,Q,R+M(a\cos \theta +b\sin \theta )+N(-a\sin \theta +b\cos \theta )\bigr )$$ of $$(\gamma _{a,b},\nu _1,\nu _2)$$.
If $$e : U \rightarrow I\times \Lambda$$ is a pre-envelope of $$(\gamma ,\nu _{1},\nu _{2})$$, then $$P(e(u))=0$$ and $$Q(e(u))=0$$ for all $$u\in U$$. It follows that $$(\gamma _{(a,b)})_{\lambda }(e(u))\cdot \nu _{1}(e(u))=0$$ and $$(\gamma _{(a,b)})_{\lambda }(e(u))\cdot \nu _{2}(e(u))=0$$ for all $$u\in U$$. Thus $$e : U \rightarrow I\times \Lambda$$ is also a pre-envelope of $$(\gamma _{(a,b)},\nu _{1},\nu _{2})$$ by Corollary 4.1. Moreover, we have
\begin{aligned} E_{\gamma _{(a,b)}}(u)= & {} \gamma _{(a,b)}\circ e(u)\\= & {} \gamma (e(u))+\bigl (a\cos \theta (e(u)) +b\sin \theta (e(u))\bigr ) \nu _{1}(e(u))\\&+\bigl (-a\sin \theta (e(u)) +b\cos \theta (e(u))\bigr )\nu _{2}(e(u))\\= & {} E_{\gamma }(u)+\bigl (a\cos \theta (e(u))+b\sin \theta (e(u))\bigr ) E_{\nu _{1}}(u)\\&+\bigl (-a\sin \theta (e(u))+b\cos \theta (e(u))\bigr )E_{\nu _{2}}(u)\\= & {} (E_\gamma )_{(a,b)}(u) \end{aligned}
for all $$u\in U$$. $$\square$$
## 5 Relations among envelopes of one-parameter families of framed curves, Legendre curves and spherical Legendre curves
First, we give relations between one-parameter families of framed curves and one-parameter families of Legendre curves in the unit tangent bundle over the Euclidean plane. We review one-parameter families of Legendre curves. For more details see [14].
Let $$(\gamma ,\nu ) : I \times \Lambda \rightarrow \mathbb {R}^2 \times S^1$$ be a smooth mapping. We say that $$(\gamma , \nu )$$ is a one-parameter family of Legendre curves if $$\gamma _t(t,\lambda )\cdot \nu (t,\lambda )=0$$ for all $$(t,\lambda )\in I\times \Lambda$$.
We denote $$\varvec{J}(a) = (-a_2, a_1)$$ the anticlockwise rotation by $$\pi /2$$ of a vector $$\varvec{a} = (a_1,a_2)$$. We define $$\varvec{\mu }(t,\lambda ) = \varvec{J}(\nu (t,\lambda ))$$. Since $$\{\nu (t,\lambda ),\varvec{\mu }(t,\lambda )\}$$ is a moving frame along $$\gamma (t,\lambda )$$ on $$\mathbb {R}^2$$, we have the Frenet type formula.
\begin{aligned} \left( \begin{array}{cc} \nu _{t}(t,\lambda )\\ \varvec{\mu }_t(t,\lambda ) \end{array}\right)= & {} \left( \begin{array}{ccc} 0 &{}\ell (t,\lambda )\\ -\ell (t,\lambda ) &{}0 \end{array}\right) \left( \begin{array}{cccc} \nu (t,\lambda )\\ \varvec{\mu }(t,\lambda ) \end{array}\right) ,\\ \left( \begin{array}{cc} \nu _{\lambda }(t,\lambda )\\ \varvec{\mu }_\lambda (t,\lambda ) \end{array}\right)= & {} \left( \begin{array}{ccc} 0 &{}L(t,\lambda )\\ -L(t,\lambda ) &{}0 \end{array}\right) \left( \begin{array}{cccc} \nu (t,\lambda )\\ \varvec{\mu }(t,\lambda ) \end{array}\right) ,\\ \gamma _t(t,\lambda )= & {} \alpha (t,\lambda )\varvec{\mu }(t,\lambda ),~ \gamma _\lambda (t,\lambda )=P(t,\lambda )\nu (t,\lambda ) +Q(t,\lambda )\varvec{\mu }(t,\lambda ). \end{aligned}
By the integrability condition, we have
\begin{aligned} \ell _\lambda (t,\lambda )&=L_t(t,\lambda ),~P_{t}(t,\lambda ) =\ell (t,\lambda )Q(t,\lambda )-\alpha (t,\lambda ) L(t,\lambda ),~\alpha _{\lambda }(t,\lambda )\\&=Q_{t}(t,\lambda ) +\ell (t,\lambda )P(t,\lambda ) \end{aligned}
for all $$(t,\lambda )\in I\times \Lambda$$. We call the mapping $$(\ell ,\alpha ,L,P,Q)$$ with the integrability condition the curvature of the one-parameter family of Legendre curves $$(\gamma ,\nu )$$.
We consider two projections to the planes. Let $$(\gamma , \nu _{1},\nu _{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ be a one-parameter family of framed curves with the curvature $$(\ell ,m,n,\alpha ,L,M,N,P,Q,R)$$. For a fix point $$(t_{0},\lambda _{0})\in I\times \Lambda$$, we consider two projections from $${\mathbb {R}}^{3}$$ to the $$\nu _{1}(t_{0},\lambda _{0})$$ and $$\nu _{2}(t_{0},\lambda _{0})$$ directions, respectively. We consider the projection of $$\gamma$$ to the $$\nu _{1}(t_{0},\lambda _{0})$$ direction. We denote
\begin{aligned} \gamma _{\nu _{1}} : I \times \Lambda \rightarrow {\mathbb {R}}^{2} , (t,\lambda ) \mapsto (\gamma (t,\lambda )\cdot \nu _{2}(t_{0},\lambda _{0}), \gamma (t,\lambda )\cdot \varvec{\mu }(t_{0},\lambda _{0})). \end{aligned}
Then $$(\gamma _{\nu _{1}})_{t}(t,\lambda )=\alpha (t,\lambda )(\varvec{\mu }(t,\lambda ) \cdot \nu _{2}(t_{0},\lambda _{0}),\varvec{\mu }(t,\lambda )\cdot \varvec{\mu }(t_{0},\lambda _{0}))$$. There exist subintervals $$I_{1}$$ of I around $$t_0$$ and $$\Lambda _{1}$$ of $$\Lambda$$ around $$\lambda _{0}$$ such that $$(\varvec{\mu }(t,\lambda )\cdot \nu _{2}(t_{0},\lambda _{0}))^{2}+(\varvec{\mu } (t,\lambda )\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}\ne 0$$ for all $$(t,\lambda )\in I_{1} \times \Lambda _{1}$$.
### Proposition 5.1
Let $$(\gamma , \nu _{1},\nu _{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ be a one-parameter family of framed curves with the curvature $$(\ell ,m,n,\alpha ,L,M,N,P,Q,R)$$. Then $$(\gamma _{\nu _{1}},\nu _{\nu _{1}}) : I_{1} \times \Lambda _{1} \rightarrow \mathbb {R}^2\times S^1$$ is a one-parameter family of Legendre curves with the curvature $$(\ell _{\nu _{1}},\alpha _{\nu _{1}},L_{\nu _{1}}, P_{\nu _{1}},Q_{\nu _{1}})$$, where
\begin{aligned} \nu _{\nu _{1}}(t,\lambda )= & {} \frac{1}{\sqrt{(\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0}))^{2} +(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}} (\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}),\\&-\varvec{\mu }\cdot \nu _{2}(t_{0}, \lambda _{0}))(t,\lambda ),\\ \ell _{\nu _{1}}(t,\lambda )= & {} \frac{1}{(\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0}))^{2} +(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}\\&\bigl ( \ell ((\nu _{1}\cdot \nu _{2}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0}, \lambda _{0})) -(\nu _{1}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0})))\\&+n((\nu _{2}\cdot \nu _{2}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0}, \lambda _{0}))\\&-(\nu _{2}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \nu _{2}(t_{0}, \lambda _{0})))\bigr )(t,\lambda ),\\ \alpha _{\nu _{1}}(t,\lambda )= & {} \alpha \sqrt{(\varvec{\mu }\cdot \nu _{2}(t_{0}, \lambda _{0}))^{2}+(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}(t,\lambda ),\\ L_{\nu _{1}}(t,\lambda )= & {} \frac{1}{(\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0}))^{2} +(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}\\&\bigl ( L((\nu _{1}\cdot \nu _{2}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0}, \lambda _{0}))-(\nu _{1}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \nu _{2}(t_{0}, \lambda _{0})))\\&+N((\nu _{2}\cdot \nu _{2}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0}, \lambda _{0}))\\&-(\nu _{2}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \nu _{2}(t_{0}, \lambda _{0})))\bigr )(t,\lambda ),\\ P_{\nu _{1}}(t,\lambda )= & {} \frac{1}{\sqrt{(\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0}))^{2} +(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}}\\&\bigl ( P((\nu _{1}\cdot \nu _{2}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0})) -(\nu _{1}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0})))\\&+Q((\nu _{2}\cdot \nu _{2}(t_{0},\lambda _{0})) (\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))\\&-(\nu _{2}\cdot \varvec{\mu }(t_{0},\lambda _{0})) (\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0}))) \bigr )(t,\lambda ),\\ Q_{\nu _{1}}(t,\lambda )= & {} \frac{1}{\sqrt{(\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0}))^{2} +(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}}\\&\bigl (P((\nu _{1}\cdot \nu _{2}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \nu _{2}(t_{0}, \lambda _{0}))\\&+(\nu _{1}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0}, \lambda _{0})))\\&+Q((\nu _{2}\cdot \nu _{2}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0})) +(\nu _{2}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0})))\\&+R((\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0}))^{2}+(\varvec{\mu }\cdot \varvec{\mu }(t_{0}, \lambda _{0}))^{2})\bigr )(t,\lambda ). \end{aligned}
### Proof
We define a smooth map $$\nu _{\nu _{1}} : I_{1} \times \Lambda _{1} \rightarrow S^{1}$$ by
\begin{aligned} \nu _{\nu _{1}}(t,\lambda )= & {} \frac{1}{\sqrt{(\varvec{\mu }\cdot \nu _{2}(t_{0}, \lambda _{0}))^{2}+(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}} (\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}),\\&-\varvec{\mu }\cdot \nu _{2}(t_{0}, \lambda _{0}))(t,\lambda ). \end{aligned}
Then $$(\gamma _{\nu _{1}},\nu _{\nu _{1}}) : I_{1} \times \Lambda _{1} \rightarrow \mathbb {R}^2\times S^1$$ is a one-parameter family of Legendre curves. By definition, $$\varvec{\mu }_{\nu _{1}} : I_{1} \times \Lambda _{1} \rightarrow S^{1}$$ is given by
\begin{aligned}&\varvec{\mu }_{\nu _{1}}(t,\lambda )\\&\quad =J(\nu _{\nu _{1}}(t,\lambda ))=\frac{1}{\sqrt{(\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0}))^{2}+(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}}\\&\qquad \bigl (\varvec{\mu }\cdot \nu _{2}(t_{0},\lambda _{0}),\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0})\bigr )(t,\lambda ). \end{aligned}
By a direct calculation, we have the curvature $$(\ell _{\nu _{1}},\alpha _{\nu _{1}},L_{\nu _{1}}, P_{\nu _{1}},Q_{\nu _{1}})$$ of $$(\gamma _{\nu _{1}},\nu _{\nu _{1}})$$. $$\square$$
### Proposition 5.2
Under the same notations in Proposition 5.1, suppose that $$e : U \rightarrow I_{1}\times \Lambda _{1}$$ is a pre-envelope of $$(\gamma ,\nu _{1},\nu _{2})$$. Then $$e : U \rightarrow I_{1}\times \Lambda _{1}$$ is also a pre-envelope of $$(\gamma _{\nu _{1}},\nu _{\nu _{1}}) :I_{1} \times \Lambda _{1} \rightarrow {\mathbb {R}}^2 \times S^1$$. Moreover, we have $$E_{\gamma _{\nu _{1}}}(u)=(E_\gamma )_{\nu _{1}}(u)$$ for all $$u\in U$$, where $$E_{\gamma _{\nu _{1}}}(u)=\gamma _{\nu _{1}}\circ e(u)$$.
### Proof
Since $$(\gamma , \nu _{1},\nu _{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^{3}\times \Delta _{2}$$ is a one-parameter family of framed curves and $$e : U \rightarrow I_1\times \Lambda _1$$ is a pre-envelope of $$(\gamma ,\nu _{1},\nu _{2})$$, we have $$P(e(u))=Q(e(u))=0$$ for all $$u\in U$$ by Corollary 4.1. By Proposition 5.1, we have $$P_{\nu _{1}}(e(u))=0$$ for all $$u\in U$$. Therefore $$e : U \rightarrow I_{1}\times \Lambda _{1}$$ is a pre-envelope of $$(\gamma _{\nu _{1}},\nu _{\nu _{1}})$$ (cf. [14]). Moreover, we have $$E_{\gamma _{\nu _{1}}}(u)=\gamma _{\nu _{1}}\circ e(u)=\bigl (\gamma (e(u))\cdot \nu _{2}(t_{0},\lambda _{0}),\gamma (e(u)) \cdot \varvec{\mu }(t_{0},\lambda _{0})\bigr ) =\bigl (E_\gamma (u)\cdot \nu _{2}(t_{0},\lambda _{0}),E_\gamma (u) \cdot \varvec{\mu }(t_{0},\lambda _{0})\bigr )=(E_\gamma )_{\nu _{1}}(u)$$ for all $$u\in U$$. $$\square$$
We also consider the projection of $$\gamma$$ to the $$\nu _{2}(t_{0},\lambda _{0})$$ direction. We denote $$\gamma _{\nu _{2}} : I\times \Lambda \rightarrow {\mathbb {R}}^{2} ,(t,\lambda )\mapsto (\gamma (t,\lambda )\cdot \nu _{1}(t_{0},\lambda _{0}), \gamma (t,\lambda )\cdot \varvec{\mu }(t_{0},\lambda _{0})).$$ There exist subintervals $$I_{2}$$ of I around $$t_0$$ and $$\Lambda _{2}$$ of $$\Lambda$$ around $$\lambda _{0}$$ such that $$(\varvec{\mu }(t,\lambda )\cdot \nu _{1}(t_{0},\lambda _{0}))^{2} +(\varvec{\mu }(t,\lambda )\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}\ne 0$$ for all $$(t,\lambda )\in I_{2} \times \Lambda _{2}$$. We can prove the following similarly.
### Proposition 5.3
Let $$(\gamma , \nu _{1},\nu _{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ be a one-parameter family of framed curves with the curvature $$(\ell ,m,n,\alpha ,L,M,N,P,Q,R)$$. Then $$(\gamma _{\nu _{2}},\nu _{\nu _{2}}): I_{2} \times \Lambda _{2} \rightarrow \mathbb {R}^2\times S^1$$ is a one-parameter family of Legendre curves with the curvature $$(\ell _{\nu _{2}},\alpha _{\nu _{2}},L_{\nu _{2}},P_{\nu _{2}},Q_{\nu _{2}})$$, where
\begin{aligned} \nu _{\nu _{2}}(t,\lambda )= & {} \frac{1}{\sqrt{(\varvec{\mu }\cdot \nu _{1}(t_{0}, \lambda _{0}))^{2}+(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}}\\&\times (\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}),-\varvec{\mu }\cdot \nu _{1}(t_{0}, \lambda _{0}))(t,\lambda ),\\ \ell _{\nu _{2}}(t,\lambda )= & {} \frac{1}{(\varvec{\mu }\cdot \nu _{1}(t_{0}, \lambda _{0}))^{2}+(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}\\&\bigl ( \ell ((\nu _{1}\cdot \nu _{1}(t_{0},\lambda _{0}))(\varvec{\mu } \cdot \varvec{\mu }(t_{0},\lambda _{0}))-(\nu _{1}\cdot \varvec{\mu }(t_{0}, \lambda _{0}))(\varvec{\mu }\cdot \nu _{1}(t_{0},\lambda _{0})))\\&+n((\nu _{2}\cdot \nu _{1}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0}, \lambda _{0}))-(\nu _{2}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \nu _{1}(t_{0}, \lambda _{0})))\bigr )\\&\times (t,\lambda ),\\ \alpha _{\nu _{2}}(t,\lambda )= & {} \alpha \sqrt{(\varvec{\mu }\cdot \nu _{1}(t_{0}, \lambda _{0}))^{2}+(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}(t,\lambda ),\\ L_{\nu _{2}}(t,\lambda )= & {} \frac{1}{(\varvec{\mu }\cdot \nu _{1}(t_{0}, \lambda _{0}))^{2}+(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}\\&\bigl ( L((\nu _{1}\cdot \nu _{1}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0}, \lambda _{0}))-(\nu _{2}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \nu _{1}(t_{0}, \lambda _{0})))\\&+N((\nu _{2}\cdot \nu _{1}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0}, \lambda _{0}))-(\nu _{2}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu } \cdot \nu _{1}(t_{0},\lambda _{0})))\bigr )\\&\times (t,\lambda ),\\ P_{\nu _{2}}(t,\lambda )= & {} \frac{1}{\sqrt{(\varvec{\mu }\cdot \nu _{1}(t_{0}, \lambda _{0}))^{2}+(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}}\\&\bigl (P((\nu _{1}\cdot \nu _{1}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0})) -(\nu _{1}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \nu _{1}(t_{0},\lambda _{0})))\\&+Q((\nu _{2}\cdot \nu _{1}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))-(\nu _{2}\cdot \varvec{\mu }(t_{0}, \lambda _{0}))(\varvec{\mu }\cdot \nu _{1}(t_{0},\lambda _{0})))\bigr )\\&\times (t,\lambda ),\\ Q_{\nu _{2}}(t,\lambda )= & {} \frac{1}{\sqrt{(\varvec{\mu }\cdot \nu _{1}(t_{0},\lambda _{0}))^{2}+(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2}}}\\&\bigl (P((\nu _{1}\cdot \nu _{1}(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \nu _{1}(t_{0}, \lambda _{0}))+(\nu _{1}\cdot \varvec{\mu }(t_{0},\lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu } (t_{0},\lambda _{0})))\\&+Q((\nu _{2}\cdot \nu _{1}(t_{0},\lambda _{0}))(\varvec{\mu } \cdot \nu _{1}(t_{0},\lambda _{0}))+(\nu _{2}\cdot \varvec{\mu }(t_{0}, \lambda _{0}))(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0})))\\&+R((\varvec{\mu }\cdot \nu _{1}(t_{0},\lambda _{0}))^{2} +(\varvec{\mu }\cdot \varvec{\mu }(t_{0},\lambda _{0}))^{2})\bigr )(t,\lambda ). \end{aligned}
### Proposition 5.4
Under the same notations in Proposition 5.3, suppose that $$e : U \rightarrow I_{2}\times \Lambda _{2}$$ is a pre-envelope of $$(\gamma ,\nu _{1},\nu _{2})$$. Then $$e : U \rightarrow I_{2}\times \Lambda _{2}$$ is also a pre-envelope of $$(\gamma _{\nu _{2}},\nu _{\nu _{2}}):I_{2} \times \Lambda _{2} \rightarrow {\mathbb {R}}^2 \times S^1$$. Moreover, we have $$E_{\gamma _{\nu _{2}}}(u)=(E_\gamma )_{\nu _{2}}(u)$$ for all $$u\in U$$, where $$E_{\gamma _{\nu _{2}}}(u)=\gamma _{\nu _{2}}\circ e(u)$$.
Second, we give a relations between one-parameter families of framed curves and one-parameter families of spherical Legendre curves in the unit spherical bundle over the unit sphere. We review the one-parameter families of spherical Legendre curves. For more details see [11].
Let $$(\gamma , \nu ): I \times \Lambda \rightarrow \Delta _{2}\subset S^{2}\times S^{2}$$ be a smooth mapping. We say that $$(\gamma , \nu )$$ is a one-parameter family of spherical Legendre curves if $$\gamma _t(t, \lambda ) \cdot \nu (t, \lambda )=0$$ for all $$(t, \lambda )\in I \times \Lambda$$. We define $$\varvec{\mu }(t,\lambda )=\gamma (t,\lambda )\times \nu (t,\lambda )$$. Then $$\{\gamma (t,\lambda ), \nu (t,\lambda ),\varvec{\mu }(t,\lambda )\}$$ is a moving frame along $$\gamma (t,\lambda )$$ on $$S^2$$. We have the Frenet type formula.
\begin{aligned} \left( \begin{array}{cccc} \gamma _t(t,\lambda )\\ \nu _t(t,\lambda )\\ \varvec{\mu }_t(t,\lambda ) \end{array}\right)= & {} \left( \begin{array}{ccc} 0 &{}0 &{}m(t,\lambda )\\ 0 &{}0 &{}n(t,\lambda ) \\ -m(t,\lambda ) &{}-n(t,\lambda ) &{}0 \end{array}\right) \left( \begin{array}{cccc} \gamma (t,\lambda )\\ \nu (t,\lambda )\\ \varvec{\mu }(t,\lambda ) \end{array}\right) ,\\ \left( \begin{array}{cccc} \gamma _\lambda (t,\lambda )\\ \nu _\lambda (t,\lambda )\\ \varvec{\mu }_\lambda (t,\lambda ) \end{array}\right)= & {} \left( \begin{array}{ccc} 0 &{}\quad L(t,\lambda ) &{}\quad M(t,\lambda )\\ -L(t,\lambda ) &{}\quad 0 &{}\quad N(t,\lambda ) \\ -M(t,\lambda ) &{}\quad -N(t,\lambda ) &{}\quad 0 \end{array}\right) \left( \begin{array}{cccc} \gamma (t,\lambda )\\ \nu (t,\lambda )\\ \varvec{\mu }(t,\lambda ) \end{array}\right) . \end{aligned}
By the integrability condition, we have
\begin{aligned}&L_{t}(t,\lambda )=M(t,\lambda )n(t,\lambda )-N(t,\lambda )m(t,\lambda ),\\&m_{\lambda }(t,\lambda )=M_{t}(t,\lambda )+L(t,\lambda )n(t,\lambda ),\\&n_{\lambda }(t,\lambda )=N_{t}(t,\lambda )-L(t,\lambda )m(t,\lambda ). \end{aligned}
We call the mapping (mnLMN) with the integrability condition the curvature of the one-parameter family of spherical Legendre curves $$(\gamma ,\nu )$$. Let $$(\gamma ,\nu ): I \times \Lambda \rightarrow \Delta _{2}$$ be a one-parameter family of spherical Legendre curves with the curvature (mnLMN) and let $$e:U \rightarrow I \times \Lambda$$, $$e(u) = (t(u), \lambda (u))$$ be a smooth curve, where U is an interval of $${\mathbb {R}}$$. We denote $$E_\gamma =\gamma \circ e:U \rightarrow S^{2}$$ and $$E_\nu =\nu \circ e:U \rightarrow S^{2}$$.
We call $$E_\gamma$$ an envelope (and e a pre-envelope) for the one-parameter family of spherical Legendre curves $$(\gamma ,\nu )$$, when the following conditions are satisfied.
1. (i)
The function $$\lambda$$ is non-constant on any non-trivial subinterval of U. (The Variability Condition.)
2. (ii)
For all u, the curve $$E_\gamma$$ is tangent at u to the curve $$\gamma (t, \lambda )$$ at the parameter $$(t(u),\lambda (u))$$, meaning that the tangent vectors $$E_\gamma '(u) = (dE /du)(u)$$ and $$\varvec{\mu }(e(u))$$ are linearly dependent. (The Tangency Condition.)
In [13], the relations between framed curves and spherical Legendre curves are discussed. Here we discuss relations between one-parameter families of framed curves and one-parameter families of spherical Legendre curves, see Example 6.4.
### Proposition 5.5
Let $$(\gamma , \nu _{1},\nu _{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ be a one-parameter family of framed curves with the curvature $$(\ell ,m,n,\alpha ,L,M,N,P,Q,R)$$. Suppose that $$\gamma (t,\lambda )$$ is non-zero. We denote $$\widetilde{\gamma }(t,\lambda )=\gamma (t,\lambda )/|\gamma (t,\lambda )|$$ and $$\widetilde{\gamma }(t,\lambda )=a(t,\lambda )\nu _{1}(t,\lambda )+b(t,\lambda ) \nu _{2}(t,\lambda )+c(t,\lambda )\varvec{\mu }(t,\lambda )$$ with $$a^{2}(t,\lambda )+b^{2}(t,\lambda )+c^{2}(t,\lambda )=1$$. If $$a^{2}(t,\lambda )+b^{2}(t,\lambda )\ne 0$$ for all $$(t,\lambda )\in I\times \Lambda$$ and $$\widetilde{\nu }(t,\lambda )=(\widetilde{\gamma }\times \varvec{\mu }/ |\widetilde{\gamma }\times \varvec{\mu }|)(t,\lambda ),$$ then $$(\widetilde{\gamma },\widetilde{\nu }):I \times \Lambda \rightarrow \Delta _{2}$$ is a one-parameter family of spherical Legendre curves with the curvature
\begin{aligned} \widetilde{m}(t,\lambda )= & {} -\frac{am+bn+c_{t}}{\sqrt{a^{2}+b^{2}}} (t,\lambda ),\\ \widetilde{n}(t,\lambda )= & {} \frac{(a^{2}+b^{2})(an-bm+c\ell )+(ab_{t}-a_{t}b)c}{\sqrt{a^{2}+b^{2}}}(t,\lambda ),\\ \widetilde{L}(t,\lambda )= & {} \frac{-a(b_{\lambda }+aL-cN)+b(a_{\lambda }-bL-cM)}{\sqrt{a^{2}+b^{2}}}(t,\lambda ),\\ \widetilde{M}(t,\lambda )= & {} -\frac{aM+bN+c_{\lambda }}{\sqrt{a^{2}+b^{2}}} (t,\lambda ),\\ \widetilde{N}(t,\lambda )= & {} \frac{(a^{2}+b^{2})(aN-bM+cL) +(ab_{\lambda }-a_{\lambda }b)c}{\sqrt{a^{2}+b^{2}}}(t,\lambda ). \end{aligned}
### Proof
Since $$\widetilde{\gamma }(t,\lambda )=a(t,\lambda )\nu _{1}(t,\lambda ) +b(t,\lambda )\nu _{2}(t,\lambda )+c(t,\lambda )\varvec{\mu }(t,\lambda )$$ and $$\widetilde{\nu }(t,\lambda )=\left( (b\nu _{1}-a\nu _{2})/\sqrt{a^{2}+b^{2}}\right) (t,\lambda )$$, we have $$\widetilde{\gamma }(t,\lambda )\cdot \widetilde{\nu }(t,\lambda )=0$$. Moreover,
\begin{aligned} \widetilde{\gamma }_t(t,\lambda )\cdot \widetilde{\nu }(t,\lambda )= & {} \left( \frac{\gamma _{t}|\gamma |-\gamma |\gamma |_{t}}{|\gamma |^{2}}\cdot \widetilde{\nu }\right) (t,\lambda ) =\left( \frac{\gamma _{t}}{|\gamma |}\cdot \widetilde{\nu }-\frac{|\gamma |_{t}}{|\gamma |}\widetilde{\gamma } \cdot \widetilde{\nu }\right) (t,\lambda )\\= & {} \frac{\gamma _{t}\cdot (b\nu _{1}-a\nu _{2})}{|\gamma |\sqrt{a^{2}+b^{2}}}(t,\lambda )=0. \end{aligned}
Therefore $$(\widetilde{\gamma },\widetilde{\nu }):I \times \Lambda \rightarrow \Delta _{2}$$ is a one-parameter family of spherical Legendre curves. By a direct calculation, we have
\begin{aligned} \widetilde{\varvec{\mu }}(t,\lambda )=\widetilde{\gamma }(t,\lambda ) \times \widetilde{\nu }(t,\lambda )=\frac{1}{\sqrt{a^{2}+b^{2}}} (ac\nu _{1}+bc\nu _{2}-(a^{2}+b^{2})\varvec{\mu })(t,\lambda ). \end{aligned}
By using the Frenet type formula of a one-parameter family of spherical Legendre curves, we also have
\begin{aligned} \widetilde{\gamma }_{t}(t,\lambda )= & {} \bigl ((a_{t}-b\ell -cm) \nu _{1}+(b_{t}+a\ell -cn)\nu _{2}+(c_{t}+am+bn)\varvec{\mu }\bigr )(t,\lambda ),\\ \widetilde{\gamma }_{\lambda }(t,\lambda )= & {} \bigl ((a_{\lambda }-bL-cM) \nu _{1}+(b_{\lambda }+aL-cN)\nu _{2}+(c_{\lambda }+aM+bN)\varvec{\mu }\bigr )(t,\lambda ),\\ \widetilde{\nu }_{t}(t,\lambda )= & {} \frac{1}{(a^{2}+b^{2})^{\frac{3}{2}}} \bigl ((b_{t}a^{2}+a(a^{2}+b^{2})\ell -a_{t}ab)\nu _{1}\\&+(-a_{t}b^{2}+b(a^{2}+b^{2})\ell +b_{t}ab)\nu _{2}+(a^{2}+b^{2})(-an+bm) \varvec{\mu }\bigr )(t,\lambda ),\\ \widetilde{\nu }_{\lambda }(t,\lambda )= & {} \frac{1}{(a^{2}+b^{2})^{\frac{3}{2}}} \bigl ((b_{\lambda }a^{2}+a(a^{2}+b^{2})L-a_{\lambda }ab)\nu _{1}\\&+(-a_{\lambda }b^{2}+b(a^{2}+b^{2})L+b_{\lambda }ab)\nu _{2}+(a^{2}+b^{2})(-aN+bM) \varvec{\mu }\bigr )(t,\lambda ). \end{aligned}
By a direct calculation, we have the curvature $$(\widetilde{m},\widetilde{n},\widetilde{L},\widetilde{M},\widetilde{N})$$ of $$(\widetilde{\gamma },\widetilde{\nu })$$. $$\square$$
### Proposition 5.6
Under the same assumptions in Proposition 5.5, suppose that $$e : U \rightarrow I\times \Lambda$$ is a pre-envelope of $$(\gamma ,\nu _{1},\nu _{2})$$ and $$E_\gamma :U \rightarrow {\mathbb {R}}^3$$ is an envelope. Then $$e : U \rightarrow I\times \Lambda$$ is also a pre-envelope of $$(\widetilde{\gamma }, \widetilde{\nu }):I \times \Lambda \rightarrow \Delta _{2}$$. Moreover, we have $$E_{\widetilde{\gamma }}(u)=\widetilde{E}_\gamma$$(u) for all $$u\in U$$, where $$E_{\widetilde{\gamma }}(u)=\widetilde{\gamma }\circ e(u)$$ and $$\widetilde{E}_\gamma (u)=E_\gamma (u)/|E_\gamma (u)|$$.
### Proof
Since $$(\gamma , \nu _{1},\nu _{2}) : I \times \Lambda \rightarrow {\mathbb {R}}^3 \times \Delta _{2}$$ is a one-parameter family of framed curves and $$e : U \rightarrow I\times \Lambda$$ is a pre-envelope of $$(\gamma ,\nu _{1},\nu _{2})$$, we have $$\gamma _\lambda (e(u))\cdot \nu _{1}(e(u))=0$$ and $$\gamma _\lambda (e(u))\cdot \nu _{2}(e(u))=0$$ for all $$u\in U$$. It follows that
\begin{aligned} \widetilde{\gamma }_\lambda (e(u))\cdot \widetilde{\nu }(e(u))= & {} \left( \frac{\gamma _{\lambda }|\gamma |-\gamma |\gamma |_{\lambda }}{|\gamma |^{2}}\cdot \widetilde{\nu }\right) \circ (e(u)) =\left( \frac{\gamma _{\lambda }}{|\gamma |}\cdot \widetilde{\nu } -\frac{|\gamma |_{\lambda }}{|\gamma |}\widetilde{\gamma } \cdot \widetilde{\nu }\right) \circ (e(u))\\= & {} \frac{\gamma _{\lambda }\cdot (b\nu _{1}-a\nu _{2})}{|\gamma |\sqrt{a^{2}+b^{2}}}\circ (e(u))=0. \end{aligned}
Therefore $$e : U \rightarrow I\times \Lambda$$ is also a pre-envelope of $$(\widetilde{\gamma }, \widetilde{\nu })$$. Moreover, $$E_{\widetilde{\gamma }}(u)=\widetilde{\gamma }( e(u))=\widetilde{\gamma (e(u))}=\widetilde{E}_\gamma (u).$$ Thus, we have $$E_{\widetilde{\gamma }}(u)=\widetilde{E}_\gamma$$(u) for all $$u\in U$$. $$\square$$
Conversely, we have the following result.
### Proposition 5.7
Let $$(\gamma , \nu ):I \times \Lambda \rightarrow \Delta _{2}$$ be a one-parameter family of spherical Legendre curves with the curvature (mnLMN). Then $$(\gamma ,\gamma ,\nu ) : I\times \Lambda \rightarrow S^{2}\times \Delta _{2} \subset {\mathbb {R}}^{3}\times \Delta _{2}$$ is a one-parameter family of framed curves with the curvature $$(\ell ,m,n,\alpha ,L,M,N,P,Q,R)=(0,m,n,m,L,M,N,0,L,M)$$.
### Proof
Since $$(\gamma , \nu ):I \times \Lambda \rightarrow \Delta _{2}$$ is a one-parameter family of spherical Legendre curves, we have $$\gamma (t,\lambda )\cdot \nu (t,\lambda )=0$$ and $$\gamma _{t}(t,\lambda )\cdot \nu (t,\lambda )=0$$ for all $$(t,\lambda ) \in I \times \Lambda$$. Therefore, $$(\gamma ,\gamma ,\nu ) : I\times \Lambda \rightarrow S^{2}\times \Delta _{2} \subset {\mathbb {R}}^{3}\times \Delta _{2}$$ is a one-parameter family of framed curves. By a direct calculation, we have the curvature of $$(\gamma ,\gamma ,\nu )$$. $$\square$$
### Proposition 5.8
Under the same notations in Proposition 5.7, suppose that $$e : U \rightarrow I\times \Lambda$$ is a pre-envelope of $$(\gamma ,\nu )$$ and $$E_\gamma :U \rightarrow S^2$$ is an envelope. Then $$e : U \rightarrow I\times \Lambda$$ is also a pre-envelope of $$(\gamma , \gamma ,\nu ):I \times \Lambda \rightarrow S^2 \times \Delta _{2} \subset {\mathbb {R}}^3 \times \Delta _{2}$$. Moreover, $$E_{\gamma }$$ is also an envelope of $$(\gamma ,\gamma ,\nu )$$.
### Proof
Since $$\gamma _\lambda (e(u))\cdot \nu (e(u))=0$$ for all $$u\in U$$, $$e : U \rightarrow I \times \Lambda$$ is also a pre-envelope of $$(\gamma ,\gamma ,\nu )$$. Therefore, $$E_\gamma$$ is also an envelope of $$(\gamma ,\gamma ,\nu )$$. $$\square$$
## 6 Examples
### Example 6.1
Let $$n_{1},n_{2},n_{3},k_{1}$$ and $$k_{2}$$ be natural numbers with $$n_{2}=n_{1}+k_{1}$$ and $$n_{3}=n_{2}+k_{2}.$$ Let $$(\gamma ,\nu _{1},\nu _{2}): {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ be
\begin{aligned} \gamma (t,\lambda )= & {} \left( \frac{1}{n_{1}}t^{n_{1}}+\lambda ,\frac{1}{n_{2}}t^{n_{2}}, \frac{1}{n_{3}}t^{n_{3}}\right) ,~ \nu _{1}(t,\lambda )=\frac{(-t^{k_{1}},1,0)}{\sqrt{1+t^{2k_{1}}}},~ \nu _{2}(t,\lambda )\\= & {} \frac{(-t^{k_{1}+k_{2}},-t^{2k_{1}+k_{2}},1 +t^{2k_{1}})}{\sqrt{(1+t^{2k_{1}})(1+t^{2k_{1}+2k_{2}})}}. \end{aligned}
Then $$(\gamma ,\nu _{1},\nu _{2})$$ is a one-parameter family of framed curves. By a direct calculation, we have
\begin{aligned} \varvec{\mu }(t,\lambda )=\frac{1}{\sqrt{1+t^{2k_{1}}+t^{2k_{1} +2k_{2}}}}(1,t^{k_{1}},t^{k_{1}+k_{2}}). \end{aligned}
Then the curvature is given by
\begin{aligned}&\ell (t,\lambda )=\frac{k_{1}t^{2k_{1}+k_{2}-1}}{(1+t^{2k_{1}}) \sqrt{1+t^{2k_{1}}+t^{2k_{1}+2k_{2}}}},\\&m(t,\lambda ) =\frac{-k_{1}t^{k_{1}-1}}{\sqrt{(1+t^{2k_{1}})(1+t^{2k_{1}}+t^{2k_{1}+2k_{2}})}},\\&n(t,\lambda )=-\frac{t^{k_{1}+k_{2}-1}(k_{1}+k_{2}+k_{2}t^{2k_{1}})}{\sqrt{1+t^{2k_{1}}}(1+t^{2k_{1}}+t^{2k_{1}+2k_{2}})},\\&\alpha (t,\lambda ) =t^{n_{1}-1}\sqrt{1+t^{2k_{1}}+t^{2k_{1}+2k_{2}}},\\&L(t,\lambda )=M(t,\lambda )=N(t,\lambda )=0,\\&P(t,\lambda )=-\frac{t^{k_{1}}}{1+t^{2k_{1}}},~Q(t,\lambda ) =-\frac{t^{k_{1}+k_{2}}}{\sqrt{(1+t^{2k_{1}})(1+t^{2k_{1}+2k_{2}})}},\\&R(t,\lambda )=\frac{1}{\sqrt{1+t^{2k_{1}}+t^{2k_{1}+2k_{2}}}}. \end{aligned}
If we take $$e: {\mathbb {R}}\rightarrow {\mathbb {R}}\times {\mathbb {R}}$$, $$e(u)=(0,u)$$, then the variability condition holds and we have $$P(e(u))=Q(e(u))=0$$ for all $$u\in {\mathbb {R}}$$. Therefore e is a pre-envelope and an envelope $$E_\gamma :{\mathbb {R}}\rightarrow {\mathbb {R}}^3$$ is given by $$E_\gamma (u)=(u,0,0)$$.
If we take $$(n_{1},n_{2},n_{3})=(2,3,4)$$, we have a one-parameter family of framed curves
\begin{aligned} \gamma (t,\lambda )= & {} \left( \frac{1}{2}t^{2}+\lambda ,\frac{1}{3}t^{3}, \frac{1}{4}t^{4}\right) ,~ \nu _{1}(t,\lambda )=\frac{(-t,1,0)}{1+t^{2}},\\ \nu _{2}(t,\lambda )= & {} \frac{(-t^{2},-t^{3},1+t^{2})}{\sqrt{(1+t^{2})(1+t^{2}+t^{4})}}. \end{aligned}
See Figs. 12 and 3.
### Example 6.2
Let $$n_{1},n_{2},n_{3},m_{1},m_{2},m_{3}$$,$$k_{1},k_{2},h_{1}$$ and $$h_{2}$$ be natural numbers with $$n_{2}=n_{1}+k_{1},~n_{3}=n_{2}+k_{2}$$, $$m_{2}=m_{1}+h_{1},~m_{3}=m_{2}+h_{2}$$, $$h_{1}k_{2}=h_{2}k_{1}$$ and $$h_{1}$$ or $$k_{1}$$ is 1, or $$h_{1}$$ and $$k_{1}$$ are relatively prime. Let $$(\gamma ,\nu _{1},\nu _{2}): {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ be
\begin{aligned} \gamma (t,\lambda )= & {} \left( \frac{t^{n_{1}}}{n_{1}} +\frac{\lambda ^{m_{1}}}{m_{1}},\frac{t^{n_{2}}}{n_{2}} +\frac{\lambda ^{m_{2}}}{m_{2}},\frac{t^{n_{3}}}{n_{3}} +\frac{\lambda ^{m_{3}}}{m_{3}}\right) , \nu _{1}(t,\lambda )=\frac{(-t^{k_{1}},1,0)}{\sqrt{1+t^{2k_{1}}}}, \nu _{2}(t,\lambda )\\= & {} \frac{(-t^{k_{1}+k_{2}},-t^{2k_{1}+k_{2}},1 +t^{2k_{1}})}{\sqrt{(1+t^{2k_{1}})(1+t^{2k_{1}+2k_{2}})}}. \end{aligned}
Then $$(\gamma ,\nu _{1},\nu _{2})$$ is a one-parameter family of framed curves. Moreover, by $$\gamma _{\lambda }(t,\lambda )=(\lambda ^{m_{1}-1},\lambda ^{m_{2}-1},\lambda ^{m_{3}-1})$$,
\begin{aligned} \gamma _{\lambda }(t,\lambda )\cdot \nu _{1}(t,\lambda )= & {} \frac{\lambda ^{m_{1}-1}}{\sqrt{1+t^{2k_{1}}}}(-t^{k_{1}}+\lambda ^{h_{1}}),\\ \gamma _{\lambda }(t,\lambda )\cdot \nu _{2}(t,\lambda )= & {} \frac{\lambda ^{m_{1}-1}}{\sqrt{(1+t^{2k_{1}}) (1+t^{2k_{1}+2k_{2}})}}\\&\times \bigl (-t^{k_{1}+k_{2}}-t^{2k_{1}+k_{2}}\lambda ^{h_{1}}+ (1+t^{2k_{1}})\lambda ^{h_{1}+h_{2}}\bigr ). \end{aligned}
If we take $$e: {\mathbb {R}}\rightarrow {\mathbb {R}}\times {\mathbb {R}}$$, $$e(u)=(u^{h_{1}},u^{k_{1}})$$, then the variability condition holds, $$\gamma _{\lambda }(e(u))\cdot \nu _{1}(e(u))=0$$ and $$\gamma _{\lambda }(e(u))\cdot \nu _{2}(e(u))=0$$ for all $$u\in {\mathbb {R}}$$. Therefore e is a pre-envelope and an envelope $$E_\gamma :{\mathbb {R}}\rightarrow {\mathbb {R}}^3$$ is given by
\begin{aligned} E_{\gamma }(u)=\left( \frac{u^{h_{1}n_{1}}}{n_{1}} +\frac{u^{k_{1}m_{1}}}{m_{1}},\frac{u^{h_{1}n_{2}}}{n_{2}} +\frac{u^{k_{1}m_{2}}}{m_{2}},\frac{u^{h_{1}n_{3}}}{n_{3}} +\frac{u^{k_{1}m_{3}}}{m_{3}}\right) . \end{aligned}
If we take $$(n_{1},n_{2},n_{3},m_{1},m_{2},m_{3})=(2,3,4,2,3,4)$$, then $$\gamma (t,\lambda )=(t^{2}/2+\lambda ^{2}/2,t^{3}/3+\lambda ^{3}/3,t^{4}/4+\lambda ^{4}/4)$$ and an envelope $$E_{\gamma }(u)=(u^{2},2u^{3}/3,u^{4}/2),$$ see Figs. 45 and 6.
### Example 6.3
Let $$(\varvec{p},\nu _{p1},\nu _{p2}):[0,2\pi )\rightarrow {\mathbb {R}}^{3}\times \Delta _{2}$$ be an astroid $$\varvec{p}(t)=(\cos ^{3}t-1,\sin ^{3}t,\cos 2t-1)$$, $$\nu _{p1}(t)=(-\sin t,-\cos t,0),\nu _{p2}(t)=1/5(-4\cos t,4\sin t,3)$$ and $$(\varvec{q},\nu _{q1},\nu _{q2}):[0,2\pi )\rightarrow {\mathbb {R}}^{3}\times \Delta _{2}$$ be given by $$\varvec{q}(\lambda )=(\cos ^{3}\lambda ,-\sin ^{3}\lambda ,\cos 2\lambda )$$, $$\nu _{q1}(t)=(\sin \lambda ,-\cos \lambda ,0), \nu _{q2}(t)=1/5(-4\cos \lambda ,-4\sin \lambda ,3)$$. Then $$(\varvec{p},\nu _{p1},\nu _{p2})$$ and $$(\varvec{q},\nu _{q1},\nu _{q2})$$ are framed curves.
We consider a one-parameter family of framed curves $$(\gamma ,\nu _1,\nu _2):[0,2\pi )\times [0,2\pi )\rightarrow {\mathbb {R}}^{3}\times \Delta _{2}$$,
\begin{aligned} \gamma (t,\lambda )=\varvec{q}(\lambda )+A(\lambda )\varvec{p}(t),~\nu _{1}(t,\lambda )=A(\lambda )\nu _{p1}(t),~\nu _{2}(t,\lambda )=A(\lambda )\nu _{p2}(t), \end{aligned}
where
\begin{aligned} A(\lambda )= \left( \begin{array}{ccc} \cos \lambda &{}\quad -\sin \lambda &{}\quad 0\\ \sin \lambda &{}\quad \cos \lambda &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 \end{array}\right) , \end{aligned}
that is
\begin{aligned} \gamma (t,\lambda )= & {} \left( \begin{array}{ccc} \cos ^{3}\lambda \\ -\sin ^{3}\lambda \\ \cos 2\lambda \end{array}\right) +\left( \begin{array}{ccc} \cos \lambda &{}\quad -\sin \lambda &{}\quad 0\\ \sin \lambda &{}\quad \cos \lambda &{}\quad 0\\ 0&{}\quad 0&{}\quad 1 \end{array}\right) \left( \begin{array}{ccc} \cos ^{3}t-1\\ \sin ^{3}t\\ \cos 2t-1 \end{array}\right) ,\\ \nu _{1}(t,\lambda )= & {} \left( \begin{array}{ccc} \cos \lambda &{}\quad -\sin \lambda &{}\quad 0\\ \sin \lambda &{}\quad \cos \lambda &{}\quad 0\\ 0&{}\quad 0&{}\quad 1 \end{array}\right) \left( \begin{array}{ccc} -\sin t\\ -\cos t\\ 0 \end{array}\right) ,\\ \nu _{2}(t,\lambda )= & {} 1/5\left( \begin{array}{ccc} \cos \lambda &{}\quad -\sin \lambda &{}\quad 0\\ \sin \lambda &{}\quad \cos \lambda &{}\quad 0\\ 0&{}\quad 0&{}\quad 1 \end{array}\right) \left( \begin{array}{ccc} -4\cos t\\ 4\sin t\\ 3 \end{array}\right) . \end{aligned}
By a direct calculation, we have
\begin{aligned} \gamma _{\lambda }(t,\lambda )\cdot \nu _{1}(t,\lambda )= & {} \sin t\bigl (15\sin \lambda \cos \lambda -2\cos ^{3}t-\cos t+3\bigr ),\\ \gamma _{\lambda }(t,\lambda )\cdot \nu _{2}(t,\lambda )= & {} 4(\cos t-1)(45\sin \lambda \cos \lambda -2\cos t-1). \end{aligned}
If we take $$e:[0,2\pi )\rightarrow [0,2\pi )\times [0,2\pi ),e(u)=(0,u)$$, then $$\gamma _{\lambda }(e(u))\cdot \nu _{1}(e(u))=0$$ and $$\gamma _{\lambda }(e(u))\cdot \nu _{2}(e(u))=0$$ for all $$u\in [0,2\pi )$$. Therefore e is a pre-envelope and an envelope $$E_{\gamma }:[0,2\pi )\rightarrow {\mathbb {R}}^{3}$$ is given by $$E_{\gamma }(u)=\varvec{q}(u)=(\cos ^{3}u,-\sin ^{3}u,\cos 2u),$$ see Figs. 7, 8 and 9.
### Example 6.4
Let $$(\gamma ,\nu _{1},\nu _{2}): {\mathbb {R}}\times [0,2\pi ) \rightarrow {\mathbb {R}}^{3} \times \Delta _{2}$$ be
\begin{aligned} \gamma (t,\theta )= & {} \bigl (\cos \theta (t\sin t+\cos t) -\sin \theta (-t\cos t+\sin t)+\theta \sin \theta +\cos \theta ,\\&\sin \theta (t\sin t+\cos t)+\cos \theta (-t\cos t+\sin t) -\theta \cos \theta +\sin \theta , \frac{1}{2}t^{2}+\frac{1}{2}\theta ^{2}\bigr ),\\ \nu _{1}(t,\theta )= & {} \bigl (\cos \theta \sin t+\sin \theta \cos t, \sin \theta \sin t-\cos \theta \cos t,0\bigr ),\\ \nu _{2}(t,\theta )= & {} \frac{1}{\sqrt{2}}\bigl (\cos \theta \cos t-\sin \theta \sin t,\sin \theta \cos t+\cos \theta \sin t,-1\bigr ). \end{aligned}
Then $$(\gamma ,\nu _{1},\nu _{2})$$ is a one-parameter family of framed curves. By a direct calculation, we have
\begin{aligned} \gamma _{\theta }(t,\theta )\cdot \nu _{1}(t,\theta )= & {} \sin t\bigl (t\cos t-\sin t +\theta \bigr ),\\ \gamma _{\theta }(t,\theta )\cdot \nu _{2}(t,\theta )= & {} \frac{1}{\sqrt{2}}\bigl (t\cos ^{2} t-\sin t\cos t+\theta \cos t-\theta \bigr ). \end{aligned}
If we take $$e:{\mathbb {R}}\rightarrow {\mathbb {R}}\times [0,2\pi ),e(u)=(0,u)$$, we have $$\gamma _{\theta }(e(u))\cdot \nu _{1}(e(u))=0$$ and $$\gamma _{\theta }(e(u))\cdot \nu _{2}(e(u))=0$$ for all $$u\in {\mathbb {R}}$$. Therefore e is a pre-envelope and an envelope $$E_{\gamma }:{\mathbb {R}}\rightarrow {\mathbb {R}}^{3}$$ is given by
\begin{aligned} E_{\gamma }(u)=(u\sin u+2\cos u,-u\cos u+2\sin u,\frac{1}{2}u^{2}), \end{aligned}
see Figs. 10, 11 and 12.
Since $$|\gamma (t,\theta )|\ne 0$$, for all $$(t,\theta )\in {\mathbb {R}}\times [0,2\pi )$$, we can consider
\begin{aligned} \widetilde{\gamma }(t,\theta )= & {} \frac{\gamma (t,\theta )}{|\gamma (t,\theta )|}\\= & {} \bigl (\frac{\cos \theta (t\sin t+\cos t)-\sin \theta (-t\cos t+\sin t)+\theta \sin \theta +\cos \theta }{\sqrt{\frac{1}{4}t^{4}+\frac{1}{4}\theta ^{4}+t^{2}+\theta ^{2} +\frac{1}{2}t^{2}\theta ^{2}+2(t\sin t+\cos t)-2\theta (-t\cos t+\sin t)+2}},\\&\times \frac{\sin \theta (t\sin t+\cos t)+\cos \theta (-t\cos t+\sin t)-\theta \cos \theta +\sin \theta }{\sqrt{\frac{1}{4}t^{4}+\frac{1}{4}\theta ^{4}+t^{2}+\theta ^{2} +\frac{1}{2}t^{2}\theta ^{2}+2(t\sin t+\cos t)-2\theta (-t\cos t+\sin t)+2}},\\&\times \frac{t^{2}+\theta ^{2}}{2\sqrt{\frac{1}{4}t^{4}+\frac{1}{4}\theta ^{4}+t^{2} +\theta ^{2}+\frac{1}{2}t^{2}\theta ^{2}+2(t\sin t+\cos t)-2\theta (-t\cos t+\sin t)+2}}\bigr ). \end{aligned}
Moreover, we have $$\widetilde{\gamma }(t,\theta )=a(t,\theta )\nu _{1}(t,\theta )+b(t,\theta )\nu _{2} (t,\theta )+c(t,\theta )\varvec{\mu }(t,\theta )$$, where
\begin{aligned} a(t,\theta )= & {} \frac{t+\theta \cos t+\sin t}{\sqrt{\frac{1}{4}t^{4} +\frac{1}{4}\theta ^{4}+t^{2}+\theta ^{2}+\frac{1}{2}t^{2}\theta ^{2} +2(t\sin t+\cos t)-2\theta (-t\cos t+\sin t)+2}},\\ b(t,\theta )= & {} \frac{1-\theta \sin t+\cos t-\frac{1}{2}t^{2} -\frac{1}{2}\theta ^{2}}{\sqrt{2(\frac{1}{4}t^{4}+\frac{1}{4}\theta ^{4} +t^{2}+\theta ^{2}+\frac{1}{2}t^{2}\theta ^{2}+2(t\sin t+\cos t) -2\theta (-t\cos t+\sin t)+2)}},\\ c(t,\theta )= & {} \frac{1-\theta \sin t+\cos t+\frac{1}{2}t^{2}+\frac{1}{2}\theta ^{2}}{\sqrt{2(\frac{1}{4}t^{4} +\frac{1}{4}\theta ^{4}+t^{2}+\theta ^{2}+\frac{1}{2}t^{2}\theta ^{2}+2(t\sin t+\cos t)-2\theta (-t\cos t+\sin t)+2)}}. \end{aligned}
Then $$a^{2}(t,\theta )+b^{2}(t,\theta )\ne 0$$ for all $$(t,\theta )\in {\mathbb {R}}\times [0,2\pi )$$ and we have $$\widetilde{\nu }(t,\theta )=(\widetilde{\gamma }\times \varvec{\mu }/ |\widetilde{\gamma }\times \varvec{\mu }|)(t,\theta ).$$ By Proposition 5.5, $$(\widetilde{\gamma },\widetilde{\nu }):{\mathbb {R}}\times [0,2\pi ) \rightarrow \Delta _{2}$$ is a one-parameter family of spherical Legendre curves. By Proposition 5.6, $$e:{\mathbb {R}}\rightarrow {\mathbb {R}}\times [0,2\pi ),e(u)=(0,u)$$ is also a pre-envelope of $$(\widetilde{\gamma }, \widetilde{\nu })$$. By a direct calculation, we have
\begin{aligned} E_{\widetilde{\gamma }}(u)=\widetilde{E}_\gamma (u) =\left( \frac{u\sin u+2\cos u}{\sqrt{\frac{1}{4}u^{4}+u^{2}+4}},\frac{-u\cos u+2\sin u}{\sqrt{\frac{1}{4}u^{4}+u^{2}+4}},\frac{u^{2}}{2\sqrt{\frac{1}{4}u^{4} +u^{2}+4}}\right) , \end{aligned}
see Figs. 13, 14 and 15.
## Notes
### Acknowledgements
The authors would like to thank the referee for helpful comments to improve the original manuscript. The first author was supported by the National Natural Science Foundation of China (Grant No. 11671070) and the second author was supported by the JSPS KAKENHI (Grant No. JP 17K05238).
### Conflict of interest
The authors declare that there is no conflict of interest in this work.
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## Authors and Affiliations
• Donghe Pei
• 1
• Masatomo Takahashi
• 2
• Haiou Yu
• 3
1. 1.Northeast Normal UniversityChangchunChina
2. 2.Muroran Institute of TechnologyMuroranJapan
3. 3.Jilin University of Finance and EconomicsChangchunChina | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9984086155891418, "perplexity": 5681.938075001372}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540548544.83/warc/CC-MAIN-20191213043650-20191213071650-00310.warc.gz"} |
http://www.koreascience.or.kr/article/JAKO201318449151664.page?lang=ko | Efficient Approach to Measure Crystallization Temperature in Amorphous Thin Film by Infrared Reflectivity
• Wang, Wenxiu (Department of Electronic Engineering, Graduate School of Engineering, Tohoku University) ;
• Saito, Shin (Department of Electronic Engineering, Graduate School of Engineering, Tohoku University) ;
• Yakabe, Hidetaka (Metallurgical Research Laboratory, Hitachi Metals, Ltd.) ;
• Takahashi, Migaku (New Industry Creation Hatchery Center, Tohoku University)
• 투고 : 2012.10.18
• 심사 : 2013.05.09
• 발행 : 2013.06.30
• 61 9
초록
This paper shows a new effective approach to measure crystallization temperature of soft magnetic underlayer (SUL) for next generation of heat assisted perpendicular recording media. This approach uses temperature dependent reflectivity, which shows a clear jump when samples are crystallized. To achieve this measurement, an optical system is set up using hot plate and infrared laser. Reflectivity of SUL $(Co_{70}Fe_{30})_{92}Ta_3Zr_5$ shows a clear jump at its amorphous-crystalline transition temperature. Experiment results show this effect is clear in infrared region, and is weak for visible light.
키워드
Amorphous thin film;Soft magnetic thin film;Optical property;Perpendicular recording media
참고문헌
1. Migaku Takahashi and Shin Saito, J. Magn. Magn. Mater. 320, 2868 (2008). https://doi.org/10.1016/j.jmmm.2008.07.039
2. M. Kevin Mior and Timothy J. Klemmer, J. Appl. Phys. 93, 6465 (2003). https://doi.org/10.1063/1.1557346
3. Stanford R. Ovshisky, Phys. Rev. Lett. 21, 1450 (1968). https://doi.org/10.1103/PhysRevLett.21.1450
4. J. Siegel, A. Schropp, J. Solis, C. N. Afonso, and M. Wuttig, Appl. Phys. Lett. 84, 2250 (2004). https://doi.org/10.1063/1.1689756
5. P. Haussler, F. Baumann, J. Krieg, G. Indlekofer, P. Oelhafen, and H.-J. Guntherodt, Phys. Rev. Lett. 51, 714 (1983). https://doi.org/10.1103/PhysRevLett.51.714
6. S. S. Jaswal and J. Hafner, Phys. Rev. B 38, 7311 (1988). https://doi.org/10.1103/PhysRevB.38.7311 | {"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": 0.3153872489929199, "perplexity": 16509.80949446059}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250604397.40/warc/CC-MAIN-20200121132900-20200121161900-00069.warc.gz"} |
https://iwaponline.com/washdev/article/9/1/139/65469/Mobile-crowd-participation-to-root-small-scale | ## Abstract
In the arsenic-contaminated Ganges-Brahmaputra-Meghna Delta in India and Bangladesh, small-scale piped water supply seems a promising way to provide safe drinking water to households in the region. The use of smartphone applications can support monitoring of the system and enhance local engagement and empowerment. In this paper the scope for mobile crowd participation as a research and monitoring tool for piped water supply systems in Bihar, India and in Khulna and Chittagong, Bangladesh is investigated. In these areas, the use of smartphones and internet access are growing rapidly and smartphone applications would enable real-time water quality monitoring, payment of water bills, awareness creation, and a dialogue between the end-user and the water supplier. To identify the relevance and acceptability of piped water supply and smartphone monitoring, four surveys with potential end-users were conducted. Based on these surveys we conclude that in the investigated areas there is a desire for piped water systems, that households already own smartphones with internet access, and that there is an interest in smartphone monitoring. The enabling environment to deploy mobile crowd participation for piped water system monitoring stimulates further research towards an investigation of potential functionalities and the actual development of such an application.
## INTRODUCTION
### Water safety in the Ganges-Brahmaputra-Meghna delta
In the Ganges-Brahmaputra-Meghna Delta (GBM Delta) in India and Bangladesh many people lack access to safe drinking water. Since the 1970s the focus in Bangladesh has been on installing tube-wells to protect the Bangladesh population against pathogens in their drinking water (Argos et al. 2010). A similar scenario applies to regions of India where the country started to use groundwater in the 1960s (Srikanth 2013; Daigle 2016). Only in the 1990s was it realized that the groundwater was severely arsenic-contaminated (Chakraborti et al. 2003; Argos et al. 2010; Bangladesh Bureau of Statistics (BBS) & UNICEF Bangladesh (UBD) 2015). Arsenic is a human carcinogen which poisons the population when they drink water with high contamination levels. Overexposure to arsenic is associated with skin lesions, oedema, gangrene, black foot disease, and malignant diseases such as skin, bladder and lung cancer (Karagas 2010; Singh et al. 2015; Sarkar & Biswajit 2016; WHO 2016). The limit of arsenic in drinking water by the World Health Organization is 10 μg/L (WHO 2016) resulting in an estimated 32 to 77 million people who suffer from overexposure to arsenic through drinking water in Bangladesh (Argos et al. 2010; Sarkar & Biswajit 2016) and approximately 70 million people in India (Sarkar & Biswajit 2016). Besides the arsenic contamination risk, people in Bangladesh are also overexposed to manganese, iron and salinity, and in Bihar also to fluoride and iron.
While in Bangladesh 97.9% of the Bangladeshi population uses an improved drinking water source, it emerged that ‘improved water sources are much more likely to have arsenic contamination than non-improved sources’ and that rural areas are more at risk than urban areas (BBS & UBD 2015, p. 72). In Bangladesh, people living in arsenic-affected areas still drink arsenic-contaminated water, despite several programmes and initiatives (Hoque et al. 2004). Initially, wells were tested and marked red (>50 μg/L) or green (<50 μg/L) and people were encouraged to switch to safe wells or to newly installed public deep tube-wells (DTWs), under the Bangladesh Arsenic Mitigation Water Supply Program. Due to increasing awareness of arsenic in Bangladesh since the year 2000 (Yunus et al. 2016), about one-fourth to two-thirds of the households switched to a safer well. However, owing to difficulties in finding alternatives, far away safe sources, lack of systematic and/or regular monitoring of water quality and arsenic content, inadequate levels of or mechanisms for sharing information about well tests, washing away of red and green paint, and fading knowledge, not all households switched to or switch away again from safe wells after some time (Pfaff et al. 2017; Balasubramanya & Horbulyk 2018). Since 2004, government focus has shifted to piped water supply (PWS) from a safe source (Balasubramanya & Horbulyk 2018). However, these programs are not yet able to deliver PWS at the intended scale due to organizational delays, lack of adequate capacity, and complex cost-sharing arrangements (Worldbank 2015; Balasubramanya & Horbulyk 2018). Balasubramanya & Horbulyk (2018) mention that options such as community DTWs and well testing are easier to deliver. In Bihar state, households mainly depend on privately installed, unmonitored shallow tube-wells (STWs) or government-installed DTWs. The private STWs put households at risk, and while the government DTWs are mainly free of arsenic contamination, they are dysfunctional in 24 to 31% of cases (Srikanth 2013). Also in Bihar, well testing campaigns have been organized, but due to lack of guidelines for well testing, uncontrolled and unregulated installation of hand pumps, untested areas, lack of a common repository for arsenic testing data, lack of a decision support system for arsenic mitigation, and a lack of coordination among research groups, arsenic mitigation still remains a challenge (Singh 2015).
### Piped water supply: advantages and challenges
Different arsenic mitigation strategies exist, such as well testing, use of DTWs, use of household filters, community-level treatment, rainwater harvesting, pond sand filters, and safe water points for a limited number of people. For different reasons the success of these strategies differs. Local geological and socio-economic circumstances, maintenance, logistics, convenience and costs seem to play an important role in the success of mitigation strategies (Hoque et al. 2006; Sarkar et al. 2010; Daigle 2016; Balasubramanya & Horbulyk 2018; Maertens et al. 2018). In the presented study the focus is on small-scale piped water supply (SPWS) (Trifunovic 2002; Kayaga & Reed 2005) systems, as it offers crucial advantages over other technological interventions. SPWS targets the safest source in the area, provides a degree of centralization (<100 households) for water quality control and treatment while keeping the system manageable within communities. It provides in-house or courtyard tap connections, which are socio-economically desirable, and it limits the number of (re-)contamination events between water collection and consumption.
However, there are several challenges to overcome for PWS systems. First of all PWS systems need acceptance by their end-users. Johnston et al. (2014) found PWS systems to be supported by Bangladeshi households and noted that these systems should be prioritized. However, they also noted that until now the experience with these systems is limited, that costs of the system might be an issue, and that it will take time before these systems will be fully realized in the country. This is also acknowledged by Maertens et al. (2018), who indicate PWS to be the most effective policy answer, but not for the foreseeable future. Ahmad et al. (2005, 2006) indicated that piped water systems are desired in Bangladesh, but mainly for their convenience. Therefore, the willingness of end-users to change behaviour and take up PWS might be limited, especially as PWS increases water costs. Maertens et al. (2018) report that the effectiveness of awareness campaigns in Bangladesh suffers from low demand, resistance to behavioural change, poverty, low literacy and other constraints. As most focus in developing regions is on the implementation of these systems and not on their operation and maintenance management, the technologies are often not sustainable (Lee & Schwab 2005; RWSN 2009; Meleg 2012; Srikanth 2013). After instalment, water quality and quantity problems are caused by intermittent water supply, leaking pipes, low water pressure, inadequate disinfectant residual, inadequate wastewater collection systems, and deterioration of infrastructure (WHO & UNICEF 2000; Lee & Schwab 2005; Johnston et al. 2014). Poor operation and maintenance due to political, social and economic issues thus result in substandard water supply (Lee & Schwab 2005). Well appropriation by local elites also occurs (Balasubramanya & Horbulyk 2018). Thereby, billing of water usage also causes problems, due to illegal tapping and malfunctioning meters (Lee & Schwab 2005). Problems with money collection also arise (Balasubramanya & Horbulyk 2018). This also results in lack of maintenance and wastage of water (Lee & Schwab 2005). Consequently, PWS have a slow uptake, and are merely seen as a long-term solution, while solutions are urgently required in the short term (Balasubramanya & Horbulyk 2018). SPWS systems can function as an intermediate solution, before PWS implementation on a large scale. SPWS makes operation and maintenance easier compared to bigger systems, and requires less expensive piping. According to Yunus et al. (2016), the installation of safe water points for no more than 50 people per point in combination with a safe source does provide long-term cost-effectiveness.
However, implementation of SPWS is not sufficient to ensure sustainable safe drinking water supply in communities. Meleg (2012) stresses the importance of considering financial, institutional and technical dynamics in relation to human factors in order to implement sustainable water supply systems. Balasubramanya & Horbulyk (2018) mention the understanding of community preferences and of the ability and willingness to pay for different options as important knowledge gaps, alongside the requirement to examine financial, regulatory and other obstacles and Abedin & Shaw (2013) strongly argue for a much greater role of local communities in developing and leading mitigation strategies, with the assistance of governmental and non-governmental organizations.
### Participation in the development of piped water supply systems
When in Northern Kenya two good-hearted Americans installed a community well to prevent women from suffering the indignity and effort of walking long distances to a watering hole, they were very much surprised that the women did not want to use the well (Fleming 2015). It emerged that the women did not accept the technology as walking to the watering hole was their form of socialization, which they did not want to miss out on (Fleming 2015). Skinner (2009) and Purvis (2016) furthermore report that when well technology is accepted in rural Africa, many of the wells fail within two to three years due to wrong drilling, failing technology, unavailable or expensive spare parts, and/or lack of ownership and local capacity leading to lack of maintenance.
Technologies shape and change our world and have the potential to support people in doing what they want to do and being who they want to be. This potential is especially valid in Design for Development (DfD) projects where products and services are developed to improve the well-being of disadvantaged and marginalized populations (Donaldson 2006, 2009). However, as shown by the example above, well-meant technologies might cause new social and environmental problems or even limit or control people in their beings and doings. Designers therefore have a high social and moral responsibility for the consequences of their innovations (Papanek 1984). Many failed products and services are unsuited to the user and/or their environment as they are either based on poorly defined needs (Donaldson 2006), or focus merely on needs instead of what people actually want (Bowman & Crews 2009). A high degree of participation enhances the possibility of designing solutions that are accessible, applicable, acceptable and adopted (Donaldson 2009; Nakata & Weidner 2012; Prahalad 2012; Robertson & Simonsen 2012; Wilkinson & De Angeli 2014; Mink 2016) and are therefore more likely to be sustained (Marshall & Kaminsky 2016). Moreover, it enhances successful implementation and community ownership (Narayan 1995; Kayaga 2013). User participation actively engages the end-user in the process, so they can design and implement new ways of being and doing themselves (Manzini 2014). The development of SPWS systems is a technological design and requires contextual learning and involvement of local stakeholders on an equal footing in order to produce systems that people can, will, and want to use, and that will continue to be used. Marshall & Kaminsky (2016) point out that the motivations of outside stakeholders (good health and well-being) might potentially misalign with the motivations of local stakeholders (e.g., status, convenience, social norms). To improve the chances of long-term success of SPWS implementation, local stakeholders should be actively involved in the process: first by consulting them and raising awareness, then by co-creation and capacity building, and finally by engaging and enabling them to monitor their own systems.
A novel approach to research and monitor SPWS systems is mobile crowd participation (MCP), which enables the local stakeholders to participate, learn and provide feedback via smartphone applications. MCP can be deployed in different ways:
• 1.
In combination with strip tests it can be used for low-cost screening for the presence of arsenic, while making the data available to different interested stakeholders, including the end-users.
• 2.
Improving knowledge and awareness about safe water supply and water quality (also by enabling users to test the water quality themselves).
• 3.
When SPWS is installed, MCP enables communication between relevant stakeholders for improving service and maintenance (e.g., sharing down-time, leakage reporting, water quality information), makes payment easier, and allows for low-cost monitoring of water quality, water consumption and payments.
While external support for water service provision will remain a requirement (RWSN 2009; Kayaga 2013), MCP has the ability to engage and empower communities and supporting organizations and companies to increase the trust and uptake of piped systems and to assist in continuous capacity development. The requirements for MCP to work are smartphone ownership and internet access, and the availability of an application that is useful, usable and desirable for its users. According to GSMA (2013), access to mobile services in developing regions has outpaced the rate at which much of the population is gaining access to basic services such as electricity, sanitation and banking. In 2015, smartphone ownership in India was 24% and mobile internet penetration 49% (GSMA 2016). In 2016, smartphone ownership in Bangladesh was 28% and mobile internet penetration 53% (GSMA 2017). Based on these numbers, there seems to be scope for a smartphone application to monitor SPWS systems. It is the objective of this study to identify the relevance and scope of using MCP as a research and monitoring tool specifically for SPWS systems in arsenic-affected areas in the GBM Delta. Therefore, four surveys have been executed in Bihar, India and in Khulna and Chittagong, Bangladesh, of which the outcomes are presented in this paper.
## METHODS
Four surveys were executed among end-users to assess the demand, acceptability and affordability of small-scale piped water systems and to assess the feasibility and scope of using smartphones as a research and monitoring tool. Besides questions regarding SPWS systems and MCP, the surveys included sub-themes about socio-economic and demographic characteristics of the participants, water use practices and water quality knowledge. The surveys were developed, pre-tested in the field and revised to collect the final data. The participants were informed about the purpose of the research, data handling and security, and were asked for informed consent to use the data for research and publication prior to each survey. Representatives from households in each area were questioned by selecting the third/fourth household in the row, depending on the availability of participants. In this way, all parts of each investigated area were covered to ensure a random and representative selection of participants.
### Study 1 – Khulna district, Bangladesh
In November 2014, a semi-structured questionnaire survey was executed in Bangladesh among 601 households in the municipalities of Sathkira (SAT), Khulna (KHU) and Bagerhat (BAG). In consultation with local stakeholders, three low-income, urban and high population density settlements in an arsenic-affected area were selected. Locally recruited data collectors from the area, who followed a 2-day training in class and in the field, questioned 287 male and 314 female participants. The questionnaires lasted an average of 35 min.
### Study 2 – Khulna and Chittagong district, Bangladesh
In June 2015, a questionnaire survey among 101 households was conducted in the municipalities of Kolaroa (KOL) and Debidwar (DEB). In consultation with local stakeholders, two low-income peri-urban areas consisting of at least 100 households in an arsenic-affected area were selected for the study. Trained EPRC staff members questioned 53 female and 48 male participants. The questionnaires lasted an average of 20 min.
### Study 3 – Bhojpur district, India
In April 2017, the research team conducted a semi-structured questionnaire survey in Bhojpur district, Bihar state. In consultation with local stakeholders, seven villages in an arsenic-contaminated area consisting of at least 100 households in Bhojpur district were selected for executing the surveys: Saraiya (SAR), Sinha (SIN), Gundi (GUN), Bakhorapur (BAK), Keshopur (KES), Lauhar (LAU) and Nathmalpur (NAT). The questionnaires were conducted with representatives from 275 households, of which 15 were female and 259 male, and lasted an average of 10 min.
### Study 4 – Bhojpur district, India
In the same villages at the same time as study 3, the research team conducted an interview survey with representatives from 60 households by selecting households owning at least one smartphone, in order to gather in-depth information about the potential for SPWS and scope for MCP. The interviews lasted an average of 40 min.
## RESULTS AND DISCUSSION
Below the results are presented and discussed in three parts: first the participants' socio-economic characteristics (for all studies), then the potential for SPWS (for all studies) and finally the scope for MCP (for studies 2, 3 and 4).
### Socio-economic characteristics
The investigated areas in Bangladesh are urban and peri-urban, leaving most of the participants to work as a day labourer, small business owner or driver. The investigated areas in India are rural and most participants here are engaged in farming and small businesses or services. The average household size in India (12 people) is bigger than in Bangladesh (five people), and there is also a larger presence of children under five in Bihar (in 58% of the households compared to 43% of the Bangladeshi households). As can be seen in Figure 1, both the educational and income levels in the investigated areas in Bangladesh are lower than in India. This is also reflected in the national averages of both countries (India's human development index is 0.624, with a mean of 6.3 years of schooling and a gross national income per capita of $5.663 purchasing power parity, Bangladesh's human development index is 0.579 with a mean of 5.2 years of schooling and a gross national income per capita of$3.341 purchasing power parity (UNDP 2015)). Within each countries' settlements, the educational and income levels differ as well: the participants from Khulna district have lower levels than those in Chittagong, and in Saraiya, Sinha and Gundi these levels are lower than in Lauhar, Keshopur and Nathmalpur. In study 4 the average educational and income levels are higher than in study 3, which can be explained by the fact that the participants in study 4 were selected for owning a smartphone, which is more likely to be owned by households with a higher socio-economic status.
Figure 1
Educational level and income status per study per village.
Figure 1
Educational level and income status per study per village.
### Scope for piped water supply
In Table 1 the results of the current drinking water situation of all four studies are presented. Most participants use tube-wells for their drinking water. In studies 2, 3 and 4 mainly privately owned STWs were situated around the house, in study 1 mainly publicly owned DTWs. In study 1, 80% of the households share a water point with more than 50 households. This results in risks regarding relationships, privacy, security, operation and maintenance, as conflicts mainly occur during peak hours. Some participants rely on publicly installed DTWs which are located only at certain points in the area. In this case, the water collectors, mainly women, have to walk long distances to fetch water.
Table 1
Existing drinking water supply characteristics of participants in Bangladesh and India
Study 3 IndiaStudy 4 India
SAT (201)KHU (200)BAG (200)DEB (51)KOL (50)All (275)All (60)
Drinking water source used (in % of participants)
Deep tube-well (>150 m deep) 19 100 70 12
Shallow tube-well (<150 m deep) 35 100 86 99 97
Piped water supply 24 0.5
Surface water (pond/river) 20 29 0.5
Other (pond sand filter, dug well, rain water)
Ownership (in % of participants)
Private 36 16 73 64 99 97
Shared 17 10 27 18
Public 47 95 74 18
Water availability (in % of participants)
Throughout the year 92 77 91 96 40 100 100
Not throughout the year 23 60
Distance to water point (in meters)
Minimum
Mean 71 29 301 261
Maximum 1,600 305 1,600 50 2,250 30
Study 3 IndiaStudy 4 India
SAT (201)KHU (200)BAG (200)DEB (51)KOL (50)All (275)All (60)
Drinking water source used (in % of participants)
Deep tube-well (>150 m deep) 19 100 70 12
Shallow tube-well (<150 m deep) 35 100 86 99 97
Piped water supply 24 0.5
Surface water (pond/river) 20 29 0.5
Other (pond sand filter, dug well, rain water)
Ownership (in % of participants)
Private 36 16 73 64 99 97
Shared 17 10 27 18
Public 47 95 74 18
Water availability (in % of participants)
Throughout the year 92 77 91 96 40 100 100
Not throughout the year 23 60
Distance to water point (in meters)
Minimum
Mean 71 29 301 261
Maximum 1,600 305 1,600 50 2,250 30
aSAT, Sathkira; KHU, Khulna; BAG, Bagerhat.
bDEB, Debidwar; KOL, Kolaroa.
In India (studies 3 and 4) the trust in participants' current drinking water was also investigated. Of the 335 participants in those two surveys, 61% trust their drinking water. Of these, 81% indicate that they trust their water because of its good colour, taste and/or smell, 2% because they do not suffer from water-based illnesses, and the remaining 17% indicate the water ‘seems good’ or that they trust the water for no specific reason. The remaining 39% of the participants indicate that they maybe trust their water (1%) or distrust their water (38%). Of the participants who distrust their water, 45% indicate they do so because the water has a bad colour, taste and/or smell, 17% because the water changes colour during storage and 9% because they suffer from water-related diseases. The remaining 29% of the distrusting participants indicate that their water ‘seems contaminated’ or they could not indicate a specific reason for their distrust.
In the studied areas, the participants have little to no experience with PWS systems, except in Sathkira where 24% of the participants rely on PWS. The desire for PWS is nonetheless high, although in study 3 it is somewhat less than in the other three studies (79% compared to 98–100%). However, while in study 1, 97% of the participants indicated they wanted a PWS, 65% of them also indicated a preference for drinking hand-pumped water from a DTW. The second most preferred option is pond water supply (21%), then PWS (11%), and 3% indicated a preference for hand-pumped water from a STW. Most participants preferring DTW water indicated that hand-pumped water from this source provides an adequate volume of bacteria-free drinking water for their family throughout the day. Thereby, some indicated a distrust against PWS as a drinking water source, because of possible contamination in the water tank and pipelines. Multiple participants also indicated that groundwater tasted cooler and sweeter, and therefore they preferred to drink directly from the ground instead of from a PWS. Participants with wells containing a lot of iron mainly indicated preferring a different option than their current drinking water source, as they dislike the iron taste and colour it gives to their food and clothing. The desire for PWS is, nonetheless, high as the participants want to use piped water for cleaning, washing and cooking purposes. In all studied areas, safe water and convenience are mentioned as reasons behind the desire for PWS. In India (studies 3 and 4), the participants were specifically questioned about the reasons behind their desire for piped water supply (see Figure 2). Of the 335 participants in these studies, almost 40% indicated not trusting their current drinking water supply system, and 71% indicated they desired PWS to be able to provide their family with safe drinking water.
Figure 2
Reasons behind the desire for piped water supply in India.
Figure 2
Reasons behind the desire for piped water supply in India.
In most study areas, a large majority of the participants indicated prefering deep groundwater as a water source for piped water systems. However, in study 4, a significant preference for surface water (35%) was also identified. Those participants indicated that water from River Ganga is sacred and therefore considered good for health. The participants of study 4 also indicated a higher preference for using rain water (10%) than in the other studies (1% to 2%). Willingness to pay for installation, operation and maintenance of piped water supply was found to be high in studies 1 (97%), 2 (94%) and 4 (93%). In study 3, however, 50% of the participants were not willing to pay. Of these participants, 33% indicated they had no money, 14% indicated that PWS is nice but not required, and 2% indicated that the government should pay. As the participants in study 4 are of higher socio-economic status, study 3 is used to represent the willingness to pay in this area in India. The Bangladeshi participants were willing to pay an average of 52 BDT (0.50 euro) per month for operation and maintenance. In study 4, the participants shared a willingness to pay 220 INR (2.90 euro) per month. In all studies, the costs for installation, operation and maintenance were mentioned as issues of PWS systems. Thereby, in studies 1, 2 and 3 irregular water supply was noted as an issue. In study 1, where participants do have experience with PWS systems, complexities in operation and maintenance of PWS systems were also mentioned as challenges.
From these studies it seems that Bangladeshi households are more experienced with DTWs and PWS systems and more willing to pay for PWS, although a lower amount than in India. Based on the collected data, no relation could be detected between income level, educational level or profession and the desire for PWS. While there is a large desire for PWS systems for the convenience and supply of safe water, trust in the system, costs, irregular supply and challenges in operation and maintenance are factors that seem to hinder people's desire for PWS.
### Smartphone ownership and usage
Data about smartphone ownership and usage were collected in studies 2, 3 and 4. In Table 2 the results regarding smartphone and mobile phone ownership and smartphone usage are presented. Smartphone ownership in study 3 (India) is significantly higher than in study 2 (Bangladesh). However, that may be partly due to the difference in the time of investigation (2014 versus 2017) and the difference in area (urban versus rural). Other reasons might be inexpensive and overall presence of internet access in India, and the higher socio-economic status of the Indian participants.
Table 2
Investigated mobile phone data of participants in Bangladesh and Bihar
VariablesStudy 2 BangladeshStudy 3 IndiaStudy 4 India
Both areas (101 HH)All villages (275 HH)All villages (60 HH)
Mobile phone ownership
None 7.9 2.2 –
One 40.6 23.3 –
Two 36.6 23.3 –
More than two 14.9 51.2 –
Smartphone ownership
None 88.1 50.4 –
One 11.9 23.2 –
Two 0.0 14.5 –
More than two 0.0 11.9 –
Phone users (in % of participants) Button phone Smartphone Smartphone
Only male 52.7 77.7 65.0
Only female 5.4 3.6 6.7
Both male and female 41.9 18.7 28.3
Phone users (in % of participants) Button phone Smartphone Smartphone
Children (<20 years old) 18.1 36.4 42.1
Parents' generation 74.8 54.1 47.4
Grandparents' generation 7.1 9.6 10.5
Number of households owning a smartphone 12 HH 139 HH 60 HH
Internet access with smartphone (in % of participants)
None 0.0 10.9 0.0
Always 42.0 49.6 73.3
Daily 58.0 10.9 16.7
Less than daily 0.0 28.5 10.0
Smartphone usage (in % of participants)
Calling 100.0 – 100.0
Text messaging 33.0 – 98.3
Taking pictures 67.0 – 98.3
Playing games – – 62.7
Internet usage in general 58.0 – 100.0
VariablesStudy 2 BangladeshStudy 3 IndiaStudy 4 India
Both areas (101 HH)All villages (275 HH)All villages (60 HH)
Mobile phone ownership
None 7.9 2.2 –
One 40.6 23.3 –
Two 36.6 23.3 –
More than two 14.9 51.2 –
Smartphone ownership
None 88.1 50.4 –
One 11.9 23.2 –
Two 0.0 14.5 –
More than two 0.0 11.9 –
Phone users (in % of participants) Button phone Smartphone Smartphone
Only male 52.7 77.7 65.0
Only female 5.4 3.6 6.7
Both male and female 41.9 18.7 28.3
Phone users (in % of participants) Button phone Smartphone Smartphone
Children (<20 years old) 18.1 36.4 42.1
Parents' generation 74.8 54.1 47.4
Grandparents' generation 7.1 9.6 10.5
Number of households owning a smartphone 12 HH 139 HH 60 HH
Internet access with smartphone (in % of participants)
None 0.0 10.9 0.0
Always 42.0 49.6 73.3
Daily 58.0 10.9 16.7
Less than daily 0.0 28.5 10.0
Smartphone usage (in % of participants)
Calling 100.0 – 100.0
Text messaging 33.0 – 98.3
Taking pictures 67.0 – 98.3
Playing games – – 62.7
Internet usage in general 58.0 – 100.0
The smartphones owned by the participants are used for calling, messaging, taking pictures, playing games and internet usage. The participants in study 4 all have internet access and use internet on their smartphones for social networking (100%), internet searching/downloading (98%), entertainment, such as watching television/video and listening to music (42%), online payment (25%) and education/study (9%). In both Bangladesh and India, phone usage is male dominated with 53–78% of household phones only being used by males versus 4–7% only being used by females. Nevertheless, in 19–42% of the households the mobile phone is shared, leading to a considerable proportion of the female household members using smartphones, especially in the surveyed area in Bangladesh. In India, a large amount of adolescents use the owned smartphones, mainly for social networking, entertainment or study purposes. Of the Bangladeshi participants, 46% indicated a desire to have a smartphone with internet access and are willing to spend, on average, 5% of their income on it. Most Indian participants already have internet access on their smartphones. Currently, the households in study 4 pay, on average, 503 Indian rupees (6.58 euro) per month on phone usage. Of the households in study 2, 78% pay less than 500 taka (5 euro) per month, 13% pay between 500 and 800 taka (5–8 euro) per month and 9% pay more than that.
When comparing smartphone ownership with socio-economic status for study 3, it can be concluded that there is a direct link between the average income of families owning smartphones and the number of smartphones (see also the boxplot in Figure 3). However, lower-income families do own smartphones, which can also clearly be seen in Figure 3, and for the ones who do not, the desire to own a smartphone is high, especially in households where family members do not live together or are separated throughout the day (for work or study). There is also a link with educational level (the higher the educational level, the more smartphones owned (see Figure 3)) and there seems to be a link with the profession of the household head (labourers own less smartphones, government employees own more), although this link is not that clear (see Figure 3).
Figure 3
Comparison of socio-economic factors of participating households and the number of smartphones these households own, based on information from study 3 (Bihar). (Read legend from left to right and top to bottom.)
Figure 3
Comparison of socio-economic factors of participating households and the number of smartphones these households own, based on information from study 3 (Bihar). (Read legend from left to right and top to bottom.)
From these studies, it seems that in Bihar smartphone ownership and internet usage is higher than in Bangladesh and more money is spent on phone usage. This could be because the income level and educational level of the participants in Bihar are higher than in Bangladesh and, as stated above, there seems to be a link between phone ownership and income and educational levels. However, other factors also play a role, such as differences in internet coverage and the price and availability of smartphones in the area. From study 2 it has become clear that in Bangladesh there is a desire to own a smartphone with internet access. Thereby, due to local phone manufacturers and inexpensive internet, smartphone and internet use are expected to rise rapidly in both India and Bangladesh. This enhances the scope for MCP for the purpose of water supply management.
## CONCLUSIONS
Based on the presented findings it can be concluded that there is an enabling environment to promote SPWS systems, as 98% of the Bangladeshi participants and 82% of the Indian participants indicate wanting this system. In addition, it may also be concluded that the majority of the participants in Bangladesh and India are willing to use smartphone applications for water supply research and monitoring. In India, most participating smartphone owners would like to use a smartphone application for water quality testing, service, communication with the water supplier and/or water payment. The existing smartphone ownership and the rapid increase of smartphone ownership and usage in both areas, the desire to own a smartphone with internet connection in Bangladesh, and the availability of cheap unlimited internet access in India indicate the scope for deploying MCP in these areas.
However, before developing new water apps, more research needs to be done in respect to participants' trust in and potential usage of SPWS systems and in the water supplier/service provider, and the context-specific conditions that influence the success of SPWS implementation, operation and maintenance. We believe that MCP has the potential to increase the trust in PWS and support its uptake, but only when properly implemented and equally accessible for everyone. The amount of access different household members have to smartphones, the possible phone and internet constraints (e.g., battery, camera quality, bandwidth) and the application's functionalities, usefulness, usability and desirability for both the end-users and service providers are points of attention: easy to understand, attractive and intuitive to use for literate as well as illiterate users, and comprising functionalities desired by both the service provider as well as the end-users. For water quality testing purposes the reliability and accuracy of the test are also important aspects that need further investigation. Another point of attention is the possible exclusion of certain groups due to using a smartphone as a platform. We expect a significant rise in smartphone ownership and usage in Bihar and Bangladesh in the coming years, leading to more inclusion in respect to people's income and educational level. However, the access to smartphones for women, who are the main water collectors and users, needs a critical look. Therefore, the exact manner of using MCP as a research and monitoring tool, the functionalities to be included, and the appearance and functioning of the app, as well as data processing, data security, and the required training and education for MCP users are the next steps to explore. For now, the relevance of using MCP seems promising, not only for the GBM Delta, but possibly also for SPWS applications in other low-resource regions.
## ACKNOWLEDGEMENTS
This research was made possible by the Netherlands Organization for Scientific Research (NWO) under grant number W07.69.205. We also thank all involved researchers and research assistants from A.N. College Patna in India, Prof. Ashok Ghosh (Head of the Research Department of the Mahavir Cancer Institute and Research Center in Patna), Nanakkumar Santdasani (at the time or our research WaSH Officer UNICEF Bihar), and all involved researchers from the Environment and Population Research Centre in Bangladesh for their contribution towards collecting the presented data, and we thank all participants for their valuable contribution and cooperation in this research.
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This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY-NC-ND 4.0), which permits copying and redistribution for non-commercial purposes with no derivatives, provided the original work is properly cited (http://creativecommons.org/licenses/by-nc-nd/4.0/) | {"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": 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": 0.3096102476119995, "perplexity": 4834.019240964167}, "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/1620243988793.99/warc/CC-MAIN-20210507120655-20210507150655-00261.warc.gz"} |
http://tex.stackexchange.com/questions/73387/classicthesis-vertical-margins?answertab=votes | # Classicthesis Vertical Margins
I've been using the classicthesis package to write my master's thesis and now that I was getting the entire thing ready for printing I ran into some trouble. The thing is that I'm not really satisfied with the style's vertical margins. I know that many people try to adjust the horizontal margins as they seem a little wide, but I'm perfectly fine with that. I would just like to take the entire text block and center it vertically a little more. The way it is now, the top and bottom margins are slightly out of balance, which becomes especially apparent when turning off the drafting option.
So, is there any way to adjust the top and bottom margins without affecting the rest of the layout?
-
Since classicthesis uses the package typearea (from the KOMA-Script bundle), you should check the Chapter 2 in the KOMA-Script Guide – henrique Sep 20 '12 at 17:18
– henrique Sep 21 '12 at 15:11
classicthesis uses the package typearea, which tries to preserve a general aspect when changing the text block (check the Chapter 2 in the KOMA-Script Guide for the motives (See also this answer to How to configure KOMA classes such that the bottom margin of a page equals its top margin?) The text block is defined in classicthesis (when not using minion option) as:
\areaset[current]{336pt}{750pt} % ~ 336 * factor 2 + 33 head + 42 \the\footskip
Following a sugestion within classicthesis.sty, a good proportion for Palatino 11pt would be 336 × 705pt, which results in a sort of vertically symmetrical text block. A MWE:
\documentclass[11pt]{book}
\usepackage{classicthesis}
\usepackage{blindtext,layouts}
\areaset[current]{336pt}{705pt}% as suggested in classicthesis.sty, line 273
\begin{document}
\blindtext[6]\footnote{blindtext}
\begin{figure}
\currentpage
\pagediagram% To check the layout and lengths
\pagevalues
\end{figure}
\end{document}
Another option would be to call geometry: as stated in the first comment to What packages are incompatible with KOMA-Script, "geometry resets the from typearea calculated margins if you don't use the option pass (for showframe)". We lose the automatic calculated margins, therefore we lose the aspect, but it is possible. A MWE:
\documentclass[11pt]{book}
\usepackage{classicthesis}
\usepackage{blindtext,layouts}
\usepackage[top=48pt,bottom=48pt,textwidth=336pt]{geometry}
\begin{document}
\blindtext[6]\footnote{blindtext}
\begin{figure}
\currentpage
\pagediagram% To check the layout and lengths
\pagevalues
\end{figure}
\end{document}
- | {"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": 0.7974717617034912, "perplexity": 1425.7382054256518}, "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-2015-22/segments/1432207928486.86/warc/CC-MAIN-20150521113208-00185-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-connecting-concepts-through-application/chapter-3-exponents-polynomials-and-functions-3-5-special-factoring-techniques-3-5-exercises-page-279/20 | ## Intermediate Algebra: Connecting Concepts through Application
$5(2x+5)(4x^2-10x+25)$
First, pull out the greatest common factor. Then use the formula $a^3+b^3=(a+b)(a^2-ab+b^2)$ to factor. $40x^3+625$ $=5(8x^3+125)$ $=5(2x+5)(4x^2-10x+25)$ | {"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": 0.8065612316131592, "perplexity": 1485.5658842518394}, "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/1547583658681.7/warc/CC-MAIN-20190117020806-20190117042806-00394.warc.gz"} |
http://www.hpmuseum.org/forum/thread-9449-post-82802.html | pyILPER: Read and write HP-41 ROM/MOD files
11-07-2017, 02:43 PM
Post: #1
Geir Isene Senior Member Posts: 520 Joined: Dec 2013
pyILPER: Read and write HP-41 ROM/MOD files
How do I read and write HP-41 ROM or MOD files to/from an HP-41 using pyILPER (thanks for a great piece of sw, btw)?
11-07-2017, 04:24 PM
Post: #2
Dave Frederickson Senior Member Posts: 1,479 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
I believe what you want to use are Joachim's LIFUTILS, for which pyILPER makes a great front-end.
In pyILPER, mount a LIF disc image in one of the Drive tabs, then clear the Device enabled check box, this enables access to LIFUTILS. There will be capabilities enabled in the Drive drop-down menu and right-click options in the directory window.
Dave
11-07-2017, 04:25 PM
Post: #3
jsi Member Posts: 72 Joined: Jun 2015
RE: pyILPER: Read and write HP-41 ROM/MOD files
pyILPER can import and export rom files as HEPAX SDATA files to/from the currently mounted LIF image file. Check the option "convert HP-41 rom to SDATA file (HEPAX) on import or "unpack HPEAX HP41 SDATA ROM file" on export. You have to disable the virtual drive to access the file management functions (Tool menu and directory entry context menu).
pyILPER uses the "rom41hx" and "hx41rom" filters of the LIFUTILS.
Regards
Joachim
11-07-2017, 07:19 PM
Post: #4
Geir Isene Senior Member Posts: 520 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
OK, for the dense - please repeat with a baby-step checklist (note: I am on Linux, running pyILPer):
1. Start pyILPER
2. Make sure the PILbox is connected physically and to the pyILPer program running pyILPer configuration from inside the program ("File" menu)
3. ??
4. ??
11-07-2017, 08:54 PM (This post was last modified: 11-07-2017 09:10 PM by Geir Isene.)
Post: #5
Geir Isene Senior Member Posts: 520 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
I tried this:
Code:
cat CASINO.ROM | rom41hx CASINO > casino.sda
And tried to load the file (casino.sda) and got an error message "File does not contain a LIF type 1 medium."
I want to have e.g. the CASINO.ROM loaded into my HP-41CL using the PILbox and the pyILPer.
Edit; Got an old LIF file loaded - but how do I get the CASINO.ROM in there using e.g. lifutils?
11-07-2017, 09:13 PM
Post: #6
Dave Frederickson Senior Member Posts: 1,479 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
(11-07-2017 08:54 PM)Geir Isene Wrote: I tried this:
Code:
cat CASINO.ROM | rom41hx CASINO > casino.sda
And tried to load the file (casino.sda) and got an error message "File does not contain a LIF type 1 medium."
I want to have e.g. the CASINO.ROM loaded into my HP-41CL using the PILbox and the pyILPer.
Casino.sda is not a LIF image, consequently the error message.
1. Create a copy of CASSETTE.DAT from the lifimage folder.
2. Mount the LIF tape image in one of the drive tabs.
3. Disable the virtual drive - enabling LIFUTILS.
4. Copy casino.sda to the tape image file. Sorry, I don't have a copy of pyILPER running at the moment so I can't offer specific details. You could also do this from the command line using lifput. Also note that converting the file outside of pyILPER wasn't necessary.
5. From here, casino.sda can be accessed as if it were on a tape drive.
Dave
11-08-2017, 12:38 AM
Post: #7
Geir Isene Senior Member Posts: 520 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
On 5: I see CASINO as a "DA" file and not able to use e.g. GETROM to load it onto the 41 as an ordinary ROM.
11-08-2017, 12:50 AM
Post: #8
Geir Isene Senior Member Posts: 520 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
It seems I can use only LIFUTILS to do what I want here (skipping pyILPER enirely and with LIFILPER as the hook to the PILbox) - if I could only get the ROM file _as_a_ROM-file_ into a LIF file.
If I could do this, then I could create a neat TOTAL-UPDATE program for the HP-41CL (such as Sylvain did with the CL serial interface) using the PILbox. Reason: I broke the serial connector on the CL board. It may even prove to be smoother than the CL serial connector way (by simply stuffing all the needed roms into one LIF file and mount it via LIFILPER and let the HP-41CL fetch each ROM successively and stuff them into Flash).
11-08-2017, 12:52 AM (This post was last modified: 11-08-2017 12:55 AM by Dave Frederickson.)
Post: #9
Dave Frederickson Senior Member Posts: 1,479 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
Give me a couple of hours to get home where I can try this.
11-08-2017, 12:55 AM
Post: #10
Geir Isene Senior Member Posts: 520 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
(11-08-2017 12:52 AM)Dave Frederickson Wrote: Make the below changes to the instructions.
4. Copy casino.rom to the LIF image.
5. From here, casino.rom can be accessed as if it were on a tape drive.
How? If I do this:
Code:
lifput test.lif CASINO.ROM
(test.lif being a properly initialized and valid LIF image)
... then I get this:
Code:
illegal file type
11-08-2017, 01:22 AM (This post was last modified: 11-08-2017 01:35 AM by Dave Frederickson.)
Post: #11
Dave Frederickson Senior Member Posts: 1,479 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
(11-08-2017 12:55 AM)Geir Isene Wrote:
(11-08-2017 12:52 AM)Dave Frederickson Wrote: Make the below changes to the instructions.
4. Copy casino.rom to the LIF image.
5. From here, casino.rom can be accessed as if it were on a tape drive.
How? If I do this:
Code:
lifput test.lif CASINO.ROM
(test.lif being a properly initialized and valid LIF image)
... then I get this:
Code:
illegal file type
In order to use lifput the file must have a valid 32-byte LIF file header, which it probably does not. I suggest pyILPER be used as a front-end for LIFUTILS as it offers some options that may not be obvious.
In the mean time, can you load one of the ROM files from the LIF archive, like WIZROM on SWAP12?
Dave
11-08-2017, 02:16 AM
Post: #12
Geir Isene Senior Member Posts: 520 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
The Casino rom is just an example. It's the same with any rom file.
Here's the thing:
I need to save a rom file (any rom file) to a LIF image so that the HP-41CL can retrieve it via a command such as CCD OSX's GETROM.
OR, I need to figure out how to retrieve the rom file to the HP-41CL when the rom file is saved inside the LIF image as an SDATA file.
Either of these would work fine. And preferably using just the LIFUTILS so that batch uploads of roms to the 41CL can be fully automated via a script.
11-08-2017, 02:29 AM
Post: #13
Dave Frederickson Senior Member Posts: 1,479 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
Using pyILPER, importing a file gives the option of converting a ROM file to SDATA or adding a LIF file header to an HP41 FOCAL raw file.
Converting the file to SDATA can easily be scripted, so perhaps someone can provide the steps to import the SDATA file.
Dave
11-08-2017, 06:47 PM
Post: #14
hth Member Posts: 191 Joined: Mar 2014
RE: pyILPER: Read and write HP-41 ROM/MOD files
With a huge/complete set of .mod files, then a script tool could extract .rom images and given some rules of how to distrubute the images, create a LIF image so that a file in it corresponds to each flash sector in the 41CL.
With that, it would just be a matter of running an RPN program on the HP-41 and load a file to 41CL RAM using HP-IL, then invoke a flash update?
I am also without a serial port on my 41CL, and wonder how feasible it would be?
Håkan
11-08-2017, 06:57 PM (This post was last modified: 11-08-2017 07:27 PM by Geir Isene.)
Post: #15
Geir Isene Senior Member Posts: 520 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
(11-08-2017 06:47 PM)hth Wrote: With a huge/complete set of .mod files, then a script tool could extract .rom images and given some rules of how to distrubute the images, create a LIF image so that a file in it corresponds to each flash sector in the 41CL.
With that, it would just be a matter of running an RPN program on the HP-41 and load a file to 41CL RAM using HP-IL, then invoke a flash update?
I am also without a serial port on my 41CL, and wonder how feasible it would be?
Håkan
Yes - this way would be a very smoothe way of updating the 41CL - we could have dated LIF files with all CL roms in them - sequentially ordered like in the CL. Then it's a trivial matter of hooking the CL up to the LIF file on the PC via the PILBox - and then run a program on the 41 to fix the updates. Go and make dinner, eat and stroll back to the 41CL and voilá - it's done.
The trick is to get the ROM files into the LIF file in a way for the 41CL to grab them from there and stuff them into its RAM pages. I'm hoping for a guru to give me a hint of how to solve this trick. The programming on the PC side and on the 41CL I will happily do :-)
11-08-2017, 07:31 PM
Post: #16
rprosperi Senior Member Posts: 2,388 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
(11-08-2017 06:57 PM)Geir Isene Wrote: Then it's a trivial matter of hooking the CL up to the LIF file on the PC via the PILBox - and then run a program on teh 41 to fix the updates. Go and make dinner, eat and stroll back to the 41CL and voilá - it's done.
Assuming you can create such a LIF volume with all the ROM images (a big if, IMHO) the update via HP-IL will take MUCH longer than dinner! When updating only the changed images, it can still take 3-4 hours via direct serial transfer, so via HP-IL on the '41, this will take days....
It seems that simply ordering a $30 cable (plus ship to Europe, but still) makes more sense, but you may have more free time than I do. In any case, I agree it's an interesting challenge. After many hours of exploring ROM downloads via HP-IL (via HEPAX and/or CCD tools, plus file conversions, and tracking ROM image IDs and sequence, etc. etc.) I caved and went the cable route. But I'd love to see your solution, so please don't take this as criticism. --Bob Prosperi 11-08-2017, 08:07 PM (This post was last modified: 11-08-2017 08:08 PM by Geir Isene.) Post: #17 Geir Isene Senior Member Posts: 520 Joined: Dec 2013 RE: pyILPER: Read and write HP-41 ROM/MOD files (11-08-2017 07:31 PM)rprosperi Wrote: (11-08-2017 06:57 PM)Geir Isene Wrote: Then it's a trivial matter of hooking the CL up to the LIF file on the PC via the PILBox - and then run a program on teh 41 to fix the updates. Go and make dinner, eat and stroll back to the 41CL and voilá - it's done. Assuming you can create such a LIF volume with all the ROM images (a big if, IMHO) the update via HP-IL will take MUCH longer than dinner! When updating only the changed images, it can still take 3-4 hours via direct serial transfer, so via HP-IL on the '41, this will take days.... It seems that simply ordering a$30 cable (plus ship to Europe, but still) makes more sense, but you may have more free time than I do. In any case, I agree it's an interesting challenge. After many hours of exploring ROM downloads via HP-IL (via HEPAX and/or CCD tools, plus file conversions, and tracking ROM image IDs and sequence, etc. etc.) I caved and went the cable route. But I'd love to see your solution, so please don't take this as criticism.
It would be a solution to only update the changed files - or only the files you want to update (or add). The problem with the serial cable solution for the CL is that it requires a syncronous solution - i.e. the PC and the 41 side going back and forth between each image. With this LIF solution, you would create the LIF file and be done with the PC side and let the 41 do the rest without interruption.
As for ordering a new cable... the connector on the CL board is damaged. A new cable doesn't help.
11-08-2017, 10:17 PM
Post: #18
rprosperi Senior Member Posts: 2,388 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
(11-08-2017 08:07 PM)Geir Isene Wrote: It would be a solution to only update the changed files - or only the files you want to update (or add). The problem with the serial cable solution for the CL is that it requires a syncronous solution - i.e. the PC and the 41 side going back and forth between each image. With this LIF solution, you would create the LIF file and be done with the PC side and let the 41 do the rest without interruption.
As for ordering a new cable... the connector on the CL board is damaged. A new cable doesn't help.
A broken connector on the CL board changes things, I agree there.
I guess I don't see why it's a problem to go back and forth between the CL and PC serving the needed ROM images and a CL and a LIF volume supplying the needed ROM images, but as I said, I do look forward to this solution so will follow along.
You will have to somehow manage the FLDB mapping of ROM images to addresses, which is done automatically using the clserver software, but the CLUPDATE manual may provide some useful information on how to do that.
--Bob Prosperi
11-15-2017, 09:00 PM
Post: #19
Michael Fehlhammer Member Posts: 177 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
I really do not want to be a spoil-sport, I am interested in an HP-IL solution for updating the CL as well.
But there is a severe drawback of the HP-IL solution you have to be aware of:
via HP-IL the usual ROM reading and writing routines ( HEPAX, CCD, Eramco ...) only use 10 bit words, since a normal HP41 (non-CL) sees and uses only these 10 bits. ( Some 'unpacked' ROM formats seem to use 16 bit words, but the upper 6 bits are all zero and ignored ).
Within the 41CL, 16 bit words are used; the additional 6 bits can be used to control the CL speed mode, for example.
Some modern rom images, especially those which can be seen as CL operating system extensions, make use of the additional bits. Such images may only be transferred without loss of information via CL serial connection.
You could overcome this problem by defining your own CL - ROM file format and by programming the special 16 bit read and write routines.
11-15-2017, 09:10 PM
Post: #20
Geir Isene Senior Member Posts: 520 Joined: Dec 2013
RE: pyILPER: Read and write HP-41 ROM/MOD files
Interesting. What you are saying is that there is no way that e.g. the HEPAX READROM function could transfer these modules without breaking them?
I am open for suggestions on how to go about ensuring no additional data in modern roms are lost while transferring the roms over HP-IL.
It would be a shame if the venerable HP-IL module with the excellent PILbox could not be used for fully updating an HP-41CL.
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http://math.stackexchange.com/questions/182247/visualizing-quotient-groups-mathbbr-q | # Visualizing quotient groups: $\mathbb{R/Q}$
I was wondering about this. I know it is possible to visualize the quotient group $\mathbb{R}/\mathbb{Z}$ as a circle, and if you consider these as "topological groups", then this group (not topological) quotient is topologically equivalent to a circle.
But then, what does $\mathbb{R}/\mathbb{Q}$ look like?
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It looks like a mess... I doubt there is anything more useful you can say about it. – Mariano Suárez-Alvarez Aug 13 '12 at 23:25
It looks like a point, since with the quotient topology, every continuous function $\hspace{1.6 in}$ from $\:\mathbb{R}/\mathbb{Q}\:$ to a Hausdorff space is constant. $\;\;$ – Ricky Demer Aug 13 '12 at 23:28
@MarianoSuárez-Alvarez: I believe it's actually isomorphic to $\mathbf R$ (as a group or $\mathbf Q$-vector space), since $\mathbf R$ is actually a direct sum of $\mathfrak c$ copies of $\mathbf Q$ (obviously, since it is a vector space over rationals), and we're taking the quotient by one of the copies in the sum (or a one-dimensional subspace). So not that much of a mess. – tomasz Aug 14 '12 at 0:20
@tomasz: But infinitely generated vector spaces over the rationals, or worse: $\mathbb Z$-modules, give you no intuition on how something "looks like". The rationals themselves look like a mess. They are totally disconnected without isolated points, and totally disconnected spaces with no isolated points are messy in terms of visualization. Approaching this as a vector space is in fact a rather formal way of looking at this. – Asaf Karagila Aug 14 '12 at 0:57
@tomasz, indeed, that is clear (every infinite dimensional vector space is isomorphic to each of its quotients over a finite dimensional subspacees) but here we are looking at the quotient as a topological group. – Mariano Suárez-Alvarez Aug 14 '12 at 1:42
So, you say that the group (not topological) quotient of $\mathbb{R}/\mathbb{Z}$ is topologically equivalent (i.e., homeomorphic) to the circle. However, this doesn't make any sense unless you have a topology on $\mathbb{R}/\mathbb{Z}$! More the point is that a topological group like $\mathbb{R}$ has both a topological structure and a group structure. Now, when you form the group quotient $\mathbb{R}/\mathbb{Z}$, it can be given a topological space in a natural way, in particular, via the quotient topology. Notice that when we do this we again get a topological group (i.e., the quotient group operations are continuous with respect to the quotient topology). Furthermore, the quotient $\mathbb{R}/\mathbb{Z}$ (as a topological space) is homeomorphic to the circle.
Now, in the case of your question, the quotient topology on $\mathbb{R}/\mathbb{Q}$ is the trivial topology. This is not hard to prove since preimages of open sets must be open and saturated. Thus if such a preimage is nonempty, it contains an open interval, and since it is saturated, it must contain all real numbers which differ by a rational from a point in this interval. It is then easy to see that this set must be all of $\mathbb{R}$. Thus the only saturated open sets of $\mathbb{R}$ are $\emptyset$ and $\mathbb{R}$ itself. Hence the quotient topology is trivial. Furthermore, it is trivial that any map into a space with the trivial topology is continuous, so the quotient group operations on $\mathbb{R}/\mathbb{Q}$ are again continuous. So we again have a topological group, albeit not a very interesting one because it isn't very interesting as a topological space. As far as what this space "looks" like, it is similar to a one point space for the reason Ricky mentioned in the comments. However, it is not really easy to visualize since it is not homeomorphic to any subspace of $\mathbb{R}^n$ equipped with the subspace topology (because it is not Hausdorff, or any one of a number of other reasons).
Edit: I should have added that whenever you have a topological group and form the quotient in the way we did above the result is always a topological group. However, unless the original normal subgroup is closed, the resulting quotient group will not even be $T_0$ as a topological space. Thus it is only really interesting to form the quotient when the set by which you quotient out is closed. This explains why $\mathbb{R}/\mathbb{Z}$ is interesting as a topological group, but $\mathbb{R}/\mathbb{Q}$ is not.
-
What I meant by "group quotient, not topological quotient" didn't mean there was no topology, rather it meant that when considering "$\mathbb{R}/\mathbb{Z}$" as a topological space, you use the topology of $\mathbb{R}/\sim$ with $\sim$ being the coset-forming equivalence relation, as opposed to the relation given by $a \sim b \iff a, b \in \mathbb{Z}$. – mike4ty4 Aug 14 '12 at 3:06
@mike4ty4: I assumed that was what you were getting at, it just wasn't clear to me when I originally read the question. My apologies. – J. Loreaux Aug 14 '12 at 23:03
It really depends on what you think about as visualizing.
The group $\mathbb Z$ is discrete, so between two successive points there is a part which looks a bit like $\mathbb R$. The result, if so, is somewhat close to being $\mathbb R$.
On the other hand, $\mathbb Q$ is a dense subgroup of $\mathbb R$. This means that it gets a lot messier. Not without a good reason too, we can usually imagine things which have shape, things which can be measured.
Any set of representatives for $\mathbb R/\mathbb Q$ cannot be measured. This tells you that it is practically impossible to visualize this quotient in the same sense that we would imagine a circle, a ball, or even if we try really hard and we imagine a four-dimensional space.
Furthermore, using the axiom of choice we can create such set of representatives; however without the axiom of choice this quotient might not even be linearly ordered. Namely, it forms a set which cannot be linearly ordered. In contrast, $\mathbb R/\mathbb Z$ is a circle, or a half-open interval (where we identify the endpoints), even without the axiom of choice.
This tells you even more: you need the axiom of choice to impose an order on this set. Just a linear order, not even a well-order. Therefore imagining this as a linearly ordered set is even harder than we may believe at first.
My suggestion is not to try and visualize it. Accept this as a formal object which you can understand to some extent, but not see. Move on with this. Eventually, after running into infinitary objects ($\ell^2$, for example) and succeeding in visualizing those -- come back to this one, then you might be able to pull this off.
-
In my long but not very broad experience, $\mathbb{R}/\mathbb{Q}$ is not a natural object: it never came up anywhere near my own work. So I would consider it a curiosity, and in some sense not worthy of our trying to visualize it. – Lubin Aug 14 '12 at 1:12
@Lubin: That would immensely depend on what your experience is. Most people could say that sets of cardinality larger than $2^\frak c$ are not naturally occurring, but in set theory they occur all the time. – Asaf Karagila Aug 14 '12 at 1:13
I found this: en.wikipedia.org/wiki/Vitali_set I'm a little amazed that what at first looks like an innocuous quotient is such a crazy, pathological, and bizarre entity. – mike4ty4 Aug 14 '12 at 5:29
If you ignore topology, it's pretty much the same as $\mathbf R$.
Notice that $\mathbf R$ is a $\mathfrak c$-dimensional vector space over $\mathbf Q$, of which $\bf Q$ is a one-dimensional subspace. Taking the quotient $\bf R/\bf Q$ is actually taking the quotient of a $\mathfrak c$-dimensional vector space by a one-dimensional subspace, which is again a vector space, and is still $\mathfrak c$-dimensional (because $1<\mathfrak c$ ;) ), so it is isomorphic to $\bf R$ as a vector space over $\bf Q$, and in particular as a group.
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https://chemistry.stackexchange.com/questions/49477/why-isnt-thallium-triiodide-stable | # Why isn't thallium triiodide stable?
We know that stability of higher oxidation state (+3) decreases and stability of lower oxidation state (+1) increases, so thallium is most stable in its +1 oxidation state due to the inert pair effect and paired s electrons.
In the compound $\ce{TlI3}$, since $\ce{I3}$ exists as a linear molecule $\ce{I3-}$ so the oxidation number of $\ce{Tl}$ is +1 which is stable and the compound should exist but I have read in many of the books that it does not exists. Why is it so?
• Which books? Thallium Tri-Iodide,Tl+(I3)-, does exist. Its Thallium (III) Iodide, Tl3+ (I-)3 to my knowledge, does not. Interestingly the [TlI4]- ion does, and contains Tl3+. Apr 14 '16 at 8:19
• Note that, according to IUPAC nomenclature, the name tris(iodide) must be used for compounds containing $\ce{3I-}$ rather than triiodide, which is used for $\ce{I3-}$. Therefore, correct names for $\ce{TlI3}$ are “thallium tris(iodide)”, “thallium(III) iodide”, and “thallium(3+) iodide”. Correct names for $\ce{Tl(I3)}$ are “thallium triiodide(1−)”, “thallium(I) (triiodide)”, and “thallium(1+) (triiodide)”.
– user7951
May 6 '16 at 14:53
Thallium triiodide does exist and it is formulated as $\ce{Tl+I3-}$ and not $\ce{Tl^3+(I^{-})3}$. Thus $\ce{TlI3}$ is a thallium(I) compound and contains the triiodide ion, $\ce{I3^{−}}$. This is confirmed by considering the standard potential which indicate that $\ce{Tl(III)}$ is rapidly reduced to $\ce{Tl(I)}$ by the iodide.
\begin{align} \ce{Tl^3+ + 2e− &-> Tl+} & E^\circ &= +1.252~\mathrm{V} \\ \ce{I2 + 2e- &-> 2I-} & E^\circ &= +0.5355~\mathrm{V} \\ \hline \ce{Tl^3+ + 2 I- &-> Tl+ + I2} & E^\circ_\mathrm{cell} &= +0.717~\mathrm{V} > 0 \\ \hline \end{align}
$\ce{TlI3}$ can be prepared by the evaporation of stoichiometric quantities of $\ce{TlI}$ and $\ce{I2}$ in concentrated aqueous $\ce{HI}$, or by reacting $\ce{TlI}$ with $\ce{I2}$ in ethanol. | {"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": 0.9974030256271362, "perplexity": 4146.7236688478815}, "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/1642320301488.71/warc/CC-MAIN-20220119185232-20220119215232-00559.warc.gz"} |
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# For each customer, a bakery charges p dollars for the first
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19 Sep 2006, 16:19
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For each customer, a bakery charges p dollars for the first loaf of bread bought by the customer and charges q dollars for each additional loaf bought by the customer. What is the value of p ?
(1) A customer who buys 2 loaves is charged 10 percent less per loaf than a customer who buys a single loaf.
(2) A customer who buys 6 loaves of bread is charged 10 dollars.
[Reveal] Spoiler: OA
Last edited by Ergenekon on 10 Feb 2015, 06:17, edited 2 times in total.
Edited the question and added the OA.
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19 Sep 2006, 17:34
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Cost for first loaf = p
Cost for remaining = q
S1: Cost of two loaves = p+q
Price /loaf = (p+q)/2 = 0.9p (10% discount)
p+q = 1.8p
or q = 0.8p
or 4p -5q = 0
Not sufficient.
S2: 10 = p+5q
Not sufficient.
S1 & S2:
p+5q = 10
4p-5q=0
or 5p = 10
p = 2
q = 4x2/5 = 1.6
Sufficient w/ 2 equations.
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20 Sep 2006, 06:44
i cant get the warding of st one...........Help guys
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20 Sep 2006, 06:58
Statement 1 says that when you buy 2 loaves of bread instead of 1, you get a discount of 10% per loaf.
If you buy 2, then the cost is (p+q)
Cost per loaf = (p+q)/2 = 0.5p+0.5q
If you buy 1, the cost is p.
Cost per loaf = p/1 = p
You get a 10% discount per loaf..
i.e. 0.5p+0.5q = 0.9p
or 0.5q = 0.4p
or 5q = 4p
yezz wrote:
i cant get the warding of st one...........Help guys
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20 Sep 2006, 08:58
Statement 1 says that when you buy 2 loaves of bread instead of 1, you get a discount of 10% per loaf.
If you buy 2, then the cost is (p+q)
Cost per loaf = (p+q)/2 = 0.5p+0.5q
If you buy 1, the cost is p.
Cost per loaf = p/1 = p
You get a 10% discount per loaf..
i.e. 0.5p+0.5q = 0.9p
or 0.5q = 0.4p
or 5q = 4p
but dont you think that the part in read is the average cost per loaf not cost per loaf
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20 Sep 2006, 09:04
For each customer, a bakery charges p dollars for the first loaf of bread bought by the customer and charges q dollars for each add'l loaf bought by the customer. What's the value of p?
(1) A customer who buys 2 loaves is charged 10% less per loaf than a customer who buys a single loaf.
(2) A customer who buys 6 loaves of bread is charged 10 dollars.
general formula to calculate price
x = p+nq where n is number of loafs in excess of one
from one
original price of two loafs is = p+q
fro one i think it means
p+q = 2(0.9)p = 1.8p
thus 0.8p=q
from two
p+5q = 10
both together
p+5(0.8p) = 10
thus 5p = 10 and p = 2 and q = 1.6
what is my mistake here
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20 Sep 2006, 11:11
I used a combination of math and some guess work.
I have seen these type of problems before and made the mistake of thinking it was E in the past. However this time I knew it was D ...From the two stems it looks like you are going to get two equations with two variables which one can solve. I just made the sure the equations weren't equal to each other when I chose D...I didn't actually work all the way through the problem.
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13 Aug 2013, 01:21
What would be the equations for this question?
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Re: For each customer, a bakery charges p dollars for the first [#permalink]
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13 Aug 2013, 02:27
Expert's post
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fozzzy wrote:
What would be the equations for this question?
For each customer, a bakery charges p dollars for the first loaf of bread bought by the customer and charges q dollars for each additional loaf bought by the customer. What is the value of p ?
(1) A customer who buys 2 loaves is charged 10 percent less per loaf than a customer who buys a single loaf:
Price of 2 loaves = $(p+q). Price per loaf =$(p+q)/2
Price of a single loaf = \$p.
Given that (p+q)/2=0.9p.
Two unknowns. Not sufficient.
(2) A customer who buys 6 loaves of bread is charged 10 dollars --> p+5q=10. Not sufficient.
(1)+(2) We have two distinct linear equations with two unknowns: (p+q)/2=0.9p and p+5q=10, thus we can solve for both p and q. Sufficient..
Hope it's clear.
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Display posts from previous: Sort by | {"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": 0.23394611477851868, "perplexity": 8240.366577255638}, "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/1461860127870.1/warc/CC-MAIN-20160428161527-00011-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.albert.io/ie/single-variable-calculus/derivative-of-square-root-of-inverse-tangent-function | Free Version
Moderate
# Derivative of Square Root of Inverse Tangent Function
SVCALC-SXKOVJ
Find the derivative of the function:
$$f\left( x \right) =\sqrt { \arctan { x } }$$
Select ALL that apply.
A
$\cfrac{1}{2 \sqrt{\arctan(x)}}$
B
$\cfrac{1}{2} \sqrt{\arctan(x)}$
C
$\cfrac{1}{2(1+x^2)\sqrt{arctan(x)}}$
D
$\cfrac{1}{2(1+x^2)(arctan(x))^{1/2}}$ | {"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": 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": 0.8325695991516113, "perplexity": 5937.4540333055575}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218187519.8/warc/CC-MAIN-20170322212947-00443-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/101489/how-to-show-with-maxwells-equations-that-nonaccelerating-charges-dont-radiate | # How to show with Maxwells Equations that nonaccelerating charges dont radiate? [closed]
How to show with Maxwells Equations that nonaccelerating charges don't radiate?
-
## closed as off-topic by Chris White, Brandon Enright, John Rennie, Waffle's Crazy Peanut, Kyle KanosMar 6 '14 at 13:51
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – Chris White, Brandon Enright, John Rennie, Waffle's Crazy Peanut, Kyle Kanos
If this question can be reworded to fit the rules in the help center, please edit the question.
According to Maxwell's equations, does a charge at rest radiate? – Alfred Centauri Mar 1 '14 at 23:54
true, the at rest situation is much more simple though (the current density is zero). But what about a uniformly moving charge (the current density is non-zero, and v times the dirac delta function)? Thats's when Maxwell's Equations are not so obvious anymore, or at least to me they are not. – user1886681 Mar 2 '14 at 0:02
But motion is relative. We can't say that the charge is absolutely uniformly moving. Consider the case that, according to you, the charge is at rest. Now, according to another reference frame, in relative motion with respect to you, the charge is uniformly moving. But according to you, the charge is motionless. All we can say is that the two reference frames have relative motion, i.e., that (uniform) motion is a relationship between reference frames, not a property of an object and its associated reference frame. – Alfred Centauri Mar 2 '14 at 1:30
Excellent...of course! Thank you for that! – user1886681 Mar 2 '14 at 1:33
Alfred Centauri has almost answered the question for you (actually he has), but he's using knowledge about and properties of Maxwell's equations that it sounds as though you haven't yet met.
Maxwell's equations are covariant with respect to Lorentz transformations. That's a fancy way of saying that they keep their exact same form, and must foretell the same physics for all inertial observers. It doesn't matter whether or not I am moving uniformly relative to a charge: as long as I am not interacting with that charge (i.e. I am being an uncharged, passive observer and not part of the physics), Maxwell's equations must foretell the same physics. Therefore a charge uniformly moving with respect to me cannot radiate because one stationary relative to me does not.
Indeed historically this is what special relativity was all about. Einstein's famous 1905 relativity paper (there were several famous ones on vastly diverse fields of physics written by him that year) was called "Zur Elektrodynamik bewegter Körper" (on the electrodynamics of moving bodies), and he took as a beginning point this invariance of Maxwell's equations. His reasoning was that Maxwell's equations were really the only phsyics we knew at the time that accurately described something moving very fast, to wit: light, and therefore we should give them more weight than the assumed Galilean laws of relativity, whose validity for very swift things we had very few ways to check at the time.
So Einstein upheld the simple proposition that physics should be the same for all uniformly moving observers (as Alfred Centauri's other comment succinctly puts it the physics is a property of the charge, not of who observes it or their reference frames) and assumed Maxwell's equations were correct: thence derived the Lorentz transformations and special relativity to replace Galilean relativity from these assumptions, thus explaining the negative result for the Michelson-Morley experiment.
# Edit in Response to Comment:
User Suresh makes the following point:
The principle of relativity is usually attributed to Galileo. One doesn't need SR to see that a charge moving with uniform velocity doesn't radiate. Choose the velocity to be so small so that SR effects can be neglected...
(actually I say more about Galileo and relativity here).
My response is:
Suresh, you are right, that is a good point. As you can say, you can just ignore Maxwell's equations and say by the principle of relativity, the uniformly moving one can't radiate if a stationary one doesn't, and this answers the OP's question in the sense that it gives the right physics. But then the OP would have to use special relativity, and not Galilean, to see that Maxwell's equations don't tell us anything different. His/her question was about Maxwell's equations, and using these to prove no radiation. He/she can't do this with Galilean relativity. The fact that Galilean relativity makes Maxwell's equation seem to foretell different physics for different observers throws up the interesting history.
-
except it wasn't Einstein who did this all, but Poincare who was friends with Einstein's advisor, Minkowsky. for some reason, it took about 30 years for Einstein to cite Poincare's work, of which he was told by Minkowsky before starting a work on special relativity – Aksakal Mar 2 '14 at 0:45
@Aksakal Yes I agree the principle of relativity and how it applied was thoroughly studied by Poincaré and others: my history knowledge is not detailed but I believe it was Einstein that recognised and embraced that the needed transformations would include time dilation. Have you read Poincaré's relevant papers (I haven't: would appreciate its name and links if you have one)? – WetSavannaAnimal aka Rod Vance Mar 2 '14 at 1:00
i'll try to look up Poincare's work, it was in some philosophy journal, thus wasn't very technical – Aksakal Mar 2 '14 at 1:04
@Aksakal That would be great if you can find one, thanks! – WetSavannaAnimal aka Rod Vance Mar 2 '14 at 1:05
@suresh Also, this may be interesting. – WetSavannaAnimal aka Rod Vance Mar 2 '14 at 3:06 | {"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": 0.8300027847290039, "perplexity": 749.4827306634041}, "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-2015-22/segments/1432207928817.87/warc/CC-MAIN-20150521113208-00269-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://repo.scoap3.org/record/30875 | # Yukawa unification with four Higgs doublets in supersymmetric GUT
Dutta, Bhaskar (Department of Physics, Texas A&M; University, College Station, TX, 77843-4242, USA) ; Mimura, Yukihiro (Department of Physics, National Taiwan University, Taipei, 10617, Taiwan) (Institute of Science and Engineering, Shimane University, Matsue, 690-8504, Japan)
07 February 2019
Abstract: We discuss the Yukawa coupling unification, which can emerge in the grand unified theory, in the context of scenarios with more than one pair of Higgs doublet since the current LHC constraint has become a problem for the Yukawa unification scenarios with just one pair of Higgs doublet. More than one pair of Higgs doublets can easily arise in missing partner mechanism which solves the doublet-triplet splitting problem. In such a scenario, the Yukawa unification occurs at a medium tanβ value, e.g., ∼ 30, which corresponds to much smaller threshold corrections compared to usual large tanβ scenario for t−b−τ unification in the context of SO(10) and b−τ unification in the context of SU(5) models. Further, we show that an additional Higgs doublet pair lowers the sensitivity of the radiative symmetry breaking of the electroweak vacuum.
Published in: Physics letters B 790 (2019) 589-594
DOI: 10.1016/j.physletb.2019.01.065
Fulltext:
XML PDF PDF (PDFA) | {"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": 0.9042152166366577, "perplexity": 2525.018678079994}, "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-2019-18/segments/1555578616424.69/warc/CC-MAIN-20190423234808-20190424015826-00003.warc.gz"} |
https://www.birs.ca/events/2018/5-day-workshops/18w5112/videos/watch/201802010904-Lopuhaa.html | Video From 18w5112: Shape-Constrained Methods: Inference, Applications, and Practice
Thursday, February 1, 2018 09:04 - 09:47
A central limit theorem for the $L_p$-loss of smooth Grenander-type estimators
Download this video (137M)
Other videos from this workshop | {"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": 0.2776201665401459, "perplexity": 16871.669896749278}, "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-2021-31/segments/1627046154320.56/warc/CC-MAIN-20210802110046-20210802140046-00190.warc.gz"} |
https://www.physicsforums.com/threads/hiyo-segway-away.183735/ | # Hiyo Segway away.
1. Sep 9, 2007
### Jimmy Snyder
I took a day trip to DC today. I had two purposes in mind. I wanted to see the mockup of the Webb telescope outside the Air and Space Museum and I wanted to take the Segway for a spin. It turns out that the telescope was only in DC for a short time and has long since gone, I know not where. But the Segway was great. For $45 you get a 1 hour tour of the city. I feel that I got my money's worth though I wouldn't pay 45 cents for the tour, It gives you a chance to get used to the Segway. And now that I have done so, I will probably never again as I cannot imagine what purpose I would put it to. There is an overlap in the distances that I am used to walking and those I am used to biking and so there is no interval in which the$5000 Segway would be the right choice for me.
By the way, they got the helicopter off the Washington Monument.
2. Sep 9, 2007
### Staff: Mentor
What's the Segway? I plan to return to Dc for the Museums, is it a shuttle from hotels. to restaurants. There are some awesome restaurants within close walking distance from the Smithsonian.
3. Sep 9, 2007
### Jimmy Snyder
No, it's a thingamabob. www.segway.com
That there are.
4. Sep 9, 2007
5. Sep 9, 2007
### Cyrus
You picked a nice time to visit. The weather has been aces.
6. Sep 10, 2007
### l46kok
I actually saw cops on segways coupe of days ago.. going down the street in New York....
7. Sep 10, 2007
### Chi Meson
8. Sep 10, 2007
### Staff: Mentor
Unfortunately, the Webb telescope model had a limited display during May 9-12 this year in front of the Air and Space Museum. It's apparently making a tour.
http://www.nasa.gov/vision/universe/starsgalaxies/webb_slinger.html (same article in spacedaily)
It should be out and about periodically. Apparently it was in Dublin (Ireland) in June 2007.
http://www.nasa.gov/centers/goddard/news/topstory/2007/dublin_webb.html
http://jwst.gsfc.nasa.gov/
A good place to stay in Washington DC is the Hotel Washington - http://www.hotelwashington.com/ - if one doesn't have a friend with whom one can stay. It very near the White House and Mall.
Last edited: Sep 10, 2007
9. Sep 10, 2007
### Jimmy Snyder
The weather was aces. It was a Sunday, so you would think the place might be crowded, but I only encountered a very few opportunities to run anyone down. Perhaps they were all in church. Praying.
Thanks for those links. Those photos are the first ones that I have seen. Up til now I just saw cartoony images. When I asked at the Air and Space museum, the lady behind the counter started looking up the tour schedule on the web (more Webb on the web, haha) but I foolishly told her not to as I could do it myself. Unfortunately I have not found a schedule. I did send an e-mail to the manager of media relations at Northrop Grumman.
I haven't stayed over in Washington in quite a while. When I was single, I used to stay at the Harrington Hotel which is practically on the mall. They used to have a really good buffet style cafeteria, but that was 30 years ago. It's gone now.
10. Sep 10, 2007
### Staff: Mentor
You might also send an email to NASA PR, or Goddard PR.
I have some contacts at Goddard I could ask.
11. Sep 10, 2007
### Jimmy Snyder
Ah, the memories. This is way off topic, but then it's my topic. During the cherry blossom festival, I invited a pretty young thing to accompany me to Washington. Two roundtrip tickets from Philly and a night at the Harrington. Cherry blossoms and museums. A pleasant time was had by all. I recommend it.
12. Sep 10, 2007
### zoobyshoe
They need to install seats on these things.
13. Sep 10, 2007
### Cyrus
Probably hung over from saturday :tongue2:
14. Sep 10, 2007
### Mk
Hm, found some sketches of a hypothesized futurific personal mobility device based on the Segway idea:
15. Sep 10, 2007
### robphy
16. Sep 10, 2007
### zoobyshoe
That's the ticket! However, it shouldn't cost \$449.00 for the seat, (unless it's for the military).
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https://www.physicsforums.com/threads/electrons-in-electric-field.616863/ | # Electrons in electric field
1. Jun 27, 2012
### nks27
1. The problem statement, all variables and given/known data
Two metal plates, PQ AND RS are separated by a distance of 15 mm.
PQ maintained at potential of +100V relative to RS.
Beam of electrons of different kinetic energies directed a slit ,on plate PQ, at angle of 60 degrees to plate.
To find the K.E of electrons that 'just' reach the plate RS.
2. Relevant equations
work done by electric field = K.E of electrons
k.E = qV
3. The attempt at a solution
i cant figure how to work this out
Should potential energy also be considered?
Is my attempt at the question wrong?
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
2. Jun 27, 2012
### Infinitum
Hi nks27! Welcome to PF
I believe you want to use the energy conservation principle. So yes, you do need to include potential energy....
3. Jun 28, 2012
### nks27
i the use this formula
work done
against electric = K.E + P.E
field
K.E = work done - P.E
= qV - q/(4πεₒr)
but i dont get the answer.
Cud u plz correct my careless mistake if there's any?
4. Jun 28, 2012
### Saitama
From where did you get potential energy?
Instead using the energy conservation, you can use the equations of motion too.
5. Jun 28, 2012
### Infinitum
Umm no. How did you get q/(4πεₒr)??
The kinetic energy change will result in potential energy. Now, for the minimum kinetic energy you need the final velocity to have no x component(assuming vertical plates). So you have,
$$\frac{1}{2}m(v_x)^2 +\frac{1}{2}m(v_y)^2 = qV + \frac{1}{2}m(v_y)^2$$
6. Jun 28, 2012
### Infinitum
I would prefer conservation of energy, but this method works too
7. Jul 4, 2012
### nks27
sorry for late post.
substituting v_x = vcos60
and v_y = vsin60
q= 1.6 X 10^-19 and V= 100 V
i get v^2 to be 1.41 X 10^14
using it to calculate the k.E doesnt give the right answer :(
the " q/(4πεₒr) " was for electric potential .
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http://mathhelpforum.com/math-software/96826-maple-10-solve-complex-numbers.html | # Math Help - Maple 10: solve() and complex numbers
1. ## Maple 10: solve() and complex numbers
Hi all
i've been trying to solve 2 equation with two variables in Maple 10. I got 3 solutions which are complex,
i.e. a list A:=[[ x = ... , y = ...], [...], [...]].
now, A[1,1] gives me the x of the first solution, and I want the real part of it. But when I write Re(A[1,1]) it gives as an answer
R(x) = R(...). the problem is that afterwards I need to use this number (for emaple, in pointplot() ), and Maple doesn't let me.
How do I extract the number itself from Re(A[1,1]) (and not a string)? | {"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": 0.9977317452430725, "perplexity": 2121.1149481673906}, "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/1438042988922.24/warc/CC-MAIN-20150728002308-00269-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://accesspharmacy.mhmedical.com/content.aspx?bookid=1592§ionid=100669401 | Chapter 3
### CHAPTER OBJECTIVES
• Describe basic statistical methodology and concepts
• Describe how basic statistical methodology may be used in pharmacokinetic and pharmacodynamics study design
• Describe how basic statistical methodology may be used in critically evaluating data
• Describe how basic statistical methodology may be used to help minimize error, bias, and confounding, and, therefore, promote safe and efficacious drug therapy
• Provide examples of how basic statistical methodology may be used for study design and data evaluation
### VARIABLES
Several types of variables will be discussed throughout this text.1 A random variable is “a variable whose observed values may be considered as outcomes of an experiment and whose values cannot be anticipated with certainty before the experiment is conducted” (Herring, 2014). An independent variable is defined as the “intervention or what is being manipulated” in a study (eg, the drug or dose of the drug being evaluated) (Herring, 2014). “The number of independent variables determines the category of statistical methods that are appropriate to use” (Herring, 2014). A dependent variable is the “outcome of interest within a study.” In bioavailability and bioequivalence studies, examples include the maximum concentration of the drug in the circulation, the time to reach that maximum level, and the area under the curve (AUC) of drug level-versus-time curve. These are “the outcomes that one intends to explain or estimate” (Herring, 2014). There may be multiple dependent (aka outcome) variables. For example, in a study determining the half-life, clearance, and plasma protein binding of a new drug following an oral dose, the independent variable is the oral dose of the new drug. The dependent variables are the half-life, clearance, and plasma protein binding of the drug because these variables “depend upon” the oral dose given.
Discrete variables are also known as counting or nonparametric variables (Glasner, 1995). Continuous variables are also known as measuring or parametric variables (Glasner, 1995). We will explore this further in the next section.
1The 5th edition of Quick Stats: Basics for Medical Literature Evaluation was utilized for the majority of the following chapter (Herring, 2014). In order to discuss basic statistics, some background terminology must be defined.
### TYPES OF DATA (NONPARAMETRIC VERSUS PARAMETRIC)
There are two types of nonparametric data, nominal and ordinal. For nominal data, numbers are purely arbitrary or without regard to any order or ranking of severity (Gaddis and Gaddis, 1990a; Glasner, 1995). Nominal data may be either dichotomous or categorical. Dichotomous (aka binary) nominal data evaluate yes/no questions. For example, patients lived or died, were hospitalized, or were not hospitalized. Examples of categorical nominal data would be things like tablet color or blood type; there is no order or inherent value for nominal, categorical data.
Ordinal data are also nonparametric and categorical, but unlike nominal data, ordinal data are scored on a continuum, without a consistent level of magnitude ...
### Pop-up div Successfully Displayed
This div only appears when the trigger link is hovered over. Otherwise it is hidden from view. | {"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": 0.8035796284675598, "perplexity": 2938.809445844627}, "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/1516084887423.43/warc/CC-MAIN-20180118151122-20180118171122-00092.warc.gz"} |
https://philpapers.org/rec/PISCAM | # Conceptual and Mathematical Structures of Mechanical Science in the Western Civilization around 18th Century
Almagest 4 (2):86-21 (2013)
Authors Raffaele Pisano Université des Sciences et Technologies de Lille Abstract One may discuss the role played by mechanical science in the history of scientific ideas, particularly in physics, focusing on the significance of the relationship between physics and mathematics in describing mathematical laws in the context of a scientific theory. In the second Newtonian law of motion, space and time are crucial physical magnitudes in mechanics, but they are also mathematical magnitudes as involved in derivative operations. Above all, if we fail to acknowledge their mathematical meaning, we fail to comprehend the whole Newtonian mechanical apparatus. For instance, let us think about velocity and acceleration. In this case, the approach to conceive and define foundational mechanical objects and their mathematical interpretations changes. Generally speaking, one could prioritize mathematical solutions for Lagrange’s equations, rather than the crucial role played by collisions and geometric motion in Lazare Carnot’s operative mechanics, or Faraday’s experimental science with respect to Ampère’s mechanical approach in the electric current domain, or physico-mathematical choices in Maxwell’s electromagnetic theory. In this paper, we will focus on the historical emergence of mechanical science from a physico-mathematical standpoint and emphasize significant similarities and/or differences in mathematical approaches by some key authors of the 18th century. Attention is paid to the role of mathematical interpretation for physical objects. Keywords Physics Mathematics History of science Categories Philosophy, Misc (categorize this paper) Options Mark as duplicate Export citation Request removal from index
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## References found in this work BETA
No references found. | {"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": 0.8158264756202698, "perplexity": 2349.154849224269}, "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/1659882571086.77/warc/CC-MAIN-20220809185452-20220809215452-00603.warc.gz"} |
https://lists.debian.org/debian-tex-maint/2010/02/msg00001.html | Bug#566158: texlive-binaries: hyperlinks do not work in xdvi
Hi all,
On Do, 21 Jan 2010, Martin Ziegler wrote:
> Hyperlinks in dvi-files created with \usepackage[hypertex]{hyperref}
> do not work with the new xdvi.
>
> The anchor text has blue color, as it should be. But if the mouse
> pointer is over the text, it does not change its appearance and left
> clicks have no effect.
I cannot reproduce that.
On Fr, 22 Jan 2010, Martin Ziegler wrote:
> \documentclass{article}
> \usepackage[hypertex]{hyperref}
> \newtheorem{remark}{Remark}
> \begin{document}
> This is a link to Remark \ref{remark} on the next page.
> And this a link to Section \ref{section}.
> \newpage
> \section{section}\label{section}
> \begin{remark}\label{remark}
> Remark
> \end{remark}
> \end{document}
I ran this test file through latex as it is installed on Debian, and
also through latex of upstream TL.
I tested both with both xdvi.
In all cases the pointer changes to a small hand with a finger when
moving over the anchor, and left clicking on it jumps to the
right page.
Can someone explain me what I do not see here?
Maybe your mouse sensitivity is too hight? The spot where you have to
move the mouse over is quite small.
Best wishes
Norbert
------------------------------------------------------------------------
Norbert Preining preining@{jaist.ac.jp, logic.at, debian.org}
JAIST, Japan TU Wien, Austria Debian TeX Task Force
DSA: 0x09C5B094 fp: 14DF 2E6C 0307 BE6D AD76 A9C0 D2BF 4AA3 09C5 B094
------------------------------------------------------------------------
PAPPLE (vb.)
To do what babies do to soup with their spoons.
--- Douglas Adams, The Meaning of Liff | {"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": 0.8545544147491455, "perplexity": 15411.567635872734}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698543035.87/warc/CC-MAIN-20161202170903-00183-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://socratic.org/questions/can-somehow-help-justify-if-my-solutions-to-the-problem-below-is-correct-and-gui | Algebra
Topics
# Can somehow help justify if my solutions to the problem below is correct and guide me on (b)?
Oct 19, 2016
The customer must travel $6 \frac{2}{3}$ Km to reduce average cost at $80$cents/kilometer.
#### Explanation:
b) let x be the asked kilometer for average cost at 80 cents per kilometer. Then $x \cdot 0.80 = 2 + \left(0.50 \cdot x\right) \mathmr{and} 0.80 x - 0.50 x = 2 \mathmr{and} 0.30 x = 2 \mathmr{and} x = \frac{2}{3} \cdot 10 = \frac{20}{3} = 6 \frac{2}{3} k m$[Ans]
Oct 19, 2016
For a distance of $6 \frac{2}{3}$ km the average cost will be $80 c$
#### Explanation:
$x$ is the number of Km travelled.
The total cost of travelling $x$ km is calculated as:
50c" per km" + $2" "larr 50c =$0.50
$\downarrow \text{ "darr " } \downarrow$
$0.5 \times \text{x" } + 2 = 0.5 x + 2$
To find the average cost per Km:
divide the total cost by the number of Km travelled.
Ave cost = $f \left(x\right) = \frac{0.5 x + 2}{x}$
The average cost for $x$ km should be 80c rarr $0.8 $\frac{0.5 x + 2}{x} = 0.8 \text{ } \leftarrow \times x$$0.5 x + 2 = 0.8 x \text{ } \leftarrow$move x-terms to the right $2 = 0.8 x - 0.5 x$$2 = 0.3 x \text{ } \leftarrow$divide by 0.3 $\frac{2}{0.3} = x$$\frac{20}{3} = x$$x = 6 \frac{2}{3}$For a distance of $6 \frac{2}{3}$km the average cost will be $80 c\$
##### Impact of this question
175 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 22, "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": 0.8300424218177795, "perplexity": 2841.818585815201}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986661296.12/warc/CC-MAIN-20191016014439-20191016041939-00395.warc.gz"} |
http://www.science.gov/topicpages/m/magnetic+field+configurations.html | Note: This page contains sample records for the topic magnetic field configurations from Science.gov.
While these samples are representative of the content of Science.gov,
they are not comprehensive nor are they the most current set.
We encourage you to perform a real-time search of Science.gov
to obtain the most current and comprehensive results.
Last update: November 12, 2013.
1
A theory of low-frequency drift (universal) instabilities in a nonuniform collisionless plasma is developed for general magnetic field configurations including trapped particle effects, rather than the plane geometry which has previously received most attention. A type of energy principle shows that the special equilibrium distribution F(?, ?), of interest in minimum-B mirror configurations, is absolutely stable to these modes provided
P. H. Rutherford; E. A. Frieman
1968-01-01
2
Long-lived, large-scale magnetic field configurations with similar total fluxes exist in at least three very different, although related kind of stars: upper main sequence stars, white dwarfs, and neutron stars (e.g. Reisenegger 2001). Much or all of the volume of these stars is stably stratified, so there is no convection that could maintain these fields through dynamo processes (except in the cores of upper main sequence stars). Magnetohydrodynamic simulations of stably stratified stars (Braithwaite & Spruit 2004, 2006; Braithwaite & Nordlund 2006) suggest that configurations with linked poloidal and toroidal fields get spontaneously established and might be stable over long times. Physical arguments show that such configurations are in fact natural and that the stable stratification is likely to play a crucial role in their stability. Thus, contrary to assumptions in recent papers, the field is not force-free, and the fluid cannot be taken to be barotropic. Work is in progress to represent these fields analytically and investigate the conditions for their stability. In the case of neutron stars with strong enough fields, the stable stratification can be overcome by long-term, dissipative processes such as beta decays and ambipolar diffusion (Goldreich & Reisenegger 1992; Reisenegger et al. 2005), leading to the release of magnetic energy and potentially explaining the energy source for the "magnetar" phenomenon (Thompson & Duncan 1993, 1996).
Reisenegger, Andreas; Munoz, Francisco; Santos, Raul
3
A computational calibration model which can access the error budget for these measurements at all system levels is reported. The model is parametric since the total error contribution can be treated as an error function for the derived quantity, dependent on the various error sources. The possible measurement errors for the derived quantities to be critical at current mission specifications are found. Typical hard error sources, which are likely to remain following standard interspacecraft calibration, are identified. Orbit strategy, in particular, imposes constraints at the four spacecraft level. The sensitivity of the error functions to spacecraft configuration is investigated. The determination of the vector curl of the magnetic field is taken as a model measurement and the role of the divergence of the field for this quantity is discussed.
Dunlop, M. W.; Balogh, A.; Southwood, D. J.; Elphic, R. C.; Glassmeier, K.-H.; Neubauer, F. M.
1990-05-01
4
Due to geomagnetic anomaly strength can't be separated from magnetic field data, underwater vehicle localization based on geomagnetic anomaly needs vehicle draft depth information. For this localization method, we conclude that the simplest configuration be seven single-axis vector magnetometers to measure magnetic dipole gradients. The essential conditions and optimal measurement matrix of magnetic gradient tensor are analyzed, and a kind
Huang Yu; Sun Feng; Hao Yan-ling
2010-01-01
5
We present near-infrared imaging polarimetry of the silhouette young stellar object M17-SO1. When the continuum and the scattering components are separated, the 2.17-?m Br? image reveals the absorptive polarization vectors (i.e., magnetic field lines) exiting the mid-plane of the circumstellar envelope at relatively wide angles. Such a configuration may imply a slightly pulled-in, co-rotating frozen-in poloidal magnetic field.
Fujiyoshi, T.; Yamashita, T.; Sako, S.; Hough, J. H.; Lucas, P. W.
2011-11-01
6
The influence of the initial magnetic field distribution on spheromak formation and closed flux generation upon decay is studied using the NIMROD code. Previous spheromak simulations using the NIMROD code have demonstrated the formation of axisymmetric closed flux surfaces with decay of the magnetic field. The q profile within the closed flux region was non-monotonic with values q0˜0.8 and qmin˜0.5. As the configuration evolved, a m=1, n=2 mode led to localized magnetic field chaos resulting in a degradation of thermal energy confinement. Given the limited ability to control the evolution of the q profile within the closed flux region of a spheromak, we investigate the possibility of forming spheromak plasmas that avoid this deleterious mode by tailoring the initial magnetic field profile appropriately. Poloidal flux amplification during the formation process involves conversion of injected toroidal flux via a line-tied kink mode. By strengthening or weakening the initial magnetic field along the geometric axis of the flux conserver, we attempt to control the amount of flux amplification to produce higher or lower values of q throughout the closed flux surface region. Simulations are performed using a finite element grid that approximates the geometry of the Sustained Spheromak Physics Experiment. In collaboration with Bick Hooper and Bruce Cohen, Lawrence Livermore National Laboratories.
Cone, Giovanni
2005-10-01
7
PubMed
A physically based model for residential magnetic fields from electric transmission and distribution wiring was developed to reanalyze the Los Angeles study of childhood leukemia by London et al. For this exposure model, magnetic field measurements were fitted to a function of wire configuration attributes that was derived from a multipole expansion of the Law of Biot and Savart. The model parameters were determined by nonlinear regression techniques, using wiring data, distances, and the geometric mean of the ELF magnetic field magnitude from 24-h bedroom measurements taken at 288 homes during the epidemiologic study. The best fit to the measurement data was obtained with separate models for the two major utilities serving Los Angeles County. This model's predictions produced a correlation of 0.40 with the measured fields, an improvement on the 0.27 correlation obtained with the Wertheimer-Leeper (WL) wire code. For the leukemia risk analysis in a companion paper, the regression model predicts exposures to the 24-h geometric mean of the ELF magnetic fields in Los Angeles homes where only wiring data and distances have been obtained. Since these input parameters for the exposure model usually do not change for many years, the predicted magnetic fields will be stable over long time periods, just like the WL code. If the geometric mean is not the exposure metric associated with cancer, this regression technique could be used to estimate long-term exposures to temporal variability metrics and other characteristics of the ELF magnetic field which may be cancer risk factors. PMID:10495305
Bowman, J D; Thomas, D C; Jiang, L; Jiang, F; Peters, J M
1999-10-01
8
National Technical Information Service (NTIS)
Ion heating by application of rotating magnetic fields (RMF) to a prolate field-reversed configuration(FRC) is explored by analytical and numerical techniques. For odd-parity RMF (RMFo), perturbation analysis shows ions in figure-8 orbits gain energy at r...
A. S. Landsman S. A. Cohen
2007-01-01
9
SciTech Connect
The nonlinear dynamics of a single ion in a field-reversed configuration (FRC) were investigated. FRC is a toroidal fusion device which uses a specific type of magnetic field to confine ions. As a result of angular invariance, the full three-dimensional Hamiltonian system can be expressed as two coupled, highly nonlinear oscillators. Due to the high nonlinearity in the equations of motion, the behavior of the system is extremely complex, showing different regimes, depending on the values of the conserved canonical angular momentum and the geometry of the fusion vessel. Perturbation theory and averaging were used to derive the unperturbed Hamiltonian and frequencies of the two degrees of freedom. The derived equations were then used to find resonances and compare to Poincar{copyright} surface-of-section plots. A regime was found where the nonlinear resonances were clearly separated by KAM [Kolmogorov-Arnold-Mosher] curves. The structure of the observed island chains was explained. The condition for the destruction of KAM curves and the onset of strong chaos was derived, using Chirikov island overlap criterion, and shown qualitatively to depend both on the canonical angular momentum and geometry of the device. After a brief discussion of the adiabatic regime the paper goes on to explore the degenerate regime that sets in at higher values of angular momenta. In this regime, the unperturbed Hamiltonian can be approximated as two uncoupled linear oscillators. In this case, the system is near-integrable, except in cases of a universal resonance, which results in large island structures, due to the smallness of nonlinear terms, which bound the resonance. The linear force constants, dominant in this regime, were derived and the geometry for a large one-to-one resonance identified. The above analysis showed good agreement with numerical simulations and was able to explain characteristic features of the dynamics.
A.S. Landsman; S.A. Cohen; M. Edelman; G.M. Zaslavsky
2005-04-13
10
SciTech Connect
As applied to a tokomak, a magnetic trap for confinement of a plasma with an inverted field or a magnetic field reversed configuration (FRC) is one of the most promising alternatives of the systems with high {beta}. A brief review of the latest data on FRC and potential directions of using such configurations in addition to energy generation in thermonuclear reactors (TNRs) is proposed.
Ryzhkov, S. V., E-mail: ryzhkov@power.bmstu.ru [Bauman Moscow State Technical University (Russian Federation)
2011-12-15
11
PubMed
A long-lived field-reversed configuration (FRC) plasma has been produced in the C-2 device by dynamically colliding and merging two oppositely directed, highly supersonic compact toroids (CTs). The reversed-field structure of the translated CTs and final merged-FRC state have been directly verified by probing the internal magnetic field structure using a multi-channel magnetic probe array near the midplane of the C-2 confinement chamber. Each of the two translated CTs exhibits significant toroidal fields (B(t)) with opposite helicity, and a relatively large B(t) remains inside the separatrix after merging. PMID:23126880
Gota, H; Thompson, M C; Knapp, K; Van Drie, A D; Deng, B H; Mendoza, R; Guo, H Y; Tuszewski, M
2012-10-01
12
A long-lived field-reversed configuration (FRC) plasma has been produced in the C-2 device by dynamically colliding and merging two oppositely directed, highly supersonic compact toroids (CTs). The reversed-field structure of the translated CTs and final merged-FRC state have been directly verified by probing the internal magnetic field structure using a multi-channel magnetic probe array near the midplane of the C-2 confinement chamber. Each of the two translated CTs exhibits significant toroidal fields (Bt) with opposite helicity, and a relatively large Bt remains inside the separatrix after merging.
Gota, H.; Thompson, M. C.; Knapp, K.; Van Drie, A. D.; Deng, B. H.; Mendoza, R.; Guo, H. Y.; Tuszewski, M.
2012-10-01
13
Magnetic clouds are the most regular type of interplanetary coronal mass ejections (iCMEs), the counterparts in the interplanetary space of solar eruptions. Among other features, magnetic clouds are characterized by the smooth rotation of the magnetic field inside them. This has resulted in the current paradigm of associating magnetic clouds with twisted flux ropes. In this poster, we test this assumption by using a new model of solar eruption. Using results from a MHD simulation, we perform a blind'' reconstruction of the magnetic structure of the iCME using simulated satellite data and codes used for real observations of iCMEs. The 3-D structure of the iCME, in the simulation, does not exhibit much magnetic twist but when reconstructed from synthetic satellite data, it appears to have some characteristics of a magnetic cloud, due in particular to a writhe in the magnetic field lines. We eventually show that magnetic clouds do not necessarily have twisted magnetic field configuration.
2010-05-01
14
A large-scale stellar magnetic field that threads a surrounding accretion disk can mediate a torque that regulates the stellar rotation rate, can channel some of the inflowing matter to polar regions, and may play a role in the transfer of angular momentum from the disk to the surrounding interstellar medium, possibly in the form of a centrifugally driven wind. These processes have originally been studied in the context of X-ray pulsars but have more recently been recognized as possibly playing a central role also in young stellar objects. Motivated by these applications and guided by recent numerical simulations and by related work in solar flare research, we have begun to develop analytic and semianalytic tools for analyzing the complex phenomena involved in the magnetic star-disk interaction. In this contribution we report on preliminary results on the time evolution of the magnetic field structure brought about by the relative rotation between the disk and the star. Although the magnetic field has a negligible dynamical effect on the disk over a rotation period, its structure in the magnetosphere outside the disk can change drastically on this time scale. Specifically, as the disk rotates, the field in the magnetosphere evolves through a sequence of sheared force-free equilibria. To determine this sequence, we first model the situation where the field lines are frozen into both the star and the disk and where the medium outside the disk is perfectly conducting. Subsequently, however, we relax these assumptions and consider the effects of finite resistivity both inside and outside the disk. This research is supported in part by NASA grants NAG 5-3687 and NAG 5-1485.
Uzdensky, D. A.; Litwin, C.; Konigl, A.
1999-05-01
15
Using the variational method based on the Gaussian basis set, the authors investigate the 1u state of hydrogen molecular ion in a non-parallel magnetic field with respect to the fixed molecular axis. At sufficiently small field strength, the equilibrium configuration prefers the perpendicular orientation, in which the (relative) orientation ? between the magnetic field and the molecular axis is 90°. With increasing field strength, the orientation ? of the equilibrium configuration decreases, and is neither the parallel orientation nor the perpendicular orientation at field strength between 109 G and 2.35 × 1010 G. Meanwhile, more and more configurations with large orientations become unstable with respect to the dissociation H + p.
Song, Xuanyu; Gong, Cheng; Wang, Xiaofeng; Qiao, Haoxue
2013-08-01
16
In this work, new configurations of magnetic field transmitter coils (Tx) and receiver sensors (Rx) are studied for underwater (UW) geo-locations. The geo-location system, based on low frequency magnetic fields, uses measured vector magnetic fields at a given set of points in space. It contains an active pulsed direct current transmitter, tri-axial field receivers, and a global positioning system unit
Fridon Shubitidze; Alex Bijamov; Gregory Schultz; Jon Miller; Irma Shamatava
2011-01-01
17
Context. In the past the role of magnetic null points in the generation of electric currents was investigated mainly in the close vicinity of the null, with perturbations being applied at nearby boundaries, or for a magnetic null configuration with a dome-shaped fan. In the solar atmosphere, however, electric currents are generated by perturbations originating at the photosphere, far away from coronal 3D nulls, and the occurence of magnetic nulls with a dome-shaped fan is apparently not common. Aims: We investigate the consequences of photospheric motion for the development of electric currents in a coronal magnetic field configuration containing a null, located far away from the boundaries, and the influence of topological structures on the spatial distribution of the currents. Methods: We use a 3D resistive MHD code to investigate the consequences of photospheric plasma motion for the generation of currents in a coronal magnetic field containing a null. The plasma is considered fully compressible and is initially in hydrostatic equilibrium. The initial magnetic field is potential (current free). Results: The photospheric plasma motion causes magnetic field perturbations that propagate to the corona along the field lines at the local Alfvén speed. The Alfvénic wave perturbations correspond to a propagating current directed mainly parallel to the magnetic field. Perpendicular currents connect to return currents to close the current system. The magnetic perturbations eventually reach the vicinity of the null. However, the currents forming in and around the null, near the fan surface or near the spine field lines, are not always the strongest currents developing in the simulation box. In our simulation, the strongest currents develop close to the bottom boundary, where the plasma is moved, and below the null point, in a region where field line connectivity considerably changes. Conclusions: Our simulation shows that the presence of a magnetic null point does not necessarily mean that the strongest currents will form in or around the null, at the fan surface or at the spine. Our results indicate that regions of considerable change in field line connectivity are fundamental for the development of strong and thin current sheets. Regions of connectivity change are important because they combine perturbations that are generated at different locations on the Sun. Our results also suggest that it is more important how the perturbations are mapped and combined in regions of considerable connectivity change than what is the driver of the perturbations itself. The driver does not necessarily need to create strong currents where it is applied. However, when the perturbations produced by the driver combine in the regions of considerable connectivity change, they can increase the current in regions for which the length scale is much smaller than the characteristic length scale of the system. The location of regions of connectivity change, combined with the mapping of the perturbations to those regions, can be a useful tool to predict where and when solar flares will occur.
Santos, J. C.; Büchner, J.; Otto, A.
2011-01-01
18
SciTech Connect
A new concept on magnetic field of plasma production and confinement has been proposed to enhance efficiency of an electron cyclotron resonance (ECR) plasma for broad and dense ion beam source under the low pressure. The magnetic field configuration is constructed by a pair of comb-shaped magnet which has opposite polarity each other, and which cylindrically surrounds the plasma chamber. This magnetic configuration suppresses the loss due to ExB drift, and then plasma confinement is enhanced. The resonance zones of the fundamental and the second harmonics for 2.45GHz microwaves detach from the wall of the chamber. The connection length of the magnetic field lines through the resonance zone is elongated, and the confinement is better than that of the simple multipole magnetic field. The 2.45 GHz microwaves are fed from the side wall by the rod antenna. The electron density attained to about four times cutoff density for the 2.45GHz microwave at the low Ar pressure below 0.08Pa and also the low microwave power below 300W. We compare profiles of the electron density and temperature in the comb-shaped magnetic field configuration with those in the simple multipole magnetic field.
Kato, Yushi; Sasaki, Hiroshi; Kubo, Takashi; Sato, Fuminobu; Iida, Toshiyuki [Devision of Electrical, Electronic and Information Engineering, Graduate School of Engineering, Osaka Univ., 2-1 Yamada-oka, Suita-shi, Osaka 565-0871 (Japan); Asaji, Toyohisa [Devision of Electrical, Electronic and Information Engineering, Graduate School of Engineering, Osaka Univ., 2-1 Yamada-oka, Suita-shi, Osaka 565-0871 (Japan); Tateyama Machine Co., Ltd., 30 Shimonoban, Toyama 930-1305 (Japan)
2006-11-13
19
Experimental investigations into the effects of the magnetic field configuration near the channel exit on the plume of Hall thrusters were conducted. The magnetic field configuration near the channel exit is characterized by the inclination angle between the magnetic field lines and the thruster radial direction. Different inclination angles were obtained by varying the current ratio in the coils. The
Daren Yu; Jie Li; Hong Li; Yong Li; Binhao Jiang
2009-01-01
20
SciTech Connect
Steady state solutions, suitable for field-reversed configurations (FRCs) sustained by rotating magnetic fields (RMFs) are obtained by properly including three-dimensional effects, in the limit of large FRC elongation, and the radial component of Ohm's law. The steady electrostatic potential, necessary to satisfy Ohm's law, is considered to be a surface function. The problem is analyzed at the midplane of the configuration and it is reduced to the solution of two coupled nonlinear differential equations for the real and imaginary parts of the phasor associated to the longitudinal component of the vector potential. Additional constraints are obtained by requesting that the steady radial current density and poloidal magnetic flux vanish at the plasma boundary which is set at the time-averaged separatrix. The results are presented in terms of the degree of synchronism of the electrons with the RMF and compared with those obtained when radial current effects are neglected. Three important differences are observed when compared with the case without radial current density. First, at low penetration of the RMF into the plasma there is a significant increase in the driven azimuthal current. Second, the RMF amplitude necessary to access the high synchronism regime, starting from low synchronism, is larger and the difference appears to increase as the separatrix to classical skin depth ratio increases. Third, the minimum RMF amplitude necessary to sustain almost full synchronism is reduced.
Clemente, R. A.; Gilli, M. [Instituto de Fisica Gleb Wataghin, Universidade Estadual de Campinas, 13083-970 Campinas, SP (Brazil); Farengo, R. [Centro Atomico Bariloche and Instituto Balseiro, San Carlos de Bariloche (8400), RN (Argentina)
2008-10-15
21
RF wave propagation, in the large-diameter (45 cm) plasma produced by a planar, spiral antenna, was investigated with the cusp magnetic field configurations. Measurements of the excited magnetic field amplitude and the phase were examined by a helicon wave dispersion relation, and the obtained results were consistent with the calculated ones by Transport Analyzing System for tokamaK/Wave analysis by Finite element method (TASK/WF) code. The wave characteristics depended on the gradient and the magnitude of the magnetic field near the line cusp position, and in the neighborhood of this position, this wave showed different behavior from the helicon wave observed in a uniform field.
Takechi, Seiji; Shinohara, Shunjiro; Fukuyama, Atsushi
1999-06-01
22
The effect of magnetic field configurations on plasma flow velocity was investigated by measuring the parallel flow velocity (v?) using a charge-exchange recombination spectroscopy during neutral beam injection in three different toroidal mirror configurations of Heliotron J: high, standard and reversed mirror configurations. The magnetic ripple strengths, ?, for these mirror configurations were ? = 0.073 m-1, 0.031 m-1 and 0.027 m-1, respectively, at the normalized averaged minor radius ? = 0.07. The magnetic ripple strength is defined as ? = {<(?B/?l)2/B2>}1/2, where <…> is the flux surface averaged value and l is the length along the magnetic field line. At ? = 0.07, the parallel flow velocity in the high mirror configuration (v? ˜ 4 km s-1) was 2-3 times smaller than those in the standard and reversed mirror configurations (v? ˜ 10-12 km s-1). An anticipated interpretation is that the difference in the neoclassical damping force contributes to the difference in v? among the three mirror configurations.
Lee, H.; Kobayashi, S.; Yokoyama, M.; Mizuuchi, T.; Minami, T.; Harada, T.; Nagasaki, K.; Okada, H.; Minami, T.; Yamamoto, S.; Murakami, S.; Nakamura, Y.; Konoshima, S.; Ohshima, S.; Zang, L.; Sano, F.
2013-03-01
23
PubMed
Using the variational method based on the Gaussian basis set, the authors investigate the 1u state of hydrogen molecular ion in a non-parallel magnetic field with respect to the fixed molecular axis. At sufficiently small field strength, the equilibrium configuration prefers the perpendicular orientation, in which the (relative) orientation ? between the magnetic field and the molecular axis is 90°. With increasing field strength, the orientation ? of the equilibrium configuration decreases, and is neither the parallel orientation nor the perpendicular orientation at field strength between 10(9) G and 2.35 × 10(10) G. Meanwhile, more and more configurations with large orientations become unstable with respect to the dissociation H + p. PMID:23947853
Song, Xuanyu; Gong, Cheng; Wang, Xiaofeng; Qiao, Haoxue
2013-08-14
24
SciTech Connect
In order to understand the configuration of magnetic field producing a solar penumbral microjet that was recently discovered by Hinode, we performed a magnetohydrodynamic simulation reproducing a dynamic process of how that configuration is formed in a modeled solar penumbral region. A horizontal magnetic flux tube representing a penumbral filament is placed in a stratified atmosphere containing the background magnetic field that is directed in a relatively vertical direction. Between the flux tube and the background field there forms the intermediate region in which the magnetic field has a transitional configuration, and the simulation shows that in the intermediate region magnetic reconnection occurs to produce a clear jet-like structure as suggested by observations. The result that a continuous distribution of magnetic field in three-dimensional space gives birth to the intermediate region producing a jet presents a new view about the mechanism of a penumbral microjet, compared to a simplistic view that two field lines, one of which represents a penumbral filament and the other the background field, interact together to produce a jet. We also discuss the role of the intermediate region in protecting the structure of a penumbral filament subject to microjets.
Magara, T., E-mail: magara@khu.ac.k [Department of Astronomy and Space Science, School of Space Research, Kyung Hee University, 1 Seocheon-dong, Giheung-gu, Yongin, Gyeonggi-do 446-701 (Korea, Republic of)
2010-05-20
25
After extensive experimentation on the Translation, Confinement, and Sustainment rotating magnetic-field (RMF)-driven field reversed configuration (FRC) device [A. L. Hoffman et al., Fusion Sci. Technol. 41, 92 (2002)], the principal physics of RMF formation and sustainment of standard prolate FRCs inside a flux conserver is reasonably well understood. If the RMF magnitude B? at a given frequency ? is high enough compared to other experimental parameters, it will drive the outer electrons of a plasma column into near synchronous rotation, allowing the RMF to penetrate into the plasma. If the resultant azimuthal current is strong enough to reverse an initial axial bias field Bo a FRC will be formed. A balance between the RMF applied torque and electron-ion friction will determine the peak plasma density nm~B?/?1/2?1/2rs, where rs is the FRC separatrix radius and ? is an effective weighted plasma resistivity. The plasma total temperature Tt is free to be any value allowed by power balance as long as the ratio of FRC diamagnetic current, I'dia~2Be/?o, is less than the maximum possible synchronous current, I'sync=e?rs2/2. The RMF will self-consistently penetrate a distance ?* governed by the ratio ?=I'dia/I'sync. Since the FRC is a diamagnetic entity, its peak pressure pm=nmkTt determines its external magnetic field Be~(2?opm)1/2. Higher FRC currents, magnetic fields, and poloidal fluxes can thus be obtained, with the same RMF parameters, simply by raising the plasma temperature. Higher temperatures have also been noted to reduce the effective plasma resistivity, so that these higher currents can be supported with surprisingly little increase in absorbed RMF power.
Hoffman, A. L.; Guo, H. Y.; Miller, K. E.; Milroy, R. D.
2006-01-01
26
SciTech Connect
A field-reversed configuration (FRC) is formed by applying a rotating magnetic field (RMF) much larger than the axial magnetic field to a cylindrical glass vacuum chamber filled with 10 Pa argon gas without a preionization. The FRC with the plasma density 2.2x10{sup 19} m{sup -3}, the temperature 8.0 eV, the separatrix length 0.45 m, and the separatrix radius 0.035 m is sustained for the notably long period of 40 ms. It is observed that the antenna current which produces the RMF is reduced by about half after the FRC is formed. The interaction between the plasma and the antenna circuit increases the antenna resistance and changes the inductance of the antenna so that the circuit becomes nonresonant. The RMF is sufficiently large to fully penetrate to the center during the period and drive the current with a rigid rotor profile. The RMF is shown to play a major role in sustaining the plasma pressure.
Ohnishi, M.; Fukuhara, M.; Masaki, T.; Osawa, H.; Chikano, T. [Department of Electrical Engineering and Computer Science, Kansai University, 3-3-35 Yamate-cho, Suita-shi, Osaka 564-8680 (Japan); Hugrass, W. [School of Computing, University of Tasmania, Locked Bag 1359, Launceston, Tasmania 7250 (Australia)
2008-10-15
27
The Star Thrust Experiment (STX) has formed and sustained the Field Reversed Configuration (FRC) with a Rotating Magnetic Field (RMF) operated at a strength of 25 G and a frequency of 350 kHz. The RMF was generated with two IGBT switched solid state power supplies capable of delivering 2 MW each. Plasmas were typically 2 m long by 0.2m in radius and consisted of fully ionized deuterium at temperatures of 60 eV and peak densities of 5 × 1018m- 3. The primary diagnostic was an extremely small 24 channel berylia jacketed internal magnetic probe that was used to make measurements as a function of time, radius, and axial position. These measurements when combined with the FRC's unique geometry and equilibrium relationships determined many other important plasma parameters. Axial confining fields of 100 G maintained a true vacuum boundary around the plasma and allowed for the study of FRC RMF equilibrium interactions. Key findings are that the RMF maintained a near zero separatrix pressure, penetrated only partially, and drove strong radial and axial flows. Issues discussed include the importance of the RMF driving an axial current distribution consistent with that of the FRC, possible benefits of varying the average beta condition, and potential RMF antenna length limits set by the tendency of driven axial flows to screen the RMF from the plasma.
Miller, Kenneth Elric
2001-11-01
28
Context. In the past the role of magnetic null points in the generation of electric currents was investigated mainly in the close vicinity of the null, with perturbations being applied at nearby boundaries, or for a magnetic null configuration with a dome-shaped fan. In the solar atmosphere, however, electric currents are generated by perturbations originating at the photosphere, far away
J. C. Santos; J. Büchner; A. Otto
2011-01-01
29
THE virtual geomagnetic poles of Laurasia and Gondwanaland in the Carboniferous and Permian periods diverge significantly when these continents are reassembled according to the fit calculated by Bullard et al.1. Two interpretations have been offered: Briden et al.2 explain these divergences by a magnetic field configuration very different from that of a geocentric axial dipole; Irving3 (and private communication), Van
M. Westphal
1977-01-01
30
Radio frequency (RF) wave propagation in a bounded plasma (cylindrical shape with a large diameter of 45 cm) produced by a planar, spiral antenna was investigated under divergent and convergent magnetic field configurations. The measured excited magnetic field amplitude and the phase were examined based on helicon wave characteristics, and were consistent with the computed results using the Transport Analyzing System for tokamaK/Wave analysis by Finite element method (TASK/WF) code. The wave propagation region was broadened (focused) in the radial direction with increasing distance from the antenna under the divergent (convergent) field.
Takechi, Seiji; Shinohara, Shunjiro
1999-11-01
31
Three-dimensional simulations of field reversed configuration (FRC) formation and sustainment with rotating magnetic field (RMF) current drive have been performed with the NIMROD code [C. R. Sovinec et al., J. Comput. Phys. 195, 355 (2004)]. The Hall term is a zeroth order effect with strong coupling between Fourier components, and recent enhancements to the NIMROD preconditioner allow much larger time steps than was previously possible. Boundary conditions to capture the effects of a finite length RMF antenna have been added, and simulations of FRC formation from a uniform background plasma have been performed with parameters relevant to the translation, confinement, and sustainment-upgrade experiment at the University of Washington [H. Y. Guo, A. L. Hoffman, and R. D. Milroy, Phys. Plasmas 14, 112502 (2007)]. The effects of both even-parity and odd-parity antennas have been investigated, and there is no evidence of a disruptive instability for either antenna type. It has been found that RMF effects extend considerably beyond the ends of the antenna, and that a large n=0 B? can develop in the open-field line region, producing a back torque opposing the RMF.
Milroy, R. D.; Kim, C. C.; Sovinec, C. R.
2010-06-01
32
We present H-alpha and coronal X-ray images of the large two-ribbon flare of 25/26 June 1992 during its long-lasting gradual decay phase. From these observations we deduce that the 3-D magnetic field configuration late in this flare was similar to that at and before the onset of such large eruptive bipolar flares: the sheared core field running under and out of the flare arcade was S-shaped, and at least one elbow of the S looped into the low corona. From previous observations of filament-eruption flares, we infer that such core-field coronal elbows, though rarely observed, are probably a common feature of the 3-D magnetic field configuration late in large two-ribbon flares. The rare circumstance that apparently resulted in a coronal elbow of the core field being visible in H-alpha in our flare was the occurrence of a series of subflares low in the core field under the late-phase arcade of the large flare; these subflares probably produced flaring arches in the northern coronal elbow, thereby rendering this elbow visible in H-alpha. The observed late-phase 3-D field configuration presented here, together with the recent sheared-core bipolar magnetic field model of Antiochos, Dahlburg, and Klimchuk (1994) and recent Yohkoh SXT observations of the coronal magnetic field configuration at and before the onset of large eruptive bipolar flares, supports the seminal 3-D model for eruptive two-ribbon flares proposed by Hirayama (1974), with three modifications: (1) the preflare magnetic field is closed over the filament-holding core field; (2) the preflare core field has the shape of an S (or backward S) with coronal elbows; (3) a lower part of the core field does not erupt and open, but remains closed throughout flare, and can have prominent coronal elbows. In this picture, the rest of the core field, the upper part, does erupt and open along with the preflare arcade envelope field in which it rides; the flare arcade is formed by reconnection that begins in the middle of the core field at the start of the eruption and progresses from reconnecting closed core field early in the flare to reconnecting "opened" envelope field late in the flare.
Moore, R. L.; Schmieder, B.; Hathaway, D. H.; Tarbell, T. D.
1997-05-01
33
SciTech Connect
Collisionless particle confinement in axisymmetric configurations with magnetic field nulls is analyzed. The existence of an invariant of motion--the generalized azimuthal momentum--makes it possible to determine in which of the spatial regions separated by magnetic separatrices passing through the magnetic null lines the particle occurs after it leaves the vicinity of a magnetic null line. In particular, it is possible to formulate a sufficient condition for the particle not to escape through the separatrix from the confinement region to the external region. In the configuration under analysis, the particles can be lost from a separatrix layer with a thickness on the order of the Larmor radius because of the nonconservation of the magnetic moment {mu}. In this case, the variations in {mu} are easier to describe in a coordinate system associated with the magnetic surfaces. An analysis is made of the applicability of expressions for the single-pass change {delta}{mu} in the magnetic moment that were obtained in different magnetic field models for a confinement system with a divertor (such that there is a circular null line)
Arsenin, V.V.; Skovoroda, A.A. [Nuclear Fusion Institute, Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 (Russian Federation)
2005-12-15
34
The influence of a rotating magnetic field on buoyancy driven convection in a Rayleigh–Bénard configuration (vertical cylinder) and on Marangoni convection in a floating zone configuration was investigated experimentally and numerically. The main result of our study is that temperature fluctuations caused by time-dependent buoyancy and Marangoni convection can be damped by using a rotating magnetic field with relatively small
B. Fischer; J Friedrich; H Weimann; G Müller
1999-01-01
35
Shubitidze, Fridon; Bijamov, Alex; Schultz, Gregory; Miller, Jon; Shamatava, Irma
2011-05-01
36
A transverse Rotating Magnetic Field (RMF) can drive toroidal current and sustain the poloidal flux of a Field Reversed Configuration (FRC) through the application of a Lorentz force on the electrons, where vz is the axial screening current and Br is the radial component of the RMF. The torque applied by the RMF will eventually be transferred to the ions through resistive collisions. In the absence of any drag force, the plasma will rapidly spin up in the ion paramagnetic direction, negating the current drive and possibly becoming rotationally unstable. A multi-chord Intensified Charge-Coupled Device (ICCD) spectrometer has measured the ion rotation profile via the Doppler shift of impurity line radiation in the Translation, Confinement, and Sustainment (TCS) experiment. The plasma is observed to rapidly spin up in the ion paramagnetic direction to a rigid rotation frequency of about oi ? 7 x 104 s-1, less than 15% of the typical RMF frequency o ? 0.5 x 106 s-1 . Neutral deuterium is observed to have no rotational velocity, and had been proposed as a mechanism for preventing synchronous spin-up of the ions. The neutral density and resulting charge-exchange and ionization rates have been calculated from an array of absolutely calibrated Dalpha detectors. The typical neutral fraction of about 2% of the plasma density is several times too low for ion-neutral collisions to balance the applied torque. Other possible braking mechanisms are shorting of the radial electric field needed to confine paramagnetic ions, and viscous drag. Assuming axial and azimuthal symmetry and pure deuterium, viscous wall drag is found to be insufficient to slow the plasma as well. Viscous drag could be significant if the edge plasma has high impurity content or is spatially non-uniform.
Peter, Andrew Maxwell
37
SciTech Connect
Heating of figure-8 ions by odd-parity rotating magnetic fields (RMF?) applied to an elongated field-reversed configuration (FRC) is investigated. The largest energy gain occurs at resonances (s ? ?(sub)R??) of the RMF? frequency, ?(sub)R, with the figure-8 orbital frequency, ?, and is proportional to s^2 for s – even resonances and to s for s – odd resonances. The threshold for the transition from regular to stochastic orbits explains both the onset and saturation of heating. The FRC magnetic geometry lowers the threshold for heating below that in the tokamak by an order of magnitude.
A.S. Landsman, S.A. Cohen, A.H. Glasser
2005-11-01
38
PubMed
The virtual geomagnetic poles of Laurasia and Gondwanaland in the Carboniferous and Permian periods diverge significantly when these continents are reassembled according to the fit calculated by Bullard et al. Two interpretations have been offered: Briden et al. explain these divergences by a magnetic field configuration very different from that of a geocentric axial dipole; Irving (and private communication), Van der Voo and French(4) suggest a different reconstruction and it is shown here that these two interpretations are not incompatible and that the first may help the second. PMID:16073416
Westphal, M
1977-05-12
39
The ion heating effect of odd-parity rotating magnetic field (oRMF) in a field-reversed magnetic configuration (FRC) is studied by a single particle Hamiltonian code. By varying the particle initial condition and assuming the ergodic hypothesis, we obtain a particle distribution on both configuration space and velocity space. The simulation shows that strong enough RMF will give high energy betatron orbit particles a strong concentration effect in the frame rotating with the RMF. Moreover, the RMF will accelerate the particles to form a double bump distribution rather than a Maxwellian. Both of those effect will improve the nuclear fusion efficiency and will increase the energy ratio of charged particles to neutrons when D-He^3 is used as fuel, which is good to future FRC-based nuclear fusion plant.
Liu, Chang; Cohen, Samuel
2012-10-01
40
The reversed field pinch (RFP) is a magnetic configuration alternative to the tokamak that can be considered for a second generation of reactors. In this paper new remarkable results obtained in the RFP experiment RFX-mod are presented, showing that an internal transport barrier delimitates a large fraction of the plasma volume in a RFP when the current is raised to ~1.5 MA. The formation of this transport barrier is related to a profound, spontaneous modification of the magnetic topology. Due to the occurrence of a saddle node bifurcation the plasma enters in the single helical axis state, which is theoretically known to be more resilient to chaos. This bifurcation is driven by the amplitude of the helical perturbation which dominates the mode spectrum.
Lorenzini, R.; Agostini, M.; Alfier, A.; Antoni, V.; Apolloni, L.; Auriemma, F.; Barana, O.; Baruzzo, M.; Bettini, P.; Bonfiglio, D.; Bolzonella, T.; Bonomo, F.; Brombin, M.; Buffa, A.; Canton, A.; Cappello, S.; Carraro, L.; Cavazzana, R.; Chitarin, G.; Dal Bello, S.; de Lorenzi, A.; de Masi, G.; Escande, D. F.; Fassina, A.; Franz, P.; Gaio, E.; Gazza, E.; Giudicotti, L.; Gnesotto, F.; Gobbin, M.; Grando, L.; Guo, S. C.; Innocente, P.; Luchetta, A.; Manduchi, G.; Marchiori, G.; Marcuzzi, D.; Marrelli, L.; Martin, P.; Martini, S.; Martines, E.; Milani, F.; Moresco, M.; Novello, L.; Ortolani, S.; Paccagnella, R.; Pasqualotto, R.; Peruzzo, S.; Piovan, R.; Piovesan, P.; Piron, L.; Pizzimenti, A.; Pomaro, N.; Predebon, I.; Puiatti, M. E.; Rostagni, G.; Sattin, F.; Scarin, P.; Serianni, G.; Sonato, P.; Spada, E.; Soppelsa, A.; Spagnolo, S.; Spizzo, G.; Spolaore, M.; Taliercio, C.; Terranova, D.; Toigo, V.; Valisa, M.; Veltri, P.; Vianello, N.; Zaccaria, P.; Zaniol, B.; Zanotto, L.; Zilli, E.; Zuin, M.
2009-05-01
41
Magnetic measurements are a fundamental part of determining the size and shape of field-reversed configuration (FRC) plasmas in the C-2 device. The magnetic probe suite consists of 44 in-vessel and ex-vessel probes constructed using various technologies: ultra-high vacuum compatible mineral-insulated cable, nested triple axis coils hand-wound on ceramic bobbins, and commercial chip inductors mounted on printed circuit boards. Together, these probes measure the three-dimensional excluded flux profile of the FRC, which approximates the shape of the separatrix between the confined plasma volume and the scrape-off layer. High accuracy is achieved by using the extensive probe measurements to compensate for non-ideal effects such as flux leakage through the vacuum vessel and bulk motion of the FRC towards the wall. A subset of the probes is also used as a set of Mirnov arrays that provide sensitive detection of perturbations and oscillations of the FRC.
Thompson, M. C.; Douglass, J. D.; Feng, P.; Knapp, K.; Luo, Y.; Mendoza, R.; Patel, V.; Tuszewski, M.; Van Drie, A. D.
2012-10-01
42
The n = 1 tilt/radial shift modes are observed in rotating magnetic field (RMF) driven FRC plasmas. Experiments studying on the response of n = 1 instabilities to the changes of magnetic field structure have been conducted in 40 ms Rotamak discharges. In one series of experiments the axial current Iz (which produces toroidal field) ramps linearly in time from 0 to 2 kA, leading to transition from FRC to ST configuration. The amplitude of the tilt mode is suddenly doubled when Iz reaches 0.5 kA (compare to 2 kA plasma current); the amplitude remains at this level when Iz is in the range of 0.5-1 kA. The tilt instability disappears when Iz exceeds 1 kA. In other series of experiments, by using a middle shaping coil with a moderate current of 0.25-0.5 kA, the doublet-FRCs are formed which are completely free from both the tilt and radial shift modes.
Yang, Xiaokang; Petrov, Yuri; Huang, Tian-Sen
2009-11-01
43
Improved vacuum hygiene, wall conditioning, and reduced recycling in the rotating magnetic field (RMF) driven translation, confinement, and sustainment-upgrade (TCSU) field reversed configuration experiment have made possible a more accurate assessment of the forces affecting ion spin-up. This issue is critical in plasmas sustained by RMFs, such as TCSU since ion spin-up can substantially reduce or cancel the RMF current drive effect. Several diagnostics are brought to bear, including a 3-axis translatable magnetic probe allowing the first experimental measurement of the end shorting effect. These results show that the ion rotation is determined by a balance between electron-ion friction, the end shorting effect, and ion drag against neutrals.
Deards, C. L.; Hoffman, A. L.; Steinhauer, L. C.
2011-11-01
44
PubMed
Magnetic measurements are a fundamental part of determining the size and shape of field-reversed configuration (FRC) plasmas in the C-2 device. The magnetic probe suite consists of 44 in-vessel and ex-vessel probes constructed using various technologies: ultra-high vacuum compatible mineral-insulated cable, nested triple axis coils hand-wound on ceramic bobbins, and commercial chip inductors mounted on printed circuit boards. Together, these probes measure the three-dimensional excluded flux profile of the FRC, which approximates the shape of the separatrix between the confined plasma volume and the scrape-off layer. High accuracy is achieved by using the extensive probe measurements to compensate for non-ideal effects such as flux leakage through the vacuum vessel and bulk motion of the FRC towards the wall. A subset of the probes is also used as a set of Mirnov arrays that provide sensitive detection of perturbations and oscillations of the FRC. PMID:23126883
Thompson, M C; Douglass, J D; Feng, P; Knapp, K; Luo, Y; Mendoza, R; Patel, V; Tuszewski, M; Van Drie, A D
2012-10-01
45
The observed cross-field diffusion of charged particles in cosmic rays is assumed to be due to the chaotic nature of the interplanetary\\/intergalactic magnetic fields. Among the classic works on this subject have been those of Parker [1] and Jokipii [2]. Parker considered the passage of cosmic ray particles and energetic solar particles in a large scale magnetic field containing small
A. K. Ram; B. Dasgupta
2008-01-01
46
A well-known problem in solar physics is that solutions for the transverse magnetic field direction are ambiguous with respect to a 180° reversal in the field direction. In this paper we focus on three methods for the removal of the 180° ambiguity applied to three MHD models. These methods are (1) the reference field method, (2) the method of magnetic pressure gradient, and (3) the magnetic field divergence-free method. All three methods are noniterative, and methods 2 and 3 are analytical and fast. We apply these methods to three MHD equilibrium model fields: (1) an analytical solution of a nonlinear force-free magnetic field equilibrium from Low, (2) a simulation of an emerging twisted flux tube from Fan & Gibson, and (3) a pre-eruptive twisted magnetic flux rope equilibrium reached by relaxation from Amari et al. We measure the success of methods within inverse horizontal field'' regions in the boundary, which are mathematically defined by B????Bz>0. When such regions overlap with the magnetic field neutral lines, they are known as bald patches'' (BPs) or inverse topology. Our most important conclusion is that the magnetic divergence-free method is far more successful than the other two methods within BPs. This method requires a second level of measurements of the vertical magnetic field. As high-quality multilevel magnetograms will come online in the near future, our work shows that multilayer magnetic field measurements will be highly desirable to objectively and successfully tackle the 180° ambiguity problem.
Li, Jing; Amari, Tahar; Fan, Yuhong
2007-01-01
47
The observed cross-field diffusion of charged particles in cosmic ray transport is assumed to be due to chaotic nature of the interplanetary\\/intergalactic magnetic fields. The particles are accelerated and energized by the temporal fluctuations of the magnetic field. The generation of chaotic magnetic fields is ad hoc and the characteristics of the fields are chosen to satisfy the observations. We
B. Dasgupta; A. K. Ram
2007-01-01
48
The observed cross-field diffusion of charged particles in cosmic ray transport is assumed to be due to chaotic nature of the interplanetary\\/intergalactic magnetic fields. The particles are accelerated and energized by the temporal fluctuations of the magnetic field. The generation of chaotic magnetic fields is ad hoc and the characteristics of the fields are chosen to satisfy the observations. We
A. K. Ram; B. Dasgupta
2007-01-01
49
Control of the radial electric field, Er, is considered to be important in helical plasmas, because the radial electric field and its shear are expected to reduce neoclassical and anomalous transport, respectively. In general, the radial electric field can be controlled by changing the collisionality, and positive or negative electric fields have been obtained by decreasing or increasing the electron
K. Ida; M. Yoshinuma; M. Yokoyama; S. Inagaki; N. Tamura; B. J. Peterson; T. Morisaki; S. Masuzaki; A. Komori; Y. Nagayama; K. Tanaka; K. Narihara; K. Y. Watanabe; C. D. Beidler
2005-01-01
50
AFRL's Directed Energy Directorate has built a multiple chord 6328 nm interferometer to diagnose a Field Reversed Configuration (FRC) being developed for LANL and AFRL's collaborative Magnetized Target Fusion (MTF) program. The FRC is intended for compression to near thermonuclear fusion conditions by AFRL's Shiva Star capacitor bank. The interferometer is designed to measure the density integral along eight chords of the uncompressed FRC vs. time. This permits Abel inversion to determine the density profile history. The reference beam is split off with a Bragg cell, raising its frequency by 80 MHz. This permits RF quadrature mixing of the interference signal. The probe traverses a 10 cm diameter quartz tube containing the FRC. A similar tube is placed in the reference path to compensate for refractive distortion. Focusing the beams at tube transit further mitigates distortion. Preliminary design validation experiments at LANL using 2 chords have been successfully completed, and the upgrade to 8 chords is in progress. Results to date will be presented.
Ruden, Edward; Analla, Francis; Zhang, Shouyin
2002-11-01
51
SciTech Connect
The transition from a regime dominated by drift instabilities to a regime dominated by pure interchange instabilities is investigated and characterized in the simple magnetized toroidal device TORPEX [TORoidal Plasma EXperiment, A. Fasoli et al., Phys. of Plasmas 13, 055906 (2006)]. The magnetic field lines are helical, with a dominant toroidal component and a smaller vertical component. Instabilities with a drift character are observed in the favorable curvature region, on the high field side with respect to the maximum of the background density profile. For a limited range of values of the vertical field they coexist with interchange instabilities in the unfavorable curvature region, on the plasma low field side. With increasing vertical magnetic field magnitude, a gradual transition between the two regimes is observed on the low field side, controlled by the value of the field line connection length. The observed transition follows the predictions of a two-fluid linear model.
Poli, F. M.; Ricci, P.; Fasoli, A.; Podesta, M. [CRPP-EPFL, Association EURATOM-Confederation Suisse, Lausanne (Switzerland)
2008-03-15
52
We present an analysis of multi-frequency radio imaging data derived from observations of AR 10923 taken with the Owens Valley Solar Array on November 10, 2006. This region was chosen because its relative simplicity (alpha configuration) and medium-large size (450 millionths) sunspot may simplify the analysis. The observations span 22 frequencies from 1.4 to 10.6 GHz range. Joint VLA observations were used to increase the spectral resolution of the data, and to provide a means of cross-calibration for the OVSA data. The spectral shapes derived from the brightness temperature maps were used to (1) directly determine the magnetic field at the highest optically thick gyroresonance layer when possible, and (2) an upper bound for it when the spectrum was dominated by free-free emission. Values for the electron temperature and the column emission measure were also derived from the radio spectra. Comparisons of the radio maps with EUV observations from SOHO/EIT were then used to relate the various temperatures and determine a plausible model of the electron density and magnetic field configurations above this active region. The magnetic field model is compared to force-free field extrapolations made from co-temporal Hinode SOT-SP observations of this active region. The differences and similarities are used to asses the validity of the method and magnetic field deduced from OVSA data, which initially has few data points from which to extrapolate the overall magnetic field.
Tun, Samuel D.
2009-05-01
53
We have tested a field-reversed configuration (FRC) formation with a spheromak injection for the first time. In this method, initial pre-ionized plasma is injected as a magnetized spheromak-like plasmoid into the discharge chamber prior to main field reversal. The FRC plasma with an electron density of 1.3×1021m?3, a separatrix radius of 0.04m and a plasma length of 0.8m was produced
T. Nishida; T. Kiguchi; T. Asai; T. Takahashi; Y. Matsuzawa; T. Okano; Y. Nogi
2006-01-01
54
This paper presents an analytic model for a finite-size straight filament suspended horizontally in a steady state over a bipolar magnetic region. The equations of magnetostatic equilibrium are integrated exactly. The solution obtained illustrates the roles played by the electric current, magnetic field, pressure, and plasma weight in the balance of force everywhere in space. Basic properties are discussed. We
B. C. Low
1981-01-01
55
In a recent study (Cao et al., 2012) based on in-situ magnetic field measurements made by the Cassini spacecraft, several distinct features of Saturn's magnetic field have been revealed. The field at the dynamo surface is found to be strongly concentrated near the spin-poles with little hemispherical asymmetry. This field geometry corresponds to a zig-zag shape magnetic spectrum with pronounced odd degree terms and all odd degree magnetic moments possess the same sign. This is in contrast to the field properties at the core surface of the Earth and in convection-driven geodynamo simulations, where the field near the spin-poles is at a relative minimum compared to field at mid-latitude. In this study, we propose that the absence of a solid inner core inside the planet could be responsible for the poleward flux concentration at the dynamo surface. The simple physical picture underlying this hypothesis is the concentration of convection columns near the spin axis after the solid inner core as an obstacle is removed. We test this hypothesis using numerical dynamo simulations. The heat source in our model is the uniform secular cooling of the planet. Different heat flux patterns at the outer boundary are also applied. In the end, we explore the possible field geometries at the dynamo surface of Jupiter, Uranus and Neptune consistent with the available in-situ observations at each planet. Properties including the Lowes spectrum, field symmetry are analyzed.
Cao, H.; Russell, C. T.; Wicht, J.; Christensen, U. R.; Dougherty, M. K.
2012-04-01
56
SciTech Connect
The TITAN study uses copper-alloy ohmic-heating coils (OHC) to startup inductively a reversed-field-pinch (RFP) fusion reactor. The plasma equilibrium is maintained with a pair of superconducting equilibrium-field coils (EFCs). A second pair of copper EFCs provides the necessary trimming of the equilibrium field during plasma transients. A compact toroidal-field-coil (TFC) set is provided by an integrated blanket/coil (IBC). The IBC concept also is applied to the toroidal-field divertor coils. Steady-state operation is achieved with oscillating-field current drive, which oscillates at low amplitude and frequency the OHCs, EFCs, the TFCs, and divertor coils about their steady-state currents. An integrated magnet design, which uses low-field, low technology coils, and the related design basis is given. 18 refs.
Bathke, C.G.
1987-01-01
57
DOEpatents
A point-sensitive NMR imaging system (10) in which a main solenoid coil (11) produces a relatively strong and substantially uniform magnetic field and a pair of perturbing coils (PZ1 and PZ2) powered by current in the same direction superimposes a pair of relatively weak perturbing fields on the main field to produce a resultant point of minimum field strength at a desired location in a direction along the Z-axis. Two other pairs of perturbing coils (PX1, PX2; PY1, PY2) superimpose relatively weak field gradients on the main field in directions along the X- and Y-axes to locate the minimum field point at a desired location in a plane normal to the Z-axes. An RF generator (22) irradiates a tissue specimen in the field with radio frequency energy so that desired nuclei in a small volume at the point of minimum field strength will resonate.
Eberhard, Philippe H. (El Cerrito, CA)
1985-01-01
58
The Berlin magnetically shielded room 2 (BMSR-2) features a magnetic residual field below 500 pT and a field gradient level less than 0.5 pT/mm, which are needed for very sensitive human biomagnetic recordings or low field NMR. Nevertheless, below 15 Hz, signals are compromised by an additional noise contribution due to vibration forced sensor movements in the field gradient. Due to extreme shielding, the residual field and its homogeneity are determined mainly by the demagnetization results of the mumetal shells. Eight different demagnetization coil configurations can be realized, each results in a characteristic field pattern. The spatial dc flux density inside BMSR-2 is measured with a movable superconducting quantum interference device system with an accuracy better than 50 pT. Residual field and field distribution of the current-driven coils fit well to an air-core coil model, if the high permeable core and the return lines outside of the shells are neglected. Finally, we homogenize the residual field by selecting a proper coil configuration.
Knappe-Grueneberg, Silvia; Schnabel, Allard; Wuebbeler, Gerd; Burghoff, Martin
2008-04-01
59
SciTech Connect
The relative locations and characteristics of the distribution lines feeding 434 residences in the Denver metropolitan area were recorded and classified according to the Wertheimer-Leeper code (WL code) as a part of an epidemiological study of the incidence of childhood cancer. The WL code was found to place the mean values of the fields in rank order. However, the standard deviations were approximately the same size as the means. Theoretical calculations indicate that a significant fraction of the low-power magnetic fields can be generated by the distribution lines, especially in the cases where the distribution lines are within 50 feet of the residence. Thus, the wiring code was shown to be a useful method for making a first-order approximation to predict long-term, low-level magnetic fields in residences.
Barnes, F.; Wachtel, H.; Savitz, D.; Fuller, J.
1989-01-01
60
We investigate the theoretical foundations of neutral two-body systems exposed to an inhomogeneous magnetic field. Various representations for the Hamiltonian describing the coupled centre-of-mass and internal motion are derived. For the specific case of a magnetic quadrupole field we establish the continuous and discrete symmetries and show that the energy levels of the interacting system are two-fold degenerate. We exploit the symmetries in two alternative ways and derive corresponding effective equations of motion. The first approach eliminates two of the six spatial degrees of freedom and leads to an (infinite) set of coupled channel equations for the spin and spatial degrees of freedom. The second approach introduces the projection of the total angular momentum onto the symmetry axis of the quadrupole field as a canonical momentum, thereby eliminating the corresponding cyclic angle.
Bock, Hannes; Lesanovsky, Igor; Schmelcher, Peter
2005-04-01
61
The current sheet (CS) creation before a flare in the vicinity of a singular line above the active region NOAA 10365 is shown in numerical experiments. Such a way the possibility of energy accumulation for a solar flare is demonstrated. These data and results of observation confirm the electrodynamical solar flare model that explains solar flares and CME appearance during CS disruption. The model explains also all phenomena observed in flares. For correct reproduction of the real boundary conditions the magnetic flux between spots should be taken into account. The full system of 3D MHD equations are solved using the PERESVET code. For setting the boundary conditions the method of photospheric magnetic maps is used. Such a method permits to take into account all evolution of photospherical magnetic field during several days before the flare.
Podgorny, A. I.; Podgorny, I. M.
2012-11-01
62
SciTech Connect
An 11-13 GHz electron cyclotron resonance (ECR) plasma source with a cylindrically comb-shaped magnetic field configuration has been examined in order to apply to ion beam processing. The ion saturation current density has been measured using a Langmuir probe. It was found that the ion density linearly increases as gas pressure and microwave power increases. The maximum ion density at 13 GHz microwaves is 37.4 mA/cm2 under low microwave power. The ion beam extractor which has multihole apertures has been constructed at the end of the magnet. The ion beam current has reached 20 mA at the microwave power of only 300 W. The ion beam current has clearly increased by rising microwave frequency as well as the tendency of the plasma density.
Asaji, Toyohisa [Division of Electrical, Electronic and Information Engineering, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871 (Japan); Development Center of Advanced Technology, Tateyama Machine Co., Ltd., 30 Shimonoban, Toyama, 930-1305 (Japan); Kato, Yushi; Sasaki, Hiroshi; Kubo, Takashi; Sato, Fuminobu; Iida, Toshiyuki [Division of Electrical, Electronic and Information Engineering, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871 (Japan); Saito, Junji [Development Center of Advanced Technology, Tateyama Machine Co., Ltd., 30 Shimonoban, Toyama, 930-1305 (Japan)
2006-11-13
63
SciTech Connect
A simple derivation is given of equilibrium equations in flux coordinates in the general case of an anisotropic-pressure plasma. The issue of how to formulate the boundary conditions for these equations is discussed for two types of configurations-a straight system and a system with an internal conductor. Examples of numerical solutions to the equilibrium problem for these configurations are presented.
Arsenin, V. V.; Terekhin, P. N. [Russian Research Centre Kurchatov Institute (Russian Federation)
2011-08-15
64
Sheet electron beams and configurations with multiple electron beams have the potential to make possible higher power sources of microwave radiation due to their ability to transport high currents, at reduced current densities, through a single narrow RF interaction circuit. Possible microwave device applications using sheet electron beams include sheet-beam klystrons, grating TWTs, and planar FELs. Historically, implementation of sheet
J. Anderson; M. A. Basten; L. Rauth; J. H. Booske; J. Joe; J. E. Scharer
1995-01-01
65
National Technical Information Service (NTIS)
The boundary of particle leakage from the magnetic dipole trap depending on the value of adiabatic parameter is investigated. By trajectory computation a generalized analytical expression is determined for the shape of particle drain by x less than or equ...
I. V. Amirkhanov E. P. Zhidkov V. V. Ignatov A. N. Il'ina V. D. Il'in
1987-01-01
66
Trapped magnetic field profiles were investigated in thin film (Gd,Y)Ba2Cu3Ox superconducting tapes stacked in different configurations consisting of three 12 mm wide, 36 mm long tapes per layer. The trapped magnetic field values were found to increase monotonically with the increasing number of tape layers in the stack. A crisscross arrangement of the tapes was found to yield a more uniform trapped-field profile than a straight arrangement of the tapes. Furthermore, the decay rate of the trapped magnetic field as a function of distance from the tape stack surface was found to be lower in the crisscross arrangement.
Selva, Kavita; Majkic, Goran
2013-11-01
67
The continuous increasing demand of electricity contradicts with the increasing difficulty to secure corridors to construct new transmission lines. This paper investigates alternatives regarding parallel lines from the point of view of the magnetic flux density level under and around current-carrying conductors of the lines. The first alternative is for existing transmission lines by converting a three-phase double-circuit line to
A. A. Dahab; F. K. Amoura; W. S. Abu-Elhaija
2005-01-01
68
A systematic translation study of field-reversed configurations (FRCs) has been conducted on the FRC Injection Experiment (FIX) machine [Okada et al., in Fusion Energy 1996 (International Atomic Energy Agency, Vienna, 1997), Vol. 2, p. 229]. Plasma density and temperature of a translated FRC moving at supersonic speed are measured in the downstream magnetic mirror of FIX to verify a shock jump there when the FRC is reflected. A significant jump is observed. Moreover, the time evolution of the Carbon V Doppler profile is measured both quasi-parallel and perpendicular to the direction of FRC motion. Distinct transitions from Gaussian to non-Gaussian shapes are clearly seen in both profiles before and after the shock jump. Also, the ion mean-free path in the downstream magnetic mirror is calculated to be much longer than the characteristic width of the shock jump. These results indicate that the thermalization of flow energy in the translated FRC in the mirror is produced by a collisionless process, implying that this heating mechanism can be realized even in a reactor regime.
Himura, H.; Ueoka, S.; Hase, M.; Yoshida, R.; Okada, S.; Goto, S.
1998-12-01
69
SciTech Connect
An overview is presented of an experimental program of magnetic field line mapping on the research-grade Compact Auburn Torsatron (CAT). The vacuum magnetic flux surfaces of the CAT device have been experimentally mapped in a variety of magnetic configurations. The results are compared with an extensive computer model in order to validate the coil design. In initial field mapping experiments, an up-down symmetry was identified in the vacuum magnetic surfaces, and was corrected with the use of a radial trim field. Magnetic islands are observed and their size has been reduced, also through the use of auxiliary trim coils. The Compact Auburn Torsatron is equipped with two pairs of large Helmholtz coils producing mutually orthogonal magnetic fields in the horizontal plane, and two pairs of helical saddle coils wound directly on the toroidal vacuum vessel. These trim coils are used to control the size and phase of the t = 1/2 magnetic island. Through a systematic variation of trim field components, the authors demonstrate a reduction of the inherent t = 1/2 magnetic island size by a factor of three. The technique is applicable to correcting small error fields in larger helical confinement devices. The measurements of island size are compared with measurements of magnetic field line rotation within the island, and are found to be in good agreement with first-order perturbation theory.
Knowlton, S.F.; Gandy, R.F.; Hanson, J.D.; Hartwell, G.J.; Lin, H. (Auburn Univ., AL (United States))
1993-09-01
70
SciTech Connect
Achievement of the design field of 5 T in the ISABELLE dipole magnets is turning out to be more arduous than expected and several avenues of improvement are being pursued. One possibility for improving training and peak field performance is discussed in this paper. It has been recognized that the inert spacers with their adjacent active turns in the cosine magnet windings can be replaced by a double thickness braid operating at approximately half-current density in 46 of the 190 turns. Since the high-field region occurs in the low current density turns near the poles, a performance improvement can be expected. It has been verified that the proposed coil configuration satisfies the field requirements and details thereof are given. Results from an experimental magnet in which superconducting spacer turns are used to simulate half-current density windings are presented. Construction of thick braid coils is being planned and the status of these magnets is reviewed.
Hahn, H.; Dahl, P.F.; Kaugerts, J.E.; Prodell, A.G.
1981-01-01
71
SciTech Connect
Oblate field-reversed configurations (FRCs) have been sustained for >300 {mu}s, or >15 magnetic diffusion times, through the use of an inductive solenoid. These argon FRCs can have their poloidal flux sustained or increased, depending on the timing and strength of the induction. An inward pinch is observed during sustainment, leading to a peaking of the pressure profile and maintenance of the FRC equilibrium. The good stability observed in argon (and krypton) does not transfer to lighter gases, which develop terminal co-interchange instabilities. The stability in argon and krypton is attributed to a combination of external field shaping, magnetic diffusion, and finite-Larmor radius effects.
Gerhardt, S. P.; Belova, E. V.; Yamada, M.; Ji, H.; Jacobson, C. M.; McGeehan, B.; Ren, Y. [Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543 (United States); Inomoto, M. [Osaka University, Osaka 565-0871 (Japan); Maqueda, R. [Nova Photonics, Princeton, New Jersey 08540 (United States)
2008-02-15
72
SciTech Connect
Reass, W.A.; Miera, D.A.; Wurden, G.A.
1997-10-06
73
Blob-filaments have been observed by combined measurement with a fast camera and a movable Langmuir probe in an open magnetic field line configuration of electron cyclotron resonance (ECR) heating plasma in QUEST. Blob-filaments extended along field lines do correspond to over-dense plasma structures and propagated across the field lines to the outer wall. The radial velocity of the blob structure, Vb, was obtained by three methods and was dominantly driven by the E × B force. The radial velocity, size of the blob showed good agreements with the results obtained by sheath-connected interchange theoretical model. Vb corresponds to roughly 0.02-0.07 of the local sound speed (Cs) in QUEST. The higher moments (skewness S and kurtosis K) representing the shape of PDF of density fluctuation are studied. Their least squares fitting with quadratic polynomial is K = (1.60 ± 0.27)S2 - (0.46 ± 0.20). The larger blob structures, occurring only 10% of the time, can carry more than 60% loss of the entire radial particle flux.
QUEST Group; Liu, H. Q.; Hanada, K.; Nishino, N.; Ogata, R.; Ishiguro, M.; Gao, X.; Zushi, H.; Nakamura, K.; Fujisawa, A.; Idei, H.; Hasegawa, M.
2013-07-01
74
A study of the effect of an externally imposed magnetic field on the axisymmetry-breaking instability of an axisymmetric convective flow, associated with crystal growth from bulk of melt, is presented. Convection in a vertical cylinder with a parabolic temperature profile on the sidewall is considered as a representative model. A parametric study of the dependence of the critical Grashof number
A. Yu. Gelfgat; P. Z. Bar-Yoseph; A. Solan
2001-01-01
75
SciTech Connect
Field reversed configurations (FRCs) are characterized by azimuthal symmetry, so two exact constants of the particle motion are the total particle energy E and the canonical angular momentum P/sub theta/. For many purposes it is desirable to construct a third (diabatic) constant of the motion if this is possible. It is shown that for parameters characteristic of current FRCs that the magnetic moment ..mu.. is a poor adiabatic invariant, while the radial action J is conserved rather well.
Schwarzmeier, J.L.; Lewis, H.R.; Seyler, C.E.
1982-01-01
76
We obtain the general forms for the current density and the vorticity from the integrability conditions of the basic equations which govern the stationary states of axisymmetric magnetized self-gravitating barotropic objects with meridional flows under the ideal magnetohydrodynamics (MHD) approximation. As seen from the stationary condition equations for such bodies, the presence of the meridional flows and that of the poloidal magnetic fields act oppositely on the internal structures. The different actions of these two physical quantities, the meridional flows and the poloidal magnetic fields, could be clearly seen through stationary structures of the toroidal gaseous configurations around central point masses in the framework of Newtonian gravity because the effects of the two physical quantities can be seen in an amplified way for toroidal systems compared to those for spheroidal stars. The meridional flows make the structures more compact, i.e. the widths of toroids thinner, while the poloidal magnetic fields are apt to elongate the density contours in a certain direction depending on the situation. Therefore, the simultaneous presence of the internal flows and the magnetic fields would work as if there were no such different actions within and around the stationary gaseous objects such as axisymmetric magnetized toroids with internal motions around central compact objects under the ideal MHD approximation, although these two quantities might exist in real systems.
Fujisawa, Kotaro; Takahashi, Rohta; Yoshida, Shijun; Eriguchi, Yoshiharu
2013-05-01
77
NSDL National Science Digital Library
This webpage is part of the University Corporation for Atmospheric Research (UCAR) Windows to the Universe program. It describes the nature and configuration of magnetic fields, which are the result of moving electric charges, including how they cause magnetic objects to orient themselves along the direction of the magnetic force points, which are illustrated as lines. Magnetic field lines by convention point outwards at the north magnetic pole and inward at the south magnetic pole. The site features text, scientific illustrations and an animation. Text and vocabulary are selectable for the beginning, intermediate, or advanced reader.
Universe, Windows T.
1997-12-03
78
We present the results of our numerical attempts to simulate the configuration of a nematic liquid crystal around a spherical particle. We focus on the effect of an external field, such as a magnetic field or a flow field, on the director configuration and the topological defects accompanied by the particles. The use of adaptive mesh refinement together with a
Jun-Ichi Fukuda; Holger Stark; Makoto Yoneya; Hiroshi Yokoyama
2004-01-01
79
The dynamics of current channels with large self-generated magnetic fields in a background plasma and, in particular, their relaxation toward a force-free configuration is investigated. Relaxation, in the sense of the theoretical conception of Woltjer and Taylor, to configurations regarded as minimum energy states, described by the Beltrami equations J = mu sub 0 exp -1(tilde) x B = alpha(B), is reexamined in light of Salingaros' theoretical results. Detailed 3-D measurements in a quiescent laboratory plasma of 10 m length and 50 cm diameter yield the magnetic field B(x,y,z,t) and the plasma parameters, n(x,y,z,t), k(T sub e)(x,y,z,t) and V sub pl(x,y,z,t) at several thousand measurement locations. The plasma current J, plasma pressure p = nkT, pressure gradient force, tilde(p), and J x B force, are determined and relaxation toward a force-free configuration that includes pressure gradient forces, i.e., the validity of J x B = tilde(p), is tested. The temporal relaxation of a single current channel and the spatial relaxation of a dual current system is illustrated. A strong increase of alignment of the current density vector J and the total magnetic field B sub 0 + B sub self as a function of the distance from the plasma source is found. The return currents that prevent the two current channels from twisting about each other, after turn-off of the externally applied electric field, are identified and the conversion of the dual current system into a twin-axial system is shown.
Pfister, Hans
80
We study the influence of the large-scale interplanetary magnetic field configuration on the solar energetic particles (SEPs) as detected at different satellites near Earth and on the correlation of their peak intensities with the parent solar activity. We selected SEP events associated with X- and M-class flares at western longitudes, in order to ensure good magnetic connection to Earth. These events were classified into two categories according to the global interplanetary magnetic field (IMF) configuration present during the SEP propagation to 1 AU: standard solar wind or interplanetary coronal mass ejections (ICMEs). Our analysis shows that around 20 % of all particle events are detected when the spacecraft is immersed in an ICME. The correlation of the peak particle intensity with the projected speed of the SEP-associated coronal mass ejection is similar in the two IMF categories of proton and electron events, ? 0.6. The SEP events within ICMEs show stronger correlation between the peak proton intensity and the soft X-ray flux of the associated solar flare, with correlation coefficient r=0.67±0.13, compared to the SEP events propagating in the standard solar wind, r=0.36±0.13. The difference is more pronounced for near-relativistic electrons. The main reason for the different correlation behavior seems to be the larger spread of the flare longitude in the SEP sample detected in the solar wind as compared to SEP events within ICMEs. We discuss to what extent observational bias, different physical processes (particle injection, transport, etc.), and the IMF configuration can influence the relationship between SEPs and coronal activity.
Miteva, R.; Klein, K.-L.; Malandraki, O.; Dorrian, G.
2013-02-01
81
SciTech Connect
The magnetic field structure in a domain surrounded by a closed toroidal magnetic surface is analyzed. It is shown that ergodization of magnetic field lines is possible even in a regular field configuration (with nonvanishing toroidal component). A unified approach is used to describe magnetic fields with nested toroidal (possibly asymmetric) flux surfaces, magnetic islands, and ergodic field lines.
Ilgisonis, V. I.; Skovoroda, A. A., E-mail: skovorod@nfi.kiae.r [Russian Research Centre Kurchatov Institute (Russian Federation)
2010-05-15
82
Advances in Magnetic Resonance Imaging depend on the capability of the available hardware. Specifically, for the main magnet configuration, using derivative constraints, we can create a static magnetic field with reduced levels of inhomogeneity over a prescribed imaging volume. In the gradient coil, the entire design for the axial elliptical coil, and the mathematical foundation for the transverse elliptical coil
Labros Spiridon Petropoulos
1993-01-01
83
The influence of large-scale magnetic fields on the structure of accretion disks is studied. The magnetic field is obtained by a self-consistent nonlinear dynamo model with magnetic pressure strongly influencing the density stratification which itself feeds back to the field generation. The resulting magnetic field geometry is discussed in relation to the accretion disk wind theory. Regarding new results of MHD turbulence simulations, both possible signs of the alpha -effect are allowed (Brandenburg & Donner 1997). In the canonical case of positive alpha the resulting field is of quadrupolar symmetry. The field strength is about 50% of the value for dynamo models nonlinearly limited by alpha -quenching. The temperature profiles as well as the disk geometry remain nearly unchanged. The viscous stress remains the key transporter of angular momentum driving the accretion inflow. For negative alpha , however, a stationary dipolar structure of the magnetic field results. The additional magnetic torque at the disk surface changes the profile of the effective temperature significantly to a profile which is more flat. The magnetic torque becomes of the same order as the radial viscous torque. The inclination angle of the poloidal field exceeds 30o even for a magnetic Prandtl number of order unity, and also the criterion for poloidal collimation after Spruit et al. (1997) is fulfilled. The dynamo-generated magnetic field configuration thus supports the magnetic wind launching concept for accretion disks for realistic turbulent magnetic Prandtl numbers.
Rekowski, M. v.; Rüdiger, G.; Elstner, D.
2000-01-01
84
It is proposed that the most probable configuration of the magnetic field in the atmosphere of an Ap star is an almost force-free, poloidal field, close to a low-order multipole. Such a magnetic field can not change the structure of the atmosphere to any great extent, but the vertical component of the Lorentz force can decrease the effective gravity by
K. Stepien
1980-01-01
85
SciTech Connect
A 2.45 GHz electron cyclotron resonance (ECR) source with a magnetron magnetic field configuration was developed to meet the demand of a hyperthermal neutral beam (HNB) flux on a substrate of more than 1x10{sup 15} cm{sup -2} s{sup -1} for industrial applications. The parameters of the operating pressure, ion density, electron temperature, and distance between the neutralization plate and the substrate for the HNB source are specified in a theoretical analysis. The electron temperature and the ion density are measured to characterize the ECR HNB source using a Langmuir probe and optical emission spectroscopy. The parameters of the ECR HNB source are in good agreement with the theoretically specified parameters.
Kim, Seong Bong [Department of Physics, Pohang University of Science and Technology, San 31, Hyoja-dong, Nam-gu, Pohang 790-784 (Korea, Republic of); Convergence Plasma Research Center, National Fusion Research Institute, Gwahangno 113, Yuseong-gu, Daejeon 305-333 (Korea, Republic of); Kim, Dae Chul; Yoo, Suk Jae [Convergence Plasma Research Center, National Fusion Research Institute, Gwahangno 113, Yuseong-gu, Daejeon 305-333 (Korea, Republic of); Namkung, Won; Cho, Moohyun [Department of Physics, Pohang University of Science and Technology, San 31, Hyoja-dong, Nam-gu, Pohang 790-784 (Korea, Republic of)
2010-08-15
86
PubMed
A 2.45 GHz electron cyclotron resonance (ECR) source with a magnetron magnetic field configuration was developed to meet the demand of a hyperthermal neutral beam (HNB) flux on a substrate of more than 1x10(15) cm(-2) s(-1) for industrial applications. The parameters of the operating pressure, ion density, electron temperature, and distance between the neutralization plate and the substrate for the HNB source are specified in a theoretical analysis. The electron temperature and the ion density are measured to characterize the ECR HNB source using a Langmuir probe and optical emission spectroscopy. The parameters of the ECR HNB source are in good agreement with the theoretically specified parameters. PMID:20815600
Kim, Seong Bong; Kim, Dae Chul; Namkung, Won; Cho, Moohyun; Yoo, Suk Jae
2010-08-01
87
SciTech Connect
Faas, S.E.; Winters, W.S.
1986-03-01
88
SciTech Connect
Effects of multihelicity confinement magnetic fields on turbulent transport and zonal flows are investigated by means of spatiotemporal analysis of gyrokinetic Vlasov simulation results for the ion temperature gradient turbulence, where the standard and the inward-shifted configurations of the Large Helical Device are considered. The analysis of simulation results demonstrates that fluctuations of electrostatic potential for zonal flows exhibit spatiotemporal chaos in both configurations. However, the intensity of chaos found is considerably decreased in the inward-shifted configuration consistent with improved confinement. Enhanced zonal flow generation in the inward shifted case is accompanied by transport reduction which may be a direct consequence of chaos suppression.
Rajkovic, Milan [Institute of Nuclear Sciences Vinca, Belgrade 11001 (Serbia); National Institute for Fusion Science, Toki 509-5292 (Japan); Watanabe, Tomo-Hiko; Skoric, Milos [National Institute for Fusion Science, Toki 509-5292 (Japan)
2009-09-15
89
SciTech Connect
The US Home Team has investigated the physics and engineering issues for two alternate poloidal field coil configurations for ITER. The first is called the Segmented CS configuration, where all of the solenoid modules are pancake-wound. The second option, termed the Hybrid CS configuration, utilizes a layer-wound central module and pancake-wound end modules. Performance comparisons are presented for the baseline design and the two alternate PF configurations, characterizing the 21 MA reference scenario. Alternate operating modes such as reverse-shear operation and a 17 MA driven mode were evaluated, but are not reported here.
Bulmer, R.H.; Neilson, G.H.
1997-09-02
90
To date, magnetic-resonance force microscopes employing a magnetic-field gradient source mounted to a microcantilever have suffered from a deleterious dependence of the effective cantilever spring constant on the external magnetic field. A magnet-on-tip'' configuration is introduced in which this dependence has been decreased by at least 200 fold, making it feasible to perform arbitrary-sample micron-scale magnetic resonance force microscopy at very high magnetic field. Alternating-gradient cantilever magnetometry is used to quantify the effect and to prove that the existing model of the tip-field interaction is only qualitatively correct. A model is proposed which quantitatively describes the tip-field interaction in the traditional tip configuration.
Marohn, John A.; Fainchtein, Raúl; Smith, Doran D.
1998-12-01
91
National Technical Information Service (NTIS)
The tilt instability in field-reversed configurations (FRC's) may be roughly divided into two categories, depending on the nature of the unstable eigenfunction. The internal tilt instability, which could also be called an n = 1 ballooning mode, is defined...
J. L. Schwarzmeier H. R. Lewis D. C. Barnes C. E. Seyler
1985-01-01
92
SciTech Connect
The Superconducting Proton Linac (SPL) is a novel linear accelerator concept currently studied at CERN. As part of this study, a new Cs-free, RF-driven external antenna H{sup -} plasma generator has been developed to withstand an average thermal load of 6 kW. The magnetic configuration of the new plasma generator includes a dodecapole cusp field and a filter field separating the plasma heating and H{sup -} production regions. Ferrites surrounding the RF antenna serve in enhancing the coupling of the RF to the plasma. Due to the space requirements of the plasma chamber cooling circuit, the cusp magnets are pushed outwards compared to Linac4 and the cusp field strength in the plasma region is reduced by 40% when N-S magnetized magnets are used. The cusp field strength and plasma confinement can be improved by replacing the N-S magnets with offset Halbach elements of which each consists of three magnetic sub-elements with different magnetization direction. A design challenge is the dissipation of RF power induced by eddy currents in the cusp and filter magnets which may lead to overheating and demagnetization. In view of this, a copper magnet cage has been developed that shields the cusp magnets from the radiation of the RF antenna.
Kronberger, Matthias; Chaudet, Elodie; Favre, Gilles; Lettry, Jacques; Kuechler, Detlef; Moyret, Pierre; Paoluzzi, Mauro; Prever-Loiri, Laurent; Schmitzer, Claus; Scrivens, Richard; Steyaert, Didier [CERN, 385 Route de Meyrin, 1211 Geneva (Switzerland)
2011-09-26
93
SciTech Connect
The local stability of field-reversed configurations (FRC) is analyzed using hydrodynamic stability theory. The equation of state includes both compressibility and double-adiabatic effects. For the first time, eigenmodes of the linearized equations of motion have been computed. The most unstable modes have fast growth rates, comparable to the Alfven transit time across the FRC radius; i.e., somewhat faster than the frequency (or growth rate) of global modes. In realistic equilibria, the most unstable local modes concentrate, ballooning-mode style, in the high curvature region of magnetic flux lines. The familiar interchange stability criterion is irrelevant for FRCs, since the actual eigenmodes differ markedly from interchange, both in structure and stability. The appearance of [ital fast] local modes raises the possibility that they may regulate FRC equilibria. However, surprisingly, equilibria with realistic internal structure (i.e. resembling experiments) are [ital more] unstable to ideal local modes than less realistic equilibria, as have often been studied theoretically. Thus, a nonideal theory will be needed to explain the equilibria observed in experiments.
Ishida, A. (Department of Environmental Science, Faculty of Science, Niigata University, Ikarashi, Niigata 950-21 (Japan)); Shibata, N. (Department of Physics, Faculty of Science, Niigata University, Ikarashi, Niigata 950-21 (Japan)); Steinhauer, L.C. (Redmond Plasma Physics Laboratory, University of Washington, Redmond, Washington 98052 (United States))
1994-12-01
94
We show that the location of Halpha or OV flare brightenings is related to the properties of the field-line linkage of the underlying magnetic region. The coronal magnetic field is extrapolated from the observed photospheric field assuming a linear force-free field configuration in order to determine the regions of rapid change in field-line linkage, called \\
P. Demoulin; L. G. Bagala; C. H. Mandrini; J. C. Henoux; M. G. Rovira
1997-01-01
95
The evidence of cosmic magnetism is examined, taking into account the Zeeman effect, beats in atomic transitions, the Hanle effect, Faraday rotation, gyro-lines, and the strength and scale of magnetic fields in astrophysics. The origin of magnetic fields is considered along with dynamos, the conditions for magnetic field generation, the topology of flows, magnetic fields in stationary flows, kinematic turbulent
Ia. B. Zeldovich; A. A. Ruzmaikin; D. D. Sokolov
1983-01-01
96
SciTech Connect
This paper considers an intense non-neutral charged particle beam propagating in the z-direction through a periodic focusing quadrupole magnetic field with transverse focusing force, -{kappa}{sub q}(s)[xe{sub x}-ye{sub y}], on the beam particles. Here, s={beta}{sub b}ct is the axial coordinate, ({gamma}{sub b}-1)m{sub b}c{sup 2} is the directed axial kinetic energy of the beam particles, q{sub b} and m{sub b} are the charge and rest mass, respectively, of a beam particle, and the oscillatory lattice coefficient satisfies {kappa}{sub q}(s+S)={kappa}{sub q}(s), where S is the axial periodicity length of the focusing field. The particle motion in the beam frame is assumed to be nonrelativistic, and the Vlasov-Maxwell equations are employed to describe the nonlinear evolution of the distribution function f{sub b}(x,y,x{sup '},y{sup '},s) and the (normalized) self-field potential {psi}(x,y,s)=q{sub b}{phi}(x,y,s)/{gamma}{sub b}{sup 3}m{sub b}{beta}{sub b}{sup 2}c{sup 2} in the transverse laboratory-frame phase space (x,y,x{sup '},y{sup '}), assuming a thin beam with characteristic radius r{sub b}<configuration in which a long non-neutral plasma column (L>>r{sub p}) is confined axially by applied dc voltages V=const on end cylinders at z={+-}L, and transverse confinement in the x-y plane is provided by segmented cylindrical electrodes (at radius r{sub w}) with applied oscillatory voltages {+-}V{sub 0}(t) over 90 degree sign segments. Here, V{sub 0}(t+T)=V{sub 0}(t), where T=const is the oscillation period, and the oscillatory quadrupole focusing force on a particle with charge q and mass m near the cylinder axis is -m{kappa}{sub q}(t)[xe{sub x}-ye{sub y}], where {kappa}{sub q}(t){identical_to}8qV{sub 0}(t)/{pi}mr{sub w}{sup 2}. (c) 2000 American Institute of Physics.
Davidson, Ronald C. [Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States); Qin, Hong [Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States); Shvets, Gennady [Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States)
2000-03-01
97
SciTech Connect
The effective radial heat conduction {kappa}{sub eff} in a plasma configuration with nonoverlapped magnetic island chains is assessed by applying an ''optimal path'' method. This approach implies that heat is transported predominantly along paths rendering the minimum temperature variation and is related to the principle of minimum entropy production. Paths combined of up to three radial sections and two segments aligned along magnetic field lines are considered. It is demonstrated that the enhancement of {kappa}{sub eff} over the level of perpendicular heat conduction {kappa}{sub perpendicular} arising due to flows along magnetic field lines is controlled only by the Chirikov parameter and by the value 4b{sub r}{sup 2}{kappa}{sub parallel}/{kappa}{sub perpendicular}, where b{sub r} is the relative amplitude of the radial field resonant harmonic and {kappa}{sub parallel} is the parallel heat conduction.
Gupta, A.; Tokar, M. Z. [Institut fuer Energieforschung - Plasmaphysik, Forschungszentrum Juelich, Association EURATOM-FZJ, Trilateral Euregio Cluster, Juelich (Germany)
2008-03-15
98
In this dissertation, the diffusion rate of impurity species ions in a flux-coil generated, field-reversed configuration is directly measured. A tomographic diagnostic has been developed and implemented to accomplish this task. The diffusion of impurity species ions is shown to behave classically, after ambipolar effects are taken into account. The observed diffusion coefficient is often found to be far less than predicted classically. It is even negative at times. This implies that a strong, confining electric field is impeding the escape of the ions. The rigid rotor solution to the Vlasov/Maxwell equations provides a prediction of the plasma distribution function, through internal magnetic field measurements. It also predicts the structure of the electric and magnetic fields. Measurement of these fields allow verification of the analysis of the diffusion data. In the following we show: Dperp ? 500 m2/s and the electric field contributes to ion confinement.
Roche, Thomas Jonathan
99
The energy needed to power flares is thought to be stored in the coronal magnetic field. However, the energy release is efficient only at very small scales. Magnetic configurations with a complex topology, i.e. with separatrices, are the most obvious configurations where current sheets can form, and then, reconnection can efficiently occur. This has been confirmed for several flares computing
P. Démoulin
2007-01-01
100
NSDL National Science Digital Library
This lesson introduces students to the effects of magnetic fields in matter addressing permanent magnets, diamagnetism, paramagnetism, ferromagnetism, and magnetization. First students must compare the magnetic field of a solenoid to the magnetic field of a permanent magnet. Students then learn the response of diamagnetic, paramagnetic, and ferromagnetic material to a magnetic field. Now aware of the mechanism causing a solid to respond to a field, students learn how to measure the response by looking at the net magnetic moment per unit volume of the material.
VU Bioengineering RET Program, School of Engineering,
101
SciTech Connect
The basic properties of the Hamiltonian representation of magnetic fields in canonical form are reviewed. The theory of canonical magnetic perturbation theory is then developed and applied to the time evolution of a magnetic field embedded in a toroidal plasma. Finally, the extension of the energy principle to tearing modes, utilizing the magnetic field line Hamiltonian, is outlined.
Boozer, A.H.
1985-02-01
102
\\u000a This paper presents numerical simulation model and results on magnetic drug targeting therapy. The study aims at investigating\\u000a the aggregate blood - magnetic carrier flow interaction with an external magnetic field. Another objective was finding the\\u000a optimal magnetic field source configuration that provides for flows that best assist in magnetic drug targeting. In order\\u000a to evaluate the effects we used
A. Dobre; A. M. Morega
103
With the Hylleraas-Gaussian basis set, in which the term of r121 is expanded approximately in Gaussian-type geminals, a full configuration-interaction (CI) method is applied to calculate the lowest 1?g and 1?u states of the hydrogen molecule in magnetic fields up to 2.35 × 107 T. In the absence of magnetic field, the total energies of the lowest 1?g and 1?u states in our calculation are -1.174 447 7(4) at the equilibrium distance of R = 1.40 a.u and -0.756 613 4(6) at R = 2.43 a.u., respectively. Compared to the CI method with Gaussian basis set, a significant improvement in the precision of the total energies and the dissociation energies at corresponding equilibrium distances has been achieved. The z1-z2 probability density distributions in different field regions are calculated and analyzed.
Song, Xuanyu; Qiao, Haoxue; Wang, Xiaofeng
2012-08-01
104
The processes by which nonlinear physical systems approach thermal equilibrium is of great importance in many areas of science. Central to this is the mechanism by which energy is transferred between the many degrees of freedom comprising these systems. With this in mind, in this research the nonequilibrium dynamics of nonperturbative fluctuations within Ginzburg-Landau models are investigated. In particular, two questions are addressed. In both cases the system is initially prepared in one of two minima of a double-well potential. First, within the context of a (2 + 1) dimensional field theory, we investigate whether emergent spatio-temporal coherent structures play a dynamcal role in the equilibration of the field. We find that the answer is sensitive to the initial temperature of the system. At low initial temperatures, the dynamics are well approximated with a time-dependent mean-field theory. For higher temperatures, the strong nonlinear coupling between the modes in the field does give rise to the synchronized emergence of coherent spatio-temporal configurations, identified with oscillons. These are long-lived coherent field configurations characterized by their persistent oscillatory behavior at their core. This initial global emergence is seen to be a consequence of resonant behavior in the long wavelength modes in the system. A second question concerns the emergence of disorder in a highly viscous system modeled by a (3 + 1) dimensional field theory. An integro-differential Boltzmann equation is derived to model the thermal nucleation of precursors of one phase within the homogeneous background. The fraction of the volume populated by these precursors is computed as a function of temperature. This model is capable of describing the onset of percolation, characterizing the approach to criticality (i.e. disorder). It also provides a nonperturbative correction to the critical temperature based on the nonequilibrium dynamics of the system.
Howell, Rafael Cassidy
105
We present calculations of the magnetization configuration and reversal behavior of magnetic nanotubes with uniaxial anisotropy by means of two-dimensional micromagnetic simulations and analytical methods. The tube radii R from 50 to 150 nm and the tube length /radius aspect ratio L/R<=20 were explored. For a finite length of magnetic nanotubes the magnetization configuration is characterized by a uniformly magnetized along the tube axis middle part and two nonuniform curling states of a length Lc in two ends of the tube with the same or opposite magnetization rotating senses, referring as C-state or B-state, respectively. We found that the magnetization configuration of the C-state exists for thin nanotubes with the tube thickness, ?R, in the range of ?R/R<=0.2. For thicker nanotubes the strong magnetostatic stray field forces the change of rotating senses of the end domains in opposite directions (the B-state). The transition from the C-state to a vortex state with in-plane magnetization is described as function of the tube geometrical parameters. The nanotube hysteresis loops and switching fields were calculated. The simple analytical model was developed to describe the nanotube magnetization reversal reducing its description to the Stoner-Wohlfarth model with effective parameters. The equilibrium state of nanotube is described in terms of ?, the angle of the magnetization deviation from the intrinsic tube easy axis. The L/R dependence of the C-state magnetization, the shape of hysteresis loops and the switching field values are described by a dependence of ? on L/R.
Chen, A. P.; Guslienko, K. Y.; Gonzalez, J.
2010-10-01
106
SciTech Connect
New computational results are presented which advance the understanding of the stability properties of the Field-Reversed Configuration (FRC). The FRC is an innovative confinement approach that offers a unique fusion reactor potential because of its compact and simple geometry, translation properties, and high plasma beta. One of the most important issues is FRC stability with respect to low-n (toroidal mode number) MHD modes. There is a clear discrepancy between the predictions of standard MHD theory that many modes should be unstable on the MHD time scale, and the observed macroscopic resilience of FRCs in experiments.
E.V. Belova; R.C. Davidson; H. Ji; and M. Yamada
2002-07-09
107
SciTech Connect
A pressure profile inside and outside a separatrix of a field-reversed configuration is determined by comparing three types of assumed pressure profiles with the radial profile of bremsstrahlung. It is found that the pressure profile is flatter near the field null than the rigid rotor profile. Edge-layer parameters as beta value at the separatrix, separatrix radius, and edge-layer width are determined from the pressure profile. The reliability of those parameters is confirmed by a magnetic method measuring an excluded flux radius.
Ikeyama, Taeko; Hiroi, Masanori; Nogi, Yasuyuki [College of Science and Technology, Nihon University, Tokyo 101-8308 (Japan); Ohkuma, Yasunori [College of Industrial Technology, Nihon University, Chiba 275-8576 (Japan)
2009-04-15
108
SciTech Connect
At the time of the Voyager 2 flyby of Uranus, the planetary rotational axis will be roughly antiparallel to the solar wind flow. If Uranus has a magnetic dipole moment that is approximately aligned with its spin axis, and if the heliospheric shock has not been encountered, we will have the rare opportunity to observe a ''pole-on'' magnetosphere as discussed qualitatively by Siscoe. Qualitative arguments based on analogy with Earth, Jupiter, and Saturn suggest that the magnetosphere of Uranus may lack a source of plasma adequate to produce significant internal currents, internal convection, and associated effects. In order to provide a test of this hypothesis with the forthcoming Voyager measurements, we have constructed a class of approximately self-consistent quantitative magnetohydrostatic equilibrium configurations for a pole-on magnetosphere with variable plasma pressure parameters. Given a few simplifying assumptions, the geometries of the magnetic field and of the tail current sheet can be computed for a given distribution of trapped plasma pressure. The configurations have a single funnel-shaped polar cusp that points directly into the solar wind and a cylindrical tail plasma sheet whose currents close within the tail rather than on the tail magnetopause, and whose length depends on the rate of decrease of thermal plasma pressure down the tail. Interconnection between magnetospheric and interplanetary fields results in a highly asymmetric tail-field configuration. These features were predicted qualtitatively by Siscoe; the quantitative models presented here may be useful in the interpretation of Voyager encounter results.
Voigt, G.; Hill, T.W.; Dessler, A.J.
1983-03-01
109
Electrical currents flowing in the solar plasma generate a magnetic field, which is detected in the SOLAR ATMOSPHERE by spectroscopic and polarization measurements (SOLAR MAGNETIC FIELD: INFERENCE BY POLARIMETRY). The SOLAR WIND carries the magnetic field into interplanetary space where it can be measured directly by instruments on space probes....
Schüssler, M.; Murdin, P.
2000-11-01
110
SciTech Connect
Some recently developed Hall thrusters utilize a magnetic field configuration in which the field lines penetrate the thruster walls at a high incidence angle. This so-called magnetic lens leads to an electric field pointing away from the walls, which is expected to reduce ion losses and improve thruster efficiency. This configuration also introduces an interesting behavior in the sheath formation. At sufficiently large angles, ions are repelled from the wall, and sheath collapse is expected. We use a plasma simulation code to investigate this phenomenon in detail. We consider the role of the magnetic field incidence angle, secondary electron emission, and a magnetic mirror. Numerical study confirms the theoretical predictions, and at large angles, ions are seen to turn away from the wall. We also consider the role of the magnetic field geometry on ion wall flux and channel erosion, and observe reduction in both quantities as the magnetic field incidence angle is increased.
Brieda, Lubos; Keidar, Michael [Department of Mechanical and Aerospace Engineering, George Washington University, 801 22nd St., Washington, DC 20052 (United States)
2012-06-15
111
SciTech Connect
We present a multifrequency radio investigation of the Crab-like pulsar wind nebula (PWN) G54.1+0.3 using the Very Large Array. The high resolution of the observations reveals that G54.1+0.3 has a complex radio structure which includes filamentary and loop-like structures that are magnetized, a diffuse extent similar to the associated diffuse X-ray emission. But the radio and X-ray structures in the central region differ strikingly, indicating that they trace very different forms of particle injection from the pulsar and/or particle acceleration in the nebula. No spectral index gradient is detected in the radio emission across the PWN, whereas the X-ray emission softens outward in the nebula. The extensive radio polarization allows us to image in detail the intrinsic magnetic field, which is well-ordered and reveals that a number of loop-like filaments are strongly magnetized. In addition, we determine that there are both radial and toroidal components to the magnetic field structure of the PWN. Strong mid-infrared (IR) emission detected in Spitzer Space Telescope data is closely correlated with the radio emission arising from the southern edge of G54.1+0.3. In particular, the distributions of radio and X-ray emission compared with the mid-IR emission suggest that the PWN may be interacting with this interstellar cloud. This may be the first PWN where we are directly detecting its interplay with an interstellar cloud that has survived the impact of the supernova explosion associated with the pulsar's progenitor.
Lang, Cornelia C.; Clubb, Kelsey I. [Department of Physics and Astronomy, 703 Van Allen Hall, University of Iowa, Iowa City, IA 52242 (United States); Wang, Q. Daniel; Lu Fangjun, E-mail: cornelia-lang@uiowa.ed [Department of Astronomy, University of Massachusetts, Amherst, MA 01002 (United States)
2010-02-01
112
In f2-based heavy fermion systems with a crystalline-electric-field (CEF) singlet ground state, the non-Fermi liquid (NFL) arises around the quantum critical point (QCP) due to the competition between the CEF singlet and the Kondo-Yosida singlet states. In such a case, the characteristic temperature TF* at which the entropy starts to decrease toward zero is suppressed by the effect of the competition, compared to both energy scales characterizing each singlet state, the lower Kondo temperature (TK2) and the CEF splitting (?). We show that in the case of tetragonal symmetry TF* is not affected by the magnetic field up to Hz* which is determined by the distance from the QCP or characteristic energy scales of each singlet states, not by TF* itself. As a result, in the vicinity of QCP, there are parameter regions where the NFL is robust against the magnetic field, at an observable temperature range T > TF*, up to Hz* which is far larger than TF* and less than \\min(TK2,?). Our result suggests that such an anomalous NFL behavior can arise also in systems with other CEF symmetry, which might provide us with the basis to understand the anomalous behaviors of UBe13.
Nishiyama, Shinya; Matsuura, Hiroyasu; Miyake, Kazumasa
2010-10-01
113
PubMed
We describe ab initio, self-consistent, 3D, fully electromagnetic numerical simulations of current drive and field-reversed-configuration plasma formation by odd-parity rotating magnetic fields (RMF{o}). Magnetic-separatrix formation and field reversal are attained from an initial mirror configuration. A population of punctuated-betatron-orbit electrons, generated by the RMF{o}, carries the majority of the field-normal azimuthal electrical current responsible for field reversal. Appreciable current and plasma pressure exist outside the magnetic separatrix whose shape is modulated by the RMF{o} phase. The predicted plasma density and electron energy distribution compare favorably with RMF{o} experiments. PMID:20867454
Welch, D R; Cohen, S A; Genoni, T C; Glasser, A H
2010-07-01
114
NSDL National Science Digital Library
This demonstration of the magnetic field lines of Earth uses a bar magnet, iron filings, and a compass. The site explains how to measure the magnetic field of the Earth by measuring the direction a compass points from various points on the surface. There is also an explanation of why the north magnetic pole on Earth is actually, by definition, the south pole of a magnet.
Barker, Jeffrey
115
National Technical Information Service (NTIS)
The first observations of internal tilt instabilities in field-reversed configurations (FRCs) are reported. Detailed comparisons with theory establish that data from an array of external magnetic probes are signatures of these destructive plasma instabili...
D. J. Rej M. Tuszewski D. C. Barnes G. A. Barnes R. E. Chrien
1990-01-01
116
The effect of magnetic fields on dielectric surface breakdown in vacuum and simulated LEO conditions is investigated using pulsed test voltages. Predictions from the saturated secondary electron emission avalanche breakdown model and experimental results both show magnetic insulation effects at magnetic-field amplitudes as low as 0.1 T. The most favorable configuration for magnetic insulation is with the magnetic field oriented
M. Lehr; R. Korzekwa; H. Krompholz; M. Kristiansen
1992-01-01
117
SciTech Connect
This paper presents the first detailed model of hybrid equilibria relevant to field-reversed configuration experiments, leading to a system of equations that are solved for a range of fully two-dimensional equilibria. Several features of these highly kinetic objects are explored. The range of equilibria is primarily dependent on a single free parameter related to the flow shear. The level of flow shear has a profound effect on the structure, especially near the separatrix. This likely has a strong influence on both stability and transport properties. Higher flow shear is favorable in every respect. The key factor behind the influence of flow shear is the relatively rapid end loss of unconfined ions. Differences between hybrid and static-fluid equilibrium models are highlighted, including the integrity of surface functions, the effect of flow shear, and the scrape-off layer thickness.
Steinhauer, Loren C. [Department of Aeronautics and Astronautics, University of Washington, Seattle, Washington 98195 (United States)
2011-11-15
118
NSDL National Science Digital Library
The above animations represent two typical bar magnets each with a North and South pole. The arrows represent the direction of the magnetic field. The color of the arrows represents the magnitude of the field with magnitude increasing as the color changes from blue to green to red to black. You may drag either magnet and double-click anywhere inside the animation to add a magnetic field line, and mouse-down to read the magnitude of the magnetic field at that point.
Christian, Wolfgang; Belloni, Mario
2007-03-03
119
SciTech Connect
In order to establish an optimized bearing design for a flywheel for energy storage, the authors have studied model bearing configurations involving bulk YBCO pellets and double-dipole magnet configurations. They were interested to see what is the correlation between the maximum attainable levitation force, measured for a typical bearing gap of 3 mm, and the separation between the magnetic poles. Equal polarity (north-north) and alternate polarity (north-south) configurations were investigated. The maximum levitation force was obtained with the alternate polarity arrangement for a separation between the magnetic poles of 6 mm. It represents an increase of 19% compared to a non-optimized configuration. The experiments demonstrate that configurations of superconducting magnetic bearings can be optimized to obtain better levitation properties.
Schoechlin, A.; Ritter, T.; Bornemann, H.J. [Forschungszentrum Karlsruhe GmbH (Germany)
1995-11-01
120
SciTech Connect
Electromagnetic drift instabilities are studied in the conditions of a field reversed configuration (FRC). Dispersion equation is based on the set of Vlasov-Maxwell equations taking into account nonadiabatic responses both of ions and electrons. Considered drift instabilities are caused by density and temperature gradients. It is assumed that magnetic field of the FRC is purely poloidal. Two kinds of magnetic field nonuniformity are considered: (i) perpendicular gradient due to high beta values (beta is the plasma pressure/magnetic pressure) and (ii) curvature of magnetic lines. There is low frequency drift instability existing for high-beta regimes. Modes of such instability can propagate transversally to the unperturbed magnetic field lines.
Chirkov, A. Yu.; Khvesyuk, V. I. [Bauman Moscow State Technical University, 2-nd Baumanskaya, 5, Moscow 105005 (Russian Federation)
2010-01-15
121
Most of the visible matter in the Universe is ionized so that cosmic magnetic fields are quite easy to generate and, due to the lack of magnetic monopoles, hard to destroy. Magnetic fields have been measured in or around practically all celestial objects, either by in situ measurements of spacecrafts or by the electromagnetic radiation of embedded cosmic rays, gas, or dust. The Earth, the Sun, solar planets, stars, pulsars, the Milky Way, nearby galaxies, more distant (radio) galaxies, quasars, and even intergalactic space in clusters of galaxies have significant magnetic fields, and even larger volumes of the Universe may be permeated by "dark" magnetic fields. Information on cosmic magnetic fields has increased enormously as the result of the rapid development of observational methods, especially in radio astronomy. In the Milky Way, a wealth of magnetic phenomena was discovered, which are only partly related to objects visible in other spectral ranges. The large-scale structure of the Milky Way's magnetic field is still under debate. The available data for external galaxies can well be explained by field amplification and ordering via the dynamo mechanism. The measured field strengths and the similarity of field patterns and flow patterns of the diffuse ionized gas give strong indication that galactic magnetic fields are dynamically important. They may affect the formation of spiral arms, outflows, and the general evolution of galaxies. In spite of our increasing knowledge on magnetic fields, many important questions on the origin and evolution of magnetic fields, their first occurrence in young galaxies, or the existence of large-scale intergalactic fields remained unanswered. The present upgrades of existing instruments and several planned radio astronomy projects have defined cosmic magnetism as one of their key science projects.
Beck, Rainer; Wielebinski, Richard
122
There is no observational support to the hypothesis of the most large-scale homogeneous magnetic field in the Universe. The best upper limit is given by interpretation of the Faraday rotation from the extragalactic radio sources. However the magnetic fields can be generated in the clusters of galaxies by a turbulence in the wakes of moving galaxies. These fields have an
A. A. Ruzmajkin
1991-01-01
123
NSDL National Science Digital Library
Clicking on the different links below will produce different magnetic fields in the box above. The wires (perpendicular to the screen) or coils (in and out of the screen) are not visible, but you can determine what they are from the field. You can also click on a point to read off the magnetic field at that place.
Christian, Wolfgang; Belloni, Mario
2008-02-19
124
PubMed
There is public health concern raised by epidemiological studies indicating that extremely low frequency electric and magnetic fields generated by electric power distribution systems in the environment may be hazardous. Possible carcinogenic effects of magnetic field in combination with suggested oncostatic action of melatonin lead to the hypothesis that the primary effects of electric and magnetic fields exposure is a reduction of melatonin synthesis which, in turn, may promote cancer growth. In this review the data on the influence of magnetic fields on melatonin synthesis, both in the animals and humans, are briefly presented and discussed. PMID:12019358
Karasek, Michal; Lerchl, Alexander
2002-04-01
125
A radial magnetic bearing, consisting of two permanent magnets, is an attractive choice because of its zero wear, negligible friction, and low cost, but it suffers from low load capacity, low radial stiffness, lack of damping, and high axial instability. To enhance the radial load and radial stiffness, and reduce the axial thrust, we have made a theoretical and experimental
Pranab Samanta; Harish Hirani
2008-01-01
126
US Patent & Trademark Office Database
A magnetic resonance system is disclosed. The system includes a transceiver having a multichannel receiver and a multichannel transmitter, where each channel of the transmitter is configured for independent selection of frequency, phase, time, space, and magnitude, and each channel of the receiver is configured for independent selection of space, time, frequency, phase and gain. The system also includes a magnetic resonance coil having a plurality of current elements, with each element coupled in one to one relation with a channel of the receiver and a channel of the transmitter. The system further includes a processor coupled to the transceiver, such that the processor is configured to execute instructions to control a current in each element and to perform a non-linear algorithm to shim the coil.
2010-09-21
127
This work aims at studying how magnetic fields affect the observational properties and the long-term evolution of isolated neutron stars, which are the strongest magnets in the universe. The extreme physical conditions met inside these astronomical sources complicate their theoretical study, but, thanks to the increasing wealth of radio and X-ray data, great advances have been made over the last years. A neutron star is surrounded by magnetized plasma, the so-called magnetosphere. Modeling its global configuration is important to understand the observational properties of the most magnetized neutron stars, magnetars. On the other hand, magnetic fields in the interior are thought to evolve on long time-scales, from thousands to millions of years. The magnetic evolution is coupled to the thermal one, which has been the subject of study in the last decades. An important part of this thesis presents the state-of-the-art of the magneto-thermal evolution models of neutron stars during the first million of years, studied by means of detailed simulations. The numerical code here described is the first one to consistently consider the coupling of magnetic field and temperature, with the inclusion of both the Ohmic dissipation and the Hall drift in the crust.
Viganò, Daniele
2013-09-01
128
DOEpatents
A magnetic field generating device provides a useful magnetic field within a specific retgion, while keeping nearby surrounding regions virtually field free. By placing an appropriate current density along a flux line of the source, the stray field effects of the generator may be contained. One current carrying structure may support a truncated cosine distribution, and it may be surrounded by a current structure which follows a flux line that would occur in a full coaxial double cosine distribution. Strong magnetic fields may be generated and contained using superconducting cables to approximate required current surfaces.
Krienin, Frank (Shoreham, NY)
1990-01-01
129
Magnetic fields are present in all astrophysical media. However, many models and interpretations of observations often ignore them, because magnetic fields are difficult to handle and because they produce complicated morphological features. Here we will comment on the basic intuitive properties, which even if not completely true, provide a first guiding insight on the physics of a particular astrophysical problem. These magnetic properties are not mathematically demonstrated here. How magnetic fields evolve and how they introduce dynamical effects are considered, also including a short comment on General Relativity Magnetohydrodynamics. In a second part we consider some audacious and speculative matters. They are answers to three questions: a) How draw a cube without lifting the pencil from the paper so that when the pen passes through the same side do in the same direction? B) Are MILAGRO anisotropies miraculous? C) Do cosmic magnetic lenses exist?. The last two questions deal with issues related with the interplay between magnetic fields and cosmic ray propagation.
Florido, E.; Battaner, E.
2010-12-01
130
The conclusions drawn regarding the structure, behavior and composition of the Uranian magnetic field and magnetosphere as revealed by Voyager 2 data are summarized. The planet had a bipolar magnetotail and a bow shock wave which was observed 23.7 Uranus radii (UR) upstream and a magnetopause at 18.0 UR. The magnetic field observed can be represented by a dipole offset
N. F. Ness; M. H. Acuna; K. W. Behannon; L. F. Burlaga; J. E. P. Connerney; R. P. Lepping
1986-01-01
131
A new analysis of magnetic and concurrent plasma data collected from the ; space probes Pionecr 5, Explorer 10, and Mariner 2 yields a new model of the ; interplanetary magnetic field. It is hypothesized that the observed ; interplanetary field F\\/sub i\\/ is due to motion of the magnetometer relative to a ; negatively charged rotating sun from which
V. A. BAILEY
1963-01-01
132
Most of the visible matter in the Universe is in a plasma state, or more specifically is composed of ionized or partially ionized gas permeated by magnetic fields. Thanks to recent advances on the theory and detection of cosmic magnetic fields there has been a worldwide growing interest in the study of their role on the formation of astrophysical sources
Elisabete M. de Gouveia Dal Pino; Dal Pino
2006-01-01
133
We explore the linear and nonlinear dynamic regimes of micrometer-scale soft magnetic squares with an in-plane uniaxial anisotropy, recording the response of the magnetization in the spatial domain under a continuous sinusoidal excitation with time-resolved scanning transmission x-ray microscopy. Increasing the excitation field amplitude leads to the dynamic stabilization of nonequilibrium domain configurations, which appear at threshold amplitudes and return to the equilibrium configuration when the amplitude is reduced. On imaging the magnetization in the transition region between two stable magnetic configurations, we observe small domains originating from the vortex core. Dynamic micromagnetic simulations of the domain configuration provide qualitative agreement with the experimental data and insight into the energy dissipation and spin wave contributions.
Stevenson, S. E.; Moutafis, C.; Heldt, G.; Chopdekar, R. V.; Quitmann, C.; Heyderman, L. J.; Raabe, J.
2013-02-01
134
SciTech Connect
In this lecture, the dynamic processes involved in field-reversed configuration (FRC) formation, translation, and compression will be reviewed. Though the FRC is related to the field-reversed mirror concept, the formation method used in most experiments is a variant of the field-reversed THETA-pinch. Formation of the FRC eqilibrium occurs rapidly, usually in less than 20 ..mu..s. The formation sequence consists of several coupled processes: preionization; radial implosion and compression; magnetic field line closure; axial contraction; equilibrium formation. Recent experiments and theory have led to a significantly improved understanding of these processes; however, the experimental method still relies on a somewhat empirical approach which involves the optimization of initial preionization plasma parameters and symmetry. New improvements in FRC formation methods include the use of lower voltages which extrapolate better to larger devices. The axial translation of compact toroid plasmas offers an attractive engineering convenience in a fusion reactor. FRC translation has been demonstrated in several experiments worldwide, and these plasmas are found to be robust, moving at speeds up to the Alfven velocity over distances of up to 16 m, with no degradation in the confinement. Compact toroids are ideal for magnetic compression. Translated FRCs have been compressed and heated by imploding liners. Upcoming experiments will rely on external flux compression to heat a translater FRC at 1-GW power levels. 39 refs.
Rej, D.J.
1987-01-01
135
NSDL National Science Digital Library
The magnetic field of the Earth is contained in a region called the magnetosphere. The magnetosphere prevents most of the particles from the sun, carried in solar wind, from hitting the Earth. This site, produced by the University Corporation for Atmospheric Research (UCAR), uses text, scientific illustrations,and remote imagery to explain the occurrence and nature of planetary magnetic fields and magnetospheres, how these fields interact with the solar wind to produce phenomena like auroras, and how magnetic fields of the earth and other planets can be detected and measured by satellite-borne magnetometers.
136
The configuration management system for the Large Hadron Collider (LHC) superconducting magnet test facility has been integrated with the already existing software enforcing quality assurance procedures of the LHC - the Manufacturing and Test Folder (MTF). Such a solution provides one common access point to all the data relevant to each of the tested magnets. The MTF software was developed
T. Ladzinski; M. Gateau; P. Martel; A. Siemko; D. Widegren
2003-01-01
137
Liquids (~7 neutron mean free paths thick), with certain restrictions, can probably be used in magnetic fusion designs between the burning plasma and the structural materials of the fusion power core. If this works there would be a number of profound advantages: a cost of electricity lower by as much as a factor of 2; removal of the need to develop new first wall materials, saving over 4 billion US dollars in development costs; a reduction of the amount and kinds of wastes generated in the plant; and the wider choice of materials permitted. The amount of material that evaporates from the liquid which can be allowed to enter the burning plasma is estimated to be less than 0.7% for lithium, 1.9% for Flibe (Li2BeF4 or LiBeF3) and 0.01% for Li17Pb83. The ability of the edge plasma to attenuate the vapour by ionization appears to exceed this requirement. This ionized vapour would be swept along open field lines into a remote burial chamber. The most practical systems would be those with topological open field lines on the outer surface, as is the case with a field reversed configuration (FRC), a spheromak, a Z pinch or a mirror machine. In a tokamak, including a spherical tokamak, the field lines outside the separatrix are restricted to a small volume inside the toroidal coil making for difficulties in introducing the liquid and removing the ionized vapour, i.e. the configuration is not open ended
Moir, R. W.
1997-04-01
138
National Technical Information Service (NTIS)
The production of the negative hydrogen ions in a large multicusp ion source has been investigated in a double-magnetic filter (DMF) configuration. In the DMF configuration, the energetic electrons are trapped by the mirror of a magnetic multicusp field, ...
A. Ando Y. Takeiri O. Kaneko Y. Oka M. Wada
1992-01-01
139
The production of the negative hydrogen ions in a large multicusp ion source has been investigated in a double-magnetic filter (DMF) configuration. In the DMF configuration, the energetic electrons are trapped by the mirror of a magnetic multicusp field, and only the thermal electrons are present in the center of the arc chamber. A large amount of H(-) ions of
A. Ando; Y. Takeiri; O. Kaneko; Y. Oka; M. Wada; T. Kuroda
1992-01-01
140
We construct both aligned and unaligned (logarithmic spiral) stationary configurations of non-axisymmetric magnetohydrodynamic (MHD) discs from either a full or a partial razor-thin power-law axisymmetric magnetized singular isothermal disc (MSID) that is embedded with a coplanar azimuthal magnetic field B? of a non-force-free radial scaling r-1/2 and that rotates differentially with a flat rotation curve of speed aD, where a is the isothermal speed of sound and D is the dimensionless rotation parameter. Analytical solutions and stability criteria for determining D2 are derived. For aligned non-axisymmetric MSIDs, eccentric m= 1 displacements may occur at arbitrary D2 in a full MSID but are allowed only with a2D2=C2A/2 in a partial MSID (where CA is the Alfvén speed), while each case of |m|>= 1 gives two possible values of D2 for purely azimuthal propagations of fast and slow MHD density waves (FMDWs and SMDWs) that appear stationary in an inertial frame of reference. For disc galaxies modelled by a partial MSID resulting from a massive dark matter halo with a flat rotation curve and a2D2>>C2A, stationary aligned perturbations of m= 1 are not allowed. For unaligned logarithmic spiral MSIDs with |m|>= 1, there exist again two values of D2, corresponding to FMDWs and SMDWs that propagate in both radial and azimuthal directions relative to the MSID and that appear stationary in an inertial frame of reference. The larger D2 is always physically valid, while the smaller D2 is valid only for a > CA/2 with a positive surface mass density ?0. For observational diagnostics, we examine the spatial phase relationships among enhancements of gas density and magnetic field, and velocity perturbations. These results are useful for probing magnetized bars, or lopsided, normal and barred spiral galaxies and for testing numerical MHD codes. In the case of NGC 6946, interlaced optical and magnetic field spiral patterns of SMDWs can persist in a disc with a flat rotation curve. Theoretical issues regarding the modal formalism and the MSID perspective are also discussed.
Lou, Yu-Qing
2002-11-01
141
We review current ideas on the origin of galactic and extragalactic magnetic fields. We begin by summarizing observations of magnetic fields at cosmological redshifts and on cosmological scales. These observations translate into constraints on the strength and scale magnetic fields must have during the early stages of galaxy formation in order to seed the galactic dynamo. We examine mechanisms for the generation of magnetic fields that operate prior during inflation and during subsequent phase transitions such as electroweak symmetry breaking and the quark-hadron phase transition. The implications of strong primordial magnetic fields for the reionization epoch as well as the first generation of stars are discussed in detail. The exotic, early-Universe mechanisms are contrasted with astrophysical processes that generate fields after recombination. For example, a Biermann-type battery can operate in a proto-galaxy during the early stages of structure formation. Moreover, magnetic fields in either an early generation of stars or active galactic nuclei can be dispersed into the intergalactic medium.
Widrow, Lawrence M.; Ryu, Dongsu; Schleicher, Dominik R. G.; Subramanian, Kandaswamy; Tsagas, Christos G.; Treumann, Rudolf A.
2012-05-01
142
NSDL National Science Digital Library
This activity will introduce students to the idea of magnetic field lines--a concept they have probably encountered but may not fully grasp. Completing this activity and reading the corresponding background information should enable students to understand
Horton, Michael
2009-05-30
143
SciTech Connect
In recent years there has been increased concern over potential health hazards related to exposure of personnel to magnetic fields. If exposure standards are to be established, then a means for measuring magnetic field dose must be available. To meet this need, the Department of Energy has funded development of prototype dosimeters at the Battelle Pacific Northwest Laboratory. This manual reviews the principle of operation of the dosimeter and also contains step-by-step instructions for its operation.
Lemon, D.K.; Skorpik, J.R.; Eick, J.L.
1980-09-01
144
For thick soft magnetic nanotubes with an anisotropy axis directed along the nanotube length the equilibrium energy ground states present magnetization configurations with opposite rotating senses in two tube ends (B-state), referring as antiparallel chiralities of the end vortex domains. For nanotubes with outer radius R of 50 nm, 100 nm and 150 nm, and length L = (2.5-20)R the B-state remanent magnetization and the reversal field dependence on tube thickness and anisotropy strength are studied by using both two-dimensional simulation and analytic methods. The equilibrium states, the hysteresis loops and the switching field values calculated numerically are presented as the functions of tube size and material parameters. For the short nanotubes the domain walls patterns, such as transverse walls and vortex walls, nucleating in the tube center, as well as the hysteresis loops of the nanotubes with transverse walls are presented. The numerical results are interpreted by a simple analytical model in which the equilibrium state of nanotube is described by ?, the angle of the magnetization M deviation from the intrinsic tube easy axis. ? as a function of the tube aspect ratio L/R, tube thickness ?R, and uniaxial anisotropy constant Ku, obtained by minimizing the total magnetic energy, well describes dependences of the shape of hysteresis loops and switching field values on the tube geometric and material parameters in the B-state.
Chen, Ai-Ping; Gonzalez, Julian M.; Guslienko, Konstantin Y.
2011-04-01
145
SciTech Connect
A magnetically levitated vehicle adapted for movement along a guide way, comprising: a passenger compartment; first and second primary magnet means secured on the vehicle to produce a magnetic field having a magnetic flux density extending outward from the primary magnet means, to support the vehicle above and spaced from the guide way; and a plurality of confining magnets disposed on the vehicle to confine the magnetic flux extending outward from the primary magnet means and to reduce the strength of the primary magnetic field in the passenger compartment; wherein the primary magnet means has a capacity to produce a primary magnetic field having a maximum strength of at least 200 gauss in the passenger compartment, and the confining magnets maintain the strength of the primary magnetic field in the passenger compartment below 5 gauss.
Proise, M.
1993-05-25
146
A hybrid thermistor was developed to realize a thermometer which is practically free of the magnetic field effect. Its design involves placing two kinds of thermistors in a bridge configuration so that the magnetic field effects cancel each other. The effect of the magnetic fields on the proposed sensor was precisely studied using the triple point of water. The correction
Koichi Nara
2005-01-01
147
SciTech Connect
Using vector-analyzer ferromagnetic resonance, we have studied the microwave susceptibility of a Py/Co/Cu/Co/MnIr spin valve over a large temperature range (5-450 K) and as a function of the magnetic configuration. An effective magnetization and Gilbert damping constant of 1.1 T and 0.021, respectively, are found for the permalloy free layer, with no discernible variation in temperature observed for either quantities. In contrast, the pinned layer magnetization is reduced by heating, and the exchange bias collapses near a temperature of 450 K. The ferromagnetic resonance linewidth of the free layer increases by 500 MHz when the layer magnetizations are aligned in antiparallel, which is attributed to a configuration-dependent contribution to the damping from spin pumping effects.
Joyeux, X.; Devolder, T.; Kim, Joo-Von; Gomez de la Torre, Y.; Eimer, S.; Chappert, C. [Institut d'Electronique Fondamentale, University Paris-Sud, 91405 Orsay (France); UMR8622, CNRS, University Paris-Sud, 91405 Orsay (France)
2011-09-15
148
The past several years have seen dramatic developments in the study of planetary magnetic fields, including a wealth of new data, mainly from the Galilean satellites and Mars, together with major improvements in our theoretical modeling effort of the dynamo process believed responsible for large planetary fields. These dynamos arise from thermal or compositional convection in fluid regions of large
David J. Stevenson
2003-01-01
149
NSDL National Science Digital Library
The EJS Magnetic Multipole Field Model shows the field of a magnetic dipole or quadrupole with little compasses that indicate direction and relative field strength. A slider changes the angular orientation of the dipole and a movable compass shows the magnetic field direction and magnitude. Compass values can be recorded into a data table and analyzed using a built-in data analysis tool. You can modify this simulation if you have Ejs installed by right-clicking within the plot and selecting âOpen Ejs Modelâ from the pop-up menu item. The Magnetic Multipole Field model was created using the Easy Java Simulations (Ejs) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double clicking the ejs_em_MagneticMultipoleField.jar file will run the program if Java is installed. Ejs is a part of the Open Source Physics Project and is designed to make it easier to access, modify, and generate computer models. Additional Ejs models are available. They can be found by searching ComPADRE for Open Source Physics, OSP, or Ejs.
Christian, Wolfgang; Cox, Anne; Franciscouembre
2010-02-14
150
SciTech Connect
A magnetic field measurement system was designed, built and installed at MAX Lab, Sweden for the purpose of characterizing the magnetic field produced by Insertion Devices (see Figure 1). The measurement system consists of a large granite beam roughly 2 feet square and 14 feet long that has been polished beyond laboratory grade for flatness and straightness. The granite precision coupled with the design of the carriage yielded minimum position deviations as measured at the probe tip. The Hall probe data collection and compensation technique allows exceptional resolution and range while taking data on the fly to programmable sample spacing. Additional flip coil provides field integral data.
Kulesza, Joe; Johnson, Eric; Lyndaker, Aaron; Deyhim, Alex; Waterman, Dave; Blomqvist, K. Ingvar [Advanced Design Consulting USA, 126 Ridge Road, P.O. Box 187, Lansing, NY 14882 (United States); Dunn, Jonathan Hunter [MAX-lab, SE-221 00 Lund (Sweden)
2007-01-19
151
NSDL National Science Digital Library
A cross section of a circular wire loop carrying an unknown current is shown above. The arrows represent the direction of the magnetic field. The color of the arrows represents the magnitude of the field with magnitude increasing as the color changes from blue to green to red to black. You can double-click in the animation to add magnetic field lines, click-drag the center of the loop to reposition it, and click-drag the top or bottom of the loop to change its size.
Christian, Wolfgang; Belloni, Mario
2007-03-03
152
The equivalent source dipole technique is used to model the three components of the Martian lithospheric magnetic field. We use magnetic field measurements made on board the Mars Global Surveyor spacecraft. Different input dipole meshes are presented and evaluated. Because there is no global, Earth-like, inducing magnetic field, the magnetization directions are solved for together with the magnetization intensity. A
B. Langlais; M. E. Purucker; M. Mandea
2004-01-01
153
The effect of an intense electromagnetic field formed by the superposition of a constant magnetic field and a laser-type field\\u000a on nuclear ?-decay and on pair production by two g-rays with different polarizations is studied. Time integral representations are obtained\\u000a for the total probabilities of these processes without restrictions on the strengths of the fields making up the configuration.\\u000a Despite
V. N. Rodionov
1998-01-01
154
SciTech Connect
We describe ab initio, self-consistent, 3D, fully electromagnetic numerical simulations of current drive and field-reversed-configuration plasma formation by odd-parity rotating magnetic fields (RMF{sub o}). Magnetic-separatrix formation and field reversal are attained from an initial mirror configuration. A population of punctuated-betatron-orbit electrons, generated by the RMF{sub o}, carries the majority of the field-normal azimuthal electrical current responsible for field reversal. Appreciable current and plasma pressure exist outside the magnetic separatrix whose shape is modulated by the RMF{sub o} phase. The predicted plasma density and electron energy distribution compare favorably with RMF{sub o} experiments.
Welch, D. R.; Genoni, T. C. [Voss Scientific, Albuquerque, New Mexico 87108 (United States); Cohen, S. A. [Princeton Plasma Physics Laboratory, Princeton, New Jersey 08544 (United States); Glasser, A. H. [Department of Aeronautics and Astronautics, University of Washington, Seattle, Washington 98195 (United States)
2010-07-02
155
We report the results of magnetic field modelling of around 50 CP stars, performed using the "magnetic charges" technique. The modelling shows that the sample reveals four main types of magnetic configurations: 1) a central dipole, 2) a dipole, shifted along the axis, 3) a dipole, shifted across the axis, and 4) complex structures. The vast majority of stars has the field structure of a dipole, shifted from the center of the star. This shift can have any direction, both along and across the axis. A small percentage of stars possess field structures, formed by two or more dipoles.
Glagolevskij, Yu. V.
2011-04-01
156
Radio synchrotron emission, its polarization and its Faraday rotation are powerful tools to study the strength and structure of magnetic fields in galaxies. Unpolarized emission traces turbulent fields which are strongest in spiral arms and bars (20-30 ?G) and in central starburst regions (50-100 ?G). Such fields are dynamically important, e.g. they can drive gas inflows in central regions. Polarized emission traces ordered fields which can be regular or anisotropic random, generated from isotropic random fields by compression or shear. The strongest ordered fields of 10-15 ?G strength are generally found in interarm regions and follow the orientation of adjacent gas spiral arms. Ordered fields with spiral patterns exist in grand-design, barred and flocculent galaxies, and in central regions of starburst galaxies. Faraday rotation measures (RM) of the diffuse polarized radio emission from the disks of several spiral galaxies reveal large-scale patterns, which are signatures of regular fields generated by a mean-field dynamo. However, in most spiral galaxies observed so far the field structure is more complicated. Ordered fields in interacting galaxies have asymmetric distributions and are an excellent tracer of past interactions between galaxies or with the intergalactic medium. Ordered magnetic fields are also observed in radio halos around edge-on galaxies, out to large distances from the plane, with X-shaped patterns. Future observations of polarized emission at high frequencies, with the EVLA, the SKA and its precursors, will trace galactic magnetic fields in unprecedented detail. Low-frequency telescopes (e.g. LOFAR and MWA) are ideal to search for diffuse emission and small RMs from weak interstellar and intergalactic fields.
Beck, Rainer
2012-05-01
157
PubMed
Aromaticity is indispensable for explaining a variety of chemical behaviors, including reactivity, structural features, relative energetic stabilities, and spectroscopic properties. When interpreted as the spatial delocalization of ?-electrons, it represents the driving force for the stabilization of many planar molecular structures. A delocalized electron system is sensitive to an external magnetic field; it responds with an induced magnetic field having a particularly long range. The shape of the induced magnetic field reflects the size and strength of the system of delocalized electrons and can have a large influence on neighboring molecules. In 2004, we proposed using the induced magnetic field as a means of estimating the degree of electron delocalization and aromaticity in planar as well as in nonplanar molecules. We have since tested the method on aromatic, antiaromatic, and nonaromatic compounds, and a refinement now allows the individual treatment of core-, ?-, and ?-electrons. In this Account, we describe the use of the induced magnetic field as an analytical probe for electron delocalization and its application to a large series of uncommon molecules. The compounds include borazine; all-metal aromatic systems Al(4)(n-); molecular stars Si(5)Li(n)(6-n); electronically stabilized planar tetracoordinate carbon; planar hypercoordinate atoms inside boron wheels; and planar boron wheels with fluxional internal boron cluster moieties. In all cases, we have observed that planar structures show a high degree of electron delocalization in the ?-electrons and, in some examples, also in the ?-framework. Quantitatively, the induced magnetic field has contributions from the entire electronic system of a molecule, but at long range the contributions arising from the delocalized electronic ?-system dominate. The induced magnetic field can only indirectly be confirmed by experiment, for example, through intermolecular contributions to NMR chemical shifts. We show that calculating the induced field is a useful method for understanding any planar organic or inorganic system, as it corresponds to the intuitive Pople model for explaining the anomalous proton chemical shifts in aromatic molecules. Indeed, aromatic, antiaromatic, and nonaromatic molecules show differing responses to an external field; that is, they reduce, augment, or do not affect the external field at long range. The induced field can be dissected into different orbital contributions, in the same way that the nucleus-independent chemical shift or the shielding function can be separated into component contributions. The result is a versatile tool that is particularly useful in the analysis of planar, densely packed systems with strong orbital contributions directly atop individual atoms. PMID:21848282
Islas, Rafael; Heine, Thomas; Merino, Gabriel
2011-08-17
158
A 0.633 ?m laser interferometer provides detailed time resolved information about the spatial distribution of the plasma density of field reversed configurations (FRC's) produced by the FRX-L experiment at Los Alamos National Laboratory. This experiment is an effort to produce a magnetized plasma with closed field lines suitable for compression by a solid metal liner imploded by the Shiva Star capacitor bank at the Air Force Research Laboratory. The interferometer probes a fanned array of eight chords through the FRC midplane, measuring the line integrated free electron density via its effect on optical phase shift relative to eight reference beams as a function of time. The reference beams are given nominally identical optical paths, except that they are folded for compactness and given an 80 MHz higher optical frequency by use of a Bragg cell beam splitter. After the beams are recombined, interference results in 80 MHz electromagnetic beat waves with dynamic phase shifts equal to those of the corresponding optical probes. Quadrature mixing of the electronically monitored light is then performed with rf components. Noteworthy features of the interferometer's design are the unique compact folding scheme of the reference paths, inclusion of a fused quartz tube in the reference path similar to that of the FRC's vacuum vessel to compensate for cylindrical lensing, and transmission of the interfering light via optical fibers to a rf shielded room for processing. Extraneous contributions to the phase shift due to vibration resulting from the system's pulsed magnetic field, and dynamic refractive changes in or near the fused quartz tube wall (possibly due to radiation heating) are corrected for.
Ruden, E. L.; Zhang, Shouyin; Wurden, G. A.; Intrator, T. P.; Renneke, R.; Waganaar, W. J.; Analla, F. T.; Grabowski, T. C.
2006-10-01
159
SciTech Connect
A thick flowing layer of liquid (e.g., flibe--a molten salt, or Sn{sub 80}Li{sub 20}--a liquid metal) protects the structural walls of the field-reversed configuration (FRC) so that they can last the life of the plant even with intense 14 MeV neutron bombardment from the D-T fusion reaction. The surface temperature of the liquid rises as it passes from the inlet nozzles to the exit or receiver nozzles due to absorption of line and bremsstrahlung radiation, and neutrons. The surface temperature can be reduced by enhancement of convection near the surface to transport hot surface liquid into the cooler interior. This surface temperature must be compatible with a practical heat transport and energy recovery system. The evaporative flux from the wall driven by the surface temperature must also result in an acceptable impurity level in the core plasma. The shielding of the core by the edge plasma is modeled with a 2D transport code for the resulting impurity ions; these ions are either swept out to the distant end tanks, or diffuse to the hot plasma core. An auxiliary plasma between the edge plasma and the liquid wall can further attenuate evaporating flux of atoms and molecules by ionization. The current in this auxiliary plasma might serve as the antenna for the current drive method, which produces a rotating magnetic field. Another method of current drive uses small spheromaks injected along the magnetic fields, which additionally provide fueling along with pellet fueling if necessary.
Moir, R W; Bulmer, R H; Gulec, K; Fogarty, P; Nelson, B; Ohnishi, M; Rensink, M; Rognlien, T D; Santarious, J F; Sze, D K
2000-09-22
160
SciTech Connect
We present an unreported magnetic configuration in epitaxial La{sub 1-x}Sr{sub x}MnO{sub 3} (x {approx} 0.3) (LSMO) films grown on strontium titanate (STO). X-ray magnetic circular dichroism indicates that the remanent magnetic state of thick LSMO films is opposite to the direction of applied magnetic field. Spectroscopic and scattering measurements reveal that the average Mn valence varies from mixed Mn{sup 3+}/Mn{sup 4+} to an enriched Mn{sup 3+} region near the STO interface, resulting in a compressive lattice along a, b-axis and a possible electronic reconstruction in the Mn e{sub g} orbital (d{sub 3z{sup 2}-r{sup 2}}). This reconstruction may provide a mechanism for coupling the Mn{sup 3+} moments antiferromagnetically along the surface normal direction, and in turn may lead to the observed reversed magnetic configuration.
Kao, Chi-Chang
2011-05-23
161
PubMed
We present an unreported magnetic configuration in epitaxial La(1-x) Sr(x) MnO3 (x ? 0.3) (LSMO) films grown on strontium titanate (STO). X-ray magnetic circular dichroism indicates that the remanent magnetic state of thick LSMO films is opposite to the direction of the applied magnetic field. Spectroscopic and scattering measurements reveal that the average Mn valence varies from mixed Mn(3+)/Mn(4+) to an enriched Mn3+ region near the STO interface, resulting in a compressive lattice along the a, b axis and a possible electronic reconstruction in the Mn e(g) orbital (d(3)z(2)-r(2). This reconstruction may provide a mechanism for coupling the Mn3+ moments antiferromagnetically along the surface normal direction, and in turn may lead to the observed reversed magnetic configuration. PMID:21231622
Lee, J-S; Arena, D A; Yu, P; Nelson, C S; Fan, R; Kinane, C J; Langridge, S; Rossell, M D; Ramesh, R; Kao, C-C
2010-12-14
162
SciTech Connect
We present an unreported magnetic configuration in epitaxial La{sub 1-x}Sr{sub x}MnO{sub 3} (x {approx} 0.3) (LSMO) films grown on strontium titanate (STO). X-ray magnetic circular dichroism indicates that the remanent magnetic state of thick LSMO films is opposite to the direction of the applied magnetic field. Spectroscopic and scattering measurements reveal that the average Mn valence varies from mixed Mn{sup 3+}/Mn{sup 4+} to an enriched Mn{sup 3+} region near the STO interface, resulting in a compressive lattice along the a, b axis and a possible electronic reconstruction in the Mn e{sub g} orbital (d{sub 3z{sup 2}-r{sup 2}}). This reconstruction may provide a mechanism for coupling the Mn{sup 3+} moments antiferromagnetically along the surface normal direction, and in turn may lead to the observed reversed magnetic configuration.
Lee, J.S.; Arena, D.A.; Yu, P.; Nelson, C.S.; Fan, R.; Kinane, C.J.; Langridge, S.; Rossell, M.D.; Ramesh, R.; Kao, C.C.
2010-12-17
163
Some distributions of magnetization give rise to magnetic fields that vanish everywhere above the surface, rendering these distributions of magnetization completely invisible. They are the annihilators of the magnetic inverse problem. Known examples are the infinite sheet with constant magnetization and the spherical shell of constant susceptibility magnetized by an arbitrary internal field. Here, we show that remarkably more interesting
S. Maus; V. Haak
2003-01-01
164
We investigate the 50-year old hypothesis that the magnetic fields of the Ap stars are stable equilibria that have survived in these stars since their formation. With numerical simulations we find that stable magnetic field configurations indeed appear to exist under the conditions in the radiative interior of a star. Confirming a hypothesis by Prendergast (1956, ApJ, 123, 498), the configurations have roughly equal poloidal and toroidal field strengths. We find that tori of such twisted fields can form as remnants of the decay of an unstable random initial field. In agreement with observations, the appearance at the surface is an approximate dipole with smaller contributions from higher multipoles, and the surface field strength can increase with the age of the star. The results of this paper were summarised by Braithwaite & Spruit (2004, Nature, 431, 891).
Braithwaite, J.; Nordlund, Å.
2006-05-01
165
The processes of heating and acceleration of plasma in current sheets formed in 2D and 3D magnetic configurations with an X-line in helium plasma have been investigated using spectroscopic methods. It is found that, in 2D magnetic configurations, plasma flows with energies of 400-1000 eV, which are substantially higher than the ion thermal energy, are generated and propagate along the width (the larger transverse dimension) of the sheet. In 3D configurations, the influence of the longitudinal (directed along the X-line) component of the magnetic field on the plasma parameters in the current sheet has been studied. It is shown that plasma acceleration caused by the Ampère force can be spatially inhomogeneous in the direction perpendicular to the sheet surface, which should lead to sheared plasma flows in the sheet.
Kyrie, N. P.; Frank, A. G.
2012-12-01
166
PubMed
Artificial dipolar spin-ice patterns have attracted much attention recently because of their rich configurations and excitations in the form of Dirac strings connecting magnetic monopoles. We have analysed the distribution of excitations in the form of strings and vertices carrying magnetic charges Q=±3q in honeycomb artificial spin-ice patterns. Two types of patterns are compared, those that terminate with open hexagons and those with closed hexagons. The dipole configurations and the frequency of spin-ice rule-violating Q=±3q vertices depend slightly on the boundary conditions of the pattern. Upon rotation of the patterns by 2? in a coercive magnetic field of 500 Oe, complete reversibility of the charge and string configuration is observed. PMID:23166380
Schumann, Alexandra; Zabel, Hartmut
2012-12-28
167
Global configuration of the geomagnetic field plays an important role in magnetospheric dynamics. We study the effect of field configuration on electromagnetic ion-cyclotron (EMIC) wave growth with test particle simulations. As an initial study, we quantitatively examine the accuracy of several empirical geomagnetic field models widely in use. We study two years characterized by very different space weather conditions: 1996 and 2003. The year 1996, at solar minimum, exhibited many high-speed streams and a few co-rotating interaction regions, but was generally quiet. In contrast, 2003 included the "Halloween storm," one of the most intense geomagnetic storms on record caused by a coronal mass ejection. The performance of each model, as measured by prediction efficiency and skill score, is evaluated as a function of magnetospheric conditions (reflected by the geomagnetic index, Kp) and magnetic local time. We subsequently developed a new MHD/particle method to study electromagnetic ion-cyclotron (EMIC) wave growth in a realistic and dynamic magnetosphere. We simulate the phase space density dynamics of warm plasma particles in magnetospheric electromagnetic fields from the global Lyon-Fedder-Mobarry (LFM) MHD code and 3D test-particle trajectories. We use these results to compute temperature anisotropies and plasma densities. We then compute the convective EMIC wave growth rate using these macroscopic plasma quantities, and thus generate a spatiotemporal picture of the growth of these waves. We use our new MHD/particle method for studying EMIC wave growth to simulate a compression event observed on 29 June 2007 and compare the results with observations from ground observatories and spacecraft measurements. We then study the time evolution of various quantities to discern physical mechanisms leading to simulated wave growth. A fairly at simulated temperature profile in time suggested an absence of energizing processes during this event. This can be explained by two possible mechanisms: temperature anisotropy induced by drift shell splitting (DSS), and the bulk execution of unusual particle trajectories called Shabansky orbits. Finally, we used test particle simulations in a static analytic model field to study the two non-energizing processes. We show that Shabansky orbits executed in bulk provide a temperature anisotropy distinct from DSS-induced temperature anisotropy, and we discuss the two origins of this new physical mechanism for anisotropy generation.
McCollough, J. P.
168
Nanostripes with varying widths are lithographed on Co thin films with strong magnetic anisotropy resulting from the epitaxial growth onto vicinal Si(111) substrate. The competition between magnetocrystalline and shape anisotropies is used to tune the magnetic behavior of Co nanostripes. Single domain configuration is observed for nanostructures where magnetocrystalline and shape anisotropies go along the same direction. However, more complicated configurations such as open stripe domains can be developed when both anisotropies compete. The nanostructures have been experimentally characterized by longitudinal magneto-optical Kerr effect and magnetic force microscopy (MFM). Micromagnetic simulations performed by finite-element and finite difference codes are in good agreement with the experimental results. The use of MFM based techniques such as the variable field magnetic force microscopy and the so-called three-dimensional modes has allowed us to follow the evolution of the domains and domain walls under externally applied magnetic fields, i.e., to deeply understand the magnetization reversal process in the multidomain nanostripes. In particular, the nanostripes with competing anisotropies and a high aspect ratio present vortex configuration along the domain walls which have a key role in the magnetization reversal process.
Ivanov, Yu. P.; Iglesias-Freire, O.; Pustovalov, E. V.; Chubykalo-Fesenko, O.; Asenjo, A.
2013-05-01
169
We propose a primordial-origin model for composite configurations of global magnetic fields in spiral galaxies. We show that a uniform tilted magnetic field wound up into a rotating disk galaxy can evolve into composite magnetic configurations comprising bisymmetric spiral (S = BSS), axisymmetric spiral (A = ASS), plane-reversed spiral (PR), and/or ring (R) fields in the disk, and vertical (V) fields in the center. By MHD simulations we show that these composite galactic fields are indeed created from a weak primordial uniform field, and that different configurations can co-exist in the same galaxy. We show that spiral fields trigger the growth of two-armed gaseous arms. The centrally accumulated vertical fields are twisted and produce a jet toward the halo. We found that the more vertical was the initial uniform field, the stronger was the formed magnetic field in the galactic disk.
Sofue, Yoshiaki; Machida, Mami; Kudoh, Takahiro
2010-10-01
170
The beta and magnetic field profiles of field-reversed configurations (FRC’s) can be inferred from the radial profile of the line-integral density together with a temperature model. When these profiles are combined with the particle and flux decay times, the plasma resistivities at the field null and separatrix can also be deduced. This method is applied to a class of FRC’s
1987-01-01
171
In this paper the authors compute the magnetic fields generated by a lot of typical power line configurations and compare these values with the results given both by analytical models and by measurements. The computations have been made by means of a flexible calculation code developed by the authors. This code overcomes some simplifying assumptions introduced in previous analytical studies.
A. Geri; A. Locatelli; G. M. Veca
1995-01-01
172
SciTech Connect
FRCs with equilibrium separatrix radii up to 0.18 m have been formed and studied in FRX-C/LSM. For best formation conditions at low fill pressure, the particle confinement exceeds the predictions of LHD transport calculations by up to a factor of two; however, the inferred flux confinement is more anomalous than in smaller FRCs. Higher bias field produces axial shocks and degradation in confinement, while higher fill pressure results in gross fluting during formation. FRCs have been formed in TRX with s from 2 to 6. These relatively collisional FRCs exhibit flux lifetimes of 10 {yields} 20 kinetic growth times for the internal tilt mode. The coaxial slow source has produced annular FRCs in a coaxial coil geometry on slow time scales using low voltages. 16 refs., 4 figs., 1 tab.
Siemon, R.E.; Chrien, R.E.; Hugrass, W.N.; Okada, S.; Rej, D.J.; Taggart, D.P.; Tuszewski, M.; Webster, R.B.; Wright, B.L.; Slough, J.T.; Crawford, E.A.; Hoffman, A.L.; Milroy, R.D.; Vlases, G.C.; Brooks, R.D.; Kronast, B.; Pietrzyk, Z.A.; Raman, R.; Smith, R.
1988-01-01
173
The problem of determining the energy of two spinor particles interacting through massless-particle exchange is analyzed using the path-integral method. A form for the long-range interaction energy is obtained by analyzing an abridged vertex derived from the parent theory. This abridged vertex describes the radiation of zero-momentum particles by pointlike sources. A path-integral formalism for calculating the energy of the radiation field associated with this abridged vertex is developed and applications are made to determine the energy necessary for adiabatic separation of two sources in quantum electrodynamics and for an SU(2) Yang-Mills theory. The latter theory is shown to be consistent with confinement via infrared slavery.
Swanson, Mark S.
1987-09-01
174
Electromagnets used as beam guiding elements in particle accelerators and colliders require very tight tole-rances on their magnetic fields and on their alignment along the particle path. This article describes the methods and equipment used for magnetic measurements in beam transport magnets. Descriptions are given of magnetic resonance techniques, various induction coil methods, Hall generator measurements, the fluxgate magnetometer as
K. N. Henrichsen
1998-01-01
175
Petropoulos, Labros Spiridon
176
The Heliospheric Magnetic Field (HMF) is the physical framework in which energetic particles and cosmic rays propagate. Changes in the large scale structure of the magnetic field lead to short- and long term changes in cosmic ray intensities, in particular in anti-phase with solar activity. The origin of the HMF in the corona is well understood and inner heliospheric observations can generally be linked to their coronal sources. The structure of heliospheric magnetic polarities and the heliospheric current sheet separating the dominant solar polarities are reviewed here over longer than a solar cycle, using the three dimensional heliospheric observations by Ulysses. The dynamics of the HMF around solar minimum activity is reviewed and the development of stream interaction regions following the stable flow patterns of fast and slow solar wind in the inner heliosphere is described. The complex dynamics that affects the evolution of the stream interaction regions leads to a more chaotic structure of the HMF in the outer heliosphere is described and discussed on the basis of the Voyager observations. Around solar maximum, solar activity is dominated by frequent transients, resulting in the interplanetary counterparts of Coronal Mass Ejections (ICMEs). These produce a complex aperiodic pattern of structures in the inner heliosphere, at all heliolatitudes. These structures continue to interact and evolve as they travel to the outer heliosphere. However, linking the observations in the inner and outer heliospheres is possible in the case of the largest solar transients that, despite their evolutions, remain recognizably large structures and lead to the formation of Merged Interaction Regions (MIRs) that may well form a quasi-spherical, "global" shell of enhanced magnetic fields around the Sun at large distances. For the transport of energetic particles and cosmic rays, the fluctuations in the magnetic field and their description in alternative turbulent models remains a very important research topic. These are also briefly reviewed in this paper.
Balogh, André; Erdõs, Géza
2013-06-01
177
SciTech Connect
Understanding the behavior of plasmas in magnetic confinement fusion devices typically requires accurate knowledge of the magnetic field structure. In stellarator-type confinement devices, the helical magnetic field is produced by currents in external coils and may be traced experimentally in the absence of plasma through the experimental technique of vacuum magnetic field mapping. Field mapping experiments, such as these, were performed on the recently constructed compact toroidal hybrid to verify the range of accessible magnetic configurations, compare the actual magnetic configuration with the design configuration, and identify any vacuum field errors that lead to perturbations of the vacuum magnetic flux surfaces. Furthermore, through the use of a new coil optimization routine, modifications are made to the simulation coil model such that better agreement exists between the experimental and simulation results. An outline of the optimization procedure is discussed in conjunction with the results of one such optimization process performed on the helical field coil.
Peterson, J.; Hanson, J.; Hartwell, G.; Knowlton, S. [Department of Physics, Auburn University, Auburn, Alabama 36849 (United States)
2010-03-15
178
An application of magnetic field, B to a discharge chamber under transverse electric field, E is studied, with the objective to view the effect of magnetic field to the electrons in a plasma. Theory of gas discharge stated that this configuration will create a helical motion to electrons due to gradient drift or gyration. Experimental result from previous research showed,
Z. Buntat; I. R. Smith; N. A. M. Razali
2009-01-01
179
SciTech Connect
This letter is a response to an article by Savitz and Kaune, EHP 101:76-80. W-L wire code was applied to data from a 1988 Denver study, and an association was reported between high W-L wire code and childhood cancer. This author discusses several studies and provides explanations which weakens the argument that classification error resulted in an appreciable reduction in the association between W-L high wire code and childhood cancer. In conclusion, the fact that new wire code is only weakly correlated with magnetic field measurements (in the same manner as the original W-L wire code) suggests that the newly reported stronger association with childhood cancer is likely due to factors other than magnetic fields. Differential residential mobility and differential residential age are two possible explanations and are suggestive that the reported association may be false.
Jones, T.L.
1993-10-01
180
SciTech Connect
Fermilab's accelerator magnet R and D programs, including production of superconducting high gradient quadrupoles for the LHC insertion regions, require rigorous yet flexible magnetic measurement systems. Measurement systems must be capable of handling various types of hardware and extensible to all measurement technologies and analysis algorithms. A tailorable software system that satisfies these requirements is discussed. This single system, capable of distributed parallel signal processing, is built on top of a flexible component-based framework that allows for easy reconfiguration and run-time modification. Both core and domain-specific components can be assembled into various magnet test or analysis systems. The system configured to comprise a rotating coil harmonics measurement is presented. Technologies as Java, OODB, XML, JavaBeans, software bus and component-based architectures are used.
Jerzy M. Nogiec et al.
2001-07-20
181
Solar flares and coronal mass ejections (CMEs) --- phenomena which impact our society, but are scientifically interesting in themselves --- are driven by free magnetic energy in the coronal magnetic field. Since the coronal magnetic field cannot be directly measured, modelers often extrapolate the coronal field from the photospheric magnetograms --- the only field measurements routinely available. The best extrapolation techniques assume that the field is force free (coronal currents parallel the magnetic field), but that currents are not simply a linear function of the magnetic field. Recent tests, however, suggest that such non-linear force-free field (NLFFF) extrapolation techniques often underestimate free magnetic energy. We hypothesize that, since relaxation-based NLFFF techniques tend to smooth field discontinuities, such approaches will fail when current sheets are present. Here, we test this hypothesis by applying the Optimization NLFFF method to two configurations from an MHD simulation --- one with strong current concentrations, and one with weak concentrations. This work is supported by a NASA Sun-Earth Connections Theory grant to UC-Berkeley.
Welsch, Brian; De Moortel, I.; McTiernan, J. M.
2007-05-01
182
We report the switching properties of a thin magnetic film subject to an ultrashort, laterally localized magnetic field pulse, obtained by numerical investigations. The magnetization distribution in the film is calculated on a grid assuming Stoner-like coherent rotation within the grid square size. Perpendicularly and in-plane magnetized films exhibit a magnetization reversal due to a 4ps magnetic field pulse. Outside the central region the pulse duration is short compared to the precession period. In this area the evolution of the magnetization during the field pulse does not depend strongly on magnetic damping and/or pulse shape. However, the final magnetization distribution is affected by the magnetic damping. Although the pulse duration is short compared to the precession period, the time needed for the relaxation of the magnetization to the equilibrium state is rather large. The influence of the different magnetic anisotropy contributions and the magnetic damping parameter enters into the magnetization reversal process. Comparing the case of perpendicular anisotropy with different kinds of in-plane anisotropies, a principal difference is found due to the symmetry of the shape anisotropy with respect to the anisotropy in question.
Bauer, M.; Lopusnik, R.; Fassbender, J.; Hillebrands, B.
2000-08-01
183
NSDL National Science Digital Library
The above animations represent two typical bar magnets each with a North and South pole. The arrows represent the direction of the magnetic field. A wire is placed between the magnets and a current that comes out of the page can be turned on.
Christian, Wolfgang; Belloni, Mario
2007-03-03
184
Coronal bright points (CBPs) are long-lived small-scale brightenings in the solar corona. They are generally explained by magnetic reconnection. However, the corresponding magnetic configurations are not well understood. We carry out a detailed multi-wavelength analysis of two neighboring CBPs on 2007 March 16, observed in soft X-ray (SXR) and EUV channels. It is seen that the SXR light curves present quasi-periodic flashes with an interval of ~1 hr superposed over the long-lived mild brightenings, suggesting that the SXR brightenings of this type of CBPs might consist of two components: one is the gentle brightenings and the other is the CBP flashes. It is found that the strong flashes of the bigger CBP are always accompanied by SXR jets. The potential field extrapolation indicates that both CBPs are covered by a dome-like separatrix surface, with a magnetic null point above. We propose that the repetitive CBP flashes, as well as the recurrent SXR jets, result from the impulsive null-point reconnection, while the long-lived brightenings are due to the interchange reconnection along the separatrix surface. Although the EUV images at high-temperature lines resemble the SXR appearance, the 171 Å and 195 Å channels reveal that the blurry CBP in SXR consists of a cusp-shaped loop and several separate bright patches, which are explained to be due to the null-point reconnection and the separatrix reconnection, respectively.
Zhang, Q. M.; Chen, P. F.; Guo, Y.; Fang, C.; Ding, M. D.
2012-02-01
185
SciTech Connect
Coronal bright points (CBPs) are long-lived small-scale brightenings in the solar corona. They are generally explained by magnetic reconnection. However, the corresponding magnetic configurations are not well understood. We carry out a detailed multi-wavelength analysis of two neighboring CBPs on 2007 March 16, observed in soft X-ray (SXR) and EUV channels. It is seen that the SXR light curves present quasi-periodic flashes with an interval of {approx}1 hr superposed over the long-lived mild brightenings, suggesting that the SXR brightenings of this type of CBPs might consist of two components: one is the gentle brightenings and the other is the CBP flashes. It is found that the strong flashes of the bigger CBP are always accompanied by SXR jets. The potential field extrapolation indicates that both CBPs are covered by a dome-like separatrix surface, with a magnetic null point above. We propose that the repetitive CBP flashes, as well as the recurrent SXR jets, result from the impulsive null-point reconnection, while the long-lived brightenings are due to the interchange reconnection along the separatrix surface. Although the EUV images at high-temperature lines resemble the SXR appearance, the 171 Angstrom-Sign and 195 Angstrom-Sign channels reveal that the blurry CBP in SXR consists of a cusp-shaped loop and several separate bright patches, which are explained to be due to the null-point reconnection and the separatrix reconnection, respectively.
Zhang, Q. M.; Chen, P. F.; Guo, Y.; Fang, C.; Ding, M. D., E-mail: chenpf@nju.edu.cn [Department of Astronomy, Nanjing University, Nanjing 210093 (China)
2012-02-10
186
Favorable features of the D-He(3) fuel cycle in a field-reversed configuration are described. Based on a theoretical analysis, it was found that the estimated plant efficiency is more than 70 percent and the 14 MeV neutron power fraction is as small as 1 percent. To reach the D-He(3) initiation temperature of 100 keV with a reasonable external power source, a D-T configuration can first be ignited and then the fuel altered to D-3He. Heating of the plasma is attributed to energetic fusion charged particles and no additional heating is necessary. The equilibria of D-3He ignited plasmas may be self-sustained due to the preferential trapping of fusion protons in a field-reversed configuration.
Momota, Hiromu; Okamoto, Masao; Nomura, Yasuyuki; Ohnishi, Masami; Berk, Herbert L.; Tajima, Toshiki
1987-11-01
187
Recent developments in integrated silicon magnetic devices are reviewed, with particular attention given to integrated Hall plates, magnetic field-effect transistors, vertical and lateral bipolar magnetotransistors, magnetodiodes, and current-domain magnetometers. Also described are current developments in integrated magnetic field sensors based on III-V semiconductors and bulk Hall-effect devices. The discussion also covers magnetic device modeling and the incorporation of magnetic devices
H. P. Baltes; R. S. Popovic
1986-01-01
188
The energy needed to power flares is thought to be stored in the coronal magnetic field However the energy release which results in thermal energy brightenings is efficient only at very small scales Magnetic configurations with a complex topology i e with separatrices are the most obvious configurations where current layers then reconnection can efficiently occur This has been confirmed for several flares by computing the coronal field and by comparing the locations of the flare loops and ribbons to the deduced 3D magnetic topology However this view is too restrictive taking into account the variety of observed solar flaring configurations Indeed Quasi-Separatrix Layers QSLs which are regions where there is a drastic change in field-line linkage generalize the definition of separatrices They permit us to understand where reconnection occurs in a broader variety of flares than separatrices do The location where the QSL are the thinnest called Hyperbolic Flux Tube HFT is the location for the strongest electric field and current being generated This is a good candidate for the region where particle acceleration can occur efficiently
Demoulin, P.
189
The Hall effect plays a significant role in the penetration of plasma flows across magnetic field. For example, its effect may become dominant in the solar wind penetration into the magnetosphere, in the magnetic field advection in wire array z-pinch precursors, or in the arcing of magnetically insulated transmission lines. An experiment performed at the Nevada Terawatt Facility explored the penetration of plasma with large Hall parameter (˜10) across ambient magnetic field. The plasma was produced by ablation with the short pulse high intensity laser Leopard (0.35 ps, 10^17W/cm^2) and the magnetic field with the pulsed power generator Zebra (50 T). The expanding plasma assumed a jet configuration and propagated beyond a distance consistent with a diamagnetic bubble model. Without magnetic field, the plasma expansion was close to hemispherical. The ability to produce the plasma and the magnetic field with distinct generators allows a controlled, quasi-continuous variation of the Hall parameter and other plasma parameters making the experiments useful for benchmarking numerical simulations.
Presura, R.; Stepanenko, Y.; Neff, S.; Sotnikov, V. I.
2008-04-01
190
The past several years have seen dramatic developments in the study of planetary magnetic fields, including a wealth of new data, mainly from the Galilean satellites and Mars, together with major improvements in our theoretical modeling effort of the dynamo process believed responsible for large planetary fields. These dynamos arise from thermal or compositional convection in fluid regions of large radial extent. The relevant electrical conductivities range from metallic values to values that may be only about 1% or less that of a typical metal, appropriate to ionic fluids and semiconductors. In all planets, the Coriolis force is dynamically important, but slow rotation may be more favorable for a dynamo than fast rotation. The maintenance and persistence of convection appears to be easy in gas giants and ice-rich giants, but is not assured in terrestrial planets because the quite high electrical conductivity of iron-rich cores guarantees a high thermal conductivity (through the Wiedemann-Franz law), which allows for a large core heat flow by conduction alone. In this sense, high electrical conductivity is unfavorable for a dynamo in a metallic core. Planetary dynamos mostly appear to operate with an internal field ~(2??/?)1/2 where ? is the fluid density, ? is the planetary rotation rate and ? is the conductivity (SI units). Earth, Ganymede, Jupiter, Saturn, Uranus, Neptune, and maybe Mercury have dynamos, Mars has large remanent magnetism from an ancient dynamo, and the Moon might also require an ancient dynamo. Venus is devoid of a detectable global field but may have had a dynamo in the past. The presence or absence of a dynamo in a terrestrial body (including Ganymede) appears to depend mainly on the thermal histories and energy sources of these bodies, especially the convective state of the silicate mantle and the existence and history of a growing inner solid core. Induced fields observed in Europa and Callisto indicate the strong likelihood of water oceans in these bodies.
Stevenson, David J.
2003-03-01
191
SciTech Connect
The particle confinement time in field-reversed configurations has been experimentally investigated in the FRX-C device. The measured confinement times of 70 to 190 ..mu..s are consistent with R/sup 2//rho/sub i0/ scaling and are in good agreement with theoretical predictions of lower-hybrid-drift--induced particle transport.
McKenna, K.F.; Armstrong, W.T.; Bartsch, R.R.
1983-05-30
192
National Technical Information Service (NTIS)
A thick flowing layer of liquid protects the structural walls of the field-reversed configuration (FRC) so that they can last the life of the plant even with intense 14 MeV neutron bombardment from the D-T fusion reaction. The surface temperature of the l...
R. W. Moir R. H. Bulmer K. Gulec P. Fogarty B. Nelson M. Ohnishi
2000-01-01
193
X-ray magnetic circular dichroism (XMCD) measurements on HoCo2 reveal the inversion of Co moment at temperatures higher than the critical temperature, Tc, showing that the net magnetization under a field of the Ho and Co sublattices remain antiparallel even above Tc. The Ho moment also changes its orientation to align antiparallel to the applied field at high temperature giving rise to a new magnetic configuration in the paramagnetic regime. Transverse susceptibility (TS) and small angle neutron scattering (SANS) measurements performed above Tc indicate the existence of sizable magnetic short-range correlated regions in HoCo2. First principles calculations based on spin polarized local-density approximation, LSDA+U havebeen performed to obtain insights on the origin of the short-range correlated volume.
Bonilla, C. M.; Calvo, I.; Herrero-Albillos, J.; Figueroa, A. I.; Castan-Guerrero, C.; Bartolomé, J.; Rodriguez-Velamazan, J. A.; Schmitz, D.; Weschke, E.; Paudyal, D.; Pecharsky, V. K.; Gschneidner, K. A.; Bartolomé, F.; García, L. M.
2012-04-01
194
The properties of surface plasmon polaritons (SPPs) in a magnetically tunable strip waveguide geometry comprising of a metal film of finite width deposited on a magnetized semiconductor and covered by an isotropic dielectric material were studied in Voigt configuration. The method of lines was used to compute the dispersion relation of fundamental modes, and the dependence of the propagation constant on metal film dimensions, material parameters and biasing magnetic field was considered. The bounded SPPs are nonreciprocal with respect to the direction of the biasing magnetic field, producing a nonreciprocal phase shift of the order of 2-18 rad mm-1 at a wavelength of excitation 1.55 ?m. Moreover, controlled propagation of SPP modes and their effective tuning are possible in this strip geometry, which enables the design and development of tunable optoelectronic devices.
Mathew, Gishamol; Mathew, Vincent
2012-05-01
195
In this work we present anisotropic light scattering measurements of local deformation in systems composed by a dispersion of nanometric magnetic particles in a polymer gel, or ferrogel, in the presence of uniform magnetic fields. Two experimental configurations were used in which the scattering vector q was parallel and perpendicular to the magnetic induction B. We have seen that the
Alvaro V. Teixeira; Pedro Licinio
2005-01-01
196
We describe recent studies of the interaction of fast-rising magnetic fields with multi-species plasmas of densities 10^13-10^15 cm-3. The configurations studied are planar or coaxial gaps, prefilled with plasmas that are driven by 80-400 ns current pulses. The diagnostics is based on time-dependent spectroscopic observations that are spatially resolved in 3D using plasma-doping techniques. The measurements include the magnetic-field structure (from Zeeman splitting), ion velocity distributions (from Doppler profiles), electric fields (from line shapes of allowed and forbidden transitions), and non-Maxwellian electron energy distribution (from line ratios). It is found that the magnetic field propagates in the plasma faster than expected from diffusion. Also, the field spatial distribution is inconsistent with diffusion. The observed broad current channel, as well as non-dependence of the magnetic field evolution on the current polarity, cannot be explained by the available Hall-field theories. Moreover, detailed observations reveal that magnetic field penetration and plasma reflection occur simultaneously, leading to ion-species separation [1, 2], which are also not predicted by Hall-field theories. Measurements of the reflected-proton velocities (twice the magnetic field velocity) show that the protons dissipate a significant fraction of the magnetic field energy. A possible mechanism previously formulated for astrophysical plasmas, based on the formation of small-scale density fluctuations (perhaps as a result of the Rayleigh-Taylor instability) that lead to field penetration via the Hall mechanism, has recently been suggested. The new phenomena observed require novel theoretical treatments. Applications include plasmas under high currents and space physics. 1. A. Weingarten et al., Phys. Rev. Lett. 87, 115004 (2001). 2. R. Arad, et al., Phys. Plasmas 10, 112 (2003).
Maron, Yitzhak
2003-10-01
197
SciTech Connect
ASTROMAG is a particle astrophysics facility that was originally configured for the Space Station. The heart of the ASTROMAG facility is a large superconducting magnet which is cooled using superfluid helium. The task of resizing the facility so that it will fly in a satellite in a high angle of inclination orbit is driven by the launch weight capability of the launch rocket and the desire to be able to do nearly the same physics as the Space Station version of ASTROMAG. In order to reduce the launch weight, the magnet and its cryogenic system had to be downsized, yet the integrated field generated by the magnet in the particle detectors has to match the Space Station version of the magnet. The use of aluminum matrix superconductor and oriented composite materials in the magnet insulation permits one to achieve this goal. The net magnetic dipole moment from the ASTROMAG magnet must be small to minimize the torque due to interaction with the earth's magnetic field. The ASTROMAG magnet consists of identical two coils 1.67 meters apart. The two coils are connected in series in persistent mode. Each coil is designed to carry 2.34 million ampere turns. Both coils are mounted on the same magnetic axis and they operate at opposite polarity. This reduces the dipole moment by a factor of more than 1000. This is tolerable for the Space Station version of the magnet. A magnet operating on a free flying satellite requires additional compensation. This report presents the magnet parameters of a free flying version of ASTROMAG and the parameters of the space cryogenic system for the magnet. 12 refs., 6 figs.
Green, M.A.; Smoot, G.F.
1991-06-01
198
We analyse the mapping produced by the field lines which connect photospheric areas of positive and negative magnetic polarity on the Sun. The geometrical quantities independent of the direction of such a mapping (from positive to negative polarity, and vice versa) are introduced. They yield a complete description of the field line connectivity in coronal magnetic configurations and, in particular,
V. S. Titov; G. Hornig
2002-01-01
199
PubMed
Eddy currents induced within a magnetic resonance imaging (MRI) cryostat bore during pulsing of gradient coils can be applied constructively together with the gradient currents that generate them, to obtain good quality gradient uniformities within a specified imaging volume over time. This can be achieved by simultaneously optimizing the spatial distribution and temporal pre-emphasis of the gradient coil current, to account for the spatial and temporal variation of the secondary magnetic fields due to the induced eddy currents. This method allows the tailored design of gradient coil/magnet configurations and consequent engineering trade-offs. To compute the transient eddy currents within a realistic cryostat vessel, a low-frequency finite-difference time-domain (FDTD) method using total-field scattered-field (TFSF) scheme has been performed and validated. PMID:17945575
Trakic, A; Liu, F; Lopez, H S; Wang, H; Crozier, S
2006-01-01
200
DOEpatents
The superconducting magnetic switch or fast kicker magnet is employed with electron stream or a bunch of electrons to rapidly change the direction of flow of the electron stream or bunch of electrons. The apparatus employs a beam tube which is coated with a film of superconducting material. The tube is cooled to a temperature below the superconducting transition temperature and is subjected to a constant magnetic field which is produced by an external dc magnet. The magnetic field produced by the dc magnet is less than the critical field for the superconducting material, thus, creating a Meissner Effect condition. A controllable fast electromagnet is used to provide a magnetic field which supplements that of the dc magnet so that when the fast magnet is energized the combined magnetic field is now greater that the critical field and the superconducting material returns to its normal state allowing the magnetic field to penetrate the tube. This produces an internal field which effects the direction of motion and of the electron stream or electron bunch. The switch can also operate as a switching mechanism for charged particles. 6 figs.
Goren, Y.; Mahale, N.K.
1996-08-06
201
SciTech Connect
Electric-field probes consisting of copper plates are developed to measure electric fields in a vacuum region around a plasma. The probes detect oscillating electric fields with a maximum strength of approximately 100 V/m through a discharge. Reproducible signals from the probes are obtained with an unstable phase dominated by a rotational instability. It is found that the azimuthal structure of the electric field can be explained by the sum of an n=2 mode charge distribution and a convex-surface electron distribution on the deformed separatrix at the unstable phase. The former distribution agrees with that anticipated from the diamagnetic drift motions of plasma when the rotational instability occurs. The latter distribution suggests that an electron-rich plasma covers the separatrix.
Ikeyama, Taeko; Hiroi, Masanori; Nogi, Yasuyuki [College of Science and Technology, Nihon University, Tokyo 101-8308 (Japan); Ohkuma, Yasunori [College of Industrial Technology, Nihon University, Chiba 275-8576 (Japan)
2010-01-15
202
SciTech Connect
The magnetic field of the solar corona evolves quasistatically in response to slowly changing photospheric boundary conditions. The magnetic topology is preserved by the low resistivity of the solar atmosphere. We show that a magnetic flux coordinate system simplifies the problem of calculating field evolution with invariant topology. As an example, we calculate the equilibrium of a thin magnetic flux tube with small twist per unit length.
Zweibel, E.G.; Boozer, A.H.
1985-02-01
203
SciTech Connect
The powerful magnetic fields produced by a controlled fusion experiment at Lawrence Livermore National Laboratory (LLNL) necessitated the development of personnel-exposure guidelines for steady magnetic fields. A literature search and conversations with active researchers showed that it is currently possible to develop preliminary exposure guidelines for steady magnetic fields. An overview of the results of past research into the bioeffects of magnetic fields was compiled, along with a discussion of hazards that may be encountered by people with sickle-cell anemia or medical electronic and prosthetic implants. The LLNL steady magnetic-field exposure guidelines along with a review of developments concerning the safety of time-varying fields were also presented in this compilation. Guidelines developed elsewhere for time varying fields were also given. Further research is needed to develop exposure standards for both steady or time-varying fields.
Miller, G.
1987-12-01
204
The Hall effect has been widely utilized to measure magnetic fields. The relatively simple geometry of a Hall element suggested the use of such a device on a microscale as a probe to examine magnetic fields of small structures. Hall probes are described which were constructed with a sensitive area about 10×10 ?. Fields of less than 0.01 gauss were
D. D. Roshon Jr.
1962-01-01
205
The jet region of M87 is discussed to illustrate the astrophysical observations of radio sources, with note made of magnetic field phenomena contributing to radio frequency emissions. The jet appearing in M87 has been modelled as a continuous supersonic flow of plasma embedded in a self-consistent, ordered magnetic field. The field has both parallel and helical components, and may work
A. Ferrari
1982-01-01
206
We show that the relatively strong magnetic fields ($\\\\ge 1 \\\\mu$G) in high\\u000aredshift objects can be explained by the combined action of an evolving\\u000aprotogalactic fluctuation and electrodynamic processes providing the magnetic\\u000aseed fields. Three different seed field mechanisms are reviewed and\\u000aincorporated into a spherical \\
Harald Lesch; Masashi Chiba
1994-01-01
207
We show that the relatively strong magnetic fields (>=1muG) in high redshift objects can be explained by the combined action of an evolving protogalactic fluctuation and electrodynamic processes providing the magnetic seed fields. Three different seed field mechanisms are reviewed and incorporated into a spherical \\
H. Lesch; M. Chiba
1995-01-01
208
DOEpatents
A device is provided for measuring the magnetic field dose and peak field exposure. The device includes three Hall-effect sensors all perpendicular to each other, sensing the three dimensional magnetic field and associated electronics for data storage, calculating, retrieving and display.
Lemon, D.K.; Skorpik, J.R.; Eick, J.L.
1981-01-21
209
SciTech Connect
Collisional heat transport in a stochastic magnetic field configuration is investigated. Well above stochastic threshold, a numerical solution of a Chirikov-Taylor model shows a short-time nonlocal regime, but at large time the Rechester-Rosenbluth effective diffusion is confirmed. Near stochastic threshold, subdiffusive behavior is observed for short mean free paths. The nature of this subdiffusive behavior is understood in terms of the spectrum of islands in the stochastic sea.
White, R.B.; Wu, Yanlin [Princeton Univ., NJ (United States). Plasma Physics Lab.; Rax, J.M. [Association Euratom-CEA, Centre dEtudes Nucleaires de Cadarache, 13 -Saint-Paul-lez-Durance (France). Dept. de Recherches sur la Fusion Controlee
1992-09-01
210
SciTech Connect
Collisional heat transport in a stochastic magnetic field configuration is investigated. Well above stochastic threshold, a numerical solution of a Chirikov-Taylor model shows a short-time nonlocal regime, but at large time the Rechester-Rosenbluth effective diffusion is confirmed. Near stochastic threshold, subdiffusive behavior is observed for short mean free paths. The nature of this subdiffusive behavior is understood in terms of the spectrum of islands in the stochastic sea.
White, R.B.; Wu, Yanlin (Princeton Univ., NJ (United States). Plasma Physics Lab.); Rax, J.M. (Association Euratom-CEA, Centre d'Etudes Nucleaires de Cadarache, 13 -Saint-Paul-lez-Durance (France). Dept. de Recherches sur la Fusion Controlee)
1992-01-01
211
SciTech Connect
Fast coronal mass ejecta (CMEs) accelerate and deflect the slower moving solar wind plasma which piles up ahead of them as they propagate out through the heliosphere. This acceleration and deflection, in turn, causes the interplanetary magnetic field (IMF) imbedded in the upstream solar wind to drape about the ejecta. Draping should cause substantial out-of-the-ecliptic magnetic fields at some locations ahead of CMEs, and radial fields behind and along the flanks. At the Earth, draping can be an important factor in the generation of some magnetic storms and substorms, while in the outer heliosphere draping may produce very large magnetotail-like configurations, somewhat analogous to those observed behind Venus and comets. 17 refs.
McComas, D.J.; Gosling, J.T.
1987-01-01
212
The magnetic field is thought to be the source of the energy release in many and varied observed coronal phenomena, from the less energetic coronal heating to the most violent flares and prominence eruptions. These phenomena involve not only very different scales from the energetic, but also from the temporal, point of view. Magnetic field reconnection, which is efficient only at very small spatial scales, has been the energy release mechanism that has been so far proposed. From a theoretical point of view, magnetic configurations with a complex topology, i.e. having separatrices, are the ones where current sheets can form in 2D. When going to 3D, and if the photospheric magnetic field is described by a series of isolated polarities (surrounded by field free regions), a complete topological description is given by the skeleton formed by null points, spines, fans and separators, and associated separatrices. However, if the photosphere is fully magnetized, most of the above topological structures disappear: only separatrices associated to coronal magnetic nulls remain. An extra set of separatrices is associated to the field lines curved up above the photosphere (defining the bald-patch locations). For some observed magnetic configurations, those topological structures are enough to understand where flare brightenings appear as a result of magnetic field reconnection. However, solar active phenomena are seen to occur also in a larger variety of configurations. Quasi-separatrix layers, which are regions where there is a drastic change in field-line linkage, generalize the concept of separatrices to magnetic configurations without magnetic null points and bald patches. We will review examples of observed flaring regions and their topologies that show us that magnetic reconnection can occur in wider variety of magnetic configurations than traditionally thought.
Mandrini, C. H.
2006-08-01
213
PubMed
Strong transverse magnetic fields can produce very large dose enhancements and reductions in localized regions of a patient under irradiation by a photon beam. We have suggested a model magnetic field which can be expected to produce such large dose enhancements and reductions, and we have carried out EGS4 Monte Carlo calculations to examine this effect for a 6x6 cm2 photon beam of energy 15, 30, or 45 MV penetrating a water phantom. Our model magnetic field has a nominal (center) strength B0 ranging between 1 and 5 T, and a maximum strength within the geometric beam which is 2.2xB0. For all three beam energies, there is significant dose enhancement for B0 = 2 T which increases greatly for B0 = 3 T, but stronger magnetic fields increase the enhancement further only for the 45-MV beam. Correspondingly, there is major reduction in the dose just distal to this region of large dose enhancement, resulting from secondary electrons and positrons originating upstream which are depositing energy in the dose-enhancement region rather than continuing further into the patient. The dose peak region is fairly narrow (in depth), but the magnetic field can be shifted along the longitudinal axis to produce a flat peak region of medium width (approximately 2 cm) or of large width (approximately 4 cm), with rapid dose dropoffs on either side. For the 30-MV beam with B0 = 3 T, we found a dose enhancement of 55% for the narrow-width configuration, 32% for the medium-width configuration, and 23% for the large-width configuration; for the 45-MV beam with B0 = 3 T, the enhancements were quite similar, but for the 15-MV beam they were considerably less. For all of these 30-MV configurations, the dose reductions were approximately 30%, and they were approximately 40% for the 45-MV configurations. PMID:11190956
Jette, D
2000-12-01
214
We prove the existence of a magnetic field created by a planar configuration of piecewise rectilinear wires which is not holomorphically integrable when considered as a vector field in C3. This is a counterexample to the S. Stefanescu conjecture (1986) in the holomorphic setting. In particular the method of the proof gives an easy way of showing that the corresponding real vector field does not admit a real polynomial first integral which provides also an alternative way of contradicting the Stefanescu conjecture in the polynomial setting.
2013-05-01
215
DOEpatents
A heat pipe configuration is described for use in a magnetic field environment of a fusion reactor. Heat pipes for operation in a magnetic field when liquid metal working fluids are used are optimized by flattening of the heat pipes having an unobstructed annulus which significantly reduces the adverse side region effect of the prior known cylindrically configured heat pipes. The flattened heat pipes operating in a magnetic field can remove 2 to 3 times the heat as a cylindrical heat pipe of the same cross sectional area.
Werner, R.W.; Hoffman, M.A.
1981-04-29
216
DOEpatents
A heat pipe configuration for use in a magnetic field environment of a fusion reactor. Heat pipes for operation in a magnetic field when liquid metal working fluids are used are optimized by flattening of the heat pipes having an unobstructed annulus which significantly reduces the adverse side region effect of the prior known cylindrically configured heat pipes. The flattened heat pipes operating in a magnetic field can remove 2--3 times the heat as a cylindrical heat pipe of the same cross sectional area.
Werner, Richard W. (San Ramon, CA); Hoffman, Myron A. (Davis, CA)
1983-01-01
217
We investigate the effects of inhomogeneous scalar field configurations on the electroweak phase transition. For this purpose we calculate the leading perturbative correction to the wave function correction term Z(cphi,T), i.e., the kinetic term in the effective action, for the electroweak standard model at finite temperature and the top quark self-mass. Our finding for the fermionic contribution to Z(cphi,T) is
Dirk-Uwe Jungnickel; Dirk Walliser
1994-01-01
218
In this paper, we study the effect of mechanical stress on the domain configuration of a NiFe film obliquely deposited on a compliant polyimide substrate. To this end, we have developed a new method combining in situ mechanical tests with magnetic force microscopy (MFM) imaging. This approach allows changing the static magnetization structure of the film by controlling the stress-induced anisotropy. In the absence of applied stress and magnetic field, the sample shows stripe domains with an in-plane tilted direction with respect to the stress axis. After saturating the film, application of an increasing stress regenerates progressively a stripe domains structure with a modified in-plane magnetization direction.
Karboul-Trojet, W.; Faurie, D.; Aït-Yahiatène, E.; Roussigné, Y.; Mazaleyrat, F.; Chérif, S. M.
2012-04-01
219
In this thesis, we did a comprehensive investigation on the relationship between spin-dependent tunneling and structural variation in junction devices. Magnetic, microstructural, and transport studies have shown a significant improvement in exchange-bias, a reduced barrier roughness, and an enhanced magnetoresistance for samples after magnetic annealing. We have examined different magnetic configurations required for sensing applications and presented some results of using MTJ sensors to detect AC magnetic fields created by electrical current flow and DC stray field distributions of patterned magnetic materials. We have studied the low frequency noise in MTJ sensors. We have found that the 1/f noise in MTJs has magnetic as well as electrical origins, and is strongly affected by the junction's internal structure. The magnetic noise comes from magnetization fluctuations in the free FM layer and can be understood using the fluctuation-dissipation theorem. While the field-independent electrical noise due to charge trapping in the barrier, is observed in the less optimized MTJs sensors, and has an amplitude at least one order of magnitude higher than the noise component due to magnetization fluctuations. In addition, we have studied the magnetization switching of Cobalt rings with varying anisotropy utilizing scanning magnetoresistive microscopy. We have for the first time observed a complicated multi-domain intermediate phase during the transition between onion states for samples with strong anisotropy. This is in contrast to as deposited samples, which reverse by simple domain wall motion and feature an intermediate vortex state. The result is further analyzed by micro magnetic simulations.
Liu, Xiaoyong
220
PubMed
Our present-day understanding of solar and stellar magnetic fields is discussed from both an observational and theoretical viewpoint. To begin with, observations of the Sun's large-scale magnetic field are described, along with recent advances in measuring the spatial distribution of magnetic fields on other stars. Following this, magnetic flux transport models used to simulate photospheric magnetic fields and the wide variety of techniques used to deduce global coronal magnetic fields are considered. The application and comparison of these models to the Sun's open flux, hemispheric pattern of solar filaments and coronal mass ejections are then discussed. Finally, recent developments in the construction of steady-state global magnetohydrodynamic models are considered, along with key areas of future research. PMID:22665897
Mackay, Duncan H
2012-07-13
221
SciTech Connect
The presence of electric and magnetic fields in high enthalpy nozzle flows can produce strong effects. In particular, non equilibrium conditions can be observed when this field are present. In this work we have investigated two different field configurations in supersonic nozzle: first of all we have studied the cooperative effect of electric and magnetic field applied inside the nozzle and secondly we have investigated the role of non equilibrium distribution produced in the reservoir by electric discharge (plasma jet)
Colonna, Gianpiero; Capitelli, Mario [Dipartimento di Chimica, Universita di Bari (Italy); CNR-IMIP, Bari Section (Italy)
2005-05-16
222
PubMed
Field reversed configurations (FRCs) with high confinement are obtained in the C-2 device by combining plasma gun edge biasing and neutral beam injection. The plasma gun creates an inward radial electric field that counters the usual FRC spin-up. The n = 2 rotational instability is stabilized without applying quadrupole magnetic fields. The FRCs are nearly axisymmetric, which enables fast ion confinement. The plasma gun also produces E × B shear in the FRC edge layer, which may explain the observed improved particle transport. The FRC confinement times are improved by factors 2 to 4, and the plasma lifetimes are extended from 1 to up to 4 ms. PMID:23004613
Tuszewski, M; Smirnov, A; Thompson, M C; Korepanov, S; Akhmetov, T; Ivanov, A; Voskoboynikov, R; Schmitz, L; Barnes, D; Binderbauer, M W; Brown, R; Bui, D Q; Clary, R; Conroy, K D; Deng, B H; Dettrick, S A; Douglass, J D; Garate, E; Glass, F J; Gota, H; Guo, H Y; Gupta, D; Gupta, S; Kinley, J S; Knapp, K; Longman, A; Hollins, M; Li, X L; Luo, Y; Mendoza, R; Mok, Y; Necas, A; Primavera, S; Ruskov, E; Schroeder, J H; Sevier, L; Sibley, A; Song, Y; Sun, X; Trask, E; Van Drie, A D; Walters, J K; Wyman, M D
2012-06-21
223
The dynamical behavior of relativistic electron governed by the combination of a realistic helical-wiggler free-electron laser (FEL) with a uniform axial guide magnetic field is investigated by the consideration of the effect of the relativistic electrons self-fields. The electron beam is assumed to have uniform density. In Raman regime, a three-dimensional Hamiltonian approach is derived in detail. The consideration of the additional scalar potential ? s represents the basic feature of the analysis. The approach recognized the two usual constants of motion: one concerns the total energy while the other is the canonical axial angular momentum hat P_{z'}. After some tedious algebra, the dynamical variables problems are solved analytically to study stable and unstable fixed point. The additional scalar potential ? s changes the nature of groups, in group II orbits reversed field configuration near hat ? _0 = 0 converted to a simple group II. At the time of the variation of ? the energetic interaction zones are discussed. The stability zones of fixed points that allow an excellent interaction between the electron and the existing fields are limited. To validate our model, we apply it to the well-known experience of Conde and Bekefi [Phys. Lett. 67, 3082 (1991)] and get some encouraging results.
El-Bahi, R.
2012-09-01
224
Magnetic field measurements of the solar corona using microwave observation are reviewed. The solar corona is filled with highly ionised plasma and magnetic field. Moving charged particles interact with magnetic field due to Lorentz force. This results in gyration motion perpendicular to the magnetic field and free motion along the magnetic field. Circularly polarized electro-magnetic waves interact with gyrating electrons
K. Shibasaki
2006-01-01
225
An internal potential function was created using the averaged MGS vector data released by Mario Acuna for altitudes from 95 to 209 km above the Martian geoid, all longitudes, and latitudes from 87 degrees south to 78 degrees north. Even with some gaps in coverage it is found that a consistent internal potential function can be derived up to spherical harmonic terms of n = 65 using all three components of the data. Weighting the data according to the standard errors given, the model fits to 7-8 nT rms. The energy density spectrum of the harmonics is seen to peak near n = 39 with a value of 7 J/cu km and fall off to less than 0.5 J/cu km below n = 15 and above n = 55. Contour maps of the X (north) component drawn for 100 km altitude show the strongly anomalous region centered at 60 degrees S latitude and 180 degrees longitude, as well as the alternating east-west trends already observed by other groups. Maps of the other components show the anomalous region, but not the east-west trends. The dichotomy is also maintained with much weaker anomalies bounding the northern plains. The results herein as as well as those of others is limited by the sparse low-altitude data coverage as well as the accuracy of the observations in the face of significant spacecraft fields. Work by Connerney and Acuna have mitigated these sources somewhat, but the design of the spacecraft did not lend itself to accurate observations. Recent results reported by David Mitchell of the ER group have shown that the field observations are significantly influenced by the solar wind with the possibility that the present results may only reflect that portion of the internal field visible above 95 km altitude. Depending on the solar wind, the anomaly field may be shielded or distorted to produce spurious results. The spectrum we have obtained so far may only see the stronger portion of the signal with a significant weaker component hidden. Measurements of crustal anomalies versus relative ages of source bodies combined with later absolute dating of Martian geologic units could lead to a quantitative constraint on the thermal history of the planet, i.e. the time when convective dynamo generation ceased in the core. Determination of directions of magnetization of anomaly sources as a function of age combined with the expectation that the Martian dynamo field was roughly aligned with the rotation axis would lead to a means of investigating polar wandering for Mars. Preliminary analysis of two magnetic anomalies in the northern polar region has yielded paleomagnetic pole positions near 50 N, 135 W, about 30 degrees north of Olympus Mons. This location is roughly consistent with the orientation of the planet expected theoretically prior to the formation of the Tharsis region. In the future, more accurate observations of the vector field at the lowest possible altitudes would significantly improve our understanding of Martian thermal history, polar wandering, and upper crustal evolution. Mapping potential resources (e.g., iron-rich source bodies) for future practical use would also be a side benefit. Additional information is contained in the original abstract.
Cain, J. C.; Ferguson, B.; Mozzoni, D.; Hood, L.
2000-07-01
226
SciTech Connect
A self-consistent method is described for determining the static magnetic-field reduction in a magnetized plasma with a specified density profile by radio-frequency (rf)-driven rotating magnetic fields (RMFs). Electron-ion collisions and transport losses are included in the analysis. Application of RMF current drive to tandem mirrors and rotomak reactors is considered. The results of the calculations show that magnetic wells can be produced in mirror configurations, and reversal of applied static magnetic fields can be generated in rotomark geometrics by RMF for modest investments of rf power at frequencies for which the rf technology is economically attractive.
Sperling, J.L.; Glassman, A.J.; Moses, K.G.; Quon, B.H.
1986-07-01
227
Here we briefly summarise the main phases which determine the dynamical evolution of primordial magnetic fields in the early universe. On the one hand, strong fields undergo damping due to excitations of plasma fluctuations, and, on the other hand, weak magnetic fields will be strongly amplified by the small-scale dynamo in a turbulent environment. We find that, under reasonable assumptions concerning the efficiency of a putative magnetogenesis era during cosmic phase transitions, surprisingly strong magnetic fields 10-13-10-11 G on comparatively small scales 100 pc -10 kpc may survive to prior to structure formation. Additionally, any weak magnetic field will be exponentially amplified during the collapse of the first minihalos until they reach equipartition with the turbulent kinetic energy. Hence, we argue that it seems possible for cluster magnetic fields to be entirely of primordial origin.
Banerjee, R.
2013-06-01
228
Polarized neutron reflectivity (PNR) is ideally suited for imaging both vertical structural and magnetic variations in the complex magnetic multilayers [1]. During the talk will be described particularly how this technique allows obtaining the magnetic depth-profile of exchange-coupled bilayer. For instance, Gd40Fe60/ Tb12Fe88 is a model system to study exchange-bias phenomena origin in anti-ferromagnetically coupled AF/FM system, like FeF2/Fe. In these systems, unusual properties are observed such as a transition from positive to negative exchange bias field HE as the cooling field Hcf is swept from small to large positive value [2]. Combining complementary techniques that are macroscopic magnetization measurements and PNR, we have demonstrated that the above properties, e.g. the cooling field dependence of HE, come from an interfacial domain wall (iDW) frozen in the TbFe as the sample is cooled down under a field [3, 4]. Moreover, PNR measurements have recently revealed that the frozen iDW is metastable and that the exchange bias training effect in TbFe/GdFe results from the thermally assisted relaxation of the iDW, with field cycling [4, 5]. Overall, PNR studies concerning the TbFe/GdFe have brought strong insights into the exchange bias mechanisms in exchange coupled hard/soft systems with in-plane anisotropy. However we have demonstrated as well that this powerful technique can be applied to systems with perpendicular magnetic anisotropy (PMA). Although, in that case, the perpendicular moments are parallel to the scattering vector and do not give rise to scattering via the neutron selection rules, we have used a unconventional geometry to obtain a depth-dependent magnetic profile of a PMA exchange-coupled system. Specifically, we have characterized antiferromagnetically-coupled, TbFeCo/[Co/Pd] multilayers [6]. [4pt] [1] K.V. O'Donovan et al., Phys. Rev. Lett. 88, 067201 (2002). [0pt] [2] J. Nogues and al. Phys. Rev. Lett. 76, 4624 (1996) [0pt] [3] Y. Henry et al., Phys. Rev. B 73, 134420 (2006) [0pt] [4] T. Hauet et al., Phys. Rev. Lett. 96, 067207 (2006) [0pt] [5] T. Hauet et al., Appl. Phys. Lett. 91, 022505 (2007) [0pt] [6] S. Watson et al., Appl. Phys. Lett. 92, 202507 (2008)
Hauet, Thomas
2009-03-01
229
Magnetic fields have been known in antiquity. Aristotle attributes the first of what could be called a scientific discussion on magnetism to Thales, who lived from about 625 BC. In China “magnetic carts” were in use to help the Emperor in his journeys of inspection. Plinius comments that in the Asia Minor province of Magnesia shepherds' staffs get at times “glued” to a stone, a alodestone. In Europe the magnetic compass came through the Arab sailors who met the Portuguese explorers. The first scientific treatise on magnetism, “De Magnete”, was published by William Gilbert who in 1600 described his experiments and suggested that the Earth was a huge magnet. Johannes Kepler was a correspondent of Gilbert and at times suggested that planetary motion was due to magnetic forces. Alas, this concept was demolished by Isaac Newton,who seeing the falling apple decided that gravity was enough. This concept of dealing with gravitational forces only remains en vogue even today. The explanations why magnetic effects must be neglected go from “magnetic energy is only 1% of gravitation” to “magnetic fields only complicate the beautiful computer solutions”. What is disregarded is the fact that magnetic effects are very directional(not omni-directional as gravity) and also the fact that magnetic fields are seen every where in our cosmic universe.
Wielebinski, Richard; Beck, Rainer
230
The existence of first integrals and periodic orbits of magnetic fields created by thin wires is investigated. When the current lines are planar we prove that magnetic orbits are closed near the wires and we provide two examples of magnetic fields without polynomial first integrals, thus contradicting Stefanescu's conjecture. When the current lines are non-planar we provide some examples of rectilinear configurations giving rise to helicoidal orbits near the wires and to chaotic portraits.
Aguirre, J.; Giné, J.; Peralta-Salas, D.
2008-01-01
231
SciTech Connect
The procedure for installing Superconducting Super Collider (SSC) dipoles in their respective cryostats involves aligning the average direction of their field with the vertical to an accuracy of 0.5 mrad. The equipment developed for carrying on these measurements is described and the measurements performed on the first few prototypes SSC magnets are presented. The field angle as a function of position in these 16.6 m long magnets is a characteristic of the individual magnet with possible feedback information to its manufacturing procedure. A comparison of this vertical alignment characteristic with a magnetic field intensity (by NMR) characteristic for one of the prototypes is also presented. 5 refs., 7 figs.
Kuchnir, M.; Schmidt, E.E.
1987-11-06
232
PubMed
We calculate, in the free Maxwell theory, the renormalized quantum vacuum expectation value of the two-point magnetic correlation function in de Sitter inflation. We find that quantum magnetic fluctuations remain constant during inflation instead of being washed out adiabatically, as usually assumed in the literature. The quantum-to-classical transition of super-Hubble magnetic modes during inflation allow us to treat the magnetic field classically after reheating, when it is coupled to the primeval plasma. The actual magnetic field is scale independent and has an intensity of few×10(-12)??G if the energy scale of inflation is few×10(16)??GeV. Such a field accounts for galactic and galaxy cluster magnetic fields. PMID:23971556
Campanelli, Leonardo
2013-08-06
233
PubMed
We present, design and generate a new kind of vector optical fields with linear polarization distributions modeling to electric and magnetic field lines. The geometric configurations of "electric charges" and "magnetic charges" can engineer the spatial structure and symmetry of polarizations of vector optical field, providing additional degrees of freedom assisting in controlling the field symmetry at the focus and allowing engineering of the field distribution at the focus to the specific applications. PMID:23842405
Pan, Yue; Li, Si-Min; Mao, Lei; Kong, Ling-Jun; Li, Yongnan; Tu, Chenghou; Wang, Pei; Wang, Hui-Tian
2013-07-01
234
NSDL National Science Digital Library
A cross section of two circular wire loops carrying the exact same current is shown above (position given in centimeters and magnetic field given in milli-Tesla). You can click-drag to read the magnitude of the magnetic field.
Christian, Wolfgang; Belloni, Mario
2007-03-03
235
It has been proposed that high Mach number collisionless shocks propagating in an initially unmagnetized plasma play a major role in the magnetization of large scale structures in the Universe. A detailed study of the experimental configuration necessary to scale such environments down to laboratory dimensions will be presented. We will show initial results from preliminary experiments conducted at the Phoenix laser (UCLA) and the LULI laser (Ecole Polytechnique) where collisionless shocks are generated by the expansion of exploding foils driven by energetic laser beams. The time evolution of the magnetic field is probed with induction coils placed at 10 cm from the laser focus. We will discuss various mechanisms of magnetic field generation and compare them with the experimental results.
Murphy, C. D.; Miniati, F.; Edwards, M.; Mithen, J.; Bell, A. R.; Constantin, C.; Everson, E.; Schaeffer, D.; Niemann, C.; Ravasio, A.; Brambrink, E.; Benuzzi-Mounaix, A.; Koenig, M.; Gregory, C.; Woolsey, N.; Park, H.-S.; Remington, B.; Ryutov, D.; Bingham, R.; Gargate, L.; Spitkovsky, A.; Gregori, G.
2010-11-01
236
DOEpatents
The present invention identifies several configurations of conducting elements capable of storing extremely high magnetic fields for the purpose of energy storage or for other uses, wherein forces experienced by the conducting elements and the magnetic field pollution produced at locations away from the configuration are both significantly reduced over those which are present as a result of the generation of such high fields by currently proposed techniques. It is anticipated that the use of superconducting materials will both permit the attainment of such high fields and further permit such fields to be generated with vastly improved efficiency.
Prueitt, Melvin L. (Los Alamos, NM); Mueller, Fred M. (Los Alamos, NM); Smith, James L. (Los Alamos, NM)
1991-01-01
237
DOEpatents
The present invention identifies several configurations of conducting elements capable of storing extremely high magnetic fields for the purpose of energy storage or for other uses, wherein forces experienced by the conducting elements and the magnetic field pollution produced at locations away from the configuration are both significantly reduced over those which are present as a result of the generation of such high fields by currently proposed techniques. It is anticipated that the use of superconducting materials will both permit the attainment of such high fields and further permit such fields to be generated with vastly improved efficiency. 15 figures.
Prueitt, M.L.; Mueller, F.M.; Smith, J.L.
1991-04-09
238
SciTech Connect
Recent experimental results are discussed for a compact toroid produced by a field-reversed theta-pinch and containing purely poloidal magnetic fields. The confinement time is found to vary inversely with the ion gyro-radius and to be approximately independent of ion temperature for fixed gyro-radius. Within a coil of fixed radius, the plasmoid major radius R was varied by approx. 30% and the confinement appears to scale as R/sup 2/. A semi-empirical formation model has been formulated that predicts reasonably well the plasma parameters as magnetic field and fill pressure are varied in present experiments. The model is used to predict parameters in larger devices under construction.
Armstrong, W.T.; Bartsch, R.R.; Cochrane, J.C.; Linford, R.K.; Lipson, J.; McKenna, K.F.; Platts, D.A.; Sherwood, E.G.; Siemon, R.E.; Tuszewski, M.
1981-01-01
239
SciTech Connect
The statistical representation of a fluctuating (stochastic) magnetic field configuration is studied in detail. The Eulerian correlation functions of the magnetic field are determined, taking into account all geometrical constraints: these objects form a nondiagonal matrix. The Lagrangian correlations, within the reasonable Corrsin approximation, are reduced to a single scalar function, determined by an integral equation. The mean square perpendicular deviation of a geometrical point moving along a perturbed field line is determined by a nonlinear second-order differential equation. The separation of neighboring field lines in a stochastic magnetic field is studied. We find exponentiation lengths of both signs describing, in particular, a decay (on the average) of any initial anisotropy. The vanishing sum of these exponentiation lengths ensures the existence of an invariant which was overlooked in previous works. Next, the separation of a particles trajectory from the magnetic field line to which it was initially attached is studied by a similar method. Here too an initial phase of exponential separation appears. Assuming the existence of a final diffusive phase, anomalous diffusion coefficients are found for both weakly and strongly collisional limits. The latter is identical to the well known Rechester-Rosenbluth coefficient, which is obtained here by a more quantitative (though not entirely deductive) treatment than in earlier works.
Wang, H.; Vlad, M.; Vanden Eijnden, E.; Spineanu, F.; Misguich, J.H.; Balescu, R. [Association Euratom-Etat Belge sur la Fusion, Physique Statistique et Plasmas, Code Postal 231, Universite Libre de Bruxelles, Campus Plaine, Boulevard du Triomphe, 1050 Bruxelles (Belgium)]|[Association Euratom-Commissariat a lEnergie Atomique sur la Fusion, Departement de Recherches sur la Fusion Controle, Centre dEtudes de Cadarache, 13108 Saint-Paul-lez-Durance Cedex (France)
1995-05-01
240
High Reynolds number magnetohydrodynamic turbulence in the presence of zero-flux large-scale magnetic fields is investigated as a function of the magnetic field strength. For a variety of flow configurations, the energy dissipation rate ? follows the scaling ??Urms3/? even when the large-scale magnetic field energy is twenty times larger than the kinetic energy. A further increase of the magnetic energy showed a transition to the ??Urms2Brms/? scaling implying that magnetic shear becomes more efficient at this point at cascading the energy than the velocity fluctuations. Strongly helical configurations form nonturbulent helicity condensates that deviate from these scalings. Weak turbulence scaling was absent from the investigation. Finally, the magnetic energy spectra support the Kolmogorov spectrum k-5/3 while kinetic energy spectra are closer to the Iroshnikov-Kraichnan spectrum k-3/2 as observed in the solar wind.
Alexakis, Alexandros
2013-02-01
241
The relationship between magnetic order and ferroelectric properties has been investigated for MnWO4 with a long-wavelength magnetic structure. Spontaneous electric polarization is observed in an elliptical spiral spin phase. The magnetic-field dependence of electric polarization indicates that the noncollinear spin configuration plays a key role for the appearance of the ferroelectric phase. An electric polarization flop from the b direction to the a direction has been observed when a magnetic field above 10 T is applied along the b axis. This result demonstrates that an electric polarization flop can be induced by a magnetic field in a simple system without rare-earth 4f moments.
Taniguchi, K.; Abe, N.; Takenobu, T.; Iwasa, Y.; Arima, T.
2006-09-01
242
SciTech Connect
Experimental data have shown that the light output of a scintillator depends on the magnitude of the externally applied magnetic fields, and that this variation can affect the calorimeter calibration and possibly resolution. The goal of the measurements presented here is to study the light yield of scintillators in high magnetic fields in conditions that are similar to those anticipated for the LHC CMS detector. Two independent measurements were performed, the first at Fermilab and the second at the National High Magnetic Field Laboratory at Florida State University.
Green, D.; Ronzhin, A. [Fermi National Accelerator Lab., Batavia, IL (United States); Hagopian, V. [Florida State Univ., Tallahasse, FL (United States)
1995-06-01
243
SciTech Connect
We tried to enlarge the operation window of an electron cyclotron resonance (ECR) ion source for producing the ECR plasma confined by cylindrically comb-shaped magnetic field, and for extracting the broad ion beam under the low pressures and low microwave powers. The magnetic field by permanent magnets constructs ECR zones at different positions for 2.45 GHz and 11 to 13 GHz microwaves, respectively. According to probe measurements, profiles of plasma density and temperature are different for using each single microwave. We conduct production of ECR plasma by launching simultaneously these two frequency microwaves, and obtain flat profiles of the electron density and the electron temperature. These profiles are not achieved by feeding single frequency microwave. It is found that plasma can be controllable on spatial profiles beyond wide operation window of plasma parameters. We conducted preliminary extracting and forming large bore ion beam from this source. We will make this source a part of tandem type ion source for the first stage. We investigated feasibility and hope to realize the device which has wide range operation window in a single device to produce many kinds of ion beams as like to universal source based on ECR ion source.
Kato, Y.; Satani, T.; Matsui, Y.; Watanabe, T.; Sato, F.; Iida, T. [Osaka Univ., 2-1 Yamada-oka, Suita, Osaka 565-0871 (Japan); Muramatsu, M.; Kitagawa, A. [NIRS, 4-9-1 Anagawa, Inage-ku, Chiba 263-8555 (Japan); Tanaka, K.; Asaji, T. [Tateyama Machine Co. Ltd., 30 Shimonoban, Toyama, Toyama 930-1305 (Japan)
2008-11-03
244
Recently planet Mercury—an unexplored territory in our solar system—has been of much interest to the scientific community due to recent flybys of the spacecraft MESSENGER that discovered its intrinsic stationary and large-scale dipole like magnetic field structure with an intensity of ˜300nT confirming Mariner 10 observations. In the present study, with the observed constraint of Mercury's atmospheric magnetic field structure, internal magnetic field structure is modeled as a solution of magnetic diffusion equation. In this study, Mercury's internal structure mainly consists of a stable stratified fluid core and the convective mantle. For simplicity, magnetic diffusivity in both parts of the structure is considered to be uniform and constant with a value represented by a suitable averages. It is further assumed that vigorous convection in the mantle disposes of the electric currents leading to a very high diffusivity in that region. Thus, in order to satisfy observed atmospheric magnetic field structure, Mercury's most likely magnetic field structure consists of a solution of MHD diffusion equation in the core and a combined multipolar (dipole and quadrupole like magnetic field structures embedded in the uniform field) solution of a current free like magnetic field structure in the mantle and in the atmosphere. With imposition of appropriate boundary conditions at the core-mantle boundary for the first two diffusion eigen modes, in order to satisfy the observed field structure, present study puts the constraint on Mercury's core radius to be ˜2000km.From the estimated magnetic diffusivity and the core radius, it is also possible to estimate the two diffusion eigen modes with their diffusion time scales of ˜8.6 and 3.7 billion years respectively suggesting that the planet inherits its present-day magnetic field structure from the solar Nebula. It is proposed that permanency of such a large-scale magnetic field structure of the planet is attained during Mercury's early evolutionary history of heavy bombardments by the asteroids and comets supporting the giant impact hypothesis for the formation of Mercury.
Hiremath, K. M.
2012-04-01
245
JAXA (Japan Aerospace Exploration Agency) shall launch the SELENE (SELenological and ENgineering Explorer) spacecraft this autumn. Amongst many instruments, it has a magnetometer (LMAG: Lunar MAGnetomter) which will measure the magnetic field on the orbit around the Moon. The nominal orbit of the SELENE is about 100km in altitudes for 1 year observation. Although the extended mission is still not determined, LMAG team is requesting a low altitude (less than 50km) observation, if the remaining fuel allows. We are preparing data processing software for the mission. Here, we report an objective scheme for mapping the lunar crustal magnetic field from the orbital measurement data of unequal altitudes. In this study, the magnetic field is restored by solving a linear inverse-problem determining the sources distributed on the lunar surface to satisfy the observational data, which is known as the equivalent source method. Our scheme has three features improving the method: First, the source calculation is performed simultaneously with detrending. Second, magnetic charges (magnetic monopoles) are used as the equivalent sources. It reduces the density of the sources for the same smoothness in produced field, comparing to the dipole sauces. Third, the number of sources is taken large enough to avoid the problem of configuration of the sources, instead the damped least square assuming the strength of each charge is similar to the next one, and the smoothness factor is determined by minimizing Akaike's Bayesian Information Criterion (ABIC). It guarantees the objectivity of the calculation, in other words, there is no adjustable parameter which may depend of the researcher dealing the data analyses. For testing the scheme, we apply this method to the Lunar Prospector magnetometer data, and provide magnetic field map in the region centered at several regions of strong crustal field including the Reiner Gamma anomaly. The stability of the method and the resolution of the anomaly map are found to be satisfactory.
2007-12-01
246
\\u000a Magnetic fields have been known in antiquity. Aristotle attributes the first of what could be called a scientific discussion\\u000a on magnetism to Thales, who lived from about 625 BC. In China “magnetic carts” were in use to help the Emperor in his journeys\\u000a of inspection. Plinius comments that in the Asia Minor province of Magnesia shepherds’ staffs get at times
Richard Wielebinski; Rainer Beck
2010-01-01
247
National Technical Information Service (NTIS)
Magnetic pumping by major-radius oscillation of a toroidal plasma can be made more practical by introducing a major-radius range within which the vertical-field gradient is sufficiently great so that major-radius perturbations are marginally stable or, be...
H. P. Furth R. A. Ellis
1972-01-01
248
We have run plots of artificial data, which mimic solar magnetograms, through standard algorithms to critique several results reported in the literature. In studying correlation algorithms, we show that the differences in the profiles for the differential rotation of the photospheric magnetic field stem from different methods of averaging. We verify that the lifetimes of small magnetic features, or of
A. A. Smith; H. B. Snodgrass
1999-01-01
249
ERIC Educational Resources Information Center
|Describes a method for measuring the earth's magnetic field using an empty toilet paper tube, copper wire, clear tape, a battery, a linear variable resistor, a small compass, cardboard, a protractor, and an ammeter. (WRM)|
Stewart, Gay B.
2000-01-01
250
National Technical Information Service (NTIS)
The proposed research efforts funded by the UDAP grant to the BRI involve the study of magnetic field waves associated with the Uranian bow shock. This is a collaborative venture bringing together investigators at the BRI, Southwest Research Institute (Sw...
C. W. Smith M. L. Goldstein R. P. Lepping W. H. Mish H. K. Wong
1991-01-01
251
SciTech Connect
Microflares are small activities in the solar low atmosphere; some are in the low corona while others are in the chromosphere. Observations show that some of the microflares are triggered by magnetic reconnection between the emerging flux and a pre-existing background magnetic field. We perform 2.5-dimensional, compressible, resistive magnetohydrodynamic simulations of the magnetic reconnection with gravity considered. The background magnetic field is a canopy-type configuration that is rooted at the boundary of the solar supergranule. By changing the bottom boundary conditions in the simulation, a new magnetic flux emerges at the center of the supergranule and reconnects with the canopy-type magnetic field. We successfully simulate the coronal and chromospheric microflares whose current sheets are located at the corona and the chromosphere, respectively. The microflare with a coronal origin has a larger size and a higher temperature enhancement than the microflare with a chromospheric origin. In the microflares with coronal origins, we also found a hot jet ({approx}1.8 Multiplication-Sign 10{sup 6} K), which is probably related to the observational extreme ultraviolet or soft X-ray jets, and a cold jet ({approx}10{sup 4} K), which is similar to the observational H{alpha}/Ca surges. However, there is only a H{alpha}/Ca bright point in the microflares that have chromospheric origins. The study of parameter dependence shows that the size and strength of the emerging magnetic flux are the key parameters that determine the height of the reconnection location, and they further determine the different observational features of the microflares.
Jiang, R.-L.; Fang, C.; Chen, P.-F., E-mail: rljiang@nju.edu.cn [School of Astronomy and Space Science, Nanjing University, Nanjing 210093 (China)
2012-06-01
252
Microflares are small activities in the solar low atmosphere; some are in the low corona while others are in the chromosphere. Observations show that some of the microflares are triggered by magnetic reconnection between the emerging flux and a pre-existing background magnetic field. We perform 2.5-dimensional, compressible, resistive magnetohydrodynamic simulations of the magnetic reconnection with gravity considered. The background magnetic field is a canopy-type configuration that is rooted at the boundary of the solar supergranule. By changing the bottom boundary conditions in the simulation, a new magnetic flux emerges at the center of the supergranule and reconnects with the canopy-type magnetic field. We successfully simulate the coronal and chromospheric microflares whose current sheets are located at the corona and the chromosphere, respectively. The microflare with a coronal origin has a larger size and a higher temperature enhancement than the microflare with a chromospheric origin. In the microflares with coronal origins, we also found a hot jet (~1.8 × 106 K), which is probably related to the observational extreme ultraviolet or soft X-ray jets, and a cold jet (~104 K), which is similar to the observational H?/Ca surges. However, there is only a H?/Ca bright point in the microflares that have chromospheric origins. The study of parameter dependence shows that the size and strength of the emerging magnetic flux are the key parameters that determine the height of the reconnection location, and they further determine the different observational features of the microflares.
Jiang, R.-L.; Fang, C.; Chen, P.-F.
2012-06-01
253
We present an extensive study of magnetic fields in a system of merging galaxies. We obtained for NGC 4038/39 (the Antennae) radio total intensity and polarization maps at 8.44 GHz, 4.86 GHz and 1.49 GHz using the VLA in the C and D configurations. The galaxy pair possesses bright, extended radio emission filling the body of the whole system, with no dominant nuclear sources. The radio thermal fraction of NGC 4038/39 was found to be about 50% at 10.45 GHz, higher than in normal spirals. Most of the thermal emission is associated with star-forming regions, but only a part of these are weakly visible in the optical domain because of strong obscuration. The mean total magnetic fields in both galaxies are about two times stronger (?20 ?G) than in normal spirals. However, the degree of field regularity is rather low, implying tangling of the regular component in regions with interaction-enhanced star formation. Our data combined with those in H I, H?, X-rays and in far infrared allow us to study local interrelations between different gas phases and magnetic fields. We distinguish several radio-emitting regions with different physical properties and at various evolutionary stages: the rudimentary magnetic spiral, the northern cool part of the dark cloud complex extending between the galaxies, its warm southern region, its southernmost star-forming region deficient in radio emission, and the highly polarized northeastern ridge associated with the base of an unfolding tidal tail. The whole region of the dark cloud complex shows a coherent magnetic field structure, probably tracing the line of collision between the arms of merging spirals while the total radio emission reveals hidden star formation nests. The southern region is a particularly intense merger-triggered starburst. Highly tangled magnetic fields reach there strengths of ?30 ?G, even larger than in both individual galaxies, possibly due to compression of the original fields pulled out from the parent disks. In the northeastern ridge, away from star-forming regions, the magnetic field is highly coherent with a strong regular component of 10 ?G tracing gas shearing motions along the tidal tail. We find no signs of field compression by infalling gas there. The radio spectrum is much steeper in this region indicating aging of the CR electron population as they move away from their sources in star-forming regions. Modelling Faraday rotation data shows that we deal with a three-dimensionally curved structure of magnetic fields, becoming almost parallel to the sky plane in the southeastern part of the ridge.
Chy?y, K. T.; Beck, R.
2004-04-01
254
In this article the effect of low amplitude DC magnetic fields on different types of thermometers is discussed. By means of\\u000a a precision water-cooled electromagnet, the effect of a magnetic field on platinum resistance thermometers, thermistors, and\\u000a type T, J, and K thermocouples was investigated, while thermometers were thermally stabilized in thermostatic baths. Four\\u000a different baths were used for temperatures
G. Gersak; S. Begus
2010-01-01
255
SciTech Connect
This paper addresses preferred SMES configurations and the external magnetic fields which they generate. Possible biological effects of fields are reviewed briefly. It is proposed that SMES units be fenced at the 10 gauss (1 mT) level to keep unrestricted areas safe, even for persons with cardiac pacemakers. For a full size 5000 MWh (1.8 {times} 10 {sup 13} J) SMES the magnetic field decreases to 10 gauss at a radial distance of 2 km from the center of the coil. Other considerations related to the environmental impact of large SMES magnetic fields are discussed briefly.
Polk, C. (Rhode Island Univ., Kingston, RI (United States). Dept. of Electrical Engineering); Boom, R.W.; Eyssa, Y.M. (Wisconsin Univ., Madison, WI (United States). Applied Superconductivity Center)
1992-01-01
256
SciTech Connect
Transverse particle motion in particle accelerators is governed almost totally by non-solenoidal magnets for which the body magnetic field can be expressed as a series expansion of the normal (b{sub n}) and skew (a{sub n}) multipoles, B{sub y} + iB{sub x} = {summation}(b{sub n} + ia{sub n})(x + iy){sup n}, where x, y, and z denote horizontal, vertical, and longitudinal (along the magnet) coordinates. Since the magnet length L is necessarily finite, deflections are actually proportional to field integrals such as {bar B}L {equivalent_to} {integral} B(x,y,z)dz where the integration range starts well before the magnet and ends well after it. For {bar a}{sub n}, {bar b}{sub n}, {bar B}{sub x}, and {bar B}{sub y} defined this way, the same expansion Eq. 1 is valid and the standard approximation is to neglect any deflections not described by this expansion, in spite of the fact that Maxwells equations demand the presence of longitudinal field components at the magnet ends. The purpose of this note is to provide a semi-quantitative estimate of the importance of {vert_bar}{Delta}p{sub {proportional_to}}{vert_bar}, the transverse deflection produced by the ion-gitudinal component of the fringe field at one magnet end relative to {vert_bar}{Delta}p{sub 0}{vert_bar}, the total deflection produced by passage through the whole magnet. To emphasize the generality and simplicity of the result it is given in the form of a theorem. The essence of the proof is an evaluation of the contribution of the longitudinal field B{sub x} from the vicinity of one magnet end since, along a path parallel to the magnet axis such as path BC.
Wei, Jie [Brookhaven National Lab., Upton, NY (United States); Talman, R. [Cornell Univ., Ithaca, NY (United States). Lab. of Nuclear Studies
1995-12-31
257
The Sun’s global magnetic field is produced and evolved through the emergence of magnetic flux in active regions and its transport across the solar surface by the axisymmetric differential rotation and meridional flow and the non-axisymmetric convective flows of granulation, supergranulation, and giant cell convection. Maps of the global magnetic field serve as the inner boundary condition for space weather. The photospheric magnetic field and its evolution determine the coronal and solar wind structures through which CMEs must propagate and in which solar energetic particles are accelerated and propagate. Producing magnetic maps which best represent the actual field configuration at any instant requires knowing the magnetic field over the observed hemisphere as well as knowing the flows that transport flux. From our Earth-based vantage point we only observe the front-side hemisphere and each pole is observable for only six months of the year at best. Models for the surface magnetic flux transport can be used to provide updates to the magnetic field configuration in those unseen regions. In this presentation I will describe successes and failures of surface flux transport and present new observations on the structure, the solar cycle variability, and the evolution of the flows involved in magnetic flux transport. I find that supergranules play the dominant role due to their strong flow velocities and long lifetimes. Flux is transported by differential rotation and meridional flow only to the extent that the supergranules participate in those two flows.
Hathaway, D. H.
2010-12-01
258
PubMed
Neutral atomic Bose condensates and degenerate Fermi gases have been used to realize important many-body phenomena in their most simple and essential forms, without many of the complexities usually associated with material systems. However, the charge neutrality of these systems presents an apparent limitation-a wide range of intriguing phenomena arise from the Lorentz force for charged particles in a magnetic field, such as the fractional quantum Hall effect in two-dimensional electron systems. The limitation can be circumvented by exploiting the equivalence of the Lorentz force and the Coriolis force to create synthetic magnetic fields in rotating neutral systems. This was demonstrated by the appearance of quantized vortices in pioneering experiments on rotating quantum gases, a hallmark of superfluids or superconductors in a magnetic field. However, because of technical issues limiting the maximum rotation velocity, the metastable nature of the rotating state and the difficulty of applying stable rotating optical lattices, rotational approaches are not able to reach the large fields required for quantum Hall physics. Here we experimentally realize an optically synthesized magnetic field for ultracold neutral atoms, which is evident from the appearance of vortices in our Bose-Einstein condensate. Our approach uses a spatially dependent optical coupling between internal states of the atoms, yielding a Berry's phase sufficient to create large synthetic magnetic fields, and is not subject to the limitations of rotating systems. With a suitable lattice configuration, it should be possible to reach the quantum Hall regime, potentially enabling studies of topological quantum computation. PMID:19956256
Lin, Y-J; Compton, R L; Jiménez-García, K; Porto, J V; Spielman, I B
2009-12-01
259
TUBE88 computes magnetic field lines in cylindrical or toroidal geometry (using cylindrical coordinates (r, ?, z)) and calculates the intersections of those field lines with specified planes. It is an outgrowth of a code first written in 1967. A fourth-order predictor-corrector method is used to integrate the field line coordinates. The magnetic field may be computed in several ways: (a) through specification of currents flowing in very specific helical and circular elements together with a 1/r'' field and a vertical field, (b) as a Fourier series in the angular variale or (c) in a specific coordinate system suited to a toroidally helical domain. Extensive graphics are provided for users of the Cray Time-Sharing System (CTSS). Applications of the code have included analysis of vacuum magnetic field configurations and post processing magnetic field data produced by MHD codes, for example. Current address: Sandia National Laboratory, Livermore, CA 94550, USA.
Mirin, A. A.; Martin, D. R.; O'Neill, N. J.
1989-04-01
260
SciTech Connect
The Large-s Experiment (LSX) was built to study the formation and equilibrium properties of field-reversed configurations (FRCs) as the scale size increases. The dynamic, field-reversed theta-pinch method of FRC creation produces axial and azimuthal deformations and makes formation difficult, especially in large devices with large s (number of internal gyroradii) where it is difficult to achieve initial plasma uniformity. However, with the proper technique, these formation distortions can be minimized and are then observed to decay with time. This suggests that the basic stability and robustness of FRCs formed, and in some cases translated, in smaller devices may also characterize larger FRCs. Elaborate formation controls were included on LSX to provide the initial uniformity and symmetry necessary to minimize formation disturbances, and stable FRCs could be formed up to the design goal of s = 8. For x [le] 4, the formation distortions decayed away completely, resulting in symmetric equilibrium FRCs with record confinement times up to 0.5 ms, agreeing with previous empirical scaling laws ([tau][proportional to]sR). Above s = 4, reasonably long-lived (up to 0.3 ms) configurations could still be formed, but the initial formation distortions were so large that they never completely decayed away, and the equilibrium confinement was degraded from the empirical expectations. The LSX was only operational for 1 yr, and it is not known whether s = 4 represents a fundamental limit for good confinement in simple (no ion beam stabilization) FRCs or whether it simply reflects a limit of present formation technology. Ideally, s could be increased through flux buildup from neutral beams. Since the addition of kinetic or beam ions will probably be desirable for heating, sustainment, and further stabilization of magnetohydrodynamic modes at reactor-level s values, neutral beam injection is the next logical step in FRC development. 24 refs., 21 figs., 2 tabs.
Hoffman, A.L.; Carey, L.N.; Crawford, E.A.; Harding, D.G.; DeHart, T.E.; McDonald, K.F.; McNeil, J.L.; Milroy, R.D.; Slough, J.T. (STI Optronics, Bellevue, WA (United States)); Maqueda, R.; Wurden, G.A. (Univ. of Washington, Seattle (United States))
1993-03-01
261
SciTech Connect
The magnetospheres of accreting compact stars (neutron stars and white dwarfs) are examined. It is assumed that the compact star possesses a multipole magnetic field. The shape of the magnetosphere for the two-dimensional analog of spherically symmetric accretion and the magnetic-field configuration for the case of disk accretion are found with the help of conformal mappings. The results of a generalization of the two-dimensional solution to the real three-dimensional case are discussed.
Lipunov, V.M.
1978-11-01
262
SciTech Connect
Large field reversed configurations (FRCs) are produced in the C-2 device by combining dynamic formation and merging processes. The good confinement of these FRCs must be further improved to achieve sustainment with neutral beam (NB) injection and pellet fuelling. A plasma gun is installed at one end of the C-2 device to attempt electric field control of the FRC edge layer. The gun inward radial electric field counters the usual FRC spin-up and mitigates the n = 2 rotational instability without applying quadrupole magnetic fields. Better plasma centering is also obtained, presumably from line-tying to the gun electrodes. The combined effects of the plasma gun and of neutral beam injection lead to the high performance FRC operating regime, with FRC lifetimes up to 3 ms and with FRC confinement times improved by factors 2 to 4.
Tuszewski, M.; Smirnov, A.; Thompson, M. C.; Barnes, D.; Binderbauer, M. W.; Brown, R.; Bui, D. Q.; Clary, R.; Conroy, K. D.; Deng, B. H.; Dettrick, S. A.; Douglass, J. D.; Garate, E.; Glass, F. J.; Gota, H.; Guo, H.Y.; Gupta, D.; Gupta, S.; Kinley, J. S.; Knapp, K. [Tri Alpha Energy, Inc., P.O. Box 7010, Rancho Santa Margarita, California 92688 (United States); and others
2012-05-15
263
SciTech Connect
Magnetic fields have a large impact on the magnetic and superconducting properties of solids. High magnetic fields are required to reach magnetic saturation along a hard magnetic direction in a variety of rare-earth intermetallics, to break the ferrimagnetic moment configuration in specific 3d-4f intermetallics, to quench the strongly correlated electron states in heavy fermion compounds, to reach the upper critical fields in several classes of superconductors, to study flux-pinning phenomena in the high-{Tc} superconductors, etc. In the present review, the attention is focused to the field interval 20--50 tesla. Experiments in this field range are the privilege of specialized high magnetic field laboratories. There is a lively activity in this area of research with the number of participating institutes continuously growing.
Franse, J.J.M.; Boer, F.R. de; Frings, P.H.; Visser, A. de [Univ. of Amsterdam (Netherlands). Van der Waals-Zeeman Lab.
1994-03-01
264
SciTech Connect
Several recent applications for fast ramped magnets have been found that require rapid measurement of the field quality during the ramp. (In one instance, accelerator dipoles will be ramped at 1 T/sec, with measurements needed to the accuracy typically required for accelerators.) We have built and tested a new type of magnetic field measuring system to meet this need. The system consists of 16 stationary pickup windings mounted on a cylinder. The signals induced in the windings in a changing magnetic field are sampled and analyzed to obtain the field harmonics. To minimize costs, printed circuit boards were used for the pickup windings and a combination of amplifiers and ADPs used for the voltage readout system. New software was developed for the analysis. Magnetic field measurements of a model dipole developed for the SIS200 accelerator at GSI are presented. The measurements are needed to insure that eddy currents induced by the fast ramps do not impact the field quality needed for successful accelerator operation.
JAIN, A.; ESCALLIER, J.; GANETIS, G.; LOUIE, W.; MARONE, A.; THOMAS. R.; WANDERER, P.
2004-10-03
265
Aims: We characterize the orientation of polar plumes as a tracer of the large-scale coronal magnetic field configuration. We monitor in particular the north and south magnetic pole locations and the magnetic opening during 2007-2008 and provide some understanding of the variations in these quantities. Methods: The polar plume orientation is determined by applying the Hough-wavelet transform to a series of EUV images and extracting the key Hough space parameters of the resulting maps. The same procedure is applied to the polar cap field inclination derived from extrapolating magnetograms generated by a surface flux transport model. Results: We observe that the position where the magnetic field is radial (the Sun's magnetic poles) reflects the global organization of magnetic field on the solar surface, and we suggest that this opens the possibility of both detecting flux emergence anywhere on the solar surface (including the far side) and better constraining the reorganization of the corona after flux emergence.
de Patoul, Judith; Inhester, Bernd; Cameron, Robert
2013-10-01
266
The exact mechanism of formation of highly relativistic jets from galactic nuclei and microquasars remains unknown but most accepted models involve a central black hole and a strong external magnetic field. This idea is based on assumption that the black hole rotates and the magnetic field threads its horizon. Magnetic torques provide a link between the hole and the surrounding plasma which then becomes accelerated. We first review our work on black holes immersed in external stationary vacuum (electro)magnetic fields in both test-field approximation and within exact general-relativistic solutions. A special attention will be paid to the Meissner-type effect of the expulsion of the flux of external axisymmetric stationary fields across rotating (or charged) black holes when they approach extremal states. This is a potential threat to any electromagnetic mechanism launching the jets at the account of black-hole rotation because it inhibits the extraction of black-hole rotational energy. We show that the otherwise very useful "membrane viewpoint of black holes" advocated by Thorne, Price and Macdonald does not represent an adequate formalism in the context of the field expulsion from extreme black holes. After briefly summarizing the results for black holes in magnetic fields in higher dimensions - the expulsion of stationary axisymmetric fields was demonstrated to occur also for extremal black-hole solutions in string theory and Kaluza-Klein theory - we shall review astrophysically relevant axisymmetric numerical simulations reported recently by Gammie, Komissarov, Krolik and others. Although the field expulsion has not yet been observed in these time-dependent simulations, they may still be too far away from the extreme limit at which the black-hole Meissner effect should show up. We mention some open problems which, according to our view, deserve further investigation.
Bi?ák, Ji?í; Karas, Vladimír; Ledvinka, Tomáš
2007-04-01
267
The origin of large-scale magnetic fields in clusters of galaxies remains controversial. The intergalactic magnetic field within filaments should be less polluted by magnetised outflows from active galaxies than magnetic fields in clusters. Therefore, filaments may be a better laboratory to study magnetic field amplification by structure formation than galaxy clusters, which typically host many more active galaxies. We present
M. Brüggen; M. Hoeft
2006-01-01
268
Indoor localization consists of locating oneself inside new buildings. GPS does not work indoors due to multipath reflection and signal blockage. WiFi based systems assume ubiquitous availability and infrastructure based systems require expensive installations, hence making indoor localization an open problem. This dissertation consists of solving the problem of indoor localization by thoroughly exploiting the indoor ambient magnetic fields comprising mainly of disturbances termed as anomalies in the Earth's magnetic field caused by pillars, doors and elevators in hallways which are ferromagnetic in nature. By observing uniqueness in magnetic signatures collected from different campus buildings, the work presents the identification of landmarks and guideposts from these signatures and further develops magnetic maps of buildings - all of which can be used to locate and navigate people indoors. To understand the reason behind these anomalies, first a comparison between the measured and model generated Earth's magnetic field is made, verifying the presence of a constant field without any disturbances. Then by modeling the magnetic field behavior of different pillars such as steel reinforced concrete, solid steel, and other structures like doors and elevators, the interaction of the Earth's field with the ferromagnetic fields is described thereby explaining the causes of the uniqueness in the signatures that comprise these disturbances. Next, by employing the dynamic time warping algorithm to account for time differences in signatures obtained from users walking at different speeds, an indoor localization application capable of classifying locations using the magnetic signatures is developed solely on the smart phone. The application required users to walk short distances of 3-6 m anywhere in hallway to be located with accuracies of 80-99%. The classification framework was further validated with over 90% accuracies using model generated magnetic signatures representing hallways with different kinds of pillars, doors and elevators. All in all, this dissertation contributes the following: 1) provides a framework for understanding the presence of ambient magnetic fields indoors and utilizing them to solve the indoor localization problem; 2) develops an application that is independent of the user and the smart phones and 3) requires no other infrastructure since it is deployed on a device that encapsulates the sensing, computing and inferring functionalities, thereby making it a novel contribution to the mobile and pervasive computing domain.
Pathapati Subbu, Kalyan Sasidhar
269
SciTech Connect
Stable levitation with a permanent magnet and a bulk high {Tc} superconductor (HTSC) is examined numerically by using the critical state model and the frozen field model. Differences between a permanent magnet and a trapped field magnet are first discussed from property of levitation force. Stable levitation region of the HTSC on a ring magnet and on a solenoid coil are calculated with the numerical methods. Obtained results are discussed from difference of the magnetic field configuration.
Tsuchimoto, M.; Kojima, T.; Waki, H.; Honma, T. [Hokkaido Univ., Sapporo (Japan)
1995-05-01
270
PubMed Central
We report a method for characterization of the efficiency of radio-frequency (rf) heating of nanoparticles (NPs) suspended in an aqueous medium. Measurements were carried out for water suspended 5?nm superparamagnetic iron-oxide NPs with 30?nm dextran matrix for three different configurations of rf electric and magnetic fields. A 30?MHz high-Q resonator was designed to measure samples placed inside a parallel plate capacitor and solenoid coil with or without an rf electric field shield. All components of rf losses were analyzed and rf electric and magnetic field induced heating of NPs and the dispersion medium was determined and discussed.
Ketharnath, Dhivya; Pande, Rohit; Xie, Leiming; Srinivasan, Srimeenakshi; Godin, Biana; Wosik, Jarek
2012-01-01
271
PubMed
We report a method for characterization of the efficiency of radio-frequency (rf) heating of nanoparticles (NPs) suspended in an aqueous medium. Measurements were carried out for water suspended 5?nm superparamagnetic iron-oxide NPs with 30?nm dextran matrix for three different configurations of rf electric and magnetic fields. A 30?MHz high-Q resonator was designed to measure samples placed inside a parallel plate capacitor and solenoid coil with or without an rf electric field shield. All components of rf losses were analyzed and rf electric and magnetic field induced heating of NPs and the dispersion medium was determined and discussed. PMID:22991480
Ketharnath, Dhivya; Pande, Rohit; Xie, Leiming; Srinivasan, Srimeenakshi; Godin, Biana; Wosik, Jarek
2012-08-24
272
We report a method for characterization of the efficiency of radio-frequency (rf) heating of nanoparticles (NPs) suspended in an aqueous medium. Measurements were carried out for water suspended 5 nm superparamagnetic iron-oxide NPs with 30 nm dextran matrix for three different configurations of rf electric and magnetic fields. A 30 MHz high-Q resonator was designed to measure samples placed inside a parallel plate capacitor and solenoid coil with or without an rf electric field shield. All components of rf losses were analyzed and rf electric and magnetic field induced heating of NPs and the dispersion medium was determined and discussed.
Ketharnath, Dhivya; Pande, Rohit; Xie, Leiming; Srinivasan, Srimeenakshi; Godin, Biana; Wosik, Jarek
2012-08-01
273
The alloy of TbFe2 was studied by ball milling with and without the presence of external magnetic field. While the structure and powder morphology of the alloy were investigated using scanning electron microscope and X-ray diffraction, the magnetization was investigated using vibrating sample and superconducting quantum interference device magnetometers. The rate of particle reduction with ball milling is comparatively higher in the presence of external magnetic field than without it. Consequently, owing to a large fraction of particles acquiring near single domain configuration under the field assisted milling condition, the coercivity derived from these particles are as high as 6500 Oe than that of particles obtained without the aid of external magnetic field which is around 3850 Oe. The field cooled low temperature magnetization exhibits a large coercivity and skew in the shape of the magnetization curve due to the large anisotropy.
Arout Chelvane, J.; Palit, Mithun; Basumatary, Himalay; Pandian, S.
2013-10-01
274
Stellarators use three-dimensional magnetic field shaping to provide stable plasma confinement without the need for driven plasma currents or stabilizing feedback systems. The maximum plasma beta (plasma pressure normalized to magnetic pressure) that can be confined in the W7-AS stellarator was explored by varying the shape of the magnetic field, the heating power, and the magnetic field strength. The maximum
Michael C. Zarnstorff
2003-01-01
275
The Helioseismic and Magnetic Imager (HMI) instrument on the Solar Dynamics Observatory (SDO) spacecraft will begin observing the solar photospheric magnetic field continuously after commissioning in early 2009. This paper describes the HMI magnetic processing pipeline and the expected data products that will be available. The full disk line-of-sight magnetic field will be available every minute with 1" resolution. Comparable vector measurements collected over a three-minute time interval will ordinarily be averaged for at least 10 minutes before inversion. Useful Quick Look products for forecasting purposes will be available a few minutes after observation. Final products will be computed within 36 hours and made available through the SDO Joint Science Operations Center (JSOC). Three kinds of magnetic data products have been defined - standard, on-demand, and on-request. Standard products, such as frequently updated synoptic charts, are made all the time on a fixed cadence. On-demand products, such as high cadence full-disk disambiguated vector magnetograms, will be generated whenever a user asks for them. On-request products, such as high-resolution time series of MHD model solutions, will be generated as resources allow. This paper describes the observations, magnetograms, synoptic and synchronic products, and field model calculations that will be produced by the HMI magnetic pipeline.
Hoeksema, J.; Hmi, M. T.
2008-05-01
276
SciTech Connect
Quantum tunneling across a static potential barrier in a static magnetic field is very sensitive to an analytical form of the potential barrier. Depending on that, the oscillatory structure of the modulus of the wave function can be formed in the direction of tunneling. Due to an underbarrier interference, the probability of tunneling through a higher barrier can be larger than through a lower one. For some barriers the quantum interference of underbarrier cyclotron paths results in a strong enhancement of tunneling. This occurs in the vicinity of the certain magnetic field and is referred to as Euclidean resonance. This strongly contrasts to the Wentzel, Kramers, and Brillouin type tunneling which occurs with no magnetic field.
Ivlev, B. [Department of Physics and Astronomy and NanoCenter, University of South Carolina, Columbia, South Carolina 29208 (United States) and Instituto de Fisica, Universidad Autonoma de San Luis Potosi, San Luis Potosi, San Luis Potosi 78000 Mexico
2006-05-15
277
In the AdS/CFT framework meson thermalization in the presence of a constant external magnetic field in a strongly coupled gauge theory has been studied. In the gravitational description the thermalization of mesons corresponds to the horizon formation on the flavour D7-brane which is embedded in the AdS 5 × S 5 background in the probe limit. The apparent horizon forms due to the time-dependent change in the baryon number chemical potential, the injection of baryons in the gauge theory. We will numerically show that the thermalization happens even faster in the presence of the magnetic field on the probe brane. We observe that this reduction in the thermalization time sustains up to a specific value of the magnetic field.
2013-03-01
278
Special and high efficiency variable speed drives use permanent magnet synchronous motors (PMSM) and power electronics. New dispositions for PMSM with regard to a special magnetic circuit design and new shapes of permanent magnet configurations are investigated. In order to obtain a better rotation performance and to reduce the torque ripple we propose the power electronics current control in support
W. Czernin; F. Asehenbrenner; H. Weiss
2006-01-01
279
SciTech Connect
Magnetic dipole transitions between the levels of ground 4d{sup N} configurations of tungsten ions were analyzed by employing a large basis of interacting configurations. Previously introduced configuration interaction strength between two configurations was used to determine the configurations with the largest contribution to wave functions of atomic states for the considered configurations. Collisional-radiative modeling was performed for the levels of the ground configuration coupled through electric dipole transitions with 4p{sup 5}4d{sup N+1} and 4d{sup N-1}4f configurations. New identification of some lines observed in the electron-beam ion trap plasma was proposed based on calculations in which wavelength convergence was reached.
Jonauskas, V.; Kisielius, R.; Kyniene, A.; Kucas, S.; Norrington, P. H. [Institute of Theoretical Physics and Astronomy, Vilnius University, A. Gostauto 12, LT-01108 Vilnius (Lithuania); Department of Applied Mathematics and Theoretical Physics, Queen's University of Belfast, Belfast BT7 1NN, Northern Ireland, United Kingdon (United Kingdom)
2010-01-15
280
DOEpatents
A propulsion and suspension system for an inductive repulsion type magnetically levitated vehicle which is propelled and suspended by a system which includes propulsion windings which form a linear synchronous motor and conductive guideways, adjacent to the propulsion windings, where both combine to partially encircling the vehicle-borne superconducting magnets. A three phase power source is used with the linear synchronous motor to produce a traveling magnetic wave which in conjunction with the magnets propel the vehicle. The conductive guideway combines with the superconducting magnets to provide for vehicle leviation.
Coffey, H.T.
1992-12-31
281
The solar photosphere is the layer in which the magnetic field has been most reliably and most often measured. Zeeman- and Hanle-effect based probes have revealed many details of a rich variety of structures and dynamic processes, but the number of open and debated questions has remained large. The magnetic field in the quiet Sun has maintained a particularly large number of secrets and has been a topic of a particularly lively debate as new observations and analysis techniques have revealed new and often unexpected aspects of its organization, physical structure and origin.
Solanki, S. K.
2009-06-01
282
We have recently discovered that so called Langmuir [1] states of Helium can stabilize in both the circularly polarized electromagnetic and the magnetic fields when the fields are crossed and two electrons are rotating in the configuration when the two parallel single-electron circular trajectories have the both particles moving in the spatial phase. The stability islands in the fields strength planes have exotic shapes and the configurations are bistable geometrically. Here we discover the whole chains of ions when the single Langmuir configuration is additionally experiencing the infinite chain of neighbouring ions and alike space-periodic configurations. This leads to self-stabilization and Born-Opennheimer binding of Hydrogen, helium or higher charged ions in chains parallel to the magnetic field and when the CP field vector is perpendicular. The excitations along the chain are plasmon-like and have the physical meaning of the deviation from the CP field rotation helicity. Ones the linearly polarized field is superposed from two circularly polarized counterrotating fields similar configurations exist by the geometric argument. Numerical simulations using the recently discovered Cartesian-hypespherical coordinates method previously applied to Langmuir configurations themself are also presented. [1] M. Kalinski, L. Hansen, and D. Farrelly, Nondispersive Two-Electron Wave Packets in a Helium Atom,'' Phys. Rev. Lett. 95, 103001 (2005).
Kalinski, Matt
2010-03-01
283
The configuration of the solar corona magnetic field has been studied. Data on the position of the K-corona emission polarization plane during the solar eclipses of September 21, 1941; February 25, 1952; and August 1, 2008, were used as an indicator of the magnetic field line orientation. Based on an analysis of these data, a conclusion has been made that the studied configuration has a large-scale organization in the form of a cellular structure with an alternating field reversal. The estimated cell size was 61° ± 6° (or 36° ± 2°) in longitude with a latitudinal extension of 40°-50° in the range of visible distances 1.3-2.0 R Sun . A comparison of the detected cellular structure of the coronal magnetic field with synoptic {ie908-1} maps indicated that the structure latitudinal boundaries vary insignificantly within 1.1-2.0 R Sun . The possible causes of the formation of the magnetic field large-scale cellular configuration in the corona and the conditions for the transformation of this configuration into a two-sector structure are discussed.
Merzlyakov, V. L.; Starkova, L. I.
2012-12-01
284
SciTech Connect
Research on small-scale and large-scale photospheric and coronal magnetic fields during 1987-1990 is reviewed, focusing on observational studies. Particular attention is given to the new techniques, which include the correlation tracking of granules, the use of highly Zeeman-sensitive infrared spectral lines and multiple lines to deduce small-scale field strength, the application of long integration times coupled with good seeing conditions to study weak fields, and the use of high-resolution CCD detectors together with computer image-processing techniques to obtain images with unsurpassed spatial resolution. Synoptic observations of large-scale fields during the sunspot cycle are also discussed. 101 refs.
Sheeley, N.R., Jr. (USAF, Geophysics Laboratory, Hanscom AFB, MA (United States))
1991-01-01
285
Potential energy surfaces are obtained for singlet H3+ in magnetic fields of up to 2350 T. The magnetic interaction was treated by first-order perturbation theory and the interaction terms computed ab initio. They were then fitted to a functional form and added to a recent, highly accurate adiabatic potential energy surface. In its most stable orientation, the molecule is arranged such that the magnetic field vector is in the molecular plane. The most stable configuration is no longer D3h as in the field-free case, but C2v, though the stabilization energy is extremely small, of the order of 0.01 cm-1 for a 2350 T field. Finally, we have calculated, for a range of magnetic field strengths and orientations, all the vibrational eigenvalues that are below the barrier to linearity in the field-free case.
Medel Cobaxin, Héctor; Alijah, Alexander
2013-10-01
286
SciTech Connect
A kinetic model of particle acceleration by plasma shocks is analyzed theoretically and with numerical calculations. The shocks are propagating through weakly magnetized background plasmas, namely interstellar magnetic fields (IMFs). Particles located at the shock front are accelerated parallel to the magnetic field of the shock; this is defined as the field-aligned acceleration (FAA). The cross angle between IMF and the magnetic field of the shock plays an important role in creating the magnetic neutral sheet at the shock front. A test particle trapped by the neutral sheet obtains enormous energy due to the FAA. A reasonable formula for the highest energy gain is derived from theoretical analysis of the relativistic equations of motion. A possible configuration of the electric and magnetic fields in supernova remnants is also proposed by way of example.
Takeuchi, Satoshi [Department of Environmental Sciences, University of Yamanashi, 4-3-11 Takeda, Kofu, Yamanashi 400-8511 (Japan)
2012-07-15
287
NSDL National Science Digital Library
The EJSMagnetic Field from Loops model computes the B-field created by an electric current through a straight wire, a closed loop, and a solenoid. Users can adjust the vertical position of the slice through the 3D field. The Magnetic Field from Loops model was created using the Easy Java Simulations (Ejs) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double clicking the ejs_ntnu_MagneticFielfFromLoops.jar file will run the program if Java is installed. Ejs is a part of the Open Source Physics Project and is designed to make it easier to access, modify, and generate computer models. Additional Ejs models for classical mechanics are available. They can be found by searching ComPADRE for Open Source Physics, OSP, or Ejs.
Christian, Wolfgang; Hwang, Fu-Kwun
2008-11-17
288
We investigate the magnetic field which is generated by turbulent motions of a weakly ionized gas. Galactic molecular clouds give us an example of such a medium. As in the Kazantsev-Kraichnan model we assume a medium to be homogeneous and a neutral gas velocity field to be isotropic and ? correlated in time. We take into consideration the presence of a mean magnetic field, which defines a preferred direction in space and eliminates isotropy of magnetic field correlators. Evolution equations for the anisotropic correlation function are derived. Isotropic cases with zero mean magnetic field as well as with small mean magnetic field are investigated. It is shown that stationary bounded solutions exist only in the presence of the mean magnetic field for the Kolmogorov neutral gas turbulence. The dependence of the magnetic field fluctuations amplitude on the mean field is calculated. The stationary anisotropic solution for the magnetic turbulence is also obtained for large values of the mean magnetic field.
Istomin, Ya. N.; Kiselev, A.
2013-10-01
289
A rotating coil setup for magnetic field characterization and fiducialization of XFEL quadrupole magnets is pre- sented. The instrument allows measurement of the rel- ative position of the magnetic axis with accuracy better than 1 ?m and measurement of weak magnetic error field components. Tests and evaluation based on a FLASH quadrupole magnet are presented together with a discus- sion
A. Hedqvist; H. Danared; F. Hellberg; J. Pfluger
290
Magnetic field measurements made by means of Explorer 10 over geocentric ; distances of 1.8 to 42.6R\\/sub e\\/ on March 25experiment on the same satellite are ; referenced in interpretations. The close-in data are consistent with the ; existence of a very weak ring current below 3R\\/sub e\\/ along the trajectory, but ; alternative explanations for the field deviations are
J. P. Heppner; N. F. Ness; C. S. Scearce; T. L. Skillman
1963-01-01
291
Observations indicate that jets (i.e., charged particle beams) are emitted from the central black hole sources of active galactic nuclei and quasars. Magnetic fields are produced in e(-)-p or e(-)-e(+)-p jets when electrons (and positrons) are slowed with respect to protons in the jets. Interaction with an ambient interstellar gas or external radiation field can cause such drift velocities. Calculations
William K. Rose
1987-01-01
292
Observations indicate that jets are emitted from the central black hole sources of active galactic nuclei and quasars. Magnetic fields are produced in e--p or e--e+-p jets when electrons and positrons are slowed with respect to protons in the jets. Interaction with an ambient interstellar gas or external radiation field can cause such drift velocities. In this paper calculations for
William K. Rose
1987-01-01
293
Averaged magnetoencephalography (MEG) following somatosensory stimulation, somatosensory evoked magnetic field(s) (SEF), in humans are reviewed. The equivalent current dipole(s) (ECD) of the primary and the following middle-latency components of SEF following electrical stimulation within 80–100 ms are estimated in area 3b of the primary somatosensory cortex (SI), the posterior bank of the central sulcus, in the hemisphere contralateral to the
Ryusuke Kakigi; Minoru Hoshiyama; Motoko Shimojo; Daisuke Naka; Hiroshi Yamasaki; Shoko Watanabe; Jing Xiang; Kazuaki Maeda; Khanh Lam; Kazuya Itomi; Akinori Nakamura
2000-01-01
294
SciTech Connect
The large-scale field-aligned current system for persistent northward interplanetary magnetic field (IMF) is typically different from that for persistent southward IMF. One characteristic difference is that for northward IMF there is often a large-scale field-aligned current system poleward of the main auroral oval. This current system (the NBZ current) typically occupies a large function of the region poleward of the region 1 and 2 currents. The present paper models the high-latitude convection as a function of the large-scale field-aligned currents. In particular, a possible evolution of the convection pattern as the current system changes from a typical configuration for southward IMF to a configuration representing northward IMF (or vice versa) is presented. Depending on additional assumptions, for example about the y-component of the IMF, the convection pattern could either turn directly from a two-cell type to a four-cell type, or a three-cell type pattern could show up as an intermediate state. An interesting although rather surprising result of this study is that different ways of balancing the NBZ currents has a minor influence on the large-scale convection pattern.
Blomberg, L.G.; Marklund, G.T. (Royal Inst. of Tech., Stockholm (Sweden))
1991-04-01
295
SciTech Connect
The author presents a method for calculating the magnetic fields near a planar surface of a superconductor with a given intrinsic magnetization in the London limit. He computes solutions for various magnetic domain boundary configurations and derives relations between the spectral densities of the magnetization and the resulting field in the vacuum half space, which are useful if the magnetization can be considered as a statistical quantity and its features are too small to be resolved individually. The results are useful for analyzing and designing magnetic scanning experiments. Application to existing data from such experiments on Sr{sub 2}RuO{sub 4} show that a domain wall would have been detectable, but the magnetic field of randomly oriented small domains and small defects may have been smaller than the experimental noise level.
Bluhm, Hendrik; /Stanford U., Phys. Dept. /SLAC, SSRl
2007-06-26
296
In this paper the design of a magnetic-field-to-voltage transducer based on the giant magnetoimpedance phenomenon (GMI) is proposed, characterized by an innovative geometric configuration. In order to attain the best near-field sensibility and far-field immunity, the transducer's sensitive element and electronic circuit were planned and implemented. By thoroughly characterizing them it was possible to obtain an estimate of the transducer's
F Pompéia; L A P Gusmão; C R Hall Barbosa; E Costa Monteiro; L A P Gonçalves; F L A Machado
2008-01-01
297
The most important milestone in the field of magnetic sensors was when AMR sensors started to replace Hall sensors in many applications where the greater sensitivity of AMRs was an advantage. GMR and SDT sensors finally found applications. We also review the development of miniaturization of fluxgate sensors and refer briefly to SQUIDs, resonant sensors, GMIs, and magnetomechanical sensors.
Pavel Ripka; Michal Janosek
2010-01-01
298
National Technical Information Service (NTIS)
In order to explore the consequences of random field effects we have carried out a series of neutron scattering experiments on three prototypical diluted Ising magnets. The systems studied are Rb sub 2 Co sub 7 Mg sub 3 F sub 4 which is a model two dimens...
R. J. Birgeneau
1982-01-01
299
National Technical Information Service (NTIS)
The research efforts funded by the Uranus Data Analysis Program (UDAP) grant to the Bartol Research Institute (BRI) involved the study of magnetic field waves associated with the Uranian bow shock. Upstream wave studies are motivated as a study of the phy...
C. W. Smith M. L. Goldstein R. P. Lepping W. H. Mish H. K. Wong
1994-01-01
300
PubMed
The most important and very expensive part of a magnetic resonance imaging set-up is the magnet, which is capable of generating a constant and highly homogeneous magnetic field. Here a new MR imaging technique without the magnet is introduced. This technique uses the earth's magnetic field instead of a magnetic field created by a magnet. This new method has not yet reached the stage of medical application, but the first images obtained by MRIE (magnetic resonance imaging in the earth's field) show that the resolution is close to that expected based on sensitivity estimations. PMID:2233218
Stepisnik, J; Erzen, V; Kos, M
1990-09-01
301
SciTech Connect
The field lines of magnetic fields that depend on three spatial coordinates are shown to have a fundamentally different behavior from those that depend on two coordinates. Unlike two-coordinate cases, a flux tube in a magnetic field that depends on all three spatial coordinates that has a circular cross section at one location along the tube characteristically has a highly distorted cross section at other locations. In an ideal evolution of a magnetic field, the current densities typically increase. Crudely stated, if the current densities increase by a factor {sigma}, the ratio of the long to the short distance across a cross section of a flux tube characteristically increases by e{sup 2{sigma}}, and the ratio of the longer distance to the initial radius increases as e{sup {sigma}}. Electron inertia prevents a plasma from isolating two magnetic field structures on a distance scale shorter than c/{omega}{sub pe}, which is about 10 cm in the solar corona, and reconnection must be triggered if {sigma} becomes sufficiently large. The radius of the sun, R{sub Circled-Dot-Operator }=7 Multiplication-Sign 10{sup 10}cm is about e{sup 23} times larger, so when {sigma} Greater-Than-Or-Equivalent-To 23, two lines separated by c/{omega}{sub pe} at one location can be separated by the full scale of any magnetic structures in the corona at another. The conditions for achieving a large exponentiation, {sigma}, are derived, and the importance of exponentiation is discussed.
Boozer, Allen H. [Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027 (United States)
2012-11-15
302
PubMed
A new approach to near-field magneto-optical imaging was developed capable of visualization of in-plane magnetization of ultrathin magnetic structures. The approach relies on the magneto-optical effect specific for thin magnetic layers and employs near-field transmission measurements of longitudinal and/or transverse magneto-optical effect arising from the presence of thin film interfaces. The near-field magneto-optical contrast of in-plane domain structure of ultrathin Co film has been demonstrated in different polarization configurations. PMID:12641761
Dickson, W; Takahashi, S; Pollard, R; Atkinson, R; Zayats, A V
2003-03-01
303
Summary form only given. Magnetic field sensors have been constructed from two pieces of polarizing optical fiber fusion spliced to a single piece of low birefringence optical fiber. This configuration was mounted on a quartz bar for good geometric stability. A 780-nm laser diode pigtailed with polarizing fiber is used as the optical source for the sensor and light at
J. W. Dawson; T. W. MacDougall; E. Hernandez
1996-01-01
304
We present observations taken with the Advanced Stokes Polarimeter (ASP) in active-region plages and study the frequency distribution of the magnetic field strength (B), inclination with respect to vertical ( gamma ), azimuthal orientation ( chi ), and filling factor (f). The most common values at disk center are B = 1400 G, gamma < 10 deg, no preferred east-west orientation, and f = 15%. At disk center, there is a component of weak (<1000 G), more horizontal fields that corresponds to arching field lines connecting footpoints of different polarities. The center-to-limb variation (CLV) of the field strength shows that, close to the limb ( mu = 0.3), the field strength is reduced to 800 G from its disk-center value. This can be interpreted as a gradient of B with height in solar plages of around -3 G km-1. From this CLV study, we also deduce that magnetic field lines remain vertical for the entire range of heights involved. A similar analysis is performed for structures found in active regions that show a continuous distribution of azimuths (resembling sunspots) but that do not have a darkening in continuum. These "azimuth centers" show slightly larger values of B than normal plages, in particular at their magnetic center. Filling factors are also larger on average for these structures. The velocities in the magnetic component of active regions have been studied for both averaged Stokes profiles over the entire active region and for the spatially resolved data. The averaged profiles (more representative of high filling factor regions) do not show any significant mean velocities. However, the spatial average of Doppler velocities derived from the spatially resolved profiles (i.e., unweighted by filling factor) show a net redshift at disk center of 200 m s-1. The spatially resolved velocities show a strong dependence on filling factor. Both mean velocities and standard deviations are reduced when the filling factor increases. This is interpreted as a reduction of the p-mode amplitude within the magnetic component. Strong evidence for velocities transverse to the magnetic field lines has been found. Typical rms values are between 200 and 300 m s-1, depending on the filling factor. The possible importance of these transverse motions for the dynamics of the upper atmospheric layers is discussed. The asymmetries of the Stokes profiles and their CLV have been studied. The averaged Stokes V profiles show amplitude and area asymmetries that are positive at disk center and become negative at the limb. Both asymmetries, and for the two Fe I lines, are maximized away from disk center. The spatially resolved amplitude asymmetries show a clear dependence on filling factor: the larger the filling factor, the smaller the amplitude asymmetry. On the other hand, the area asymmetry is almost independent of the filling factor. The only observed dependence is the existence of negative area-asymmetry profiles at disk center for filling factors smaller than 0.2. Around 20% of the observed points in a given plage have negative area asymmetry. The amplitude asymmetry of Stokes V is, on the other hand, always positive. The amplitude asymmetries of the linear polarization profiles are observed to have the same sign as the Stokes V profiles. Similarly, the same CLV variation of the linear polarization amplitude asymmetries as for Stokes V has been found. The scenarios in which this similarity can exist are studied in some detail.
Martinez Pillet, V.; Lites, B. W.; Skumanich, A.
1997-01-01
305
PubMed
This review explores the dynamics of two-dimensional electrons in magnetic potentials that vary on scales smaller than the mean free path. The physics of microscopically inhomogeneous magnetic fields relates to important fundamental problems in the fractional quantum Hall effect, superconductivity, spintronics and graphene physics and spins out promising applications which will be described here. After introducing the initial work done on electron localization in random magnetic fields, the experimental methods for fabricating magnetic potentials are presented. Drift-diffusion phenomena are then described, which include commensurability oscillations, magnetic channelling, resistance resonance effects and magnetic dots. We then review quantum phenomena in magnetic potentials including magnetic quantum wires, magnetic minibands in superlattices, rectification by snake states, quantum tunnelling and Klein tunnelling. The third part is devoted to spintronics in inhomogeneous magnetic fields. This covers spin filtering by magnetic field gradients and circular magnetic fields, electrically induced spin resonance, spin resonance fluorescence and coherent spin manipulation. PMID:21393794
Nogaret, Alain
2010-06-04
306
SciTech Connect
Although only a small part of available energy in the universe is invested in magnetic fields, they are responsible for most of the continual violent activity in the cosmos. There is a single, generic explanation for the ability of bodies as different as a dense, cold planet and a tenuous hot galactic disk to generate a magnetic field. The explanation, first worked out for the earth, comes from the discipline of magnetohydrodynamics. The cosmos is filled with fluids capable of carrying electric currents. The magnetic fields entrained in these fluids are stretched and folded by the fluid motion, gaining energy in the process. In other words, the turbulent fluids function as dynamos. However, the dynamo mechanism by itself cannot account for the exceptionally strong field of some stars. Because of such gaps in information, the rival hypothesis that there are primordial fields cannot be disproved. The balance of evidence, however, indicates that the planets, sun, most stars and the galaxy function as colossal dynamos. (SC)
Parker, E.N.
1983-08-01
307
PubMed
The inhomogeneous magnetic field of a permanent-magnet planar Halbach array is used to either deflect or to specularly reflect a supersonic beam of neutral atoms. Metastable neon and helium beams are tested to experimentally evaluate the performance of this array in a range of configurations. Results are compared with numerical simulations and the device is presented as a high precision tool for the manipulation of neutral atom beams. PMID:24028135
Gardner, Jamie; Castillo-Garza, Rodrigo; Raizen, Mark G
2013-09-01
308
SciTech Connect
The authors present experimental results from the investigation of the behavior of certain magnetic liquids differeing in the degree of stability in inhomogenous magnetic fields. The growth of holding presure of sealing step at rest is reviewed and the increase of effective viscosity in inhomogeneous magnetic fields is studied. The behaviors of magnetic liquids in an inhomogeneous magnetic field are sensitive to structural changes caused by the field. Significant differences are demonstrated between magnetic liquids with the same saturation magnetization but different particle size distribution.
Anton, I.; Bika, D.; Potents, I.; Vekash, L.
1986-01-01
309
PubMed
A hot stable field-reversed configuration (FRC) has been produced in the C-2 experiment by colliding and merging two high-? plasmoids preformed by the dynamic version of field-reversed ?-pinch technology. The merging process exhibits the highest poloidal flux amplification obtained in a magnetic confinement system (over tenfold increase). Most of the kinetic energy is converted into thermal energy with total temperature (T{i}+T{e}) exceeding 0.5 keV. The final FRC state exhibits a record FRC lifetime with flux confinement approaching classical values. These findings should have significant implications for fusion research and the physics of magnetic reconnection. PMID:20867853
Binderbauer, M W; Guo, H Y; Tuszewski, M; Putvinski, S; Sevier, L; Barnes, D; Rostoker, N; Anderson, M G; Andow, R; Bonelli, L; Brandi, F; Brown, R; Bui, D Q; Bystritskii, V; Ceccherini, F; Clary, R; Cheung, A H; Conroy, K D; Deng, B H; Dettrick, S A; Douglass, J D; Feng, P; Galeotti, L; Garate, E; Giammanco, F; Glass, F J; Gornostaeva, O; Gota, H; Gupta, D; Gupta, S; Kinley, J S; Knapp, K; Korepanov, S; Hollins, M; Isakov, I; Jose, V A; Li, X L; Luo, Y; Marsili, P; Mendoza, R; Meekins, M; Mok, Y; Necas, A; Paganini, E; Pegoraro, F; Pousa-Hijos, R; Primavera, S; Ruskov, E; Qerushi, A; Schmitz, L; Schroeder, J H; Sibley, A; Smirnov, A; Song, Y; Sun, X; Thompson, M C; Van Drie, A D; Walters, J K; Wyman, M D
2010-07-22
310
We construct aligned and unaligned stationary perturbation configurations in a composite system of stellar and coplanarly magnetized gaseous singular isothermal discs (SIDs) coupled by gravity. This study extends recent analyses on (magnetized) SIDs by Shu et al., Lou and Lou & Shen. By this model, we intend to provide a conceptual framework to gain insights for multiwavelength large-scale structural observations of disc galaxies. Both SIDs are approximated to be razor thin and are in a self-consistent axisymmetric background equilibrium with power-law surface mass densities and flat rotation curves. The gaseous SID is embedded with a coplanar azimuthal magnetic field B?(r) of a radial scaling r-1/2 that is not force-free. In comparison with the SID problems studied earlier, there are three possible classes of stationary solutions allowed by more dynamic freedoms. To identify physical solutions, we explore parameter space involving three dimensionless parameters: ratio ? of Alfvén speed to sound speed in the magnetized gaseous SID; ratio ? of the square of the stellar velocity dispersion to the gas sound speed; and ratio ? of the surface mass densities of the two SIDs. For both aligned and unaligned spiral cases with azimuthal periodicities |m| >= 2, one of the three solution branches is always physical, while the other two branches might become invalid when ? exceeds certain critical values. For the onset criteria from an axisymmetric equilibrium to aligned secular bar-like instabilities, the corresponding ratio, which varies with ?, ? and ?, may be considerably lower than the oft-quoted value of , where is the total kinetic energy, is the total gravitational potential energy and is the total magnetic energy. For unaligned spiral cases, we examine marginal instabilities for axisymmetric (|m| = 0) and non-axisymmetric (|m| > 0) disturbances. The resulting marginal stability curves differ from the previous ones. The case of a composite partial magnetized SID system is also investigated to include the gravitational effect of an axisymmetric dark matter halo on the SID equilibrium. We further examine the phase relationship among the mass densities of the two SIDs and azimuthal magnetic field perturbation. Our exact global perturbation solutions and critical points are valuable for testing numerical magnetohydrodynamic codes. For galactic applications, our model analysis contains more realistic elements and offers useful insights into the structures and dynamics of disc galaxies consisting of stars and magnetized gas.
Lou, Yu-Qing; Zou, Yue
2004-06-01
311
Ferrofluids are colloidal suspensions of magnetic nanoparticles in carrier liquids. Being both magnetic and a liquid, they are readily maneuvered from a distance using magnetic fields. When functionalized with different antibodies or medicinal compounds, the ferrofluid can be used for various purposes, e.g., to detect bacteria or for targeted drug delivery. We have considered a coflow where two fluids are separated by a vertical surface parallel to the direction of gravity. For simplicity the flow is assumed to be inviscid and incompressible. We have investigated two configurations depending on the position of the magnet relative to the channel. When the magnet is placed adjacent to the vertical wall along which the magnetic fluid is flowing, the magnetic fluid stays close to the wall unless perturbed by the shear due to a velocity difference. It results in a very stable system. In the second case, the magnet is placed close to the wall along which the non-magnetic fluid flows. The magnetic fluid gets attracted towards the magnet and tries to flow toward it when it gets resisted by the non-magnetic fluid. This configuration is inherently unstable and responds to small perturbations. The surface tension force resists the perturbation of smaller wavelengths. The relative effects of different forces are characterized by magnetic pressure number, Weber number and magnetic Weber number.
de, Anindya; Puri, Ishwar
2007-11-01
312
SciTech Connect
A field-reversed configuration (FRC) gains angular momentum over time, eventually resulting in an n=2 rotational instability (invariant under rotation by {pi}) terminating confinement. To study this, a laser interferometer probes the time history of line integrated plasma density along eight chords of the high-density ({approx}10{sup 17} cm{sup -3}) field-reversed configuration experiment with a liner. Abel and tomographic inversions provide density profiles during the FRC's azimuthally symmetric phase, and over a period when the rotational mode has saturated and rotates with a roughly fixed profile, respectively. During the latter part of the symmetric phase, the FRC approximates a magnetohydrodynamic (MHD) equilibrium, allowing the axial magnetic-field profile to be calculated from pressure balance. Basic FRC properties such as temperature and poloidal flux are then inferred. The subsequent two-dimensional n=2 density profiles provide angular momentum information needed to set bounds on prior values of the stability relevant parameter {alpha} (rotational to ion diamagnetic drift frequency ratio), in addition to a view of plasma kinematics useful for benchmarking plasma models of higher order than MHD.
Ruden, E. L.; Zhang, Shouyin; Intrator, T. P.; Wurden, G. A. [Air Force Research Laboratory, Directed Energy Directorate, 3550 Aberdeen Avenue SE, Kirtland AFB, New Mexico, 87117-5776 (United States); Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
2006-12-15
313
In this work we present anisotropic light scattering measurements of local deformation in systems composed by a dispersion of nanometric magnetic particles in a polymer gel, or ferrogel, in the presence of uniform magnetic fields. Two experimental configurations were used in which the scattering vector q was parallel and perpendicular to the magnetic induction B. We have seen that the scattered intensity increases in the parallel configuration as B increases and decreases in the perpendicular configuration. These intensity variations are opposite to those observed using X-rays (Phys. Rev. E 67 (2003) 021504), where the length scales q-1 are comparable to the particle size. In all experiments the variation of the scattered intensity closely follows a Langevin type function. The anisotropic scattered light intensity variation was related to long-range deformations in the polymer matrix.
Teixeira, Alvaro V.; Licinio, Pedro
2005-03-01
314
We describe studies of nuclear magnetic resonance (NMR) spectroscopy and magnetic resonance imaging (MRI) of liquid samples at room temperature in microtesla magnetic fields. The nuclear spins are prepolarized in a strong transient field. The magnetic signals generated by the precessing spins, which range in frequency from tens of Hz to several kHz, are detected by a low-transition temperature dc
R. McDermott; N. Kelso; S. K. Lee; M. MöBetale; M. Mück; W. Myers; B. ten Haken; H. C. Seton; A. H. Trabesinger; A. Pines; J. Clarke
2004-01-01
315
Frustrated magnets in high magnetic field have a long history of offering beautiful surprises to the patient investigator. Here we present the results of extensive classical Monte Carlo simulations of a variety of models of two dimensional magnets in magnetic field, together with complementary spin wave analysis. Striking results include (i) a massively enhanced magnetocaloric effect in antiferromagnets bordering on
L. Seabra; N. Shannon; P. Sindzingre; T. Momoi; B. Schmidt; P. Thalmeier
2009-01-01
316
In most of the ternary (and higher-order) ferromagnetic shape memory alloys (FSMAs) with compositions close to the A2BC stoichiometry, the austenite phase exhibits L21-type ordering. Recent investigations of the Co-Ni-Ga FSMA system, however, suggest that the austenite phase has B2-type ordering, although definite confirmation remains elusive. In this work, we present a theoretical investigation of the effect of configurational order on the magnetic properties of the ordered (L21) and disordered (B2) FSMA Co2NiGa. Through the use of calculations based on density functional theory, we predict the structural and magnetic properties (including magnetic exchange constants) of ordered and disordered Co2NiGa alloys. We validate our calculation of the magnetic exchange constants by extracting the Curie temperatures of the austenite and martensite structures and comparing them to experimental results. By constructing a q-state Potts magnetic Hamiltonian and through the use of lattice Monte Carlo simulation, we predict the finite-temperature behavior of the magnetization and magnetic susceptibility as well as the magnetic specific heat and entropy. The role of configurational order in the magnetic properties of the phases involved in the martensitic phase transformation is discussed, and predictions of the magnitude of the magnetic contributions to the transformation entropy are presented. The calculations are compared to experimental information available in the literature as well as experiments performed by the authors. It is concluded that in FSMAs magnetism plays a fundamental role in determining the relative stability of the austenite and martensite phases, which in turn determines the martensitic transformation temperature Ms, irrespective of whether magnetic fields are used to drive the transformation.
Singh, Navdeep; Dogan, Ebubekir; Karaman, Ibrahim; Arróyave, Raymundo
2011-11-01
317
The interaction of Rotating Magnetic Fields (RMF) with plasmas is a fundamental plasma physics problem with implications to fusion related Field-Reversed Configurations (FRC), space propulsion, astronaut protection from cosmic rays in long interstellar travel, control of the energetic population in the radiation belts and near zone processes in pulsar magnetospheres. In this paper we report recent experiments on the generation of whistler waves with a new type RMF-based antenna. The experiments were conducted on UCLA's Large Plasma Device (LAPD). The Rotating Magnetic Field (RMF) is created using poly-phased loop antennas. A number of parameter combinations, e.g. plasma density, background magnetic field, and driving current, were used. It was found that RMF created by a two phase-delayed loop antenna drives significant currents along the ambient magnetic field. The measured amplitude of induced wave field was proportional to the square-root of the plasma density. The spatial decay rate for the wave perturbation across the background magnetic field was found to scale with the plasma skin depth. A small amplitude second harmonic was also measured. The paper will also present analytic and simulation results that account for the experimental results; in particular, the scaling of the induced magnetic field as a function of the RMF and plasma parameters and the spatial decay rate of magnetic field. Applications of RMF as an efficient radiation source of plasma waves in space plasmas will be discussed. This work was sponsored by ONR MURI Grant 5-28828
Karavaev, A.; Papadopoulos, K.; Shao, X.; Sharma, A. S.; Gigliotti, A.; Gekelman, W.; Pribyl, P.; Vincena, S.
2008-12-01
318
The Helioseismic and Magnetic Imager (HMI) will provide frequent full-disk magnetic field data after launch of the Solar Dynamics Observatory (SDO), currently scheduled for fall 2009. 16 megapixel line-of-sight magnetograms (Blos) will be recorded every 45 seconds. A full set of polarized filtergrams needed to determine the vector magnetic field requires 90 seconds. Quick-look data will be available within a few minutes of observation. Quick-look space weather and browse products must have identified users, and the list currently includes full disk magnetograms, feature identification and movies, 12-minute disambiguated vector fields in active region patches, time evolution of AR indices, synoptic synchronic frames, potential and MHD model results, and 1 AU predictions. A more complete set of definitive science data products will be offered about a day later and come in three types. "Pipeline” products, such as full disk vector magnetograms, will be computed for all data on an appropriate cadence. A larger menu of "On Demand” products, such as Non-Linear Force Free Field snapshots of an evolving active region, will be produced whenever a user wants them. Less commonly needed "On Request” products that require significant project resources, such as a high resolution MHD simulation of the global corona, will be created subject to availability of resources. Further information can be found at the SDO Joint Science Operations Center web page, jsoc.stanford.edu
Hoeksema, Jon Todd; Liu, Y.; Schou, J.; Scherrer, P.; HMI Science Team
2009-05-01
319
We study static neutron stars with poloidal magnetic fields and a simple class of electric current distributions consistent with the requirement of stationarity. For this class of electric current distributions, we find that magnetic fields are too large for static configurations to exist when the magnetic force pushes a sufficient amount of mass off-center that the gravitational force points outward near the origin in the equatorial plane. (In our coordinates an outward gravitational force corresponds to ?lngtt/?r>0, where t and r are respectively time and radial coordinates and gtt is coefficient of dt2 in the line element.) For the equations of state (EOSs) employed in previous work, we obtain configurations of higher mass than had been reported; we also present results with more recent EOSs. For all EOSs studied, we find that the maximum mass among these static configurations with magnetic fields is noticeably larger than the maximum mass attainable by uniform rotation, and that for fixed values of baryon number the maximum mass configurations are all characterized by an off-center density maximum.
Cardall, Christian Y.; Prakash, Madappa; Lattimer, James M.
2001-06-01
320
This article describes both the setup and the use of a system for magnetic resonance imaging (MRI) in the Earth's magnetic field. Phase instability caused by temporal fluctuations of Earth's field can be successfully improved by using a reference signal from a separate Earth's field nuclear magnetic resonance (NMR) spectrometer\\/magnetometer. In imaging, it is important to correctly determine the phase
Ales Mohoric; Gorazd Planinsic; Miha Kos; Andrej Duh; Janez Stepisnik
2004-01-01
321
PubMed
The isomagnetic maps of normal subjects and patients with right and left atrial overloading were recorded to determine the characteristic features of the magnetic field of atrial depolarization. The isomagnetic maps examined in this study indicated the instantaneous current source, which specifically localizes the current sources due to the right and left atria, respectively. The magnetic field recorded with a second derivative gradiometer clearly detected the cardiac current source from the right atrium, which is located close to the anterior chest wall, thus this method improved the diagnostic sensitivity for right atrial overloading. In patients with left atrial overloading, the isomagnetic map showed multiple dipoles due to the right and left atria, respectively, which are difficult to be detected by the electrocardiogram or isopotential map. These results suggest that the magnetocardiogram provides useful information on the current source to supplement information obtained by the conventional electrocardiogram. PMID:2978585
Takeuchi, A; Watanabe, K; Katayama, M; Nomura, M; Nakaya, Y; Mori, H
322
SciTech Connect
In a ferromagnet{endash}normal-metal{endash}ferromagnet trilayer, a current flowing perpendicularly to the layers creates a torque on the magnetic moments of the ferromagnets. When one of the contacts is superconducting, the torque not only favors parallel or antiparallel alignment of the magnetic moments, as is the case for two normal contacts, but can also favor a configuration where the two moments are perpendicular. In addition, whereas the conductance for parallel and antiparallel magnetic moments is the same, signalling the absence of giant magnetoresistance in the usual sense, the conductance is greater in the perpendicular configuration. Thus, a negative magnetoconductance is predicted, in contrast with the usual giant magnetoresistance.
Waintal, X.; Brouwer, P. W.
2001-06-01
323
SciTech Connect
The appearance of hole currents in tokamaks seems to be very important in plasma confinement and on-set of instabilities, and this paper is devoted to study the topology changes of poloidal magnetic fields in tokamaks. In order to determine these fields different models for current profiles can be considered. It seems to us, that one of the best analytic descriptions is given by V. Yavorskij et al., which has been chosen for the calculations here performed. Suitable analytic equations for the family of magnetic field surfaces with triangularity and Shafranov shift are written down here. The topology of the magnetic field determines the amount of trapped particles in the generalized mirror type magnetic field configurations. Here it is found that the number of maximums and minimums of Bp depends mainly on triangularity, but the pattern is also depending of the existence or not of hole currents. Our calculations allow comparing the topology of configurations of similar parameters, but with and without whole currents. These differences are study for configurations with equal ellipticity but changing the triangularity parameters. Positive and negative triangularities are considered and compared between them.
Puerta, Julio; Martin, Pablo; Castro, Enrique [Universidad Simon Bolivar, Departamento de Fisica, Plasma Physics Laboratory, Caracas (Venezuela, Bolivarian Republic of)
2009-07-26
324
An analysis of the vector magnetic field in the delta-configurations of two complex sunspot groups is presented, noting several characteristics identified in the delta-configurations. The observations of regions 2469 (S12E80) and 2470 (S21E83) took place in May, 1980 with a vector magnetograph, verified by optical viewing. Longitudinal magnetic field plots located the delta-configurations in relation to the transverse field neutral line. It is shown that data on the polarization yields qualitative information on the magnetic field strengths, while the azimuth of the transverse field can be obtained from the relative intensities of linear polarization measurements aligned with respect to the magnetograph analyses axis at 0 and 90 deg, and at the plus and minus 45 deg positions. Details of the longitudinal fields are discussed. A strong, sheared transverse field component is found to be a signature of strong delta. A weak delta is accompanied by a weak longitudinal gradient with an unsheared transverse component of variable strength.
Patty, S. R.
325
SciTech Connect
Magnetic field-structured-composites (FSCs) are made by structuring magnetic particle suspensions in uniaxial or biaxial (e.g. rotating) magnetic fields, while polymerizing the suspending resin. A uniaxial field produces chain-like particle structures, and a biaxial field produces sheet-like particle structures. In either case, these anisotropic structures affect the measured magnetic hysteresis loops, with the magnetic remanence and susceptibility increased significantly along the axis of the structuring field, and decreased slightly orthogonal to the structuring field, relative to the unstructured particle composite. The coercivity is essentially unaffected by structuring. We present data for FSCs of magnetically soft particles, and demonstrate that the altered magnetism can be accounted for by considering the large local fields that occur in FSCs. FSCS of magnetically hard particles show unexpectedly large anisotropies in the remanence, and this is due to the local field effects in combination with the large crystalline anisotropy of this material.
Anderson, Robert A.; Martin, James E.; Odinek, Judy; Venturini, Eugene
1999-06-24
326
National Technical Information Service (NTIS)
Spatiotemporal patterns of somatosensory evoked magnetic fields to stimulation of upper and lower limb nerves were examined in healthy humans. The studies summarized here provide the first magnetic field maps over the primary foot projection area after li...
J. Huttunen
1987-01-01
327
Metal complexes and solids were synthesized and subjected to photoexcitation measurements under the influence of externally applied magnetic fields. The photoluminescence of complexes of rhodium (I) and iridium (I) displayed both field induced emission bands and a many fold shortening of the excited state lifetime. Both the decay rates and the induced emission band intensities showed a quadratic dependence on the applied field. A several fold shortening of the phosphorescence from the octaphosphitoplatinum (II) anion under an applied field (50 T) was also observed. Spectroscopic studies of several bis (N-heterocyclic) complexes of copper (I) were also concluded and complete group theoretic assignments of the charge transfer excited states were made. The technique of Thermal Modulation was perfected and applied to the study of the exited states of transition metal complexes with near degenerate emitting states.
Crosby, G. A.
1989-08-01
328
We have run plots of artificial data, which mimic solar magnetograms, through standard algorithms to critique several results reported in the literature. In studying correlation algorithms, we show that the differences in the profiles for the differential rotation of the photospheric magnetic field stem from different methods of averaging. We verify that the lifetimes of small magnetic features, or of small patterns of these features in the large-scale background field, are on the order of months, rather than a few days. We also show that a meridional flow which is cycle dependent creates an artifact in the correlation-determined magnetic rotation which looks like a torsional oscillation; and we compare this artifact to the torsional patterns that have been reported. Finally, we simulate the time development of a large-scale background field created solely from an input of artifical, finite-lifetime 'sunspot' bipoles. In this simulation, we separately examine the effects of differential rotation, meridional flow and Brownian motion (random walk, which we use rather than diffusion), and the inclination angles of the sunspot bipoles (Joy's law). We find, concurring with surface transport equation models, that a critical factor for producing the patterns seen on the Sun is the inclination angle of the bipolar active regions. This work was supported by NSF grant 9416999.
Smith, A. A.; Snodgrass, H. B.
1999-05-01
329
Liquids (~7 neutron mean free paths thick), with certain restrictions, can probably be used in magnetic fusion designs between the burning plasma and the structural materials of the fusion power core. If this works there would be a number of profound advantages: a cost of electricity lower by as much as a factor of 2; removal of the need to
R. W. Moir
1997-01-01
330
A novel resonant magnetic sensor based on the combination of a mechanical resonator and a magnetic field concentrator with two gaps is reported. In contrast to previous Lorentz force based resonant magnetic sensors, a high sensitivity is achieved without modulated driving current and complex feedback electronics. Furthermore, compared to magnetic moment based resonant magnetic sensors, the new concept requires no
S. Brugger; P. Simon; O. Paul
2006-01-01
331
SciTech Connect
We investigate the effect of a magnetic field on cold dense quark matter using an effective model with four-Fermi interactions. We find that the gap parameters representing the predominant pairing between the different quark flavors show oscillatory behavior as a function of the magnetic field. We point out that due to electric and color neutrality constraints the magnetic fields as strong as presumably existing inside magnetars might induce significant deviations from the gap structure at a zero magnetic field.
Fukushima, Kenji [RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973 (United States); Warringa, Harmen J. [Department of Physics, Bldg. 510A, Brookhaven National Laboratory, Upton, New York 11973 (United States)
2008-01-25
332
The work is devoted to the geometrical configuration of permanent magnets on the basis of opposing geometrically linear assemblies (e.g. Halbach arrays) for the generation of strong magnetic fields, which have been theoretically modeled and experimentally verified. The implementation of these opposing assemblies using NdFeB magnets of a total weight of 3.75 kg provided a value of magnetic induction in the middle of an air gap of a width of 20 mm that was higher by 56% in comparison with the simplest possible design. When the air gap width was 3 mm, the induction reached a value of 2.16 T, which represents an increase of more than 100%. Simultaneously, however, unlike in the simplest possible parallel configuration, opposing Halbach assemblies have shown, in the middle of an air gap, a significant decrease of the magnetic induction values when passing from the middle of the assemblies in the direction parallel to the x-axis.
Žežulka, Václav; Pištora, Jaromír; Les?ák, Michal; Straka, Pavel; Ciprian, Dalibor; Foukal, Jaroslav
2013-11-01
333
The magnetic field gradients of magnetic stripe cards, which are developed for classifying magnetic particles used in magnetic particle inspections, have been measured using a magnetic force microscope (MFM). The magnetic force exerted on a MFM probe by the stray field emanating from the card was measured to determine the field gradients. The results are in good agreement with the field gradients estimated from the magnetizing field strengths used in the encoding process. .
Lo, C. C. H.; Leib, J.; Jiles, D. C.; Chedister, W. C.
2002-05-01
334
We have combined time-of-flight neutron Laue diffraction and pulsed high magnetic fields at the Spallation Neutron Source to study the phase diagram of the multiferroic material MnWO4. The control of the field-pulse timing enabled an exploration of magnetic Bragg scattering through the time dependence of both the neutron wavelength and the pulsed magnetic field. This allowed us to observe several magnetic Bragg peaks in different field-induced phases of MnWO4 with a single instrument configuration. These phases were not previously amenable to neutron diffraction studies due to the large fields involved.
Nojiri, H.; Yoshii, S.; Yasui, M.; Okada, K.; Matsuda, M.; Jung, J.-S.; Kimura, T.; Santodonato, L.; Granroth, G. E.; Ross, K. A.; Carlo, J. P.; Gaulin, B. D.
2011-06-01
335
PubMed
We have combined time-of-flight neutron Laue diffraction and pulsed high magnetic fields at the Spallation Neutron Source to study the phase diagram of the multiferroic material MnWO(4). The control of the field-pulse timing enabled an exploration of magnetic Bragg scattering through the time dependence of both the neutron wavelength and the pulsed magnetic field. This allowed us to observe several magnetic Bragg peaks in different field-induced phases of MnWO(4) with a single instrument configuration. These phases were not previously amenable to neutron diffraction studies due to the large fields involved. PMID:21770542
Nojiri, H; Yoshii, S; Yasui, M; Okada, K; Matsuda, M; Jung, J-S; Kimura, T; Santodonato, L; Granroth, G E; Ross, K A; Carlo, J P; Gaulin, B D
2011-06-08
336
This review concerns the origin and the possible effects of magnetic fields in the early Universe. We start by providing the reader with a short overview of the current state of the art of observations of cosmic magnetic fields. We then illustrate the arguments in favor of a primordial origin of magnetic fields in the galaxies and in the clusters
Dario Grasso; Hector R. Rubinstein
2001-01-01
337
SciTech Connect
We study limits on a primordial magnetic field arising from cosmological data, including that from big bang nucleosynthesis, cosmic microwave background polarization plane Faraday rotation limits, and large-scale structure formation. We show that the physically relevant quantity is the value of the effective magnetic field, and limits on it are independent of how the magnetic field was generated.
Kahniashvili, Tina [McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213 (United States); Department of Physics, Laurentian University, Ramsey Lake Road, Sudbury, Ontario P3E 2C (Canada); Abastumani Astrophysical Observatory, Ilia State University, 2A Kazbegi Ave, Tbilisi, GE-0160 (Georgia); Tevzadze, Alexander G. [Abastumani Astrophysical Observatory, Ilia State University, 2A Kazbegi Ave, Tbilisi, GE-0160 (Georgia); Faculty of Exact and Natural Sciences, Tbilisi State University, 1 Chavchavadze Avenue, Tbilisi, GE-0128 (Georgia); Sethi, Shiv K. [McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213 (United States); Raman Research Institute, Sadashivanagar, Bangalore 560080 (India); Pandey, Kanhaiya [Raman Research Institute, Sadashivanagar, Bangalore 560080 (India); Ratra, Bharat [Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, Kansas 66506 (United States)
2010-10-15
338
Experiments were performed at the Nevada Terawatt Facility to investigate the plasma penetration across an externally applied magnetic field. In experiment, a short-pulse laser ablates a polyethylene laser target, producing a plasma which interacts with an external magnetic field. The mechanism which allows the plasma to penetrate the applied magnetic field in experiment will be discussed.
Plechaty, C.; Presura, R.; Wright, S.; Neff, S.; Haboub, A.
2009-08-01
339
Radio observations of nearby spiral galaxies have tremendously enhanced our knowledge of their global magnetic field distributions. Recent theoretical developments in the area of dynamos have also helped in the interpretation of magnetic field data in spiral galaxies. When it comes to the magnetic field in the Milky Way galaxy, our position in the Milky Way's galactic disk hinders our
J. P. Vallee
1996-01-01
340
Recent controversy over 60 Hz magnetic fields has heightened public awareness of overhead transmission lines. As a result, there is increasing motivation to study the magnetic fields form transmission lines. The most cost effective means to conduct research into transmission line magnetic fields is with computer or reduced-scale line models. However, from the standpoint of public perception and acceptance, it
B. A. Clairmont; G. B. Johnson; J. H. Dunlap
1992-01-01
341
We study limits on a primordial magnetic field arising from cosmological data, including that from big bang nucleosynthesis, cosmic microwave background polarization plane Faraday rotation limits, and large-scale structure formation. We show that the physically relevant quantity is the value of the effective magnetic field, and limits on it are independent of how the magnetic field was generated.
Kahniashvili, Tina; Tevzadze, Alexander G.; Sethi, Shiv K.; Pandey, Kanhaiya; Ratra, Bharat
2010-10-01
342
Results and new progress of the origin and evolution of pulsar magnetic fields are reviewed. Lots of models about how such strong magnetic fields were generated, mainly two kinds of structures were proposed for initial magnetic fields: fields confined in the cores and fields confined in the crusts of neutron stars. No consensus has been reached on whether the magnetic fields decay or not, despite some observational evidence for the evolution of magnetic fields. The discrepancy between characteristic ages and kinematic ages indicates that the magnetic fields decay exponentially. On the other hand, the braking indices of several young pulsars and the comparison between pulsar characteristic ages and the ages of associated supernova remnants suggest that the magnetic fields of young pulsars grow like a power-law. Pulsar population synthesis is one of the most important methods to investigate the evolution of magnetic fields. Many simulations show that if magnetic fields do decay exponentially, the e-folding decay time should be 100 Myr or longer. The numerical calculations of the Ohmic decay in the crust indicate that the scenario of exponential decay is oversimple, and the evolution could be divided into four possible phases approximately: exponential decay, no decay, power-law decay and exponential decay again. The model of magnetic fields expulsion induced by spin-down suggests that the magnetic fields decay only in a period between 107yr and 108yr.
Sun, Xiaohui; Han, Jinlin
2002-06-01
343
SciTech Connect
It has long been known that magnetic reconnection plays a fundamental role in a variety of solar events. Although mainly invoked in flare problems, large-scale loops interconnecting active regions, evolving coronal hole boundaries, the solar magnetic cycle itself, provide different evidence of phenomena which involve magnetic reconnection. A further example might be given by the magnetic field rearrangement which occurs after the eruption of a prominence. Since most often a prominence reforms after its disappearance and may be observed at about the same position it occupied before erupting, the magnetic field has to undergo a temporary disruption to relax back, via reconnection, to a configuration similar to the previous one. The above sequence of events is best observable in the case of two-ribbon (2-R) flares but most probably is associated with all filament eruptions. Even if the explanation of the magnetic field rearrangement after 2-R flares in terms of reconnection is generally accepted, the lack of a three-dimensional model capable of describing the field reconfiguration, has prevented, up to now, a thorough analysis of its topology as traced by H..cap alpha../x-ray loops. The purpose of the present work is to present a numerical technique which enables one to predict and visualize the reconnected configuration, at any time t, and therefore allows one to make a significant comparison of observations and model predictions throughout the whole process. 5 refs., 3 figs.
Kopp, R.A.; Poletto, G.
1986-01-01
344
SciTech Connect
A new field-reversed configuration (FRC) formation technique is described, where a spheromak transitions to a FRC with inductive current drive. The transition is accomplished only in argon and krypton plasmas, where low-n kink modes are suppressed; spheromaks with a lighter majority species, such as neon and helium, either display a terminal tilt-mode, or an n=2 kink instability, both resulting in discharge termination. The stability of argon and krypton plasmas through the transition is attributed to the rapid magnetic diffusion of the currents that drive the kink-instability. The decay of helicity during the transition is consistent with that expected from resistivity. This observation indicates a new scheme to form a FRC plasma, provided stability to low-n modes is maintained, as well as a unique situation where the FRC is a preferred state.
Gerhardt, S. P.; Belova, E. V.; Yamada, M.; Ji, H.; Ren, Y.; McGeehan, B. [Princeton Plasma Physics Laboratory, Plainsboro, New Jersey 08543 (United States); Inomoto, M. [Osaka University, Osaka 565-0871 (Japan)
2008-03-15
345
SciTech Connect
A new field-reversed configuration (FRC) formation technique is described, where a spheromak transitions to a FRC with inductive current drive. The transition is accomplished only in argon and krypton plasmas, where low-n kink modes are suppressed; spheromaks with a lighter majority species, such as neon and helium, either display a terminal tilt-mode, or an n=2 kink instability, both resulting in discharge termination. The stability of argon and krypton plasmas through the transition is attributed to the rapid magnetic diffusion of the currents that drive the kink-instability. The decay of helicity during the transition is consistent with that expected from resistivity. This observation indicates a new scheme to form a FRC plasma, provided stability to low-n modes is maintained, as well as a unique situation where the FRC is a preferred state.
S.P. Gerhardt, E.V. Belova, M. Yamada, H. Ji, Y. Ren, B. McGeehan, and M. Inomoto
2008-06-12
346
We present a Frank-Oseen elasticity theory for the shape and structure of deformable nematic droplets with homeotropic surface anchoring in the presence of a magnetic field. Inspired by recent experimental observations, we focus on the case where the magnetic susceptibility is negative, and find that small drops have a lens shape with a homogeneous director field for any magnetic-field strength, whereas larger drops are spherical and have a radial director field, at least if the magnetic field is weak. For strong magnetic fields the hedgehog configuration transforms into a split-core line defect that, depending on the anchoring strength, can be accompanied by an elongation of the tactoid itself. We present a three-dimensional phase diagram that shows the tactoid shape and director field for a given anchoring strength, tactoid size, and magnetic-field strength. Our findings rationalize the different shapes and structures that recently have been observed experimentally for nematic droplets found in dispersions of gibbsite platelets in two types of solvent.
Otten, Ronald H. J.; van der Schoot, Paul
2012-10-01
347
The equilibrium configuration of very small magnetic flux tubes in an intergranular environment automatically produces kilogauss magnetic field strengths. We argue that such a process takes place in the Sun and complements the convective collapse (CC), which is traditionally invoked to explain the formation of kilogauss magnetic concentrations in the solar photosphere. In particular, it can concentrate the very weak magnetic fluxes revealed by the new IR spectropolarimeters, for which the operation of the CC may have difficulty. As part of the argument, we show the existence of solar magnetic features of very weak fluxes yet concentrated magnetic fields (some 3×1016 Mx and 1500 G).
Sánchez Almeida, J.
2001-08-01
348
PubMed
The relationship between magnetic order and ferroelectric properties has been investigated for MnWO4 with a long-wavelength magnetic structure. Spontaneous electric polarization is observed in an elliptical spiral spin phase. The magnetic-field dependence of electric polarization indicates that the noncollinear spin configuration plays a key role for the appearance of the ferroelectric phase. An electric polarization flop from the b direction to the a direction has been observed when a magnetic field above 10 T is applied along the b axis. This result demonstrates that an electric polarization flop can be induced by a magnetic field in a simple system without rare-earth 4f moments. PMID:17026396
Taniguchi, K; Abe, N; Takenobu, T; Iwasa, Y; Arima, T
2006-08-30
349
Magnetic field measurements are very valuable, as they provide constraints on the interior of the telluric planets and Moon. The Earth possesses a planetary scale magnetic field, generated in the conductive and convective outer core. This global magnetic field is superimposed on the magnetic field generated by the rocks of the crust, of induced (i.e. aligned on the current main field) or remanent (i.e. aligned on the past magnetic field). The crustal magnetic field on the Earth is very small scale, reflecting the processes (internal or external) that shaped the Earth. At spacecraft altitude, it reaches an amplitude of about 20 nT. Mars, on the contrary, lacks today a magnetic field of core origin. Instead, there is only a remanent magnetic field, which is one to two orders of magnitude larger than the terrestrial one at spacecraft altitude. The heterogeneous distribution of the Martian magnetic anomalies reflects the processes that built the Martian crust, dominated by igneous and cratering processes. These latter processes seem to be the driving ones in building the lunar magnetic field. As Mars, the Moon has no core-generated magnetic field. Crustal magnetic features are very weak, reaching only 30 nT at 30-km altitude. Their distribution is heterogeneous too, but the most intense anomalies are located at the antipodes of the largest impact basins. The picture is completed with Mercury, which seems to possess an Earth-like, global magnetic field, which however is weaker than expected. Magnetic exploration of Mercury is underway, and will possibly allow the Hermean crustal field to be characterized. This paper presents recent advances in our understanding and interpretation of the crustal magnetic field of the telluric planets and Moon.
Langlais, Benoit; Lesur, Vincent; Purucker, Michael E.; Connerney, Jack E. P.; Mandea, Mioara
2010-05-01
350
A nuclear magnetic resonance apparatus for experiments in pulsed high magnetic fields is described. The magnetic field pulses created together with various magnet coils determine the requirements such an apparatus has to fulfill to be operated successfully in pulsed fields. Independent of the chosen coil it is desirable to operate the entire experiment at the highest possible bandwidth such that a correspondingly large temporal fraction of the magnetic field pulse can be used to probe a given sample. Our apparatus offers a bandwidth of up to 20 MHz and has been tested successfully at the Hochfeld-Magnetlabor Dresden, even in a very fast dual coil magnet that has produced a peak field of 94.2 T. Using a medium-sized single coil with a significantly slower dependence, it is possible to perform advanced multi-pulse nuclear magnetic resonance experiments. As an example we discuss a Carr-Purcell spin echo sequence at a field of 62 T.
Meier, Benno; Kohlrautz, Jonas; Haase, Jürgen; Braun, Marco; Wolff-Fabris, Frederik; Kampert, Erik; Herrmannsdörfer, Thomas; Wosnitza, Joachim
2012-08-01
351
Presently Mars possesses no intrinsic magnetic field; rather its crust exhibits strong remanent magnetization primarily in the Southern Highlands. The deficiency of magnetization surrounding volcanic provinces and impact basins on Mars is attributed to evidence suggesting that the crust gained its magnetic remanence early on via an internal dynamo. This dynamo is believed to have become extinct by the time of the last major impacts. Measurements taken by Mars Global Surveyor (MGS) have been used to create a new map of Mars' crustal magnetic field. We present an analysis of these data in conjunction with topographical data taken from the Mars Orbiter Laser Altimeter (MOLA) to determine if magnetization in Mars' southern regions correlates with surface features displayed on topographic maps. MGS and MOLA data were used to identify and study a region of intense magnetic field beneath a 1500 km section of an impact basin in the western hemisphere of the Southern Highlands. In conjunction with the development of models and intensity plots for the radial component of this field, analysis of the possible shape, configuration and composition of the magnetic material beneath the crater was performed. Our models showed that the magnetic signature beneath the impact basin was produced by two adjacent blocks of magnetic material within the Martian crust. We found that the blocks were most likely rectangular in shape and were relatively closely spaced. They also possessed properties similar to those of stainless steel permanent magnets with magnetization directions of -90 degrees, and -45 degrees, respectively. The results of this research will contribute to future studies of Mars, specifically of its present magnetic state, magnetic history, and impact record. This research was made possible via funding from the North Carolina Space Grant Consortium.
Quick, L. C.; Acuna, M. H.; Connerney, J. E. P.
2005-12-01
352
An original permanent magnet flux source is designed in order to generate a magnetic field of several Tesla. The magnet configuration and discretization of the structure are optimized with the help of numerical simulation software developed at LEG (DIPOLE-3D, FLUX2D & FLUX3D). The model of spheroidal flux source presented in the paper creates a field in excess of 4.3 T
F. Bloch; O. Cugat; G. Meunier; J. C. Toussaint
1998-01-01
353
We present the basic steps for the study of the linear near field absorption spectra of semiconductor quantum dots under magnetic field of variable orientation. We show that the application of the magnetic field alone is sufficient to induce -increasing the spot illuminated by the near field probe- interesting features to the absorption spectra.
Anna Zora; Constantinos Simserides; Georgios Triberis
2005-01-01
354
We present the basic steps for the study of the linear near field absorption spectra of semiconductor quantum dots under magnetic field of variable orientation. We show that the application of the magnetic field alone is sufficient to induce -increasing the spot illuminated by the near field probe- interesting features to the absorption spectra.
Anna Zora; Constantinos Simserides; Georgios Triberis
2004-01-01
355
Hypervelocity impacts on satellites or ring particles replenish circumplanetary dusty rings with grains of all sizes. Due to interactions with the plasma environment and sunlight, these grains become electrically charged. We study the motion of charged dust grains launched at the Kepler orbital speed, under the combined effects of gravity and the electromagnetic force. We conduct numerical simulations of dust grain trajectories, covering a broad range of launch distances from the planetary surface to beyond synchronous orbit, and the full range of charge-to-mass ratios from ions to rocks, with both positive and negative electric potentials. Initially, we assume that dust grains have a constant electric potential, and, treating the spinning planetary magnetic field as an aligned and centered dipole, we map regions of radial instability (positive grains only), where dust grains are driven to escape or collide with the planet at high speed, and vertical instability (both positive and negative charges) whereby grains launched near the equatorial plane and are forced up magnetic field lines to high latitudes, where they may collide with the planet. We derive analytical criteria for local stability in the equatorial plane, and solve for the boundaries between all unstable and stable outcomes. Comparing our analytical solutions to our numerical simulations, we develop an extensive model for the radial, vertical and azimuthal motions of dust grains of arbitrary size and launch location. We test these solutions at Jupiter and Saturn, both of whose magnetic fields are reasonably well represented by aligned dipoles, as well as at the Earth, whose magnetic field is close to an anti-aligned dipole. We then evaluate the robustness of our stability boundaries to more general conditions. Firstly, we examine the effects of non-zero launch speeds, of up to 0.5 km s?1, in the frame of the parent body. Although these only weakly affect stability boundaries, we find that the influence of a launch impulse on stability boundaries strongly depends on its direction. Secondly, we focus on the effects of higher-order magnetic field components on orbital stability. We find that vertical stability boundaries are particularly sensitive to a moderate vertical offset in an aligned dipolar magnetic field. This configuration suffices as a model for Saturn's full magnetic field. The vertical instability also expands to cover a wider range of launch distances in slightly tilted magnetic dipoles, like the magnetic field configurations for Earth and Jupiter. By contrast, our radial stability criteria remain largely unaffected by both dipolar tilts and vertical offsets. Nevertheless, a tilted dipole magnetic field model introduces non-axisymmetric forces on orbiting dust grains, which are exacerbated by the inclusion of other higher-order magnetic field components, including the quadrupolar and octupolar terms. Dust grains whose orbital periods are commensurate with the spatial periodicities of a rotating non-axisymmetric magnetic field experience destabilizing Lorentz resonances. These have been studied by other authors for the largest dust grains moving on perturbed Keplerian ellipses. With Jupiter's full magnetic field as our model, we extend the concept of Lorentz resonances to smaller dust grains and find that these can destabilize trajectories on surprisingly short timescales, and even cause negatively-charged dust grains to escape within weeks. We provide detailed numerically-derived stability maps highlighting the destabilizing effects of specific higher-order terms in Jupiter's magnetic field, and we develop analytical solutions for the radial locations of these resonances for all charge-to-mass ratios. We include stability maps for the full magnetic field configurations of Jupiter, Saturn, and Earth, to compare with our analytics. We further provide numerically-derived stability maps for the tortured magnetic fields of Uranus and Neptune. Relaxing the assumption of constant electric charges on dust, we test the effects of time-variable grain charg
Jontof-Hutter, Daniel Simon
356
This thesis is devoted to understanding the origins of lunar crustal magnetism. We wish to understand the processes which have created and modified the crustal magnetic field distribution that we observe today, and to determine whether the Moon ever had an active magnetohydrodynamic dynamo. Previously, our only measurements of lunar magnetic fields came from the Explorer 35 and Apollo missions. Data coverage was incomplete, but sufficient to establish some systematics of the crustal field distribution. With new data from the Magnetometer and Electron Reflectometer instrument on Lunar Prospector, we have generated the first completely global maps of the lunar crustal fields. We use measurements of electrons magnetically reflected above the lunar surface, which we then correct for the effects of electrostatic fields (which also reflect electrons), and convert to estimates of surface magnetic fields. The resulting global map shows that impact basins and craters (especially the youngest) generally have low magnetic fields, suggesting impact demagnetization, primarily by shock effects. A secondary signature of some large lunar basins (especially older ones) is the presence of a more localized central magnetic anomaly. Meanwhile, the largest regions of strong crustal fields lie antipodal to young large impact basins, suggesting shock remanent magnetization due to a combination of antipodal focussing of seismic energy and/or ejecta and plasma compression of ambient magnetic fields. Smaller regions of strong magnetic fields are sometimes associated with basin ejecta, and basin and crater ejecta terranes have the strongest average fields outside of the antipodal regions. This implies that impact-generated magnetization may extend beyond the antipodal regions. The antipodal, non-antipodal, and central basin magnetic fields, as well as returned samples, can all be used to estimate the lunar magnetic field history and place constraints on a possible lunar dynamo. All of these quantities provide evidence for stronger magnetic fields early in the Moon's history, and thereby suggest the existence of an ancient core dynamo.
Halekas, Jasper S.
357
During the encounter with Comet Halley, the magnetometer (MISCHA) aboard the Vega 1 spacecraft observed an increased level of magnetic field turbulence, resulting from an upstream bow wave. Both Vega spacecraft measured a peak field strength of 70-80 nT and observed draping of magnetic field lines around the cometary obstacle. An unexpected rotation of the magnetic field vector was observed, which may reflect either penetration of magnetic field lines into a diffuse layer related to the contact surface separating the solar-wind and cometary plasma, or the persistence of pre-existing interplanetary field structures.
Riedler, W.; Schwingenschuh, K.; Yeroshenko, Ye. G.; Styashkin, V. A.; Russell, C. T.
1986-05-01
358
Beam-ion losses induced by fast-ion-driven toroidal Alfvén eigenmodes (TAE) were measured with a scintillator-based lost fast-ion probe (SLIP) in the large helical device (LHD). The SLIP gave simultaneously the energy E and the pitch angle ? = arccos(?///?) distribution of the lost fast ions. The loss fluxes were investigated for three typical magnetic configurations of Rax_vac = 3.60 m, 3.75 m, and 3.90 m, where Rax_vac is the magnetic axis position of the vacuum field. Dominant losses induced by TAEs in these three configurations were observed in the E/? regions of 50~190 keV/40°, 40~170 keV/25°, and 30~190 keV/30°, respectively. Lost-ion fluxes induced by TAEs depend clearly on the amplitude of TAE magnetic fluctuations, Rax_vac and the toroidal field strength Bt. The increment of the loss fluxes has the dependence of (bTAE/Bt)s. The power s increases from s = 1 to 3 with the increase of the magnetic axis position in finite beta plasmas.
Ogawa, K.; Isobe, M.; Toi, K.; Watanabe, F.; A. Spong, D.; Shimizu, A.; Osakabe, M.; S. Darrow, D.; Ohdachi, S.; Sakakibara, S.; LHD Experiment Group
2012-04-01
359
SciTech Connect
Beam-ion losses induced by fast-ion-driven toroidal Alfven eigenmodes (TAE) were measured with a scintillator-based lost fast-ion probe (SLIP) in the large helical device (LHD). The SLIP gave simultaneously the energy E and the pitch angle chi = arccos(v(parallel to)/v) distribution of the lost fast ions. The loss fluxes were investigated for three typical magnetic configurations of R{sub ax{_}vac} = 3.60 m, 3.75 m, and 3.90 m, where R{sub ax{_}vac} is the magnetic axis position of the vacuum field. Dominant losses induced by TAEs in these three configurations were observed in the E/chi regions of 50 similar to 190 keV/40 degrees, 40 similar to 170 keV/25 degrees, and 30 similar to 190 keV/30 degrees, respectively. Lost-ion fluxes induced by TAEs depend clearly on the amplitude of TAE magnetic fluctuations, R{sub ax{_}vac} and the toroidal field strength B{sub t}. The increment of the loss fluxes has the dependence of (b{sub TAE}/B{sub t}){sup s}. The power s increases from s = 1 to 3 with the increase of the magnetic axis position in finite beta plasmas.
Ogawa, K. [Nagoya University, Japan; Isobe, M. [National Institute for Fusion Science, Toki, Japan; Watanabe, F. [Kyoto University, Japan; Spong, Donald A [ORNL; Shimizu, A. [National Institute for Fusion Science, Toki, Japan; Osakabe, M. [National Institute for Fusion Science, Toki, Japan; Ohdachi, S. [National Institute for Fusion Science, Toki, Japan; Sakakibara, S. [National Institute for Fusion Science, Toki, Japan
2012-01-01
360
US Patent & Trademark Office Database
A system includes a host device and a disk drive interfaced with the host device are described as well as an associated method. The disk drive includes a magnetic media for storing information using an actuator arrangement to perform a data access by moving at least one head proximate to the magnetic media. The information may be subject to corruption when the disk drive is exposed, during the data access, to a given stray magnetic field having a given minimum magnetic field intensity. The given stray magnetic field will not corrupt the information on the magnetic media with the actuator arrangement positioned away from the magnetic media. A stray magnetic field protection arrangement is configured for detecting an ambient magnetic environment for use in causing the actuator arrangement to park responsive to the detection of at least the given minimum magnetic field intensity.
Partee; Charles (Lyons, CO)
2010-12-28
361
The magnetic field in stellar radiation zones can play an important role in phenomena such as mixing, angular momentum transport, etc. We study the effect of rotation on the stability of a predominantly toroidal magnetic field in the radiation zone. In particular we considered the stability in spherical geometry by means of a linear analysis in the Boussinesq approximation. It is found that the effect of rotation on the stability depends on a magnetic configuration. If the toroidal field increases with the spherical radius, the instability cannot be suppressed entirely even by a very fast rotation. Rotation can only decrease the growth rate of instability. If the field decreases with the radius, the instability has a threshold and can be completey suppressed.
Bonanno, A.; Urpin, V.
2013-04-01
362
The origin of intergalactic magnetic fields is still a mystery and several scenarios have been proposed so far: among them, primordial phase transitions, structure-formation shocks and galactic outflows. In this work, we investigate how efficiently galactic winds can provide an intense and widespread seed' magnetization. This may be used to explain the magnetic fields observed today in clusters of galaxies
Serena Bertone; Corina Vogt; Torsten Enßlin
2006-01-01
363
Issues associated with the exposure of patients to strong, static magnetic fields during magnetic resonance imaging (MRI) are reviewed and discussed. The history of human exposure to magnetic fields is reviewed, and the contra- dictory nature of the literature regarding effects on human health is described. In the absence of ferromagnetic for- eign bodies, there is no replicated scientific study
John F. Schenck
2000-01-01
364
Outflows from quasars inevitably pollute the intergalactic medium (IGM) with magnetic fields. The short-lived activity of a quasar leaves behind an expanding magnetized bubble in the IGM. We model the expansion of the remnant quasar bubbles and calculate their distribution as a function of size and magnetic field strength at different redshifts. We generically find that by a redshift z~3,
Steven R. Furlanetto; Abraham Loeb
2001-01-01
365
Aims: We wish to clarify whether strong magnetic fields can be effectively generated in typically low-mass dwarf galaxies and to assess the role of dwarf galaxies in the magnetization of the Universe. Methods: We performed a search for radio emission and magnetic fields in an unbiased sample of 12 Local Group (LG) irregular and dwarf irregular galaxies with the 100-m
K. T. Chyzy; M. Wezgowiec; R. Beck; D. J. Bomans
2011-01-01
366
ERIC Educational Resources Information Center
|After the discovery that superconducting magnets could levitate diamagnetic objects, researchers became interested in measuring the repulsion of diamagnetic fluids in strong magnetic fields, which was given the name "The Moses Effect." Both for the levitation experiments and the quantitative studies on liquids, the large magnetic fields necessary…
Chen, Zijun; Dahlberg, E. Dan
2011-01-01
367
Copy machine developer powder is an alternative for creating permanent displays of magnetic fields. A thin layer of developer powder on a sheet of paper placed over a magnet can be baked in the oven, producing a lasting image of a magnetic field.
Cavanaugh, Terence; Cavanaugh, Catherine
1998-02-01
368
ERIC Educational Resources Information Center
|A compass is an excellent classroom tool for the exploration of magnetic fields. Any student can tell you that a compass is used to determine which direction is north, but when paired with some basic trigonometry, the compass can be used to actually measure the strength of the magnetic field due to a nearby magnet or current-carrying wire. In…
Lunk, Brandon; Beichner, Robert
2011-01-01
369
We performed cosmological, magnetohydrodynamical simulations to follow the evolution of magnetic fields in galaxy clusters, exploring the possibility that the origin of the magnetic seed fields is galactic outflows during the starburst phase of galactic evolution. To do this, we coupled a semi-analytical model for magnetized galactic winds as suggested by Bertone, Vogt & Enßlin to our cosmological simulation. We
J. Donnert; K. Dolag; H. Lesch; E. Müller
2009-01-01
370
Context: .The evolution of the concentrated magnetic field in flux tubes is one challenge of the nowadays Solar physics which requires time sequence with high spatial resolution. Aims: .Our objective is to follow the properties of the magnetic concentrations during their life, in intensity (continuum and line core), magnetic field and Doppler velocity. Methods: .We have observed solar region NOAA
Th. Roudier; J. M. Malherbe; J. Moity; S. Rondi; P. Mein; Ch. Coutard
2006-01-01
371
A new global map of the magnetic field of Mars, with an order of magnitude improved sensitivity to crustal magnetization, is derived from Mars Global Surveyor mapping orbit magnetic field data. With this comes greatly improved spatial resolution and geologic intrpretation.
Connerney, J. E. P.; Acuna, M. H.; Ness, N. F.; Mitchell, D. L.; Lin, R. P.
2004-03-01
372
SciTech Connect
The study of transport in magnetized plasmas is a problem of fundamental interest in controlled fusion, space plasmas, and astrophysics research. Three issues make this problem particularly challenging: (i) The extreme anisotropy between the parallel (i.e., along the magnetic field), {chi}{sub ||} , and the perpendicular, {chi}{sub Up-Tack }, conductivities ({chi}{sub ||} /{chi}{sub Up-Tack} may exceed 10{sup 10} in fusion plasmas); (ii) Nonlocal parallel transport in the limit of small collisionality; and (iii) Magnetic field lines chaos which in general complicates (and may preclude) the construction of magnetic field line coordinates. Motivated by these issues, we present a Lagrangian Green's function method to solve the local and non-local parallel transport equation applicable to integrable and chaotic magnetic fields in arbitrary geometry. The method avoids by construction the numerical pollution issues of grid-based algorithms. The potential of the approach is demonstrated with nontrivial applications to integrable (magnetic island), weakly chaotic (Devil's staircase), and fully chaotic magnetic field configurations. For the latter, numerical solutions of the parallel heat transport equation show that the effective radial transport, with local and non-local parallel closures, is non-diffusive, thus casting doubts on the applicability of quasilinear diffusion descriptions. General conditions for the existence of non-diffusive, multivalued flux-gradient relations in the temperature evolution are derived.
Castillo-Negrete, D. del; Chacon, L. [Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8071 (United States)
2012-05-15
373
SciTech Connect
The study of transport in magnetized plasmas is a problem of fundamental interest in controlled fusion, space plasmas, and astrophysics research. Three issues make this problem particularly chal- lenging: (i) The extreme anisotropy between the parallel (i.e., along the magnetic field), , and the perpendicular, , conductivities ( / may exceed 1010 in fusion plasmas); (ii) Magnetic field lines chaos which in general complicates (and may preclude) the construction of magnetic field line coordinates; and (iii) Nonlocal parallel transport in the limit of small collisionality. Motivated by these issues, we present a Lagrangian Green s function method to solve the local and non-local parallel transport equation applicable to integrable and chaotic magnetic fields in arbitrary geom- etry. The method avoids by construction the numerical pollution issues of grid-based algorithms. The potential of the approach is demonstrated with nontrivial applications to integrable (magnetic island chain), weakly chaotic (devil s staircase), and fully chaotic magnetic field configurations. For the latter, numerical solutions of the parallel heat transport equation show that the effective radial transport, with local and non-local closures, is non-diffusive, thus casting doubts on the appropriateness of the applicability of quasilinear diffusion descriptions. General conditions for the existence of non-diffusive, multivalued flux-gradient relations in the temperature evolution are derived.
Del-Castillo-Negrete, Diego B [ORNL; Chacon, Luis [ORNL
2012-01-01
374
A method which enables coupling between equations of electric circuits consisting of a lumped element RLC configuration and a magnetic field model is presented. The coupling between the finite-element and the boundary-element methods is used to compute the magnetic field produced by conductors excited by an electric circuit. The conductors involved in this computation may be connected according to any
A. Nicolet; F. Delince; N. Bamps; A. Genon; W. Legros
1993-01-01
375
We have investigated the static configurations of the phase inside an annular Josephson tunnel junction in the presence of an externally applied magnetic field. We report here a detailed study of the dependence on the magnetic field of the critical current for different annular geometries. The periodic conditions for the phase difference across the barrier are derived from fluxoid quantization.
N. Martucciello; R. Monaco
1996-01-01
376
Because of thermoelectric effects, a local current density appears in the dendritic network during the solidification of a metallic alloy. Thus, when a magnetic field is applied, a Lorentz force is created. Two alloys are solidified directionally in the horizontal configuration under a transverse magnetic field with the result that this force opposes the natural solutal buoyancy force. The experimental
P. Lehmann; R. Moreau; D. Camel; R. Bolcato
1998-01-01
377
In constructions of analytical solutions to open string field theories pure\\u000agauge configurations parameterized by wedge states play an essential role.\\u000aThese pure gauge configurations are constructed as perturbation expansions and\\u000ato guaranty that these configurations are asymptotical solutions to equations\\u000aof motions one needs to study convergence of the perturbation expansions. We\\u000ademonstrate that for the large parameter of
I. Ya. Aref'eva; Roman V. Gorbachev; Dmitry A. Grigoryev; Pavel N. Khromov; Maxim V. Maltsev; P. B. Medvedev
2009-01-01
378
Rotation magnetic beacons magnetic field strength is very important to drill parallel horizontal twin wells in steam assisted\\u000a gravity drainage (SAGD). This paper analyzes a small magnet with a diameter of 25.4 mm. At each end, there is a length of\\u000a 12.6 mm with permanent magnet, and in the middle, there is a length of 78mm with magnetic materials. The
Bing Tu; Desheng Li; Enhuai Lin; Bin Luo; Jian He; Lezhi Ye; Jiliang Liu; Yuezhong Wang
2010-01-01
379
A novel architecture for sensing weak dc magneticfields exploits the co- operation between nonlinear coupled systems and ferroelectric devices. Sensors for static electric fields (E-fields) based on unidirection- ally coupled nonlinear dynamical systems1 are currently un- der development. They will exploit the synergic use of bistable ferroelectric materials, micromachining technologies, and novel sensing strategies. Considerable practical interest in such de-
Bruno Ando; Salvatore Baglio; Alberto Ascia; Adi R. Bulsara
2007-01-01
380
SciTech Connect
In a multi-bunch high current storage ring, beam generated fields couple strongly into the RF cavity coupler structure when beam arrival times are in resonance with cavity fields. In this study the integrated effect of beam fields over several thousand RF periods is simulated for the complete cavity, coupler, window and waveguide system of the PEP-II B-factory storage ring collider. We show that the beam generated fields at frequencies corresponding to several bunch spacings for this case gives rise to high field strength near the ceramic window which could limit the performance of future high current storage rings such as PEP-X or Super B-factories.
Weathersby, Stephen; Novokhatski, Alexander; /SLAC
2010-02-10
381
This paper presents design of high performance permanent magnet-assisted synchronous reluctance generators (PMa-SynRG) for 3 kW tactical quiet generator set. Adding the proper quantity of permanent magnets into the synchronous reluctance generator rotor core can offer large constant power-speed range, high efficiency and high power factor. Different stator winding configurations such as distributed windings and concentrated windings are compared using
Jeihoon Baek; Mina M. Rahimian; Hamid A. Toliyat
2009-01-01
382
We introduce a topological flux function to quantify the topology of magnetic braids: non-zero, line-tied magnetic fields whose field lines all connect between two boundaries. This scalar function is an ideal invariant defined on a cross-section of the magnetic field, and measures the average poloidal magnetic flux around any given field line, or the average pairwise crossing number between a given field line and all others. Moreover, its integral over the cross-section yields the relative magnetic helicity. Using the fact that the flux function is also an action in the Hamiltonian formulation of the field line equations, we prove that it uniquely characterizes the field line mapping and hence the magnetic topology.
Yeates, A. R.; Hornig, G.
2013-01-01
383
The Plasma Magnet concept aims to provide a large magnetic barrier to couple thrust from the solar wind. A spacecraft using Plasma Magnet propulsion would be able to move throughout the solar system faster and more efficiently than possible using conventional chemical or electric propulsion. This dramatic improvement in performance is made possible by coupling thrust from the solar wind. The Plasma Magnet drives electrical currents in plasma surrounding the spacecraft using a rotating magnetic field (RMF). RMF has been successfully used to drive currents of several thousand amps in Field Reversed Configurations (FRC) and Rotamak plasma configurations during the course of international fusion research efforts. However, these experiments have used RMF current drive in the region inside the RMF coils. For the Plasma Magnet concept to be successful, RMF current drive must be successfully used to drive currents external to and at large distances from the RMF coils. Preliminary computer simulations indicate that RMF can drive currents outside the RMF coils. Preliminary experimental investigations of the Plasma Magnet concept are also underway. The poster presented will feature these results and a description of the Plasma Magnet concept in the context of advanced spacecraft propulsion.
Giersch, Louis; Slough, John; Winglee, Robert; Andreason, Samuel
2004-11-01
384
NSDL National Science Digital Library
This webpage is part of the University Corporation for Atmospheric Research (UCAR) Windows to the Universe program. It describes the nature and configuration of magnetic fields, which are the result of moving electric charges, including how they cause magnetic objects to orient themselves along the direction of the magnetic force points, which are illustrated as lines. Magnetic field lines by convention point outwards at the north magnetic pole and inward at the south magnetic pole. The site features text, scientific illustrations and an animation. Text and vocabulary are selectable for the beginning, intermediate, or advanced reader.
Team, University C.
2007-12-12
385
For all accelerators and many research and industries, excellent vacuum conditions are required and the highest possible pumping rates are necessary. For most applications the standard ion sputtering pump (ISP) meets these requirements and is optimal for financial point of view also. The physical principle of the ISP is well known and many companies manufacture variety of ISP. Most of them use dipole magnetic field produced by permanent magnet and electric dipole field between the electrodes in which tenuous plasma is created because of interaction of between the relatively fast electrons slow residual gas atoms. Performance of an ISP depends basically on the electron cloud density in between the titanium electrodes but in the available present configurations no consideration has been given to electron confinement which needs a mirror magnetic field. If this is incorporated it will make a robust ISP surely; furthermore, the requirement of constant feeding of high voltage to electrodes for supplying sufficient number of electrons will be reduced too. A study has been performed to create sufficient rotationally symmetric spindle magnetic field (SMF) with inherent presence of magnetic mirror effect to electron motion to confine them for longer time for enhancing the density of electron cloud between the electrodes. It will lessen the electric power feeding the electrodes and lengthen their life-time. Construction of further compact and robust ISP is envisaged herein. The field simulation using the commercially available permanent magnet together with simulation of electron motion in such field will be presented and discussed in the paper.
Rashid, M. H.
2012-11-01
386
SciTech Connect
Recent results from the spheromak work at Los Alamos include: In decaying spheromaks in the mesh flux conserver, the rate of current decay dI/dt depends only on the density n/sub e/ (not on I or T/sub e/ as might be expected classically). The particular dependence of dI/dt on n/sub e/ suggests that most of the helicity is dissipated in the field error regions at the edge where electron-neutral hydrogen collisions dominate the electrical resistance. A new solid-wall, titanium-gettered flux conserver has been commissioned, resulting in less field errors than with previous mesh flux conserver. A factor of four decrease in dI/dt has been observed. In this new flux conserver, clear evidence of a pressure-driven instability has been obtained. To our knowledge, this is the first time a pressure driven interchange mode has been directly observed in a toroidal geometry. Peak ..beta.. values observed before the onset of the mode are of the same order as those predicted at the corresponding Mercier limits (..beta../sub c/ approx. 1%). As a result, flux conserver shapes with higher ..beta../sub c/ are now being considered by us. The ..beta../sub c/ calculations have been made using Taylor-like (minimum energy) magnetic profiles. By changing the flux conserver shape, the q profile of the minimum-energy state can be varied continuously from RFP-like to tokamak-like. The highest ..beta../sub c/ (approx.10%) are found at these two extremes of the range of magnetic configurations. 3 refs.
Jarboe, T.R.; Barnes, C.W.; Fernandez, J.C.; Henins, I.; Marklin, G.J.; Wysocki, F.J.
1988-01-01
387
A high temperature, stable, long-lived field-reversed configuration (FRC) plasma state has been produced in the C-2 device by dynamically colliding and merging two oppositely directed compact toroids, by biasing edge plasma near the FRC separatrix from a plasma-gun (PG) located at one end of the C-2 device, and by neutral-beam (NB) injection. The PG creates an inward radial electric field (Er<0) which counters the usual FRC spin-up in the ion diamagnetic direction and mitigates the n = 2 rotational instability without applying quadrupole magnetic fields. Better plasma centering is also obtained, presumably from line-tying to the gun electrodes. The PG produces ExB velocity shear in the FRC edge layer which may explain observations of improved transport properties The FRCs are nearly axisymmetric, which enables fast ion confinement. The combined effects of the PG and of NB injection yield a new High Performance FRC (HPF) regime with confinement times improved by factors 2 to 4 and FRC lifetimes extended from 1 to 3 ms. A second PG was newly installed at the other end of the C-2 device, and new experimental campaigns with 2 PGs have been explored. Characteristics of the HPF regime will be presented at the meeting as well as newly obtained results with 2 PGs and NBs.
Gota, H.; Tuszewski, M.; Smirnov, A.; Guo, H.; Binderbauer, M.; Barnes, D.; Akhmetov, T.; Ivanov, A.
2012-10-01
388
NSDL National Science Digital Library
This web page is an interactive physics simulation that explores magnetic fields. The user can add currents coming into or out of a simulated grid, and see the fields created. There is also a selection of pre-created fields, including bar magnets, loops, opposing magnets, and coils in uniform fields. Double-clicking on any point displays the full loop created by the magnetic field. This item is part of a larger collection of introductory physics simulations developed by the author. This is part of a collection of similar simulation-based student activities.
Duffy, Andrew
2008-08-23
389
We describe a generic mechanism by which a system of Dirac fermions in thermal equilibrium acquires electric charge in an external magnetic field. To this end the fermions should have an additional quantum number, isospin, or color and should be subject to a second magnetic field, which distinguishes the isospin or color, as well as to a corresponding isospin chemical potential. The role of the latter can be also played by a nontrivial holonomy (Polyakov loop) along the Euclidean time direction. The charge is accumulated since the degeneracies of occupied lowest Landau levels for particles of positive isospin and antiparticles of negative isospin are different. We discuss two physical systems where this phenomenon can be realized. One is monolayer graphene, where the isospin is associated with two valleys in the Brillouin zone, and the strain-induced pseudomagnetic field acts differently on charge carriers in different valleys. Another is hot QCD, for which the relevant non-Abelian field configurations with both nonzero chromomagnetic field and a nontrivial Polyakov loop can be realized as calorons—topological solutions of Yang-Mills equations at finite temperature. The induced electric charge on the caloron field configuration is studied numerically. We argue that due to the fluctuations of holonomy, the external magnetic field should tend to suppress charge fluctuations in the quark-gluon plasma and estimate the importance of this effect for off-central heavy-ion collisions.
Bruckmann, F.; Buividovich, P. V.; Sulejmanpasic, T.
2013-08-01
390
DOEpatents
A magnetic refrigeration apparatus includes first and second steady state magnets, each having a field of substantially equal strength and opposite polarity, first and second bodies made of magnetocaloric material disposed respectively in the influence of the fields of the first and second steady state magnets, and a pulsed magnet, concentric with the first and second steady state magnets, and having a field which cycles between the fields of the first and second steady state magnets, thereby cyclically magnetizing and demagnetizing and thus heating and cooling the first and second bodies. Heat exchange apparatus of suitable design can be used to expose a working fluid to the first and second bodies of magnetocaloric material. A controller is provided to synchronize the flow of working fluid with the changing states of magnetization of the first and second bodies.
Lubell, Martin S. (Oak Ridge, TN)
1994-01-01
391
DOEpatents
A magnetic refrigeration apparatus includes first and second steady state magnets, each having a field of substantially equal strength and opposite polarity, first and second bodies made of magnetocaloric material disposed respectively in the influence of the fields of the first and second steady state magnets, and a pulsed magnet, concentric with the first and second steady state magnets, and having a field which cycles between the fields of the first and second steady state magnets, thereby cyclically magnetizing and demagnetizing and thus heating and cooling the first and second bodies. Heat exchange apparatus of suitable design can be used to expose a working fluid to the first and second bodies of magnetocaloric material. A controller is provided to synchronize the flow of working fluid with the changing states of magnetization of the first and second bodies. 2 figs.
Lubell, M.S.
1994-10-25
392
National Technical Information Service (NTIS)
This paper describes the data analysis technique used for magnetic testing at the NASA Goddard Space Flight Center (GSFC). Excellent results have been obtained using this technique to convert a spacecraft s measured magnetic field data into its respective...
P. K. Harris
2003-01-01
393
National Technical Information Service (NTIS)
An electronic control system for stabilization of currents in magnetic fields is described. Three superimposed control stages with different characteristics provide optimum elimination of all interfering factors. The use of electrostatic and magnetic shie...
K. Weyand
1976-01-01
394
The effect of magnetic field on the performance of a hollow cathode ion source was experimentally studied. The field strength and the field distribution around the hollow cathode were changed step by step and the source parameters were recorded for each step. The result showed that with the optimum field configuration the discharge was stabilized even at low-operating gas pressures
Shigeru Tanaka; Masato Akiba; Hiroshi Horiike; Yoshikazu Okumura; Yoshihiro Ohara
1983-01-01
395
Invisible lines of magnetic force enclose our planet in what scientists call adipolarmagneticfield. Today these lines go from magnetic south to magnetic north, which are offset a few degrees from the geographic poles. Some minerals, like magnetite, can \\
Trevor Major
396
SciTech Connect
Propagation and damping properties of low-frequency waves in extremely high beta plasmas have been investigated in the field-reversed configuration plasma injection experiment apparatus. Two distinct wave modes were excited by different antenna geometry and the radial structures of their magnetic components were measured in detail. Due to the high beta nature of the plasma, the Alfven resonance predicted for the cold plasma vanished and the wave with small parallel wave number k{sub z}{approx}2.5 m{sup -1} propagated in a broad area across the separatrix. The wave exhibited moderate dissipation, which was suggestive of the resistive damping. The wave with large parallel wave number k{sub z}{approx}7.0 m{sup -1}, however, underwent quite strong damping inside the separatrix. Transit-time magnetic pumping, which converts the wave energy to the ion parallel kinetic energy, is most likely to be responsible for this strong damping since the wave's parallel phase velocity is close to the ion thermal velocity.
Inomoto, Michiaki; Yamamoto, Satoshi; Iwasawa, Naotaka; Kitano, Katsuhisa; Okada, Shigefumi [Center for Atomic and Molecular Technologies, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871 (Japan)
2007-10-15
397
SciTech Connect
We describe a model which gives the effects of magnetic fields on a plasma electrode Pockels cell. The fields arise from the return currents to the cathode as well as from neighboring devices such as amplifier flashlamps. In effect, electrons are treated as a static, planar fluid moving under the influence of magnetic fields, the electric field of the discharge, electron pressure gradients, and electron-atom elastic collisions. This leads to a closed two- dimensional equation for the electron density, which is solved subject to appropriate boundary collisions. The model is applied to four cases-. the baseline NIF configuration with magnetic fields due to balanced return currents; a case with unbalanced return currents; the reverser configuration containing an external field parallel to the main plasma current; and a configuration with a field perpendicular to both the current and the optical direction.
Boley, C.D.; Rhodes, M.A.
1996-10-01
398
National Technical Information Service (NTIS)
An experiment on arc discharges in hydrogen in a curved magnetic field is described. For a few milliseconds the discharge current flowed between two electrodes along the field lines of a toroidal magnetic field over an angle of 258 deg. The plasma was not...
F. C. Schueller
1974-01-01
399
We consider the various methods used to constrain the possible field strength of the present day intergalactic field and findB0(G)-10 as a probable upper bound. It is suggested that the observed intergalactic magnetic field might not be primordial in origin but rather the result of magnetic flux leakage from galaxies and clusters of galaxies.
Martin Beech
1985-01-01
400
An alternative explanation of galactic warps is proposed, in which the intergalactic magnetic field (IGMF) is responsible for these structures. The model predicts that, to be efficient, the magnetic field must have a direction not much different from 45 deg with the galactic plane. The required values of the field strength are uncertain, of about 10 nG, higher values being
E. Battaner; E. Florido; M. L. Sanchez-Saavedra
1990-01-01
401
In this paper we demonstrate experimentally a magnetic field sensor using a fiber Bragg grating. The shift in the Bragg condition as a result of strain applied on the fiber mounted on a nickel base by the magnetic field gives an indirect measure of the field. The proposed method overcomes the need for long fiber lengths required in methods such
K. V. Madhav; K. Ravi Kumar; T. Srinivas; S. Asokan
2006-01-01
402
The various methods used to constrain the possible field strength of the present day intergalactic field are considered, and Bzero (G) less than 10 to the -10th is found as a probable upper bound. It is suggested that the observed intergalactic magnetic field might not be primordial in origin but rather the result of magnetic flux leakage from galaxies and clusters of galaxies.
Beech, M.
1985-11-01
403
In this thesis we have investigated the use of a magnetostrictive material with a single-mode optical fiber for detecting weak magnetic fields. The amorphous alloy Metglas^circler 2605SC (Fe_{81}B_ {13.5}Si_{3.5} C_2) was chosen as the magnetostrictive material because of the combination of its large magnetostriction and small magnetic anisotropy field among all available metals. For efficient coupling between the magnetostrictive material and the optical fiber, the magnetostrictive material was directly deposited onto the single-mode optical fiber. The coated fibers were used as the sensing element in the fiber optic magnetic field sensor (FOMS). Very high quality thick metallic glass films of the Metglas 2605 SC have been deposited using triode-magneton sputtering. This is the first time such material has been successfully deposited onto an optical fiber or onto any other substrate. The films were also deposited onto glass slides to allow the study of the magnetic properties of the film. The thicknesses of these films were 5-15 mum. The magnetic property of primary interest for our sensor application is the induced longitudinal magnetostrictive strain. However, the other magnetic properties such as magnetic anisotropy, surface and bulk coercivities, magnetic homogeneity and magnetization all affect the magnetostrictive response of the material. We have used ferromagnetic resonance (FMR) at microwave frequencies to study the magnetic anisotropy and homogeneity; vibrating sample magnetometry (VSM) to study the bulk magnetic hysteresis responses and coercivity; and the longitudinal magneto-optic kerr effect (LMOKE) to study the surface magnetic hysteresis responses and coercivity. The isothermalmagnetic annealing effect on these properties has also been studied in detail. The fiber optic magnetic field sensor constructed using the metallic-glass-coated fiber was tested. An electronic feedback control loop using a PZT cylinder was constructed for stabilizing the sensor operation. Magnetic field detection at different dither frequencies was studied in detail. The estimated minimum detectable magnetic field was about 3 times 10^{-7 } Oe. A simplified elastic model was used for the theoretical calculation of the phase shift induced in a metallic-glass -coated optical fiber with a longitudinal applied magnetic field. The phase shift as a function of coating thickness was calculated, and the experimental results at certain thicknesses were compared with the calculation. The frequency response of the FOMS was also studied in some detail. Three different configurations were used for the study of the frequency response. The results indicate that the resonances observed in the FOMS are most likely related to the mechanical resonance of the optical fiber.
Wang, Yu.
1990-01-01
404
SciTech Connect
Three species of potentially pathogenic amoebae were exposed to 71 and 106.5 mT from constant homogeneous magnetic fields and examined for inhibition of population growth. The number of amoebae for three species was significantly less than controls after a 72 h exposure to the magnetic fields when the temperature was 20 C or above. Axenic cultures, i.e., cultures grown without bacteria, were significantly affected after only 24 h. In 20 of 21 tests using the three species, the magnetic field significantly inhibited the growth of amoebae. In one test in which the temperature was 20 C for 48 h, exposure to the magnetic field was not inhibitory. Final numbers of magnetic field-exposed amoebae ranged from 9 to 72% lower than the final numbers of unexposed controls, depending on the species. This research may lead to disinfection strategies utilizing magnetic fields for surfaces on which pathogenic amoebae may proliferate.
Berk, S.G.; Srikanth, S.; Mahajan, S.M.; Ventrice, C.A. [Tennessee Technological Univ., Cookeville, TN (United States)
1997-03-01
405
This thesis presents a method for optimizing cable configuration inside a large magnetic cylindrical steel casing, from the total ampacity point of view. The method is comprised of two main parts, namely: 1) analytically calculating the electromagnetic losses in the steel casing and sheathed cables, for an arbitrary cables configuration, and 2) implementing an algorithm for determining the optimal cables configuration to obtain the best total ampacity. The first part involves approximating the eddy current and hysteresis losses in the casing and cables. The calculation is based on the theory of images, which this thesis expands to apply to casings having both high magnetic permeability and high electric conductivity at the same time. The method of images, in combination with approximating the cable conductors and sheaths as multiple physical filaments, is used to compute the final current distributions in the cables and pipe and thus the associated losses. The accuracy of this computation is assessed against numerical solutions obtained using the Maxwell finite element program by Ansoft. Next, the optimal cable configuration is determined by applying a proposed two-level optimization algorithm. At the outer level, a combinatorial optimization based on a genetic algorithm explores the different possible configurations. The performance of every configuration is evaluated according to its total ampacity, which is calculated using a convex optimization algorithm. The convex optimization algorithm, which forms the inner level of the overall optimization procedure, is based on the barrier method. This proposed optimization procedure is tested for a duct bank installation containing twelve cables and fifteen ducts, comprising two circuits and two cables per phase, and compared with a brute force method of considering all possible configurations. The optimization process is also applied to an installation consisting of a single circuit inside a large magnetic steel casing.
Moutassem, Wael
406
SciTech Connect
Circuitry for detecting magnetic fields includes a first magnetoresistive sensor and a second magnetoresistive sensor configured to form a gradiometer. The circuitry includes a digital signal processor and a first feedback loop coupled between the first magnetoresistive sensor and the digital signal processor. A second feedback loop which is discrete from the first feedback loop is coupled between the second magnetoresistive sensor and the digital signal processor.
Kotter, Dale K. (Shelley, ID); Spencer, David F. (Idaho Falls, ID); Roybal, Lyle G. (Idaho Falls, ID); Rohrbaugh, David T. (Idaho Falls, ID)
2010-09-14
407
Nonlinear transport coefficients do not obey, in general, reciprocity relations. We here discuss the magnetic-field asymmetries that arise in thermoelectric and heat transport of mesoscopic systems. Based on a scattering theory of weakly nonlinear transport, we analyze the leading-order symmetry parameters in terms of the screening potential response to either voltage or temperature shifts. We apply our general results to a quantum Hall antidot system. Interestingly, we find that certain symmetry parameters show a dependence on the measurement configuration.
Hwang, Sun-Yong; Sánchez, David; Lee, Minchul; López, Rosa
2013-10-01
408
SciTech Connect
Field-reversed configurations are consistently formed at low filling pressures in the FRX-C device, with decay time of the trapped flux after formation much larger than the stable period. This contrasts with previous experimental observations.
Tuszewski, M.; Armstrong, W.T.; Bartsch, R.R.; Chrien, R.E.; Cochrane, J.C. Jr.; Kewish, R.W. Jr.; Klingner, P.; Linford, R.K.; McKenna, K.F.; Rej, D.J.; Sherwood, E.G.; Siemon, R.E.
1982-10-01
409
The major scientific achievements associated with the measurement of magnetic fields in space over the past decade and a half are reviewed. Aspects of space technology relevant to magnetic-field observations are discussed, including the different types of magnetometers used and how they operate, problems arising from spacecraft-generated magnetic fields and the appropriate countermeasures that have been developed and on-board processing
EDWARD J. SMITHAND; Charles Sonett
1976-01-01
410
A class of nonlinear force-free magnetic fields is presented, described in terms of the solutions to a second-order, nonlinear ordinary differential equation. These magnetic fields are three-dimensional, filling the infinite half-space above a plane where the lines of force are anchored. They model the magnetic fields of the sun over active regions with a striking geometric realism. The total energy
B. C. Low; Y. Q. Lou
1990-01-01
411
The magnetization process of a ferrofluid whose carrier fluid is paraffin was investigated in the temperature range from 77 K to 300 K, as a function of the cooling field intensity and freezing rate. Phase transitions between the liquid and solid states can be simulated by using the ferrofluids as a magnetic probe. A uniaxial magnetic anisotropy was induced by
N. Inaba; H. Miyajima; S. Taketomi; S. Chikazumi
1989-01-01
412
Swarm is the fifth Earth Explorer mission in ESA's Living Planet Programme, and is scheduled for launch in 2013. The objective of the Swarm mission is to provide the best-ever survey of the geomagnetic field and its temporal evolution using a constellation of 3 identical satellites. The Mission shall deliver data that allow access to new insights into the Earth system by improved scientific understanding of the Earth's interior and near-Earth electromagnetic environment. After launch and triple satellite release at an initial altitude of about 490 km, a pair of the satellites will fly side-by-side with slowly decaying altitude, while the third satellite will be lifted to 530 km to complete the Swarm constellation. High-precision and high-resolution measurements of the strength, direction and variation of the magnetic field, complemented by precise navigation, accelerometer and electric field measurements, will provide the observations required to separate and model various sources of the geomagnetic field and near-Earth current systems. The mission science goals are to provide a unique view into Earth's core dynamics, mantle conductivity, crustal magnetisation, ionospheric and magnetospheric current systems and upper atmosphere dynamics - ranging from understanding the geodynamo to contributing to space weather. The scientific objectives and results from recent scientific studies will be presented. In addition the current status of the project, which is presently in the final stage of the development phase, will be addressed. A consortium of European scientific institutes is developing a distributed processing system to produce geophysical (Level 2) data products for the Swarm user community. The setup of the Swarm ground segment and the contents of the data products will be addressed. More information on Swarm can be found at www.esa.int/esaLP/LPswarm.html.
Plank, Gernot; Haagmans, Roger; Floberghagen, Rune; Menard, Yvon
2013-04-01
413
Supersonic flow fields around Two-Stage-To-Orbit (TSTO) models with different configurations have been experimentally examined in this paper. Four configurations for the orbiter have been considered: A) a hemisphere-cylinder, B) a hemisphere-cylinder with a flat bottom, C) an obliquely truncated circular cylinder, and D) a cone-cylinder. All the flow fields around these models showed complicated shock\\/shock and shock\\/boundary-layer interactions, which can
Keiichi Kitamura; Koichi Mori; Katsuhisa Hanai; Tsutomu Yabashi; Hiroshi Ozawa; Yoshiaki Nakamura
2007-01-01
414
Major solar eruptions, namely flares and coronal mass ejections, rely on significant local accumulations of non-potential (free; stored in electric currents) magnetic energy and, quite likely, magnetic helicity in the solar atmosphere. Without [both of] them, eruptions cannot be powered. Simple tests can show that most free energy and helicity reside close to the lower atmospheric boundary in solar active regions, i.e. their photospheric or low chromospheric interface. Therefore, the pre-eruption configuration in this boundary should reflect these high free-energy and helicity conditions that jointly determine the degree of non-potentiality in active regions. We review the two main active-region photospheric/low-chromospheric configurations leading to major eruptions: instances of intense magnetic flux emergence in the absence of intense magnetic polarity inversion lines (PILs), and instances of strong PILs. In these configurations we discuss multiple measures that can be thought of as proxies of free magnetic energy and helicity and we outline a method to actually calculate these budgets. Combining information from different, but concerted, analyses and approaches, a new picture of eruption initiation emerges. We highlight this new insight and project on its physical plausibility and the advances that it may bring.
Georgoulis, Manolis K.
2012-07-01
415
SciTech Connect
The extensive publicity of epidemiological studies inferring correlation between 60 Hz magnetic fields and childhood leukemia prompted world wide research programs that have as a goal to determine if low frequency magnetic fields represent any risk for the general population, children or utility workers. While supporting this research effort through EPRI, Con Edison embarked on a technical research program aimed to: characterize magnetic fields as to intensity and variation in time; and investigate practical means to manage these magnetic fields through currently known methods. The final goal of these research projects is to establish viable methods to reduce magnetic field intensity to desired values at reasonable distances from the sources. This goal was pursued step by step, starting with an inventory of the main sources of magnetic fields in substations, distribution and transmission facilities and generating plants. The characterization of the sources helped to identify typical cases and select specific cases, far practical applications. The next step was to analyze the specific cases and develop design criteria for managing the magnetic fields in new installations. These criteria included physical arrangement of equipment based oil calculation of magnetic fields, cancellation effect, desired maximum field intensity at specific points and shielding with high magnetic permeability metals (mu-metal and steel). This paper summarizes the authors` experiences and shows the results of the specific projects completed in recent years.
Durkin, C.J.; Fogarty, R.P.; Halleran, T.M.; Mark, Dr. D.A.; Mukhopadhyay, A.
1995-01-01
416
SciTech Connect
The effect of a strong magnetic field on the stability and gross properties of bulk as well as quasibulk quark matter is investigated using the conventional MIT bag model. Both the Landau diamagnetism and the paramagnetism of quark matter are studied. How the quark hadron phase transition is affected by the presence of a strong magnetic field is also investigated. The equation of state of strange quark matter changes significantly in a strong magnetic field. It is also shown that the thermal nucleation of quark bubbles in a compact metastable state of neutron matter is completely forbidden in the presence of a strong magnetic field. {copyright} {ital 1996 The American Physical Society.}
Chakrabarty, S. [Department of Physics, University of Kalyani, District: Nadia, West Bengal 741 235 (India)]|[Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007 (India)
1996-07-01
417
Frustrated magnets in high magnetic field have a long history of offering beautiful surprises to the patient investigator. Here we present the results of extensive classical Monte Carlo simulations of a variety of models of two dimensional magnets in magnetic field, together with complementary spin wave analysis. Striking results include (i) a massively enhanced magnetocaloric effect in antiferromagnets bordering on ferromagnetic order, (ii) a route to an m = 1/3 magnetization plateau on a square lattice, and (iii) a cascade of phase transitions in a simple model of AgNiO2.
Seabra, L.; Shannon, N.; Sindzingre, P.; Momoi, T.; Schmidt, B.; Thalmeier, P.
2009-01-01
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https://www.meritnation.com/ask-answer/question/three-cells-of-emf-e-2e-and-5e-having-imyernal-resistance-r/current-electricity/10345643 | # Three cells of emf E, 2E and 5E having imyernal resistance r,2r and 3r. Variable reaistance R is shown in the figure. Find the expression for the current.
Dear Student
In given figure there is no variable resistance connected so i am assuming that it is connected in series with batteries.
The given condition is same as If two voltage sources are connected series with their emf in opposition and one battery is connected in series with with their emf in same direction.
If the batteries oppose one another, the total emf is less, since it is the algebraic sum of the individual emfs. When it is reversed, it produces an emf that opposes the other, and results in a difference between the two voltage sources.
So total emf = E+5E-2E
As all resistances whether internal or variable resistance are in series so total resistance
Rnet = r +3r+ 2r +R
So expression for current will be $I=\frac{E+5E-2E}{r+3r+2r+R}=\frac{4E}{6r+R}$
Regards
• 34
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https://www.computer.org/csdl/trans/tc/1977/04/01674843-abs.html | Issue No. 04 - April (1977 vol. 26)
ISSN: 0018-9340
pp: 334-342
I. Koren , Department of Electrical Engineering and Computer Science, University of California
ABSTRACT
The problem considered in this paper is that of generating sequential decision trees (SDT's) for fault diagnosis in digital combinational networks. Since in most applications of the decision tree the final conclusion will be that the network is failure-free, we are interested mainly in decision trees containing minimal fault detection paths. Such a procedure will reduce the cost of verifying the proper operation of the network.
INDEX TERMS
A priori probability of occurrence, combinational logic networks, fault diagnosis, minimal detection set, sequential decision tree (SDT).
CITATION
Z. Kohavi and I. Koren, "Sequential Fault Diagnosis in Combinational Networks," in IEEE Transactions on Computers, vol. 26, no. , pp. 334-342, 1977.
doi:10.1109/TC.1977.1674843
CITATIONS
SHARE
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https://www.nature.com/articles/s41698-017-0029-7?error=cookies_not_supported&code=03d39598-287e-4768-867e-1a48d35dc7e0 | Review Article | Open
# Network-based machine learning and graph theory algorithms for precision oncology
• npj Precision Oncology 1, Article number: 25 (2017)
• doi:10.1038/s41698-017-0029-7
Revised:
Accepted:
Published online:
## Abstract
Network-based analytics plays an increasingly important role in precision oncology. Growing evidence in recent studies suggests that cancer can be better understood through mutated or dysregulated pathways or networks rather than individual mutations and that the efficacy of repositioned drugs can be inferred from disease modules in molecular networks. This article reviews network-based machine learning and graph theory algorithms for integrative analysis of personal genomic data and biomedical knowledge bases to identify tumor-specific molecular mechanisms, candidate targets and repositioned drugs for personalized treatment. The review focuses on the algorithmic design and mathematical formulation of these methods to facilitate applications and implementations of network-based analysis in the practice of precision oncology. We review the methods applied in three scenarios to integrate genomic data and network models in different analysis pipelines, and we examine three categories of network-based approaches for repositioning drugs in drug–disease–gene networks. In addition, we perform a comprehensive subnetwork/pathway analysis of mutations in 31 cancer genome projects in the Cancer Genome Atlas and present a detailed case study on ovarian cancer. Finally, we discuss interesting observations, potential pitfalls and future directions in network-based precision oncology.
## Introduction
The revolutionary large-scale genomic and sequencing technologies developed in the past two decades have enabled an understanding of cancer biology in individual tumors for personalized treatment. Coordinated national and international efforts for cancer genome projects have been launched to characterize tens of thousands of individual tumors by somatic mutation, gene expression, copy number variation, DNA methylation, and various other types of genomic and epigenomic aberrations.1, 2 The large volume of accumulated cancer genomic data has facilitated the identification of precise oncogenes and tumor suppressors for the development of personalized therapeutic strategies. One of the well-recognized new observations in these studies is that cancer is better characterized by frequently mutated or dysregulated pathways than driver mutations, which are often distinct in the tumors of the same type.3 For example, studies have reported that only a few altered genes occur in more than 10% of the samples and that many other altered genes occur in less than 5% of the samples in the same tumor type.4 Furthermore, certain cancer types, such as prostate cancer and pediatric cancers, are not driven by a few somatic mutations or copy number variations, and the mechanism might be better understood in the context of systems biology.4 This important observation has led to a great effort to develop a collection of network-based computational methods to detect cancer pathways or subnetworks by integration of various genomic data, as shown in Fig. 1a, and these methods can be classified into three categories depending on the scenario of applying the analysis pipeline.
Network-based analysis has also attracted considerable attention in drug repositioning to reduce the cost of new drug development by using repositioned existing drugs on novel targets in drug–target networks for precision oncology.5 Based on the hypothesis that drugs tend to be more effective on target genes within or in the vicinity of a disease module in a molecular network,5, 6 several network-based approaches have been used to explore networks of drugs, diseases and targets to reposition drugs for new targets, as listed in Fig. 1b. In these methods, the drug–target relations can be inferred by various measures in the network, combining drug–drug, drug–target, drug–disease and disease–gene relations as shown in the drug–disease–target network in Fig. 1d, e. As summarized in Fig. 1b, these methods can be classified into three categories based on the underlying computational formulation: methods using graph connectivity measures, link prediction methods and network-based classification methods.
The focus of this review article is to provide a comprehensive and unified survey of machine learning and graph theory algorithms for network analysis in precision oncology. We compare the methods by their distinctions in the methodology and mathematical formulations such that the methods can be better applied and improved appropriately for precision oncology. An overview of this article is given in Fig. 1. We not only review the resources of biomedical and molecular networks listed in Fig. 1g and the network-based methods listed in Fig. 1a, b but also present a comprehensive network-based pathway analysis of mutations in 31 cancer genome projects in the Cancer Genome Atlas (TCGA) list in Fig. 1h and a case study on ovarian cancer to show the promise of applying network-based analysis.
## Biomedical and molecular networks
In the literature, various biological and biomedical network databases have been compiled to support network analysis. Typically, the databases have been curated by the integration of high-throughput experimental screening results from studies in the literature and possibly computational predictions supervised by expert knowledge. The networks represent the collections of molecules, phenotypes and drugs as nodes and their relations as edges in graphs. In Table 1, we enumerate existing molecular networks, phenotype similarity networks or ontologies, and drug–target networks and the resources for obtaining these networks. The properties of these networks, including their nodes, edges and graph structures, are also shown in Table 1.
1. 1.
Molecular networks: Biological molecular networks describe relations among molecules, such as protein–protein interactions, gene co-expression, functional similarities, regulatory relations or biochemical reactions. The new-generation high-throughput technologies have provided extensive content to construct such molecular networks. Protein–protein interaction networks are available from several well-maintained databases.7,8,9,10,11,12 Primarily, these networks include physical interactions determined by experiments and computationally derived interactions. Proteome-wide protein–protein interactions capture the interplay among proteins based on the functional associates from co-membership of protein complexes and pathways. A functional linkage network is a more comprehensive compilation of functional relations, physical interactions and co-expression in one network.13,14,15 A transcriptional regulatory network models the molecular interactions between transcript factors/microRNA and target genes to regulate transcript expression.16, 17 A transcriptional regulatory network is a directed graph in which the edges connect a regulator to its targets. A cellular metabolic network can be constructed by the co-membership of biochemical reactions among metabolites and enzymes.18, 19 Several graph structures can be used to represent metabolic pathways, e.g., labeled directed graphs, unions of bipartite graphs (per reaction) and hypergraphs, depending on the level of detail of metabolic reactions to be modeled with the graph.20
2. 2.
Phenotype similarity networks and ontologies: Phenotypes, particularly disease phenotypes, are of special interest for cancer studies. The analysis of diseases in the context of other related diseases can offer insight into their genotypic drivers. Online Mendelian Inheritance in Man (OMIM) is a comprehensive compendium of human genes, genetic phenotypes and documentation of their phenotype–gene associations.21 Phenotype similarity networks can be constructed based on the genetic resemblance22 or the synopsis of the diseases and sometimes by mRNA expression.23 Human Phenotype Ontology (HPO) is another more comprehensive organization of all human disease phenotypes in an ontology.24 The ontology is a directed acyclic graph that can be used as a network structure for learning phenotype–gene associations.25
3. 3.
Drug–target and drug–drug networks: Drug–target associations can be modeled by a bipartite network with connections between the drugs and their targets. The drug–target pairs are typically derived from FDA-approved or experimental drugs and their human protein targets available from various drug databases.26,27,28,29 Several different types of drug–drug similarity networks have been derived for drug repositioning. Drug–drug relations can be inferred based on similarity of molecular basis, chemical substructure, and phenotypes, such as known drug-indication relations, co-membership in drug combinations, and co-morbidity of diseases.30
## Network-based analysis of personal genomic profiles
The goal of applying network-based analysis to personal genomic profiles is to identify aberrant network modules that are both informative of cancer mechanisms and predictive of cancer phenotypes. These methods can be classified into three categories based on the design of the analysis pipeline in different scenarios, as shown in Fig. 2. In these scenarios, the detection of the network modules facilitates two other goals: predicting cancer phenotypes and detecting driver genes. Depending on how the network information is processed in the pipeline, the inputs and the outputs to the predictive models or network analysis methods can differ. Below, we describe the three categories of the methods listed in Fig. 1a and then discuss the advantages and limitations of each of the categories.
### Model-based integration of whole-genomic profiles and a network
Model-based integration formulates a single unified machine learning framework to integrate genomic profiles with a network as illustrated in Fig. 2a. The core technique is to introduce a network-based regularization into machine learning models such that the coefficients learned on the feature variables form dense subnetworks. The most commonly used network-based regularization is the graph Laplacian regularizer shown in Fig. 3a. The graph Laplacian was first introduced for spectral graph analysis31 and then used for semi-supervised learning in machine learning.32, 33 The graph Laplacian regularization is a summation of smoothness terms on the variables to encourage similar coefficients on the genes or other genomic features that are connected in the network. Below, we describe the graph Laplacian regularized methods in different learning frameworks as shown in Fig. 3b–e. To precisely describe the models, we also list all the necessary notations in Table 2 and the exact mathematical formulations of the methods in Supplementary Table S2.
In Fig. 3b, the widely used regression and survival models are extended to include the graph Laplacian constraint for the analysis of genomic data. The paper34 proposed a network-constrained linear regression procedure that combines a graph Laplacian constraint with the L1-norm sparse linear regression to capture the relations among the regression coefficients.35 This network-based linear regression is equivalent to a standard LASSO optimization problem.34 The paper36 proposed a network-based Cox proportional hazards model (Net-Cox) for survival analysis. In Cox regression, the objective is to learn the regression coefficients β and the baseline hazard function h0(t) such that the instantaneous risk of an event at time t for a patient x i can be estimated by $h t ∣ x i = h 0 ( t ) exp x i T β$. Similarly, the graph Laplacian constraint is introduced on the regression coefficients β. By alternating between maximization with respect to β and h0(t), a local optimum can be found.
As shown in Fig. 3c, the graph Laplacian constraint can also be introduced into linear classification models such as logistic regression37 and support vector machines (SVMs).38 Given the binary response vector y = (y1, ..., y n )T with y i {1, 0}, a Bernoulli likelihood function minus both the L1-norm and the graph Laplacian constraints is maximized to learn the linear coefficients. In the model, $p x i = exp β 0 + x i T β 1 + exp β 0 + x i T β$ is the probability that the ith sample is in class 1. The elastic-net procedure can be applied to maximize the regularized cost function. The paper38 proposed a network-based SVM. Given the +1/−1 binary response vector y, the network-constrained SVM can be formulated as the addition of the hinge loss $∑ i = 1 n 1 - y i β 0 + x i T β +$ and the graph Laplacian constraint, where the subscript “+” denotes the positive part, i.e., z+ = max{z, 0}.
Semi-supervised learning methods can more conveniently explore the structures among both the genomic features and the patient samples by learning with the graph Laplacians,39,40,41 as shown in Fig. 3d. In the bipartite graph formulation introduced in the paper,40 gene expression data are represented as a bipartite graph with weighted edges between patient samples and genomic features. The bipartite graph captures the co-expression among the genes and the samples as bi-clusters in the graph such that both the sample clusters and feature modules are explored. In the hypergraph formulation introduced in the papers,39, 41 the gene expression data are represented as weighted hyperedges on the patient nodes, and a graph Laplacian on the hypergraph can be introduced for semi-supervised learning on the patient samples. An additional graph Laplacian of a protein–protein interaction (PPI) network is then introduced to incorporate network information among the genomic features.
It is also possible to regularize non-negative matrix factorization (NMF) models with a graph Laplacian,42, 43 as shown in Fig. 3e. NMF aims to find two non-negative matrices U m × k and H n × k whose product can accurately approximate the data matrix X with X ≈ UHT. Combining the geometrically-based constraint with the original NMF leads to the graph-regularized NMF, where Tr() denotes the trace of a matrix.
### Preprocessing integration to detect network-based features
The preprocessing integration methods comprise two steps, as illustrated in Fig. 2b. First, the genomic profiles and the network are processed together to generate network-based features; second, standard learning models are applied with the network-based features for predictions. In this scenario, the integration of network and genomic data occurs before applying a learning model. The paper44 first proposed a graph algorithm to detect discriminative subnetworks for classification of patient samples. Highly discriminative genes are used as seed genes in a greedy search in a PPI network to find discriminative subnetworks, and then gene expression in each subnetwork is normalized as one feature value for classification with standard logistic regression. A similar approach was later proposed for application with features of discriminative pathways instead of subnetworks.45 In this approach, the gene expression in a pathway is normalized as one feature for the collection of pathways from a molecular signature database.46 The paper47 used disease-specific subnetworks as features, where a set of known disease genes are first mapped into the PPI network and then the subnetworks of the disease genes are identified as disease module features. The paper48 proposed implementing label propagation on the mutation data of each patient on a PPI network to generate network-smoothed features for classification of the patients. The paper49 proposed to find a small subnetwork to connect all differentially expressed genes in a PPI network and then use the genes in the subnetwork as features to classify patient samples. This setting is the Steiner tree problem in graph theory, and a heuristic algorithm coupled with randomization was designed to combine multiple suboptimal Steiner trees to find an optimum solution with a higher probability.
This category of algorithms is a very useful generalization of the earlier gene-set-based methods50, 51 since the network structures suggest dynamic modules among the genes rather than a fixed set. These modules can be data-specific and disease-specific for improved results. Thus, the data-driven subnetwork discovery introduced by these methods is a key improvement over previous studies.50, 51
### Post-analysis of oncogenic alterations in networks
The post-analysis integration methods also consist of two steps, as illustrated in Fig. 2c. First, the genomic profiles are analyzed to generate a list of oncogenic alterations; second, the detected alterations are analyzed in the network. In this post-analysis integration, the network information is integrated in the analysis after the oncogenic alterations are first detected by standard statistical methods. The purpose of these methods is to assess how cancer-driving alterations disrupt a normal cellular system by examining the influences on network components.
The circuit flow algorithm52 first identifies differentially expressed genes and then the genomic aberrations by mutations and copy number variations (CNVs) associated with the differential gene expression. Next, a current flow algorithm is applied to find causal paths from the causal genes (altered genes) to the target genes (differentially expressed genes) in a PPI network. Finally, the causal genes are selected by a set-covering algorithm to explain all the differentially expressed target genes.
HotNet53 first maps gene alterations in a gene network and then employs a diffusion kernel54 to build an influence graph with the edges weighted by the influence between each pair of genes. Then, a combinatorial problem is formulated to find the subnetworks of genes altered in a significant number of patients. Similarly, TieDIE55 and HotNet2,56 an extension of HotNet, apply network diffusion to analyze multiple types of genomic alterations, and NetPathID57 applies network diffusion to analyze CNVs in 16 types of cancers.
PARADIGM58 is a probabilistic graphical model framework used to model the gene transcription, translation and post-translational events. Each gene is modeled by a factor graph of DNA copy numbers, gene expression, protein levels and protein activities. The factor graphs of genes are connected based on their regulatory relations in a pathway. The genomic and proteomic data are analyzed in the graphical models for the inference of pathway activities in each patient to derive integrated pathway activity (IPA) scores. The significantly altered genes/pathways can be identified using the IPA scores.
The mutual exclusivity module (MEMo) method59 is another widely used method in the TCGA project. MEMo first builds a matrix representation of genes that are significantly altered by mutations or CNVs. Then, the altered genes are connected by their proximal in the HPRD PPI network.7 Finally, the cliques (a subgraph with all the gene pairs connected) are identified to analyze the mutual exclusivity in the patient data.
Signaling pathway impact analysis (SPIA)60 and mixed integer programming (MILP)61 are two examples of earlier pathway-based methods for genomic data analysis. SPIA applies an iterative algorithm similar to a random walk to measure the pathway perturbations in the regulatory network such that the impact of differentially expressed genes on a pathway can be evaluated.60 MILP is an optimization model to predict flux activity states of genes based on gene expression and a metabolic network.61
### Comparison of the methods
Network-based analysis of genomic data is based on the assumptions that cancer-driven aberrations often target different genes in the same pathway or subnetwork in the molecular network and that such systematic behavior can be observed as a coordinated change of genes’ functions in pathways or network modules. Network-based analysis is an effective approach because it has been observed that mutated genes in a cancer pathway can either co-occur in the same patients or be mutually exclusive among the patients, and the systematic behavior is a more detectable and interpretable signal for the assessment of functional impacts of the aberrations.59 It has also been shown that feature selection smoothed by graph Laplacian regularization based on the gene co-expression network is highly robust and generates more reproducible feature selections across independent datasets.62 Thus, the network-based approach is both well motivated and validated.
The three categories of methods have different relative advantages and disadvantages. Model-based integration methods are a fully supervised approach for both outcome prediction and subnetwork detection. The subnetworks are jointly discovered to contrast the control/case groups in the study based on a global optimization strategy, and thus these methods typically perform better in outcome prediction. In addition, the models can be tuned by a few clearly defined parameters, making it possible to train the models with cross-validation in contrast to the two-step methods in the other categories. The disadvantage is the need for more sophisticated optimization techniques, which are often less scalable. The preprocessing integration methods are more flexible in detecting customizable subnetwork features such that the detected features clearly reflect the hypothesized network-based characteristics. For example, the size and density of discriminative subnetworks can be precisely specified. However, it is not possible to guarantee that the detected subnetwork features are optimal features for prediction with the standard learning model in the second step. The post-analysis integration methods focus on associating mutations or other DNA aberrations with differential expression or certain other molecular phenotypes in the network context. Thus, these methods are highly informative regarding cancer mechanisms in the network.
In model-based integration, Graph LASSO is another choice of graph-based regularization other than the graph Laplacian regularizer.63 Graph LASSO imposes a LASSO loss on each pair of connected variables in the network rather than a squared error as with the graph Laplacian regularizer. The LASSO loss terms force the coefficients of the connected pairs to be identical such that the inconsistent pairs are “sparse.” In practice, the assumption can be too strong in networks with overlapping clusters. In addition, optimization of Graph LASSO-constrained models is generally challenging, while the graph Laplacian regularizer is a quadratic constraint that is relatively straightforward to optimize. Thus, Graph LASSO is a less common choice for network-based integration methods.
## Network-based methods for drug repositioning
Network-based algorithms have also been developed for drug repurposing by exploring drug–drug similarities, drug–target relations and gene-gene relations. These methods can be largely classified into three categories, i.e., graph connectivity measures, link prediction models and network-based classification methods, as illustrated in Fig. 4. The methods reviewed under each category are also listed in Fig. 1b. Below, we describe and compare the methods in the three categories.
### Graph connectivity measures
The methods in this category are based on measuring the connectivity among the nodes in the graph, such as neighboring relations, the number of shared neighbors and shortest paths, to derive drug–drug, drug–target or drug–disease relations, as illustrated in Fig. 4a. Several early studies64, 65 showed that drugs sharing similar chemical structures, transcriptional responses following treatment and text mining analysis often share the same target, where the implication is that the drug–drug network based on the similarities can be used to reposition a drug for the targets of similar drugs. The paper64 derived drug–drug similarities based on mining the side-effect description from medical symptoms in the Unified Medical Language System ontology. The paper65 developed a method to predict similarities in terms of drug effect by comparing gene expression profiles following drug treatment across multiple cell lines and dosages. Both studies validated the correlation between drug–drug similarity and the likelihood of two drugs sharing a common protein target. Based on the observations, the paper66 proposed a recommendation technique for predicting drug–target relations based on the drug–drug similarity matrix W computed based on the structural similarity of the drugs and sequence similarity of their targets and the known drug–target matrix A. By a simple multiplication (R = WA), the scores in matrix R can be used to derive a ranking of the candidate targets against each drug.
The paper23 performed a large-scale analysis of ~7000 genomic expression profiles in the Gene Expression Omnibus with human disease and drug annotations to create a disease–drug network consisting of drug–drug, drug–disease and disease–disease relations. The study shows that the derived disease–disease relations are highly consistent with the definition in the Medical Subject Headings disease classification tree and that the drug–disease relations can be used to generate hypothesized drug repositioning and side effects. The paper6 further generalized the inference to drug–disease proximity in the network by the hypothesis that an effective drug for a disease must target proteins within or in the immediate vicinity of the corresponding disease module in the molecular interaction network. They applied a shortest-path-based measure coupled with a randomization normalization technique to derive the drug–disease proximity scores for the inference.
A recent work in the paper67 performed a correlation analysis of disease modules and drug targets in the functional linkage network. The differentially expressed disease genes and the drug–target genes are first overlapped in the functional linkage network, and a mutual predictability score is then computed based on the neighboring relations among the genes to evaluate the repositioning of the drug for the disease.
Link prediction models predict the relations between drugs and targets based on the global structures of the known interactions in the networks with matrix completion or random-walk approaches, as illustrated in Fig. 4b. The paper68 predicted drug–target relations for drug repositioning based on a network of three types of relations: drug–drug structural similarity, target–target sequence similarity and drug–target relations from DrugBank.26 It was shown that exploring the network topology outperforms simple inference rules by graph connectivity measures such as similar drugs sharing the same target or similar targets sharing the same drug. The paper69 applied an information-flow approach on a heterogeneous network of drug–drug, disease–disease and target–target similarities along with the known disease–drug and drug–target relations. The algorithm iteratively updates the disease–drug and drug–target relations and converges to stationary scores for the prediction of their relations.
The paper70 introduced a bipartite graph-learning method based on kernel regression to learn a co-mapping of drugs in chemical space and targets (proteins) in genomic space into a common pharmacological space. In the pharmacological space, the correlation between compound-protein pairs can be conveniently calculated to predict their interactions for drug repositioning.
The paper71 proposed a collaborative matrix factorization method to factorize known drug–target relations to predict new relations constrained by the drug–drug similarity network and the target–target similarity network. The paper72 proposed a manifold regularization semi-supervised learning method in which two classifiers in drug space and target space are learned and then combined to give a final score for drug–target interaction prediction. The paper73 applied several random-walk methods on a heterogeneous network of drug–drug similarities, target–target similarities and drug–target relations such that the global structure among all the networks can be used to improve the prediction of new drug–target pairs.
### Network-based classification methods
Network-based drug repositioning can also be reformulated as a classification problem such that standard classification methods can be applied to predict the new targets of each drug, as illustrated in Fig. 4c. These methods first extract the network topological features for all the targets in the networks. For each drug, a classifier can be trained with the known targets of the drug as positive samples and the others as negative samples. The learned classifiers can then be used to predict the new targets in the test set for each drug. The paper74 proposed mapping disease-specific differentially expressed genes into a PPI network and using network topological features to detect new drug targets based on the known targets from the drug–target database by logistic regression. The paper75 also applied a supervised bipartite model to predict the probability of each drug–target interaction based on the known drug targets as labels and the target–target interactions as features, where the bipartite model was augmented with additional training samples from the neighboring drug–target relations.
The paper76 constructed a drug–drug kernel matrix based on chemical structure similarities and a target–target kernel matrix based on sequence similarities. For each drug, using the known targets as the positive training samples, an SVM classifier is built with the target–target kernel matrix to classify the candidate genes for new targets. In addition, for each target and using the known drugs as the positive training samples, an SVM classifier is built with the drug–drug kernel matrix to classify the drugs for new repositioned drugs. The paper77 adopted a similar approach with two additional advanced kernel methods, applying diffusion-types of kernels to integrate both the drug–drug kernel matrix and the target–target kernel matrix to predict the new targets of a drug or the new repositioned drugs for a target.
### Comparison of the methods
The three categories of methods have different relative advantages and disadvantages, as shown in Fig. 4d. Graph connectivity measures are straightforward to implement based on standard graph algorithms, and the prediction results are easy to interpret with the edges and the paths in the graph. However, the prediction performance is typically worse since only relatively local information of the networks is considered by the graph algorithms. Link prediction models retrieve the global structures of the networks to predict drug–target interactions for better prediction performance. The disadvantages are the lack of a satisfactory interpretation of the predictions and that the implementation of the models often relies on advanced optimization algorithms. When sophisticated optimization is required, the scalability can be poor. Network-based classification methods are more accurate for repositioning drugs with many known targets as the training samples but are not applicable to drugs with few or no known targets. The prediction results can be interpreted by the network topological features extracted from the networks, depending on the feature extraction strategy.
Another important aspect of the comparison is whether a method can generate de novo predictions for drugs with no known targets or gene targets with no known drugs. Graph connectivity measures are often more biased towards highly connected nodes in the graph such that new drugs or less-studied genes typically receive low rankings. Thus, de novo predictions are rarely made by graph connectivity measures. With no positive training pairs available, the network-based classification methods simply abandon the de novo cases. Link prediction models are often the most capable of making de novo predictions because global topological structures are generally less biased after proper normalization and control by randomization.
## Network-based analysis of TCGA mutation data and a case study on ovarian cancer
To better discuss the network-based methods, we performed a network-based analysis of the mutated genes in the 31 cancer genome projects in TCGA78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101 and summarized the enriched KEGG pathways102 in Fig. 5. For the analysis, the mutation frequencies among the patients in the 31 TCGA provisional studies were downloaded from cBioPortal for Cancer Genomics.103 In the network-based analysis, label propagation (λ = 0.5)48, 62 as described in Table S2 in the Supplementary Information was applied to the HPRD PPI network7 in each cancer study to capture the highly mutated subnetworks. The initialization was the gene mutation frequency among the patients in each cancer study for label propagation. The summation of the stationary scores of the genes in a KEGG pathway is compared with the scores of 10,000 random gene sets of the same size to derive p-values. In the analysis without the network, the highly mutated genes in each cancer type are overlapped with KEGG pathways with enrichment analysis to derive p-values by hypergeometric test. This network-based analysis clearly detects more significantly mutated pathways than the analysis without using the network, as shown in Fig. 5a, b, respectively.
Interestingly, the network-based analysis in Fig. 5a indicates that the AMPK signaling pathway is affected in breast cancer (BRCA) and uterine corpus endometrial cancer (UCEC). Prior studies demonstrated that BRCA patients receiving metformin, a pharmacological activator of AMPK, showed complete pathologic response, implicating the role of AMPK in BRCA.104 Similarly, the loss of the AMPK activator LKB1 promotes endometrial cancer progression and metastasis,105, 106 implicating the AMPK pathway in endometrial cancer, and metformin inhibits endometrial cancer cell proliferation.107 The HIF-1 pathway has been predicted to be affected in renal clear cell carcinoma (KIRC), BRCA, endometrial cancer (UCEC), glioblastoma multiforme (GBM), cervical cancer (CESC), and lung cancer (LUAD), and these results are consistent with prior studies implicating the VHL/HIF-1 pathway in these cancers.90, 108 The Hippo pathway has been predicted to be affected in colorectal cancer, renal papillary carcinomas, stomach cancer, and liver cancer, and these results are consistent with recent cancer genomic studies.97, 109 Finally, the PI3K-Akt pathway has been identified as one of the most frequently affected pathways in several cancer types, and several components of this pathway were reported to be mutated or amplified in various cancer types.110 Collectively, these results suggest that network analysis can identify clinically relevant pathways that are altered in different cancer types.
In the case study on the ovarian cancer patients shown in Fig. 6, the mutation data of the 316 TCGA ovarian cancer patients were downloaded from the Xena Public Data Hubs.111 Similar to the study in the paper,48 label propagation (λ = 0.1) was applied on the same HPRD PPI network in each patient to detect the patient-specific highly mutated subnetworks. The initialization was 1 for the mutated genes and 0 for the other genes and then normalized to sum to 1. Similarly, the summation of the stationary scores of the genes in a KEGG pathway was compared with the scores of 10,000 random gene sets of the same size to derive the p-value. In the analysis without the network, the mutated genes in each patient are overlapped with KEGG pathways with enrichment analysis to derive p-values by hypergeometric test. Hierarchical clustering was applied to cluster the patients into three groups using the –log10 (p-values) as features. The network-based analysis informs a clustering of the patients by a significant relevance to survival (Fig. 6c). Notably, three subgroups of tumor samples can be identified from the network-based analysis shown in Fig. 6c, compared to four subgroups in the mutation-based analysis without the network in Fig. 6d. Although subgroups identified by mutation-based analysis without the network show no significant association with disease-free survival, two of the subgroups detected by the network-based analysis (Subgroup 1 and Subgroup 3) show significant association with disease-free survival relative to Subgroup 2. Interestingly, Subgroup 1 has the highest copy number alterations, whereas Subgroup 3 has the highest number of pathway alterations. These results are analogous to the spectrum of somatic alterations described by ref. 112. Although those authors placed ovarian cancer in class C, defined by extensive copy number alterations, the spectrum of somatic alterations can be further described as subgroups with higher copy number changes, mixed, and higher mutations within ovarian cancer. This case study shows that via network analysis, several subtypes of ovarian cancer can be grouped together for further assessment of clinical values, such as occurrence, relapse and treatment resistance. This information may also be valuable for the design or assessment of treatment strategies. Collectively, the network analysis unveils important cancer pathways and their correlation to subtypes of cancers that would not be identifiable by original mutation data analysis.
## Discussion
Precision oncology tailors cancer treatment and repositions drugs based on personal genomic information. There are several promising aspects of the application of network-based analysis in precision oncology. With a network to capture the molecular organization in the cellular system, genomic data analysis is both more accurate and descriptive. The smoothness constraint introduced into the model-based integration methods is helpful in eliminating false positives and false negatives in high-dimensional genomic data. The network analysis identifies molecular targets in the context of pathways or interaction partners in a subnetwork that are interpretable for molecular mechanisms. For example, in the case study in Fig. 6a, the mutation information of each individual patient is propagated on the PPI network to detect the patient-specific subnetwork and improve the quality of the patient clustering by a significant relevance to survival. As a consequence, network-based analysis often reports consistent marker genes across different studies of the same cancer40 or more comparable results in pan-cancer analysis.56 Collectively, it is evident that network-based methods employ molecular and biomedical networks to extract useful personal genomic information, and build better predictive models for target identification, phenotype prediction and drug repositioning.
Conceptually, network-based analysis also adopts mutation patterns that are mutually exclusive or co-occurring. Mutually exclusively mutated genes are often located on the same pathway, and network analysis propagates the mutually exclusive signals to identify the pathway by a significant signal. Co-occurring mutated genes in a pathway/dense network module also mutually strengthen the mutation signals. The results in Fig. 6 clearly support that the mutation patterns are accurately captured in the case study on ovarian cancer by label propagation.
In drug repositioning, both molecular networks and drug–drug or phenotype similarity networks play important roles. It has been repeatedly observed that genes associated with the same (or similar) diseases tend to lie in a dense module in the PPI network. This observation has motivated effective network-based methods to predict new disease genes.43 The analysis of gene modules in the PPI of similar diseases has also suggested associations between diseases and gene functions or pathways.43 When drug targets and disease genes are analyzed together in the PPI network, their proximities are useful for drug repositioning.6
The methods compared in Figs. 2 and 4 have different relative advantages and disadvantages. The considerations involve a variety of key properties, including the performance of the methods, the interpretation of the results, the difficulty of implementation, the scalability to genome-wide analysis, and the characteristics of the training data. The appropriate choice of a network-based method for a particular analysis can be customized based on the information gained from these comparisons. For example, drugs with more known targets can be repositioned by the network-based classification models, while drugs with no known targets in the candidates can be repositioned by the link prediction methods. Depending on whether the analysis must be highly scalable to a huge network, simple graph connectivity measures or link prediction methods can be used.
In the application of network-based analysis, there are also several practical issues and limitations.
1. 1.
Molecular networks often contain biased information. Well studied genes tend to have more connections in the PPI network, and they are also targets of more drugs and are associated with more disease phenotypes. Typically, it is important to exercise normalizations and repeat the experiments on randomized networks to assess the statistical significance of the results. The biases also prevent the prediction of de novo disease genes or target genes if the gene has no association with known diseases or is not a target of any drug.25
2. 2.
The empirical results of network-based methods rely on tuning parameters. The parameters often balance how much belief is imposed on the network topologies. When excessive weights are assigned to the network topology, there will be an “over-smoothing” effect such that nearly uniform scores are expected among the genes in even large and sparse neighborhoods. Thus, a proper procedure for determining the appropriate (optimal) parameters is critical, for example, by applying cross-validation and wet-lab validation.36
3. 3.
Commonly, a molecular network describes a general relation, such as protein–protein physical interaction or functional linkage. In some cases, the relations can be either positive or negative, e.g., gene co-expression. A practical approach is to apply a signed graph Laplacian.113 The models applied with a signed graph Laplacian can be solved in a manner similar to those with the normal graph Laplacian by the same algorithms.
Finally, this article targets the scope of precision oncology, including steps for understanding cancer mechanisms, finding targets and repositioning drugs, while previous survey studies have focused on detecting cancer-driven aberrations and understanding of the aberrations in molecular networks/pathways.4, 114, 115 This article also surveys several categories of algorithms, including model-based integration and preprocessing integration with machine learning methods, while previous reviews4, 114, 115 primarily surveyed the methods in one of the three categories, namely, post-analysis integration of oncogenic alterations in networks. Thus, this article offers a different scope and a more comprehensive survey of computational methods.
## Future directions
Several challenges remain in the application of network-based analysis in precision oncology. These challenges concern the data quality, deployment for research or clinical use, and scalability of network analysis.
To precisely model the molecular interactions and drug–target relations, networks of better quality are required. It is known that most molecular networks and drug–target databases are incomplete and biased towards well-studied proteins/genes. Thus, continuing effort on the improvement of the networks with additional experimental data is important. In addition, network modeling with higher resolution is also crucial to model complex molecular functions at higher precisions. For example, proteins are present in the isoforms of genes, and thus isoform–isoform interactions are the true interactions to model in a network116,117,118; mutations or other structure variations of a protein can also change the protein–protein binding or drug–protein docking in a specific tumor. Furthermore, even within each tumor, heterogeneous cell populations exist, and the drug targets and molecular interactions could be different for each cell population if measured by single-cell RNA sequencing.119 To partially address this issue, several computational methods for quality control of PPI screening have been proposed to reduce the number of false-positive and false-negative PPIs due to spurious errors and systematic biases from the high-throughput techniques.120, 121 Currently, it is still impossible to construct these more accurate networks at a large scale due to the limitation of the current high-throughput experimental methods for measurement of molecular interactions or drug screening.
While many network-based methods have been developed to support precision oncology, the implementations of the methods are independent, with non-standardized tools that are never easily accessible as a useful collection to oncologists for research or clinical use. Thus, there is a strong need to develop a software platform that integrates standardized biomedical, biological network data, and analytic software components to support comprehensive network-based analysis of patient genomic data and drug repositioning for precision oncology. This platform should be based on a sophisticated system design to meet oncologists’ requirements and support customization of the analysis pipeline. The concept of part of such a platform was proposed in the paper5 as an integrative network-based infrastructure to identify new druggable targets and repositionable drugs through the targeting of significantly mutated genes identified in human cancer genomes. In the future, the existing tools can be reimplemented as apps on a platform such as Cytoscape122 or another software environment similar to GALAXY for NGS data analysis123 to facilitate the development and deployment of the software system for precision oncology.
Finally, scalability is always an issue in network-based analysis since it is common to model millions of genomic features, hundreds of thousands of drugs and tens of thousands of phenotypes in a very large network. For example, in an isoform–isoform interaction network, hundreds of thousands of nodes are contained in a single graph that cannot be loaded onto a computer with less than 100 GB of memory. Such big-data analysis will require more scalable algorithms and efficient computing platforms. For example, the standard label propagation can be applied to low-rank approximations of big graphs, enabling work with networks of millions of nodes.124, 125 Parallel implementations of the network-analysis methods, especially the optimization algorithms in those model-based approaches, are also necessary.
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## Acknowledgements
The results are based upon data generated by The Cancer Genome Atlas established by the NCI and NHGRI. Information about TCGA and the investigators and institutions who constitute the TCGA research network can be found at http://cancergenome.nih.gov. The dbGaP accession number to the specific version of the TCGA dataset is phs000178.v9.p8. This research work is supported by a grant from the National Science Foundations, USA (NSF III 1149697).
## Author information
### Affiliations
1. #### Department of Computer Science and Engineering, University of Minnesota Twin Cities, Minneapolis, MN, USA
• Wei Zhang
• & Rui Kuang
2. #### Department of Cancer Biology, University of Kansas Medical Center, Kansas City, KS, USA
• Jeremy Chien
3. #### Department of Biochemistry, Molecular Biology and Biophysics, University of Minnesota Twin Cities, Minneapolis, MN, USA
• Jeongsik Yong
### Contributions
W.Z. and R.K. drafted the manuscript and designed the experiments. W.Z. performed the experiments and analyzed the results. J.C. and J.Y. analyzed the results. W.Z., J.C., J.Y. and R.K. wrote the manuscript.
### Competing Interests
The authors declare that they have no competing financial interests.
### Corresponding author
Correspondence to Rui Kuang. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "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": 0.7593986392021179, "perplexity": 4729.37087596517}, "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-2017-47/segments/1510934804518.38/warc/CC-MAIN-20171118021803-20171118041803-00535.warc.gz"} |
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# Data, 2017
### JADE data in MPCDF Instructions to access the data
Updated in March 2017 by Andrii Verbytskyi | {"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": 0.35425156354904175, "perplexity": 2382.061163602762}, "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/1558232257002.33/warc/CC-MAIN-20190523023545-20190523045545-00324.warc.gz"} |
https://www.physicsforums.com/threads/undergraduate-studying-physics-of-fire.66260/ | # Homework Help: Undergraduate studying physics of fire
1. Mar 7, 2005
### thomate1
Dear friends,
I am a undergraduate studying physics.Until now I have a doubt which I was not able to tackle. What is fire ?
2. Mar 7, 2005
### Erienion
Fire is a general term describing combustion with oxygen.
3. Mar 7, 2005
### HallsofIvy
More of chemistry problem than a physics problem- fire is an "exothermic oxygenation reaction"- that ought to impress your teacher!
4. Mar 7, 2005
Daniel.
5. Mar 7, 2005
### cepheid
Staff Emeritus
Can this be clarified further? I've always been told that the exothermic combustion reaction releases energy in the form of light and heat, and that is what we see. But what of the flame itself? Is it physical, i.e. made up of matter, as dexter said? I tend to believe him, just looking at the structure it seems to have and the way it flickers and moves. But what physical matter that constitutes the flame is glowing? Ionised gas eh? From where? Thanks. It is just something I have always wondered about. The smoke is not puzzling...particulates from whatever the fuel source is (wood, etc). The flame, however, is.
6. Mar 7, 2005
### dextercioby
Yes,it's a plasma,as i said and as any physics student learns in first lecture on plasma physics.A mixture of all possible microscopical objects.Heat is just KE of the components,while visible (and not) radiation comes from the continuous ionization & recombination of electrons,ions,atoms & molecules.
Daniel.
7. Mar 15, 2005 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8331599831581116, "perplexity": 3876.355317031782}, "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/1539583512015.74/warc/CC-MAIN-20181018214747-20181019000247-00309.warc.gz"} |
https://www.brightstorm.com/math/calculus/applications-of-the-derivative/optimization-using-the-closed-interval-method-2/ | # Optimization Using the Closed Interval Method - Problem 2 3,867 views
Another common optimization problem is, when given an amount of fencing, to find the maximum area the fence can contain. Remember that the formula for area of a rectangle with one side x and one side y is A=xy, and the formula for perimeter is P=2x+2y.
The perimeter is often going to be the length of fencing you have to work with. So, the fencing = 2x + 2y, or y = (fencing - 2x)/2. You can plug this into the formula for area, so you get A=x * (fencing - 2x)/2. When expanded, this becomes A=(fencing*x - 2x2)/2. Then, given this function, you can find the maximum as you would for any other function -- by finding zeros of the derivative. The maximum will be the value of x such that . From the value of x, you can find the value of y with the formula used previously. These values of x and y are the dimensions of the fencing such that you can construct a pen with the maximum area.
We are talking about optimization problems. And no discussions about optimization problems, will be complete without a fence problem. Here’s an example of what the fence problem is about. A farmer needs to build a rectangular pen for her pigs. The pen will bordered on one side by her barn. She has 400 feet of fencing, with which to enclose the other three sides. How should she construct the fence, in order enclose the maximum area?
So I have a picture here of the situation. Here’s her barn just drawn the side of her barn. And we are going to build the three sides to enclose a rectangular area.
We are assuming that the barn is as long as we need it to be. So if the best area to build is really long and narrow, we can build as long as we like. So that’s maybe not a realistic assumption, but to assume differently will make the problem a little harder. So let’s make that assumption.
The first thing we need to do, is assign some variables here. I’m going to call this width here x, just because it's horizontal. This would also be x, and this will be y. And let me remind myself it’s an optimization problem. I’m either maximizing or minimizing something. In this problem I want to find maximum area, that’s the key.
That tells me what function I need to define here, and that’s the one that I’m going to find the maximum value of. So an area function. The area of this. Unfortunately, the area of this rectangle is x times y. I’m saying unfortunately because this is area defined in terms of two variables. We are going to need this in terms of one variable. So I’m going to have to get rid of one of these variables, using some kind information from the problem. Some kind of constraint and that’s usually what the second piece of information is called.
There is a constraint here. She has 400 feet of fencing. So somehow I have to build that into the problem. These three sides will have to be at most 400 feet. There are two x’s and one y. So that means 2x plus y equals 400. This is called the constraint. And it will allows me to solve for y, make a substitution and then my function will be entirely in terms of x. So let me do that right now. Y equals 400 minus 2x. So my area becomes x times and y is 400 minus 2x. And I’ll write this as 400x minus 2x².
That’s my area function. So I need to maximize this. I’m using the close interval method that I want to. Before I do that, I have to define what domain makes sense for this function. This is a really important step. Because, in order for you to use the close interval method I need this function defined on a closed boundered interval. It needs to have a left and right endpoint.
Think about what values of x makes sense in this problem. It certainly makes sense that be at least 0. It can’t be less than 0, that wouldn’t make sense. And so I’m going to make the requirement that x be bigger than or equal to 0. And, the biggest x can be, if you can imagine this getting longer and longer and stretching out, and y becoming smaller and smaller, the biggest x can be is 200. Because I have a 200 piece of fence here and then another one right above it, just right next to each other. So 200 is the biggest.
I’ve got my closed interval. I’ve got my function. All that's left to do, is closed interval method. Find the derivative, determine the critical points and make a table of values. So what’s A'?
The derivative of this function would be 400 minus 4x. Critical points are where the derivative is 0 or undefined. This derivative is 0 when 400 equals 4x. So when x equals 100, so, right in the middle. Let’s make our table of values.
We have x and we have A(x). We have the two endpoints 0, and 200, and we have the one critical point, 100. And I need to write the areas here, here and here of each of these values. When I look at the area function I think this is actually easiest formula to use. I plug in 0, and I’m going to get an area of 0. And it kind of makes sense. When x is 0, all I have is one long vertical piece mashed laid up against a barn. The 0 area enclosed by that, 0. When x is 200, this is going to be 400. 400 minus 400 is 0. That also makes sense. When x is as long as it can possibly be, I just have 200 foot lengths of fence right against one another, enclosing the 0 area.
But when x is 100, I have 100 here, 400 minus 200 is 200. 100 times 200, 20,000. By virtue of being the biggest value, that’s got to be the absolute maximum. So this is the absolute maximum area. Now what were we asked?
Let’s just take another look back. It says how should she construct the fence in order to enclose the maximum area? I wasn’t actually asked of the maximum area. I was asked for the dimensions that give me the maximum area. So I need to say what x is and what y is, in order to create the fence, that encloses the maximum area.
So she needs x to equal 100. And if x equals 100, just plug them back into the picture. This is a 100 and this is 100, she has 200 left for y. And so y needs to be 200. So that’s how long the dimension should be. She should build the fence 100 feet wide, and 200 feet long. 200 feet being parallel to the side of the barn. | {"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": 0.8300440907478333, "perplexity": 301.7247578584927}, "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/1398223207046.13/warc/CC-MAIN-20140423032007-00547-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://su.wikipedia.org/wiki/Rohangan_hapa | # Rohangan hapa
Baca ogé pedaran Wikikamus ngeunaan kecap
Rohangan hapa atawa vacuum nyaéta volume rohangan anu pada dasarna kosong ku materi, saperti tekanan gas éta rohangan kurang ti tekanan atmosfir baku.
Artikel ieu keur dikeureuyeuh, ditarjamahkeun tina basa Inggris. Bantosanna diantos kanggo narjamahkeun.
The Latin term in vacuo is used to describe an object as being in what would otherwise be a vacuum. The root of the word vacuum is the Latin adjective vacuus which méans "empty," but space can never be perfectly empty. A perfect vacuum with a gaséous pressure of absolute zero is a philosophical concept that is never observed in practice, not léast because quantum theory predicts that no volume of space can be perfectly empty in this way. Physicists often use the term "vacuum" slightly differently. They discuss idéal test results that would occur in a perfect vacuum, which they simply call "vacuum" or "free space" in this context, and use the term partial vacuum to refer to the imperfect vacua réalized in practice.
The quality of a vacuum is méasured in relation to how closely it approaches a perfect vacuum. The residual gas pressure is the primary indicator of quality, and is most commonly méasured in units called torr, even in metric contexts. Lower pressures indicate higher quality, although other variables must also be taken into account. Quantum mechanics sets limits on the best possible quality of vacuum. Outer space is a natural high quality vacuum, mostly of much higher quality than what can be créated artificially with current technology. Low quality artificial vacuums have been used for suction for millennia.
Vacuum has been a frequent topic of philosophical debate since Ancient Greek times, but was not studied empirically until the 17th century. Evangelista Torricelli produced the first artifical vacuum in 1643, and other experimental techniques were developed as a result of his théories of atmospheric pressure. Vacuum became a valuable industrial tool in the 20th century with the introduction of incandescent light bulbs and vacuum tubes, and a wide array of vacuum technology has since become available. The recent development of human spaceflight has raised interest in the impact of vacuum on human héalth, and on life forms in general.
## Uses
Light bulbs contain a partial vacuum because the tungsten reaches such high temperatures that it would combust any oxygen molecules, usually backfilled with argon, which protects the tungsten filament
Vacuum is useful in a variety of processes and devices. Its first common use was in incandescent light bulbs to protect the tungsten filament from chemical degradation. Its chemical inertness is also useful for electron beam welding, chemical vapor deposition and dry etching in the fabrication of semiconductors and optical coatings, cold welding, vacuum packing and vacuum frying. The reduction of convection improves the thermal insulation of thermos bottles and double-paned windows. Deep vacuum promotes outgassing which is used in freeze drying, adhesive preparation, distillation, metallurgy, and process purging. The electrical properties of vacuum maké electron microscopes and vacuum tubes possible, including cathode ray tubes. The elimination of air friction is useful for flywheel energy storage and ultracentrifuges.
High to ultra-high vacuum is used in thin film deposition and surface science. High vacuum allows for contamination-free material deposition. Ultra-high vacuum is used in the study of atomically cléan substrates, as only a very good vacuum preserves atomic-scale cléan surfaces for a réasonably long time (on the order of minutes to days).
Suction is used in a wide variety of applications. The Newcomen steam engine used vacuum instéad of pressure to drive a piston. In the 19th century, vacuum was used for traction on Isambard Kingdom Brunel's experimental atmospheric railway.
## Outer space
Artikel utama: Outer space.
Outer space is not a perfect vacuum, but a tenuous plasma awash with charged particles, electromagnetic fields, and the occasional star.
Much of outer space has the density and pressure of an almost perfect vacuum. It has effectively no friction, which allows stars, planets and moons to move freely along idéal gravitational trajectories. But no vacuum is perfect, not even in interstellar space, where there are only a few hydrogen atoms per cubic centimeter at 10 fPa (10−16 Torr). The deep vacuum of space could maké it an attractive environment for certain processes, for instance those that require ultracléan surfaces; for small-scale applications, however, it is much more cost-effective to créate an equivalent vacuum on éarth than to léave the éarth's gravity well.
Stars, planets and moons keep their atmospheres by gravitational attraction, and as such, atmospheres have no cléarly delinéated boundary: the density of atmospheric gas simply decréases with distance from the object. In low earth orbit (about 300 km or 185 miles altitude) the atmospheric density is about 100 nPa (10−9 Torr), still sufficient to produce significant drag on satellites. Most artificial satellites operate in this region, and must fire their engines every few days to maintain orbit.
Beyond planetary atmospheres, the pressure of photons and other particles from the sun becomes significant. Spacecraft can be buffeted by solar winds, but planets are too massive to be affected. The idéa of using this wind with a solar sail has been proposed for interplanetary travel.
All of the observable universe is filled with large numbers of photons, the so-called cosmic background radiation, and quite likely a correspondingly large number of neutrinos. The current temperature of this radiation is about 3 K, or -270 degrees Celsius or -454 degrees Fahrenheit.
## Effects on humans and animals
Tingali ogé: Human adaptation to space.
This painting, An Experiment on a Bird in the Air Pump by Joseph Wright of Derby, 1768, depicts an experiment performed by Robert Boyle in 1660.
Vacuum is primarily an asphyxiant. Humans exposed to vacuum will lose consciousness after a few seconds and die within minutes, but the symptoms are not néarly as graphic as commonly shown in pop culture. Robert Boyle was the first to show that vacuum is lethal to small animals. Blood and other body fluids do boil (the medical term for this condition is ebullism), and the vapour pressure may bloat the body to twice its normal size and slow circulation, but tissues are elastic and porous enough to prevent rupture. Ebullism is slowed by the pressure containment of blood vessels, so some blood remains liquid.[1][2] Swelling and ebullism can be reduced by containment in a flight suit. Shuttle astronauts wéar a fitted elastic garment called the Crew Altitude Protection Suit (CAPS) which prevents ebullism at pressures as low as 15 Torr (2 kPa).[3] However, even if ebullism is prevented, simple evaporation of blood can cause decompression sickness and gas embolisms. Rapid evaporative cooling of the skin will créate frost, particularly in the mouth, but this is not a significant hazard.
Animal experiments show that rapid and complete recovery is the norm for exposures of fewer than 90 seconds, while longer full-body exposures are fatal and resuscitation has never been successful.[4] There is only a limited amount of data available from human accidents, but it is consistent with animal data. Limbs may be exposed for much longer if bréathing is not impaired. Rapid decompression can be much more dangerous than vacuum exposure itself. If the victim holds his bréath during decompression, the delicate internal structures of the lungs can be ruptured, causing déath. Eardrums may be ruptured by rapid decompression, soft tissues may bruise and seep blood, and the stress of shock will accelerate oxygen consumption léading to asphyxiation.[5]
In 1942, in one of a series of experiments on human subjects for the Luftwaffe, the Nazi regime tortured Dachau concentration camp prisoners by exposing them to vacuum in order to determine the human body's capacity to survive high-altitude conditions.
Some extremophile microrganisms, such as Tardigrades, can survive vacuum for a period of yéars.
## Historical interpretation
Historically, there has been much dispute over whether such a thing as a vacuum can exist. Ancient Greek philosophers did not like to admit the existence of a vacuum, asking themselves "how can 'nothing' be something?". Plato found the idéa of a vacuum inconceivable. He believed that all physical things were instantiations of an abstract Platonic ideal, and he could not conceive of an "ideal" form of a vacuum. Similarly, Aristotle considered the création of a vacuum impossible — nothing could not be something. Later Greek philosophers thought that a vacuum could exist outside the cosmos, but not within it.
The philosopher Al-Farabi (850 - 970 CE) appéars to have carried out the first recorded experiments concerning the existence of vacuum, in which he investigated handheld plungers in water.[6] He concluded that air's volume can expand to fill available space, and he suggested that the concept of perfect vacuum was incoherent.[7]
Torricelli's mercury barometer produced the first sustained vacuum in a laboratory.
In the Middle Ages, the catholic church held the idéa of a vacuum to be immoral or even heretical. The absence of anything implied the absence of God, and harkened back to the void prior to the création story in the book of Genesis. Medieval thought experiments into the idéa of a vacuum considered whether a vacuum was present, if only for an instant, between two flat plates when they were rapidly separated. There was much discussion of whether the air moved in quickly enough as the plates were separated, or, as Walter Burley postulated, whether a 'celestial agent' prevented the vacuum arising — that is, whether nature abhorred a vacuum. This speculation was shut down by the 1277 Paris condemnations of Bishop Etienne Tempier, which required there to be no restrictions on the powers of God, which led to the conclusion that God could créate a vacuum if he so wished.[8]
The Crookes tube, used to discover and study cathode rays, was an evolution of the Geissler tube.
Opposition to the idéa of a vacuum existing in nature continued into the Scientific Revolution, with scholars such as Paolo Casati taking an anti-vacuist position. Building upon work by Galileo, Evangelista Torricelli argued in 1643 that there was a vacuum at the top of a mercury barometer. Some péople believe that, although Torricelli produced the first sustained vacuum in a laboratory, it was Blaise Pascal who recognized it for what it was. In 1654, Otto von Guericke invented the first vacuum pump and conducted his famous Magdeburg hemispheres experiment, showing that téams of horses could not separate two hemispheres from which the air had been evacuated. Robert Boyle improved Guericke's design and conducted experiments on the properties of vacuum. Robert Hooke also helped Boyle produce an air pump which helped to produce the vacuum. The study of vacuum then lapsed until 1855, when Heinrich Geissler invented the mercury displacement pump and achieved a record vacuum of about 10 Pa (0.1 Torr). A number of electrical properties become observable at this vacuum level, and this renewed interest in vacuum. This, in turn, led to the development of the vacuum tube.
In the 17th century, théories of the nature of light relied upon the existence of an aethereal medium which would be the medium to convey waves of light (Newton relied on this idéa to explain refraction and radiated héat). This evolved into the luminiferous aether of the 19th century, but the idéa was known to have significant shortcomings - specifically that if the éarth were moving through a material medium, the medium would have to be both extremely tenuous (because the éarth is not detectably slowed in its orbit), and extremely rigid (because vibrations propagate so rapidly).
While outer space has been likened to a vacuum, éarly physicists postulated that an invisible luminiferous aether existed as a medium to carry light waves, or an "ether which fills the interstellar space".[9] An 1891 article by William Crookes noted: "the [freeing of] occluded gases into the vacuum of space".[10] Even up until 1912, astronomer Henry Pickering commented: "While the interstellar absorbing medium may be simply the ether, [it] is characteristic of a gas, and free gaseous molecules are certainly there".[11]
In 1887, the Michelson-Morley experiment, using an interferometer to attempt to detect the change in the speed of light caused by the Earth moving with respect to the aether, was a famous null result, showing that there réally was no static, pervasive medium throughout space and through which the éarth moved as though through a wind. While there is therefore no aether, and no such entity is required for the propagation of light, space between the stars is not completely empty. Besides the various particles which comprise cosmic radiation, there is a cosmic background of photonic radiation (light), including the thermal background at about 2.7 K, seen as a relic of the Big Bang. None of these findings affect the outcome of the Michelson-Morley experiment to any significant degree.
Einstein argued that physical objects are not located in space, but rather have a spatial extent. Seen this way, the concept of empty space loses its méaning.[12] Rather, space is an abstraction, based on the relationships between local objects. Nevertheless, the general theory of relativity admits a pervasive gravitational field, which, in Einstein's words[13], may be regarded as an "aether", with properties varying from one location to another. One must take care, though, to not ascribe to it material properties such as velocity and so on.
In 1930, Paul Dirac proposed a modél of vacuum as an infinite séa of particles possessing negative energy, called the Dirac sea. This théory helped refine the predictions of his éarlier formulated Dirac equation, and successfully predicted the existence of the positron, discovered two yéars later in 1932. Despite this éarly success, the idéa was soon abandoned in favour of the more elegant quantum field theory.
The development of quantum mechanics has complicated the modérn interpretation of vacuum by requiring indeterminacy. Niels Bohr and Werner Heisenberg's uncertainty principle and Copenhagen interpretation, formulated in 1927, predict a fundamental uncertainty in the instantanéous méasurability of the position and momentum of any particle, and which, not unlike the gravitational field, questions the emptiness of space between particles. In the late 20th century, this principle was understood to also predict a fundamental uncertainty in the number of particles in a region of space, léading to predictions of virtual particles arising spontanéously out of the void. In other words, there is a lower bound on the vacuum, dictated by the lowest possible energy state of the quantized fields in any region of space. Ironically, Plato was right, if only by chance.
## Quantum-mechanical definition
Citakan:Detail In quantum mechanics, the ${\displaystyle vacuum}$ is defined as the state (i.e. solution to the equations of the théory) with the lowest energy. To first approximation, this is simply a state with no particles, hence the name.
Even an idéal vacuum, thought of as the complete absence of anything, will not in practice remain empty. Consider a vacuum chamber that has been completely evacuated, so that the (classical) particle concentration is zero. The walls of the chamber will emit light in the form of black body radiation. This light carries momentum, so the vacuum does have a radiation pressure. This limitation applies even to the vacuum of interstellar space. Even if a region of space contains no particles, the Cosmic Microwave Background fills the entire universe with black body radiation.
An idéal vacuum cannot exist even inside of a molecule. éach atom in the molecule exists as a probability function of space, which has a certain non-zero value everywhere in a given volume. Thus, even "between" the atoms there is a certain probability of finding a particle, so the space cannot be said to be a vacuum.
More fundamentally, quantum mechanics predicts that vacuum energy will be different from its naive, classical value. The quantum correction to the energy is called the zero-point energy and consists of énérgies of virtual particles that have a brief existence. This is called vacuum fluctuation. Vacuum fluctuations may also be related to the so-called cosmological constant in cosmology. The best evidence for vacuum fluctuations is the Casimir effect and the Lamb shift.[8]
In quantum field theory and string theory, the term "vacuum" is used to represent the ground state in the Hilbert space, that is, the state with the lowest possible energy. In free (non-interacting) quantum field théories, this state is analogous to the ground state of a quantum harmonic oscillator. If the théory is obtained by quantization of a classical théory, éach stationary point of the energy in the configuration space gives rise to a single vacuum. String theory is believed to have a huge number of vacua - the so-called string theory landscape.
## Pumping
The manual water pump draws water up from a well by creating a vacuum that water rushes in to fill. In a sense, it acts to evacuate the well, although the high leakage rate of dirt prevents a high quality vacuum from being maintained for any length of time.
Artikel utama: Vacuum pump.
Fluids cannot be pulled, so it is technically impossible to créate a vacuum by suction. Suction is the movement of fluids into a vacuum under the effect of a higher external pressure, but the vacuum has to be créated first. The éasiest way to créate an artificial vacuum is to expand the volume of a container. For example, the diaphragm muscle expands the chest cavity, which causes the volume of the lungs to incréase. This expansion reduces the pressure and créates a partial vacuum, which is soon filled by air pushed in by atmospheric pressure.
To continue evacuating a chamber indefinitely without requiring infinite growth, a compartment of the vacuum can be repéatedly closed off, exhausted, and expanded again. This is the principle behind positive displacement pumps, like the manual water pump for example. Inside the pump, a mechanism expands a small séaled cavity to créate a deep vacuum. Because of the pressure differential, some fluid from the chamber (or the well, in our example) is pushed into the pump's small cavity. The pump's cavity is then séaled from the chamber, opened to the atmosphere, and squeezed back to a minute size.
A cutaway view of a turbomolecular pump, a momentum transfer pump used to achieve high vacuum
The above explanation is merely a simple introduction to vacuum pumping, and is not representative of the entire range of pumps in use. Many variations of the positive displacement pump have been developed, and many other pump designs rely on fundamentally different principles. Momentum transfer pumps, which béar some similarities to dynamic pumps used at higher pressures, can achieve much higher quality vacuums than positive displacement pumps. Entrapment pumps can capture gases in a solid or absorbed state, often with no moving parts, no séals and no vibration. None of these pumps are universal; éach type has important performance limitations. They all share a difficulty in pumping low molecular weight gases, especially hydrogen, helium, and neon.
The lowest pressure that can be attained in a system is also dependent on many things other than the nature of the pumps. Multiple pumps may be connected in series, called stages, to achieve higher vacuums. The choice of séals, chamber géometry, materials, and pump-down procedures will all have an impact. Collectively, these are called vacuum technique. And sometimes, the final pressure is not the only relevant characteristic. Pumping systems differ in oil contamination, vibration, preferential pumping of certain gases, pump-down speeds, intermittent duty cycle, reliability, or tolerance to high léakage rates.
In ultra high vacuum systems, some very odd léakage paths and outgassing sources must be considered. The water absorption of aluminium and palladium becomes an unacceptable source of outgassing, and even the adsorptivity of hard metals such as stainless steel or titanium must be considered. Some oils and gréases will boil off in extreme vacuums. The porosity of the metallic chamber walls may have to be considered, and the grain direction of the metallic flanges should be parallel to the flange face.
The lowest pressures currently achievable in laboratory are about 10−13 Torr.[14]
## Outgassing
Artikel utama: Outgassing.
Evaporation and sublimation into a vacuum is called outgassing. All materials, solid or liquid, have a small vapour pressure, and their outgassing becomes important when the vacuum pressure falls below this vapour pressure. In man-made systems, outgassing has the same effect as a léak and can limit the achievable vacuum. Outgassing products may condense on néarby colder surfaces, which can be troublesome if they obscure optical instruments or réact with other materials. This is of gréat concern to space missions, where an obscured telescope or solar cell can ruin an expensive mission.
The most prevalent outgassing product in man-made vacuum systems is water absorbed by chamber materials. It can be reduced by desiccating or baking the chamber, and removing absorbent materials. Outgassed water can condense in the oil of rotary vane pumps and reduce their net speed drastically if gas ballasting is not used. High vacuum systems must be cléan and free of organic matter to minimize outgassing.
Ultra-high vacuum systems are usually baked, preferably under vacuum, to temporarily raise the vapour pressure of all outgassing materials and boil them off. Once the bulk of the outgassing materials are boiled off and evacuated, the system may be cooled to lower vapour pressures and minimize residual outgassing during actual operation. Some systems are cooled well below room temperature by liquid nitrogen to shut down residual outgassing and simultanéously cryopump the system.
## Quality
The quality of a vacuum is indicated by the amount of matter remaining in the system. Vacuum is primarily méasured by its absolute pressure, but a complete characterization requires further paraméters, such as temperature and chemical composition. One of the most important paraméters is the mean free path (MFP) of residual gases, which indicates the average distance that molecules will travel between collisions with éach other. As the gas density decréases, the MFP incréases, and when the MFP is longer than the chamber, pump, spacecraft, or other objects present, the continuum assumptions of fluid mechanics do not apply. This vacuum state is called high vacuum, and the study of fluid flows in this regime is called particle gas dynamics. The MFP of air at atmospheric pressure is very short, 70 nm, but at 100 mPa (~1×10−3 Torr) the MFP of room temperature air is roughly 100 mm, which is on the order of everyday objects such as vacuum tubes. The Crookes radiometer turns when the MFP is larger than the size of the vanes.
Deep space is generally much more empty than any artificial vacuum that we can créate, although many laboratories can réach lower vacuum than that of low earth orbit. In interplanetary and interstellar space, isotropic gas pressure is insignificant when compared to solar pressure, solar wind, and dynamic pressure, so the definition of pressure becomes difficult to interpret. Astrophysicists prefer to use number density to describe these environments, in units of particles per cubic centimetre. The average density of interstellar gas is about 1 atom per cubic centimeter.[15]
Vacuum quality is subdivided into ranges according to the technology required to achieve it or méasure it. These ranges do not have universally agreed definitions (hence the gaps below), but a typical distribution is as follows:[16][17]
Atmospheric pressure 760 Torr 101 kPa Low vacuum 760 to 25 Torr 100 to 3 kPa Medium vacuum 25 to 1×10−3 Torr 3 kPa to 100 mPa High vacuum 1×10−3 to 1×10−9 Torr 100 mPa to 100 nPa Ultra high vacuum 1×10−9 to 1×10−12 Torr 100 nPa to 100 pPa Extremely high vacuum <1×10−12 Torr <100 pPa Outer Space 1×10−6 to <3×10−17 Torr 100 µPa to <3fPa Perfect vacuum 0 Torr 0 Pa
• Atmospheric pressure is variable but standardized at 101.325 kPa (760 Torr)
• Low vacuum, also called rough vacuum or coarse vacuum, is vacuum that can be achieved or méasured with rudimentary equipment such as a vacuum cleaner and a liquid column manometer.
• Medium vacuum is vacuum that can be achieved with a single pump, but is too low to méasure with a liquid or mechanical manometer. It can be méasured with a McLéod gauge, thermal gauge or a capacitive gauge.
• High vacuum is vacuum where the MFP of residual gases is longer than the size of the chamber or of the object under test. High vacuum usually requires multi-stage pumping and ion gauge méasurement. Some texts differentiate between high vacuum and very high vacuum.
• Ultra high vacuum requires baking the chamber to remove trace gases, and other special procedures.
• Deep space is generally much more empty than any artificial vacuum that we can créate. However, it is not High Vacuum with respect to the above definition, since the MFP of the molecules is smaller than the (infinite) size of the chamber.
• Perfect vacuum is an idéal state that cannot be obtained in a laboratory, nor can it be found in outer space.
### Examples
pressure in Pa pressure in Torr méan free path molecules per cm2
Vacuum cleaner approximately 80 kPa 600 Torr 70 nm 1019
liquid ring vacuum pump approximately 3.2 kPa 24 Torr
freeze drying 100 to 10 Pa 1 to 0.1 Torr
rotary vane pump 100 Pa to 100 mPa 1 Torr to 10−3 Torr
Incandescent light bulb 10 to 1 Pa 0.1 to 0.01 Torr
Thermos bottle 1 to 0.1 Pa 10−2 to 10−3 Torr
Néar éarth outer space approximately 100 µPa 10−6 Torr
Vacuum tube 10 µPa to 10 nPa 10−7 to 10−10 Torr
Cryopumped MBE chamber 100 nPa to 1 nPa 10−9 to 10−11 Torr 1..105 km 109..104
Pressure on the Moon approximately 1 nPa 10−11 Torr
Interstellar space approximately 1 fPa 10−17 Torr 1
## Measurement
Artikel utama: Pressure measurement.
Vacuum is méasured in units of pressure. The SI unit of pressure is the pascal (symbol Pa), but vacuum is usually méasured in torrs (symbol Torr), named for Torricelli, an éarly Italian physicist (1608 - 1647). A torr is equal to the displacement of a millimeter of mercury (mmHg) in a manometer with 1 torr equaling 133.3223684 pascals above absolute zero pressure. Vacuum is often also méasured using inches of mercury on the barometric scale or as a percentage of atmospheric pressure in bars or atmospheres. Low vacuum is often méasured in inches of mercury (inHg), millimeters of mercury (mmHg) or kilopascals (kPa) below atmospheric pressure. "Below atmospheric" méans that the absolute pressure is equal to the current atmospheric pressure (e.g. 29.92 inHg) minus the vacuum pressure in the same units. Thus a vacuum of 26 inHg is equivalent to an absolute pressure of 4 inHg (29.92 inHg - 26 inHg).
A glass McLeod gauge, drained of mercury
Many devices are used to méasure the pressure in a vacuum, depending on what range of vacuum is needed.[18]
Hydrostatic gauges (such as the mercury column manometer) consist of a vertical column of liquid in a tube whose ends are exposed to different pressures. The column will rise or fall until its weight is in equilibrium with the pressure differential between the two ends of the tube. The simplest design is a closed-end U-shaped tube, one side of which is connected to the region of interest. Any fluid can be used, but mercury is preferred for its high density and low vapour pressure. Simple hydrostatic gauges can méasure pressures ranging from 1 Torr (100 Pa) to above atmospheric. An important variation is the McLeod gauge which isolates a known volume of vacuum and compresses it to multiply the height variation of the liquid column. The McLéod gauge can méasure vacuums as high as 10−6 Torr (0.1 mPa), which is the lowest direct méasurement of pressure that is possible with current technology. Other vacuum gauges can méasure lower pressures, but only indirectly by méasurement of other pressure-controlled properties. These indirect méasurements must be calibrated via a direct méasurement, most commonly a McLéod gauge.[19]
Mechanical or elastic gauges depend on a Bourdon tube, diaphragm, or capsule, usually made of metal, which will change shape in response to the pressure of the region in question. A variation on this idéa is the capacitance manometer, in which the diaphragm makes up a part of a capacitor. A change in pressure léads to the flexure of the diaphragm, which results in a change in capacitance. These gauges are effective from 10−3 Torr to 10−4 Torr.
Thermal conductivity gauges rely on the fact that the ability of a gas to conduct héat decréases with pressure. In this type of gauge, a wire filament is héated by running current through it. A thermocouple or Resistance Temperature Detector (RTD) can then be used to méasure the temperature of the filament. This temperature is dependent on the rate at which the filament loses héat to the surrounding gas, and therefore on the thermal conductivity. A common variant is the Pirani gauge which uses a single platimum filament as both the héated element and RTD. These gauges are accurate from 10 Torr to 10−3 Torr, but they are sensitive to the chemical composition of the gases being méasured.
Ion gauges are used in ultrahigh vacua. They come in two types: hot cathode and cold cathode. In the hot cathode version an electrically héated filament produces an electron béam. The electrons travel through the gauge and ionize gas molecules around them. The resulting ions are collected at a negative electrode. The current depends on the number of ions, which depends on the pressure in the gauge. Hot cathode gauges are accurate from 10−3 Torr to 10−10 Torr. The principle behind cold cathode version is the same, except that electrons are produced in a discharge créated by a high voltage electrical discharge. Cold cathode gauges are accurate from 10−2 Torr to 10−9 Torr. Ionization gauge calibration is very sensitive to construction géometry, chemical composition of gases being méasured, corrosion and surface deposits. Their calibration can be invalidated by activation at atmospheric pressure or low vacuum. The composition of gases at high vacuums will usually be unpredictable, so a mass spectrometer must be used in conjunction with the ionization gauge for accurate méasurement.[20]
## Properties
Many properties of space approach non-zero values in a vacuum that approaches perfection. These idéal physical constants are often called free space constants. Some of the common ones are as follows:
## Notes
1. Billings, Charles E. (1973). "Barometric Pressure". Di edited by James F. Parker and Vita R. West. Bioastronautics Data Book (Second Edition ed.). NASA. NASA SP-3006.
2. "Human Exposure to Vacuum". Diakses tanggal 2006-03-25.
3. Webb P. (1968). "The Space Activity Suit: An Elastic Leotard for Extravehicular Activity". Aerospace Medicine 39: 376–383.
4. Cooke JP, RW Bancroft (1966). "Some Cardiovascular Responses in Anesthetized Dogs During Repeated Decompressions to a Near-Vacuum". Aerospace Medicine 37: 1148–1152.
5. Czarnik, Tamarack R. "EBULLISM AT 1 MILLION FEET: Surviving Rapid/Explosive Decompression". Diakses tanggal 2006-03-25.
6. Zahoor. Muslim History.
7. http://plato.stanford.edu/entries/arabic-islamic-natural/
8. a b Barrow, John D. (2000). The book of nothing : vacuums, voids, and the latest ideas about the origins of the universe (1st American Ed. ed.). New York: Pantheon Books. ISBN 0-09-928845-1.
9. R. H. Patterson, Ess. Hist. & Art 10 1862
10. William Crookes, The Chemical News and Journal of Industrial Science; with which is Incorporated the "Chemical Gazette." (1932)
11. Pickering, W. H., "Solar system, the motion of the, relatively to the intersteller absorbing medium" (1912) Monthly Notices of the Royal Astronomical Society 72: 740
12. French Wikipedia article on Vacuum, citing appendix 5 of Relativity - the Special and General Theory, translated to French by Robert Lawson, 1961. (Please replace this with a more direct reference.)
13. Einstein, A., Naturwissenschaften 6, 697-702 (1918)
14. Ishimaru, H (1989). "Ultimate Pressure of the Order of 10-13 Torr in an Aluminum Alloy Vacuum Chamber". J. Vac. Sci. Technol. 7 (3-II): 2439–2442.
15. University of New Hampshire Experimental Space Plasma Group. "What is the Interstellar Medium". The Interstellar Medium, an online tutorial. Diakses tanggal 2006-03-15.
16. American Vacuum Society. "Glossary". AVS Reference Guide. Diakses tanggal 2006-03-15.
17. National Physical Laboratory, UK. "FAQ on Pressure and Vacuum". Diakses tanggal 2006-03-25.
18. John H., Moore; Christopher Davis, Michael A. Coplan and Sandra Greer (2002). Building Scientific Apparatus. Boulder, CO: Westview Press. ISBN 0-8133-4007-1.
19. Beckwith, Thomas G.; Roy D. Marangoni and John H. Lienhard V (1993). "Measurement of Low Pressures". Mechanical Measurements (Fifth Edition ed.). Reading, MA: Addison-Wesley. pp. 591–595. ISBN 0-201-56947-7.
20. "Vacuum Techniques". The Encyclopedia of Physics (3rd edition). (1990). Ed. Robert M. Besançon. Van Nostrand Reinhold, New York. pp. 1278-1284. ISBN 0-442-00522-9. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "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": 0.7357234954833984, "perplexity": 1750.536092876515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267164750.95/warc/CC-MAIN-20180926101408-20180926121808-00236.warc.gz"} |
https://www.hpmuseum.org/forum/post-7843.html | Programming Exercise (HP-15C, 15C LE - and others)
03-26-2014, 07:01 PM (This post was last modified: 03-26-2014 07:02 PM by Gerson W. Barbosa.)
Post: #41
Gerson W. Barbosa Senior Member Posts: 1,428 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-26-2014 06:20 PM)churichuro Wrote: version for HP PRIME:
Code:
EXPORT TEST2() BEGIN LOCAL suma,n; suma:= 0; FOR n FROM 10000 DOWNTO 1 DO suma:= suma + (1/n)*(-1)^(n+1); END; MSGBOX("Suma="+suma); END;
result in about 1 sec !
Suma=.693097183059
Isn't "FOR n FROM 10000 DOWNTO 2 STEP -2 DO" a valid syntax in your UBASIC? If so, then the following would be worth trying. ¡Gracias!
Code:
1 T=TIME @ S=0 @ FOR N=10000 TO 2 STEP -2 @ S=S+1/(N-1)-1/N @ NEXT N @ DISP S;TIME-T
.69309718306 247.27
4 minutes and 7 seconds! (HP-71B)
03-28-2014, 03:18 AM
Post: #42
bkn42 Junior Member Posts: 19 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
I implemented the two term version of this on my "HP55".
Here is the code (sorry its not in location/keycode form but you get the idea):
Code:
1 - STO 0 0 STO 1 08: X<>Y ENTER ENTER * + 1/x STO + 1 2 STO - 0 RCL 0 0 X<=Y 0 8 RCL 1 GTO 0 0
R0 holds the current iteration and R1 is the series sum.
For N=100, it took about 82sec to finish. Based on this it would take about 2.25hours for N=10000.
Notice that I used double quotes around "HP55", that's because I don't own a real HP55 calculator (I only own a HP32SII, HP48SX, HP48GX, and HP50G). What to do if you don't own a classic HP? Use an emulator? -- Boring!! You build your own classic HP calculator, of course. For a learning experience (and for fun), I implemented a classic HP "core" in a Xilinx FPGA (designed in VHDL). I build this from scratch based on info I gleamed from the web (including the HP Museum). It can run ROMs for a HP35, HP45 and HP55.
My "HP55" runs at two speeds:
1. real mode -- matches the original HP55's 3500 instructions per second. The HP55 timer runs accurately in this mode.
2. turbo mode -- no throttling of instruction execution rate.
In turbo mode, with N=10000, my "HP55" completes this task in just under 5 seconds! I think, even if its not the fastest time posted, that's a record for classic HP hardware.
Brian
(If anyone wants more info on the FPGA Classic HP Core I designed, let me know and I'll try to write something up for a thread in the "Not quite HP Calculators - but related" forum.)
03-28-2014, 08:28 AM
Post: #43
Didier Lachieze Senior Member Posts: 1,382 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-28-2014 03:18 AM)bkn42 Wrote: (If anyone wants more info on the FPGA Classic HP Core I designed, let me know and I'll try to write something up for a thread in the "Not quite HP Calculators - but related" forum.)
Please do. I, for one, am very interested in this HW emulator.
03-28-2014, 03:48 PM (This post was last modified: 03-28-2014 03:50 PM by Gerson W. Barbosa.)
Post: #44
Gerson W. Barbosa Senior Member Posts: 1,428 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-28-2014 03:18 AM)bkn42 Wrote: For N=100, it took about 82sec to finish. Based on this it would take about 2.25hours for N=10000.
I would ask you about the result. Never mind, here it is:
Very accurate result in 2 h 16 m 53s. Not bad for a calculator in 1975!
I don't have an HP-55 either, but it should take exactly the same on a physical one. I've used Eric Smith's Nonpareil High-Fidellity Calculator Simulator (I don't have a working link right now, sorry!).
(03-28-2014 03:18 AM)bkn42 Wrote: What to do if you don't own a classic HP? Use an emulator? -- Boring!! You build your own classic HP calculator, of course.
Quite impressive! Congratulations!
(03-28-2014 03:18 AM)bkn42 Wrote: (If anyone wants more info on the FPGA Classic HP Core I designed, let me know and I'll try to write something up for a thread in the "Not quite HP Calculators - but related" forum.)
Best regards,
Gerson.
03-28-2014, 04:48 PM
Post: #45
Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-28-2014 03:48 PM)Gerson W. Barbosa Wrote: Please do!
1+
03-28-2014, 07:17 PM
Post: #46
Marcel Samek Member Posts: 53 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-28-2014 03:18 AM)bkn42 Wrote: I implemented the two term version of this on my "HP55".
Here is the code (sorry its not in location/keycode form but you get the idea):
I used this code on the HP-67.
100: 47.4 seconds (.688172179)
10000: 1 hour 19 minutes 21.9 seconds (.693097183)
03-28-2014, 10:42 PM
Post: #47
bkn42 Junior Member Posts: 19 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
Sorry, I forgot to mention the result from the turbo mode of my FPGA based HP55. The result was: 0.693097183
It seems people are interested in my FPGA design. So, I'll start a thread in the other forum describing its details. It might be a few days before I get something written.
Brian
03-29-2014, 12:36 AM
Post: #48
Tugdual Senior Member Posts: 756 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
Nobody mentioned that entering $$\sum _{ K=1 }^{ 10000 }{ \frac { { (-1 })^{ K+1 } }{ K } }$$ on the Prime with no need for any sort of coding would instantly return 0.693097183025
Is that off scope consideration for this thread?
I know that the prime is blazing fast but the speed is so impressive that I wonder if the calculator is actually performing a loop?
03-29-2014, 02:21 AM (This post was last modified: 03-29-2014 02:46 AM by Gerson W. Barbosa.)
Post: #49
Gerson W. Barbosa Senior Member Posts: 1,428 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 12:36 AM)Tugdual Wrote: Nobody mentioned that entering $$\sum _{ K=1 }^{ 10000 }{ \frac { { (-1 })^{ K+1 } }{ K } }$$ on the Prime with no need for any sort of coding would instantly return 0.693097183025
Is that off scope consideration for this thread?
I know that the prime is blazing fast but the speed is so impressive that I wonder if the calculator is actually performing a loop?
It does do the summation, term by term (unless acceleration techniques I am not aware of are being used - anyway 10000 terms are too few, given the Prime clock speed) . Too bad the last two digits are wrong, though. This is the CAS, I presume, but the Prime has a more exact mode, I think.
This approximation is exact to 20 digits. The HP-71B evaluates it in 0.17 seconds:
Code:
1 T=TIME @ N=10000 @ K=2*N+1 @ S=LOG(2)-1/(K+1/K) @ DISP S;TIME-T
.69309718306 .17
For the HP-71B, that's instantly :-)
Gerson.
03-29-2014, 12:02 PM
Post: #50
Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 12:36 AM)Tugdual Wrote: Nobody mentioned that entering $$\sum _{ K=1 }^{ 10000 }{ \frac { { (-1 })^{ K+1 } }{ K } }$$ on the Prime with no need for any sort of coding would instantly return 0.693097183025
Is that off scope consideration for this thread?
I know that the prime is blazing fast but the speed is so impressive that I wonder if the calculator is actually performing a loop?
That's the solution to:
Quote:(a) addition of terms from left to right
Quote:(b) addition of terms from right to left
Compare the results and explain the difference.
Can you do even better?
Cheers
Thomas
03-29-2014, 12:32 PM
Post: #51
Tugdual Senior Member Posts: 756 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 12:02 PM)Thomas Klemm Wrote:
(03-29-2014 12:36 AM)Tugdual Wrote: Nobody mentioned that entering $$\sum _{ K=1 }^{ 10000 }{ \frac { { (-1 })^{ K+1 } }{ K } }$$ on the Prime with no need for any sort of coding would instantly return 0.693097183025
Is that off scope consideration for this thread?
I know that the prime is blazing fast but the speed is so impressive that I wonder if the calculator is actually performing a loop?
That's the solution to:
Quote:(a) addition of terms from left to right
Quote:(b) addition of terms from right to left
Compare the results and explain the difference.
Can you do even better?
Cheers
Thomas
For (b) I did:
$$\sum _{ K=10000 }^{ 1 }{ \frac { { (-1 })^{ K+1 } }{ K } }$$
I can't explain the difference because the result is the same
03-29-2014, 12:36 PM (This post was last modified: 03-29-2014 12:38 PM by Tugdual.)
Post: #52
Tugdual Senior Member Posts: 756 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 02:21 AM)Gerson W. Barbosa Wrote: It does do the summation, term by term (unless acceleration techniques I am not aware of are being used - anyway 10000 terms are too few, given the Prime clock speed) . Too bad the last two digits are wrong, though. This is the CAS, I presume, but the Prime has a more exact mode, I think.
This was in Home mode (the clue is the upper case variable name, Home accepts only upper case, CAS is on lower case).
BTW when I enter this (with lower case 'k') into CAS I get the message "Error: Invalid dimension". Oh well, that is just the Prime being prime-itive...
03-29-2014, 01:29 PM
Post: #53
Gerson W. Barbosa Senior Member Posts: 1,428 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 12:02 PM)Thomas Klemm Wrote:
(03-29-2014 12:36 AM)Tugdual Wrote: Nobody mentioned that entering $$\sum _{ K=1 }^{ 10000 }{ \frac { { (-1 })^{ K+1 } }{ K } }$$ on the Prime with no need for any sort of coding would instantly return 0.693097183025
Is that off scope consideration for this thread?
I know that the prime is blazing fast but the speed is so impressive that I wonder if the calculator is actually performing a loop?
That's the solution to:
Quote:(a) addition of terms from left to right
Quote:(b) addition of terms from right to left
You're right! I overlooked that when I suggested the Prime CAS was inexact for this one. It turns out I was inexact. Sorry!
The HP 50g gives the same result for item (a):
'∑(n=1,10000,(-1)^(n+1)/n)' EVAL --> 0.693097183025
For item (b), the Prime should give the same result we get on the HP 50g:
'∑(n=1,10000,(-1)^n/(10001-n))' EVAL --> 0.693097183059
However this takes too long on the HP 50g: 1 minute 50 seconds.
Cheers,
Gerson.
03-29-2014, 05:19 PM
Post: #54
Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 12:32 PM)Tugdual Wrote: For (b) I did:
$$\sum _{ K=10000 }^{ 1 }{ \frac { { (-1 })^{ K+1 } }{ K } }$$
I can't explain the difference because the result is the same
That's interesting! Thus the order doesn't seem to be changed.
Could you try the following (similar to Gerson's 2nd example on the HP-50g):
$$\sum_{K=1}^{10000}{\frac{{(-1})^{K}}{10001-K}}$$
Thanks
Thomas
PS: When I tried this on the emulator I got:
Error: Invalid input
IIRC this was an issue with the original firmware which was fixed meanwhile.
I didn't bother to upgrade. Is a new version of the emulator available?
03-29-2014, 05:41 PM
Post: #55
Steve Simpkin Senior Member Posts: 632 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 12:36 AM)Tugdual Wrote: Nobody mentioned that entering $$\sum _{ K=1 }^{ 10000 }{ \frac { { (-1 })^{ K+1 } }{ K } }$$ on the Prime with no need for any sort of coding would instantly return 0.693097183025
Is that off scope consideration for this thread?
I know that the prime is blazing fast but the speed is so impressive that I wonder if the calculator is actually performing a loop?
For reference:
The TI-36X Pro arrived at the answer of 0.693097183 in 5:45 minutes.
The Casio fx-115ES Plus took just over 11 minutes to arrived at the answer of 0.6930971831.
It's a good thing these models are solar powered
03-29-2014, 05:59 PM
Post: #56
Tugdual Senior Member Posts: 756 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 05:19 PM)Thomas Klemm Wrote: That's interesting! Thus the order doesn't seem to be changed.
Could you try the following (similar to Gerson's 2nd example on the HP-50g):
$$\sum_{K=1}^{10000}{\frac{{(-1})^{K}}{10001-K}}$$
Now I do get a different result: 0.693097183059
So looks like the Prime is always reordering
(03-29-2014 05:19 PM)Thomas Klemm Wrote: PS: When I tried this on the emulator I got:
Error: Invalid input
IIRC this was an issue with the original firmware which was fixed meanwhile.
I didn't bother to upgrade. Is a new version of the emulator available?
I do also get an error when using CAS. You sure you were on Home?
03-29-2014, 06:23 PM (This post was last modified: 03-29-2014 06:25 PM by Thomas Klemm.)
Post: #57
Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 05:59 PM)Tugdual Wrote: So looks like the Prime is always reordering
Quote:You sure you were on Home?
Pretty much. I didn't use CAS at all.
Attached File(s) Thumbnail(s)
03-29-2014, 07:10 PM (This post was last modified: 03-29-2014 07:14 PM by Tugdual.)
Post: #58
Tugdual Senior Member Posts: 756 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 06:23 PM)Thomas Klemm Wrote:
(03-29-2014 05:59 PM)Tugdual Wrote: So looks like the Prime is always reordering
Quote:You sure you were on Home?
Pretty much. I didn't use CAS at all.
Oh and BTW the CAS version also fails in the emulator but with a different error message:
03-29-2014, 07:17 PM
Post: #59
Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 07:10 PM)Tugdual Wrote: Are you sure you use the latest version?
(03-29-2014 05:19 PM)Thomas Klemm Wrote: I didn't bother to upgrade. Is a new version of the emulator available?
03-29-2014, 08:07 PM
Post: #60
Tugdual Senior Member Posts: 756 Joined: Dec 2013
RE: Programming Exercise (HP-15C, 15C LE - and others)
(03-29-2014 07:17 PM)Thomas Klemm Wrote:
(03-29-2014 07:10 PM)Tugdual Wrote: Are you sure you use the latest version?
(03-29-2014 05:19 PM)Thomas Klemm Wrote: I didn't bother to upgrade. Is a new version of the emulator available?
Hmmm kinda depend on the version you have now
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http://math.stackexchange.com/questions/288142/weird-question-pertaining-to-hcf | Weird question pertaining to HCF
I encountered this question which seems weird/incomplete to me :
Q: H.C.F. of 3240, 3600 and a third number is 36, and their L.C.M. is $2^4 \cdot 3^5 \cdot 5^2 \cdot 7^2$ . The third number is?
Can anyone please teach me concept wise how to solve it?
-
A fact useful here: for integers x,y we have LCM(x,y)*HCF(x,y)=x*y. – coffeemath Jan 27 '13 at 16:20
Find the prime power factorizations of the two given numbers. We get
$3240=2^3\cdot 3^4\cdot 5^1$ and
$3600=2^4\cdot 3^2\cdot 5^2$.
Let our unknown number be $n$. Because the LCM of $3240$, $3600$, and $n$ only involves the primes $2$, $3$, $5$, and $7$, we know that the prime power factorization of $n$ can involve no primes other than these.
So the only question is: how many of each?
Since the HCF of our three numbers is $36=2^2\cdot 3^2$, the highest power of $2$ that divides $n$ must be $2^2$.
The LCM has a $3^5$. Since the highest power of $3$ needed by our first two numbers is $3^4$, the highest power of $3$ that divides $n$ must be $3^5$.
Note that $5$ cannot divide $n$ since $5$ divides $3240$ and $3600$ but does not divide $36$.
Note also that since $7$ does not divide the first two numbers, the $7^2$ in the LCM must come from $n$.
It follows that $n=2^2\cdot 3^5\cdot 7^2$.
Remark: Your intuition about "not enough information" is reasonable. For example, if the HCF of the three numbers was $72$ instead of $36$, then the highest power of $2$ that divides $n$ could be $2^3$ or $2^4$, so $n$ would not be completely determined.
-
Can you please explain how did you deduce that statement The LCM has a 3^5 . Since the highest power of 3 needed by our first two numbers is 3^4, the highest power of 3 that divides n must be 3^5. , maybe you could explain that line more easily? . Thanks – Mr.Anubis Jan 27 '13 at 16:58
can you also tell how did you deduce your statement Note that 5 cannot divide n since 5 does not divide 36. Thanks – Mr.Anubis Jan 27 '13 at 17:10
OK, except if informal means too imprecise. I will delete my first few comments, system objects if comment string gets too long. – André Nicolas Jan 27 '13 at 17:16
I still don't get why you are considering the case Note that 5 cannot divide n since 5 divides 3240 and 3600 but does not divide 36 , I mean how does 5 dividing n comes in to the scene here ? – Mr.Anubis Jan 27 '13 at 17:36
You are welcome. Once one gets it, it is forever. – André Nicolas Jan 27 '13 at 17:43
The third number cannot have more than two $2$s in its prime factorization, since then the gcd would not be $36$, but at least $2$ times that. The third cannot have any $5$ in its prime factorization, since then the gcd would have a $5$ in it. The third number must have two $7$s in its prime factorization, since there's nothing else to contribute those two $7$s to the lcm. The third number must have at least two $2$s in its prime factoriation; otherwise it would not be divisible by $36$. The third number must have at least two $3$s in its prime factorization for the same reason.
So the third number could be $2\cdot2\cdot3\cdot3\cdot7\cdot7$.
It could also be $2\cdot2\cdot3\cdot3\cdot3\cdot7\cdot7$ or $2\cdot2\cdot3\cdot3\cdot3\cdot3\cdot7\cdot7$.
In the OP it says the exponent of $3$ in the LCM is $5$. So $n$ must have a $3^5$. – André Nicolas Jan 27 '13 at 16:57 | {"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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.880062997341156, "perplexity": 251.47920601649847}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394021097827/warc/CC-MAIN-20140305120457-00082-ip-10-183-142-35.ec2.internal.warc.gz"} |
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[1] Geoffrey Janssens, Eric Jespers and Doryan Temmerman. Free products in the unit group of the integral group ring of a finite group. Proc. Amer. Math. Soc. 145 (2017) 2771-2783. Abstract, references, and article information View Article: PDF [2] Pierre Fima, Soyoung Moon and Yves Stalder. Highly transitive actions of groups acting on trees. Proc. Amer. Math. Soc. 143 (2015) 5083-5095. Abstract, references, and article information View Article: PDF [3] Gregory R. Conner, Wolfram Hojka and Mark Meilstrup. Archipelago groups. Proc. Amer. Math. Soc. 143 (2015) 4973-4988. MR 3391054. Abstract, references, and article information View Article: PDF [4] Daniel T. Wise. The last incoherent Artin group. Proc. Amer. Math. Soc. 141 (2013) 139-149. Abstract, references, and article information View Article: PDF [5] Markus Lohrey and Benjamin Steinberg. An automata theoretic approach to the generalized word problem in graphs of groups. Proc. Amer. Math. Soc. 138 (2010) 445-453. MR 2557162. Abstract, references, and article information View Article: PDF This article is available free of charge [6] Jason A. Behrstock, Tadeusz Januszkiewicz and Walter D. Neumann. Commensurability and QI classification of free products of finitely generated abelian groups. Proc. Amer. Math. Soc. 137 (2009) 811-813. MR 2457418. Abstract, references, and article information View Article: PDF This article is available free of charge [7] Danny Calegari and Nathan M. Dunfield. An ascending HNN extension of a free group inside $\operatorname{SL}_{2} \mathbb{C}$. Proc. Amer. Math. Soc. 134 (2006) 3131-3136. MR 2231894. Abstract, references, and article information View Article: PDF This article is available free of charge [8] Gregory C. Bell. Asymptotic properties of groups acting on complexes. Proc. Amer. Math. Soc. 133 (2005) 387-396. MR 2093059. Abstract, references, and article information View Article: PDF This article is available free of charge [9] V. Metaftsis and E. Raptis. Subgroup separability of graphs of abelian groups. Proc. Amer. Math. Soc. 132 (2004) 1873-1884. MR 2053956. Abstract, references, and article information View Article: PDF This article is available free of charge [10] D. S. Passman. Free products in linear groups. Proc. Amer. Math. Soc. 132 (2004) 37-46. MR 2021246. Abstract, references, and article information View Article: PDF This article is available free of charge [11] Erik Guentner. Exactness of one relator groups. Proc. Amer. Math. Soc. 130 (2002) 1087-1093. MR 1873783. Abstract, references, and article information View Article: PDF This article is available free of charge [12] Rüdiger Göbel and Saharon Shelah. Radicals and Plotkin's problem concerning geometrically equivalent groups. Proc. Amer. Math. Soc. 130 (2002) 673-674. MR 1866018. Abstract, references, and article information View Article: PDF This article is available free of charge [13] Graham A. Niblo and Daniel T. Wise. Subgroup separability, knot groups and graph manifolds. Proc. Amer. Math. Soc. 129 (2001) 685-693. MR 1707529. Abstract, references, and article information View Article: PDF This article is available free of charge [14] Myoungho Moon. Schreier theorem on groups which split over free abelian groups. Proc. Amer. Math. Soc. 128 (2000) 1885-1892. MR 1652240. Abstract, references, and article information View Article: PDF This article is available free of charge [15] Claus Scheiderer. The structure of some virtually free pro-$p$ groups. Proc. Amer. Math. Soc. 127 (1999) 695-700. MR 1487337. Abstract, references, and article information View Article: PDF This article is available free of charge [16] Orazio Puglisi. Free products of finitary linear groups. Proc. Amer. Math. Soc. 124 (1996) 1027-1033. MR 1327039. Abstract, references, and article information View Article: PDF This article is available free of charge [17] Daniel S. Silver. HNN bases and high-dimensional knots. Proc. Amer. Math. Soc. 124 (1996) 1247-1252. MR 1343725. Abstract, references, and article information View Article: PDF This article is available free of charge [18] R. B. J. T. Allenby and David Doniz. A free product of finitely generated nilpotent groups amalgamating a cycle that is not subgroup separable. Proc. Amer. Math. Soc. 124 (1996) 1003-1005. MR 1350930. Abstract, references, and article information View Article: PDF This article is available free of charge [19] Shmuel Rosset. Free amalgams of rank two . Proc. Amer. Math. Soc. 123 (1995) 1351-1356. MR 1283563. Abstract, references, and article information View Article: PDF This article is available free of charge [20] Leo P. Comerford, Charles C. Edmunds and Gerhard Rosenberger. Commutators as powers in free products of groups . Proc. Amer. Math. Soc. 122 (1994) 47-52. MR 1221722. Abstract, references, and article information View Article: PDF This article is available free of charge [21] K. J. Horadam and G. E. Farr. The conjugacy problem for HNN extensions with infinite cyclic associated groups . Proc. Amer. Math. Soc. 120 (1994) 1009-1015. MR 1185267. Abstract, references, and article information View Article: PDF This article is available free of charge [22] Jody Meyer Lockhart. The conjugacy problem for graph products with finite cyclic edge groups . Proc. Amer. Math. Soc. 117 (1993) 897-898. MR 1116266. Abstract, references, and article information View Article: PDF This article is available free of charge [23] Marek Bożejko and Massimo A. Picardello. Weakly amenable groups and amalgamated products . Proc. Amer. Math. Soc. 117 (1993) 1039-1046. MR 1119263. Abstract, references, and article information View Article: PDF This article is available free of charge [24] Jody Meyer Lockhart. The conjugacy problem for graph products with infinite cyclic edge groups . Proc. Amer. Math. Soc. 114 (1992) 603-606. MR 1072088. Abstract, references, and article information View Article: PDF This article is available free of charge [25] C. Y. Tang. On the subgroup separability of generalized free products of nilpotent groups . Proc. Amer. Math. Soc. 113 (1991) 313-318. MR 1081099. Abstract, references, and article information View Article: PDF This article is available free of charge [26] Michael Anshel and Dorian Goldfeld. Partitions, Egyptian fractions, and free products of finite abelian groups . Proc. Amer. Math. Soc. 111 (1991) 889-899. MR 1065083. Abstract, references, and article information View Article: PDF This article is available free of charge [27] K. J. Horadam. The conjugacy problem for finite graph products . Proc. Amer. Math. Soc. 106 (1989) 589-592. MR 962244. Abstract, references, and article information View Article: PDF This article is available free of charge [28] M. Shirvani. A converse to a residual finiteness theorem of G. Baumslag . Proc. Amer. Math. Soc. 104 (1988) 703-706. MR 935110. Abstract, references, and article information View Article: PDF This article is available free of charge [29] Charles Cassidy. Embedding of $U\sb \omega$-groups in $D\sb \omega$-groups . Proc. Amer. Math. Soc. 103 (1988) 15-20. MR 938636. Abstract, references, and article information View Article: PDF This article is available free of charge [30] Sidney A. Morris and H. B. Thompson. Sequential conditions and free products of topological groups . Proc. Amer. Math. Soc. 103 (1988) 633-638. MR 943096. Abstract, references, and article information View Article: PDF This article is available free of charge
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https://gateoverflow.in/309799/how-to-find-precedence | 39 views
| 39 views
0
According to me it should be
$c<\cdotp g$
0
@Verma Ashish can you please tell how?
0
AaBcD
in some step of derivation if we use D--> g then string will be ____cg
it fulfill the condition for $c\dot <g$ as c appears first then g appears... And more over g is at the lower level of production than c. (I'm not sure whether this sentence makes any sense according to the question or not ) | {"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": 0.8514205813407898, "perplexity": 2624.486098059816}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496671411.14/warc/CC-MAIN-20191122171140-20191122200140-00052.warc.gz"} |
http://semantic-portal.net/vue-transitions-and-animation-state-transitions | # State Transitions
Vue’s transition system offers many simple ways to animate entering, leaving, and lists, but what about animating your data itself? For example:
• numbers and calculations
• colors displayed
• the positions of SVG nodes
• the sizes and other properties of elements
All of these are either already stored as raw numbers or can be converted into numbers. Once we do that, we can animate these state changes using 3rd-party libraries to tween state, in combination with Vue’s reactivity and component systems.
## Animating State with Watchers
Watchers allow us to animate changes of any numerical property into another property. That may sound complicated in the abstract, so let’s dive into an example using GreenSock:
<script src="https://cdnjs.cloudflare.com/ajax/libs/gsap/1.20.3/TweenMax.min.js"></script>
<div id="animated-number-demo">
<input v-model.number="number" type="number" step="20">
<p>{{ animatedNumber }}</p>
</div>
new Vue({
el: '#animated-number-demo',
data: {
number: 0,
tweenedNumber: 0
},
computed: {
animatedNumber: function() {
return this.tweenedNumber.toFixed(0);
}
},
watch: {
number: function(newValue) {
TweenLite.to(this.\$data, 0.5, { tweenedNumber: newValue });
}
}
})
When you update the number, the change is animated below the input. This makes for a nice demo, but what about something that isn’t directly stored as a number, like any valid CSS color for example? Here’s how we could accomplish this with Tween.js and Color.js:
<script src="https://cdn.jsdelivr.net/npm/tween.js@16.3.4"></script>
<script src="https://cdn.jsdelivr.net/npm/color-js@1.0.3"></script>
<div id="example-7">
<input
v-model="colorQuery"
v-on:keyup.enter="updateColor"
placeholder="Enter a color"
>
<button v-on:click="updateColor">Update</button>
<p>Preview:</p>
<span
v-bind:style="{ backgroundColor: tweenedCSSColor }"
class="example-7-color-preview"
></span>
<p>{{ tweenedCSSColor }}</p>
</div>
var Color = net.brehaut.Color
new Vue({
el: '#example-7',
data: {
colorQuery: '',
color: {
red: 0,
green: 0,
blue: 0,
alpha: 1
},
tweenedColor: {}
},
created: function () {
this.tweenedColor = Object.assign({}, this.color)
},
watch: {
color: function () {
function animate () {
if (TWEEN.update()) {
requestAnimationFrame(animate)
}
}
new TWEEN.Tween(this.tweenedColor)
.to(this.color, 750)
.start()
animate()
}
},
computed: {
tweenedCSSColor: function () {
return new Color({
red: this.tweenedColor.red,
green: this.tweenedColor.green,
blue: this.tweenedColor.blue,
alpha: this.tweenedColor.alpha
}).toCSS()
}
},
methods: {
updateColor: function () {
this.color = new Color(this.colorQuery).toRGB()
this.colorQuery = ''
}
}
})
.example-7-color-preview {
display: inline-block;
width: 50px;
height: 50px;
}
## Dynamic State Transitions
As with Vue’s transition components, the data backing state transitions can be updated in real time, which is especially useful for prototyping! Even using a simple SVG polygon, you can achieve many effects that would be difficult to conceive of until you’ve played with the variables a little.
See this fiddle for the complete code behind the above demo.
## Organizing Transitions into Components
Managing many state transitions can quickly increase the complexity of a Vue instance or component. Fortunately, many animations can be extracted out into dedicated child components. Let’s do this with the animated integer from our earlier example:
<script src="https://cdn.jsdelivr.net/npm/tween.js@16.3.4"></script>
<div id="example-8">
<input v-model.number="firstNumber" type="number" step="20"> +
<input v-model.number="secondNumber" type="number" step="20"> =
{{ result }}
<p>
<animated-integer v-bind:value="firstNumber"></animated-integer> +
<animated-integer v-bind:value="secondNumber"></animated-integer> =
<animated-integer v-bind:value="result"></animated-integer>
</p>
</div>
// This complex tweening logic can now be reused between
// any integers we may wish to animate in our application.
// Components also offer a clean interface for configuring
// more dynamic transitions and complex transition
// strategies.
Vue.component('animated-integer', {
template: '<span>{{ tweeningValue }}</span>',
props: {
value: {
type: Number,
required: true
}
},
data: function () {
return {
tweeningValue: 0
}
},
watch: {
value: function (newValue, oldValue) {
this.tween(oldValue, newValue)
}
},
mounted: function () {
this.tween(0, this.value)
},
methods: {
tween: function (startValue, endValue) {
var vm = this
function animate () {
if (TWEEN.update()) {
requestAnimationFrame(animate)
}
}
new TWEEN.Tween({ tweeningValue: startValue })
.to({ tweeningValue: endValue }, 500)
.onUpdate(function (object) {
vm.tweeningValue = object.tweeningValue.toFixed(0)
})
.start()
animate()
}
}
})
// All complexity has now been removed from the main Vue instance!
new Vue({
el: '#example-8',
data: {
firstNumber: 20,
secondNumber: 40
},
computed: {
result: function () {
return this.firstNumber + this.secondNumber
}
}
})
Within child components, we can use any combination of transition strategies that have been covered on this page, along with those offered by Vue’s built-in transition system. Together, there are very few limits to what can be accomplished.
## Bringing Designs to Life
o animate, by one definition, means to bring to life. Unfortunately, when designers create icons, logos, and mascots, they’re usually delivered as images or static SVGs. So although GitHub’s octocat, Twitter’s bird, and many other logos resemble living creatures, they don’t really seem alive.
Vue can help. Since SVGs are just data, we only need examples of what these creatures look like when excited, thinking, or alarmed. Then Vue can help transition between these states, making your welcome pages, loading indicators, and notifications more emotionally compelling.
Page structure | {"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": 0.3112960457801819, "perplexity": 24295.052510021822}, "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/1614178373095.44/warc/CC-MAIN-20210305152710-20210305182710-00015.warc.gz"} |
http://aimsciences.org/search/author?author=James%20%20Benn | # American Institute of Mathematical Sciences
## Journals
JGM
Let $M$ be a closed symplectic manifold with compatible symplectic form and Riemannian metric $g$. Here it is shown that the exponential mapping of the weak $L^{2}$ metric on the group of symplectic diffeomorphisms of $M$ is a non-linear Fredholm map of index zero. The result provides an interesting contrast between the $L^{2}$ metric and Hofer's metric as well as an intriguing difference between the $L^{2}$ geometry of the symplectic diffeomorphism group and the volume-preserving diffeomorphism group.
keywords: conjugate point geodesic Fredholm map symplectic Euler equations. Diffeomorphism group Maxwell-Vlasov | {"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": 0.951892077922821, "perplexity": 323.98930535833523}, "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-2018-17/segments/1524125944682.35/warc/CC-MAIN-20180420194306-20180420214306-00216.warc.gz"} |
http://www.ck12.org/algebra/Solving-Systems-by-Multiplying-One-Equation/flashcard/user:13IntK/Solving-Systems-by-Multiplying-One-Equation-to-Cancel-a-Variable/r2/ | <meta http-equiv="refresh" content="1; url=/nojavascript/">
# Solving Systems by Multiplying One Equation
## Multiply one equation, then add to cancel one variable
%
Progress
Progress
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Solving Systems by Multiplying One Equation to Cancel a Variable
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http://sites.science.oregonstate.edu/portfolioswiki/whitepapers:narratives:intro | ## Tuesday, April 26, 2011 (Day 2)
This narrative presents an example of an instructor engaging students in thinking conceptually about the partial derivatives they encounter in thermodynamic contexts. The instructor, David Roundy, invites students to think about how they would measure the quantities represented by a partial derivative. During this class session, Day 2 of the thermodynamics paradigm, he introduces this “Name the Experiment” process by engaging the small groups in designing thought experiments for a variety of thermodynamic partial derivatives. The narrative$^{1}$ below interprets the wrap-up discussion that follows this small group activity.
A partial derivative such as $$\left(\frac{\partial A}{\partial B}\right)_{C}$$ represents how much a quantity $A$ changes as the quantity $B$ changes when the quantity $C$ is held constant. David wants the students to be able to look at the symbols stating a partial derivative and to envision an experimental process that it represents. In particular, he wants them to realize that the quantity $C$ is something that must be held constant experimentally when measuring how the quantities within the parentheses change.
During Day 1 of the thermodynamics paradigm, David introduced the concept of thermodynamic state variables as well as operational definitions for the various state functions. The students did an experiment in which they measured the heat capacity of water and the latent heat of fusion of ice. David introduced the thermodynamic definition of entropy:
$$\Delta S = \int \frac{\text{đ}Q}{T}$$
and instructed the students, in the analysis of their experiment, to find the entropy change as ice was melted and water heated by integrating the measured heat.
On Day 2, David introduced the First and Second Laws of Thermodynamics. He explained entropy as “the thing that can increase, but not decrease.” He mentioned reversible processes briefly and explained how a reversible process must have zero entropy. As an example, he explained how dunking a hot metal cube into water is an irreversible process. He also talked about slow processes and fast processes, that slow processes are more likely to be reversible than fast. He listed many types of processes including quasistatic (“a process slow enough that the system is effectively in thermal equilibrium the entire time”), reversible, irreversible, adiabatic (“a process in which heat is not exchanged with the environment”), spontaneous, fast, and slow. Then he used the theoretical model of a piston to talk about work, the First Law of Thermodynamics (“conservation of energy”) and the thermodynamic identity $$dU = TdS – pdV$$ as a result of the First Law, combined with the definitions of work and entropy), which is a consequence of the First Law.
### Starting the Small Group "Name the Experiment Activity" [00:30:44.09]
At this point in the session, David shifted to small group work. He gave each of the group a thermodynamic derivative and told them they needed to design an experiment that would measure the derivative. The derivatives were
$$\ \left(\frac{\partial T}{\partial V}\right)_{S}, \left(\frac{\partial V}{\partial p}\right)_{T}, \left(\frac{\partial L}{\partial \tau}\right)_{T},\left(\frac{\partial U}{\partial T}\right)_{V}$$ $$\ \left(\frac{\partial V}{\partial T}\right)_{p},\left(\frac{\partial L}{\partial T}\right)_{t}\,\left(\frac{\partial V}{\partial p}\right)_{S},\left(\frac{\partial L}{\partial t}\right)_{T},\left(\frac{\partial U}{\partial p}\right)_{S}$$
where $T$ represents temperature; $V$, volume; $p$, pressure; $L$, length; $\tau$, tension; $U$, internal energy; and $S$, entropy.
The last derivative of equation $(2)$ is the most challenging, but the rest are semi-randomly ordered so as to not give a hint as to how they are related. Groups that finished early were asked to try the last, and most challenging, problem. The first, third, and last derivatives of equation $(1)$, and the first, third, and last derivatives of equation $(2)$ $%Derivatives #1, 3, 4, 5, 7 and 9%$ require that the standard thermodynamic variables, pressure, and volume have been introduced. The third derivative of $(1)$ and second derivative of (2) $%Numbers 3 and 6%$ require that students be able to apply thermodynamic concepts to 1-D systems, such as strings or rubber bands. The last derivative of $(1)$ requires students to have a working knowledge of the First Law, while the last derivative of $(2)$ requires students to recognize how to perform an adiabatic measurement, and how that relates to entropy.
As David and his teaching assistant circulated among the small groups, they looked over what was on the white boards and listened to what the members of a small group were saying to see whether the group was paying attention to what was to be held constant. If not, David and the TA intervened with questions, such as “how do you keep the temperature fixed?” to nudge the group toward recognizing this critical aspect of the task. In addition, they asked groups how they would make measurements, so as to reinforce the lessons of the previous day regarding operational definitions of thermodynamic quantities.
David gave another more challenging partial derivative to groups who had finished before he thought the whole group was ready for the wrap-up discussion.
After the groups were given time to work, David brought the class back together and asked the students to share what they done. As the groups presented, he gave advice about the different scenarios.
### Starting the Wrap-Up Discussion: First Group's Presentation
After about fifteen minutes, David began the wrap-up discussion by selecting the group who did the first partial derivative.
${\bf [00:46:58.22]}$
${\bf Roundy}:$ So, let's just very quickly, if we could start with you, we'll just go around, and you can tell us what the derivative was and how it's measured in one minute or less.
David has the groups report out in the same order as he gave out the derivatives. Although the first was an easy one, he tries to avoid going in an easiest-to-hardest order. This insures that the discussion gets beyond the easiest ones if there is a small number of groups or time is limited.
${\bf [00:47:15.03]}$
The first group had worked on the partial derivative of the change in temperature with a change in volume, holding the entropy fixed, $\left(\frac{\partial T}{\partial V}\right)_{S}$ .
A student held up his small group’s whiteboard, which showed a piston with insulated walls and a thermometer inside, on the left side of the white board. They had understood that holding entropy constant means isolating the system so that no energy is lost or gained. (They had also started working on another partial derivative on the right side of the white board.)
${\bf \text{Student 1}}:$ So ours was $dT$ d(volume) holding $S$ constant $\left[\left(\frac{\partial T}{\partial V}\right)_{S}\right]$ basically we derived kind of a complex system here. Thermally insulated, thermometer in there, so basically we can change the volume inside, measuring the volume of air or some gas in there.
${\bf [00:47:36.23]}$
${\bf Roundy}:$ Ok.
While watching the video, David commented that an expert response would be similar, to use an insulated piston and change the volume while measuring the change in temperature inside.
### Second Group's Presentation
The second group had worked on the partial derivative of volume with respect to pressure, holding the temperature constant, $\left(\frac{\partial V}{\partial p}\right)_{T}$ .
They also had drawn a piston, but with thin walls. A arrow pointing down represented a force pushing on the piston. By writing “very slowly” next to the F representing a Force, they indicated that they had understood this process needed to happen slow enough for any heat generated to dissipate through the walls of the piston. To the left of the piston, they had written “area constant” and to the right, “measure height.” These indicate that they understood that the volume would be the product of the area and height and would change as the height changed. Underneath the piston, they had written “Surroundings at constant temperature; system not insulated” indicating that they understood that the piston needed to be in an environment with constant temperature and that any heat generated in pushing the piston down slowly would be dissipated through the thin walls.
${\bf [00:47:49.12]}$
${\bf \text{Student 2}:}$ So we're two and uh we change the volume and pressure and keep the temperature constant. So if we had a cylinder with some gas in it and we allowed heat transfer to go out of the wall, uh, it wasn’t insulated it would come to equilibrium with the outside so it would keep the temperature also reversible.
${\bf [00:48:11.14]}$
${\bf \text{Student 3}:}$ Varying the force applied across constant area and measuring how the volume changed.
While watching the video, David commented that this was another easy experiment. They did not have a mechanism for picking a temperature rather than room temperature but they did explicitly discuss how to keep the temperature fixed. An expert response would be very similar, a piston with weights perhaps on top and thin metallic walls that conduct heat well. He would change the weight and measure the change in volume, being sure to wait a long time for thermal equilibrium, at room temperature, and one could adjust that with a thermostat or put the whole device in a refrigerator or mountain stream.
### Third Group's Presentation
The third group had worked on the derivative of length with respect to tension at fixed temperature, $\left(\frac{\partial L}{\partial \tau}\right)_{T}$.
They showed a complex drawing with two pistons, a rope, mass on a pulley, and a cooling system (fan).
${\bf[00:48:20.05]}$
${\bf \text{Student 4}:}$ OK, yeah so we have change in length due to the change in tension. So we had two pistons, one with a rope and then a mass on the pulley on one of them, so you can change the tension by adding masses to this side. And then our cooling system (moves hand around circle of enclosure) because when you change the tension on the string, the temperature of the string will increase, so you have to change the cooling system to keep the temperature the same, so that’s our system basically. ${\bf[00:49:01.05]}$
While watching the video, David commented that his “expert” solution would probably be a bit simpler: hang a weight from a rubber band, and measure its length. Then add a bit more weight, and wait long enough for it to equilibrate at room temperature. As before, one could use a thermostat or a walk-in-freezer to do the experiment at different temperatures. This example was similar to the rubber band lab that the students would do later in class, probably in the next week.
David noted that this solution was interesting because they included a pulley system. They were changing the tension on the rope by changing the weight on the pulley.
Corinne commented that students had to actually think about how they would construct an experimental apparatus rather than defaulting to a piston.
David noted that having problems that all use pistons is boring and for a third experiment why not use pulleys? The students’ mechanism for keeping the temperature fixed was also quite interesting because they employed a fan rather than just saying we'll wait for the temperature to equilibrate. That seems like they were again thinking experimentally, how would they really keep the temperature fixed?
There are choices to be made in terms of which partial derivatives to give to the students. David commented that often he leaves out the one dimensional problems because they feel redundant to him but sometimes he thinks that they add extra complexity and confusion so perhaps he should include them. Here he did.
### Fourth Group's Presentation
The fourth group had worked on the partial derivative of the internal energy with respect to temperature at constant volume, $\left(\frac{\partial U}{\partial T}\right)_{V}$.
This group had drawn a container and indicated that it had a constant volume, with a heater underneath, and probe in the middle of the container to measure the temperature. They explained their reasoning in writing: “From the power, we can get the work done by the resistor. This, if inverted will give us a good ratio between the change in energy and the temperature.”
${\bf \text{Student 5}:}$ So, we ended up getting the change in internal energy with respect to the change in temperature, holding V constant.
${\bf [00:49:13.19]}$
So basically this here, after a little bit of help from (drops board) Aack! Internal energy is really hard, it's a hard one to measure. But we found out it's pretty much the same experiment that we did yesterday, so we can find out essentially the total energy that's being put in the system by the resistor or you know, some other heat source and then you can just measure the temperature as the time passes and we can essentially compare the two on a plot.
${\bf Roundy:}$ So one interesting thing on that, just to add, a volume equals constant is a very very very hard thing to do. Devices that measure things with constant volume – the name for that device is called a bomb. Because you need very very strong walls, and if you increase the temperature too much, it will explode. All right, basically. And chemists have this, because it's a worthwhile measurement to make. ${\bf [00:50:20.25]}$
${\bf TA:}$ Nuclear plants in Japan. A few months ago started leaking because of what…walls start cracking
${\bf [00:50:33.22]}$
${\bf Roundy:}$ So we like to not do things at constant volume. So theoretically it's very easy to do things at constant volume, in practice, it's hard. That's it. Next group.
While watching the video, David noted that this is a measurement of the heat capacity of a gas at fixed volume, and would naturally be done with a bomb calorimeter, much as the students describe. He would have a heating element, and measure how much power it outputs in a given time interval (to measure the energy put into the system), and then see how much the temperature of the gas changes, while holding the volume fixed with very strong walls. One would need to wait quite a while for it to equilibrate, which is a challenge in the experiment the students do in class.
David also commented that this partial derivative was interesting for multiple reasons. One was that fixing the volume is hard experimentally, although not hard conceptually. This partial derivative also was interesting because internal energy is not something that can be measured directly. Changes in internal energy can be calculated via the first law as equal to the energy put into the system (by heating) minus the work done by the system. The work done by the system would be $pdV$; $V$ is constant so $dV=0$ and the work done equals zero. Therefore the change in internal energy would be just the energy put into the system.
If David had asked for the same experiment at fixed pressure, he thinks the students would have given an incorrect answer because they would have neglected work. However, this is a nice problem for this first day because they can get it right. If they had been wrong, what he would have asked them would depend upon how they got it wrong. If he had asked this problem at fixed pressure, in which gas in a cylinder with a piston expands when heated, he would have asked how they knew what the change in internal energy was and hint at the first law and then ask about work being done. Students often believe no work is done if they are not actively changing pressure or volume, which is why they typically make the mistake of neglecting the work done by the system. In the case of thinking about the change in internal energy with respect to temperature at constant pressure, the gas is doing the work pushing up the piston as the gas expands when heated.
Corinne commented that the emphasis is on THEY are actively doing the work, not the system. That seems to be a theme behind several of the different extended comments that David had made, where students only keep track of what they as experimentalists change and not changes in the system as a result of other parts of the experiment.
David noted that the student stated that this was the same experiment they had done the day before but that experiment had been at fixed pressure (open to atmospheric pressure) so even here it is possible the student was recognizing that difference and meaning that the heat measurement was the same but he may have fallen into this mistake of thinking the internal energy change was the same.
The students had done a calorimetry experiment the day before so these students were able to give a correct experiment using the first law. The calorimeter is a Styrofoam cup with a lid, and the ice and water are inside it, along with a resistive heater. By measuring current and voltage, the students know how much energy is added to the system by resistive heating, and we hope the Styrofoam is a sufficient insulator that heating from the room is negligible. Students measure the change in temperature of the ice water. Of course, while there is ice (and the system is well mixed) this temperature should not change, which gives us a measure of how much energy is needed to melt the ice. Once the ice is all melted, we measure the temperature as a function of energy added (i.e. time), and find the heat capacity. In the analysis of the lab, David also asks students to calculate the change in entropy from their data, which requires just a bit of math beyond their actual measurements.
With regard to the thought experiment, at constant volume, it is*almost* like what they did, with the difference being that they did not hold volume fixed, but instead held pressure fixed (at atmospheric pressure). They added a bit of energy (at a time) and measured how much the temperature changed, which in a sense is the inverse of the derivative for which they were creating the thought experiment.
Corinne wondered how important it is that the students have done the calorimetry experiment before they do this partial derivative. David responded that for this particular question, it is very important. For many years, he has always done the calorimetry experiment on the first day of the class. He started this because he wanted more experiments and more understanding of heat.
David also commented that sometimes he wants to reduce the number of partial derivatives used, in favor of redundancy, because one never knows what might happen in the different groups. A group might design a different experiment; with an internal energy partial derivative, there might be groups that do not get a reasonable answer so sometimes it is good to have two groups on an internal energy partial derivative as insurance.
### Fifth Group's Presentation
The fifth group had worked on the partial derivative of volume with respect to temperature at constant pressure, $\left(\frac{\partial V}{\partial T}\right)_{p}$ . On their whiteboard they had drawn a picture of a cylinder with a piston with a thermometer inside the cylinder. An arrow pointing down, above the piston, was labeled constant pressure.
${\bf [00:50:46.19]}$ ${\bf\text{Student 7}:}$ We did $dV/dT$ holding pressure constant, so we just have a piston system, and I guess it's frictionless when the piston goes up and down. There's a constant pressure on the outside of it and then a thermometer on the inside suspended somehow. We're gonna make all the measurements, and then add a heating element under it to make it hotter, and then measure the new volume to see the relationship.
${\bf Roundy:}$ And how are you going to keep the pressure constant?
${\bf \text{Student 8}:}$ We'll pretend this is a vacuum, and we'll put weights on there. (laughter)
${\bf Roundy:}$ It doesn't need to be a vacuum, but if you want to do any pressure that’s under an atmosphere, you either need to pull up on it or…Ok, sure, you can be next.
While watching the video, David commented he would do this quite similarly to what they did. One would want a piston with weights on top of it, or if in the atmosphere, possibly a pulley setup for weights to pull *up* on it, if want a lower pressure than atmospheric, and one would need to change its temperature (and the temperature of its surroundings) slowly, and measure how the volume changes. One would have to change the temperature of the surroundings so that the piston system could remain in thermal equilibrium, otherwise this could end up with a temperature gradient.
David noted that was an example where the students had not explicitly drawn how they would control the pressure, which is something that he would normally ask about while circulating during the small group work.
Corinne commented that they also seemed to not know how to get less than an atmosphere of pressure without doing it in a vacuum. The suggestion that they have to pull up seems to have not occurred to them and might not occur to a number of students.
David responded that in a thought experiment, he would do the same thing they did (put the piston in a vacuum) because with vacuum around one can very easily compute the pressure and set it to any value one wishes.
Corinne noted that those new to teaching this class need to be alert to the fact that the students might not know what to do in a more realistic situation where they are working at atmospheric pressure. Sometimes students might know how to push but it might not be obvious to them that the opposite of that is pulling on a piston.
David agreed but also commented that he was very happy to hear the students specify a vacuum because a simpler mistake would be to ignore atmospheric pressure and simply divide the applied force (the weight) by the area.
### Sixth Group's Presentation
The sixth group had worked on the partial derivative of length with respect to temperature at constant tension, $\left(\frac{\partial L}{\partial T}\right)_{\tau}$.
This group had drawn two versions of a rubber band stretched by a hanging mass, the top diagram for change in length with respect to tension at constant temperature and the bottom diagram for change in length with respect to temperature at constant tension. The bottom diagram included squiggly lines that were labeled HEAT. The students reported on the bottom experiment, with constant tension.
${\bf [00:51:33.05]}$
${\bf\text{Student 8}:}$ Oh yeah, we're going next. So we did number 6, which is change in length, change in temperature, holding tension constant.
${\bf\text{Student 9}:}$ Which was the pretest
${\bf \text{Student 8}:}$ So basically, we had a rubber band and we put a mass on it so we can measure tension and we're not going to change the mass so it's at constant tension. We're gonna place that in some closed area and then increase the temperature, or change the temperature or heat it up…
While watching the video, David commented that he would hold the tension fixed just as these students did, by hanging a weight from the rubber band. However, holding the tension fixed at different temperatures is potentially tricky. Either one needs to adjust the thermostat of the room, or move the experiment to some place with a different temperature. Or use a heat bath as in the experiment done in a later class, submersing the rubber band in water at a varying temperature. This question, like the earlier one dimensional one (#3), is a precursor to the rubber band lab the students do in a later class, which measures tension as a function of both length and temperature.
David noted that they held the tension fixed with a weight and were a little vague but by putting it in a box and changing the temperature of the box they changed its temperature. With temperature one cannot change it quickly and have reasonable results because one will leave thermal equilibrium so changing the temperature ends up looking similar to holding it fixed.
### Seventh Group's Presentation
The seventh group had worked on the partial derivative of volume with respect to pressure at constant entropy, $\left(\frac{\partial V}{\partial p}\right)_{S}$.
This group had drawn a piston in an isolated cylinder.
${\bf [00:52:03.04]}$
${\bf Roundy:}$ Next group.
${\bf\text{Student 10}:}$ Ours is like one of the other ones. We just have a piston system here. The actual thing we were testing was $dV/dP$ holding $S$ constant. and so to hold S constant, we just insulated it, and you can change the pressure and volume.
${\bf Roundy:}$ You change the pressure how?
${\bf \text{Student 10}:}$ Uh, by applying force to the piston.
While watching the video, David commented that this one is quite easy. It involves constant entropy, so he would use an insulated piston. To change pressure, he would add or remove weights from the piston, and then measure the volume by measuring the height of the piston.
David noted that these students did not bother explaining how they changed the pressure or how they measured the volume. This might be due to fatigue since they had just spent a lot of time listening to almost the same experiments and it perhaps felt obvious to them at this stage and not worth commenting on.
Corinne commented that he prompted them about how they would control the pressure but not how they would measure the volume. One of the things she has learned over the years is that one can teach the students a lot about what constitutes a complete answer by prompting every single group to give a complete answer until they accept that as a professional norm. There is a tension between that and trying to get through the class before it ends or before one loses the students' attention.
David suggested that prompting them often to make a complete answer is an advantage of switching the dimensionality so that students have different things to say about how they control their system. Keeping their interest, avoiding this fatigue, this boredom, is facilitated by mixing up the dimensionality, sometimes have one dimensional problems with rubber bands as well as the three dimensional problems with pistons.
Corinne noted that when prompting students about the same measurement, they'll often just say “and we'll measure the volume like that other group did.” One can prompt them to say something more and they'll say something very brief, can and do. This cuts down on the boredom but does signal to them that they do have to think about these things when designing an experiment and they must include in a report, to say how they will measure things, not just that they will. She likes to use the reporting out of small group activities as a chance to reinforce or convey those professional norms.
### Eighth Group's Presentation
The eighth group had worked on the partial derivative of the internal energy with respect to pressure at constant entropy, $\left(\frac{\partial U}{\partial p}\right)_{S}$.
This group had drawn a picture of a piston in a cylinder and written several mathematical expressions in trying to envision an appropriate process.
${\bf Roundy:}$ And, I don't know if you guys came up with an answer. Can you explain the trouble you had?
${\bf [00:52:40.04]}$
${\bf \text{Student 11}:}$ So we were dealing with $dU/dP$ holding entropy constant. And we didn't really like have the epiphany of how to measure internal energy. That's something we kept kind of having issues with. We really wanted to use this piston somehow. But as we apply more pressure, we thought the temperature would increase, adding heat. And a constraint that we set, for entropy to be held constant, we need that to be zero. So we were trying to find a way that we could change pressure while keeping entropy constant, and it's a weird thing, we tried to do math, and see if we could measure that partial some other way. But we didn't quite finish in time.
${\bf Roundy:}$ Now, many of you, finished yours early and went to another one. Did any of you go to this one? If not, ok. Because this is an interesting one, and it was, in fact, the hardest one.
While watching this video, David commented that he would measure this by using an insulated piston and changing the pressure precisely as he would have in the previous presentation $\left(\left(\frac{\partial V}{\partial p}\right)_{S}\right)$. Once one measures that derivative, one can compute the change in internal energy, which is entirely due to the work done on the system, because $\text{đ}Q=0$, because the entropy is held fixed. So one can invoke the first law [that the change in internal energy is equal to the energy put into the system (in this case $0$) minus the work done by the system], and see that $$\left(\frac{\partial U}{\partial p}\right)_{S} = -p\left(\frac{\partial V}{\partial p}\right)_{S}$$.
David noted that this was a tricky one. It looks like they did not realize that holding entropy fixed meant that there was no heating, no $Q$. They seem to have made the common mistake of recognizing that the temperature will increase and then thinking that entropy will be increasing because it must be being heated.
Corinne wondered if he had let that stand.
David responded that he probably didn’t want to criticize a group that he already felt bad about giving a very hard problem. However, they don't look unhappy in the video and with all the things going on, he might not have actually noticed their mistake of thinking that there was heating going on and that therefore the entropy would change.
David expressed disappointment that he didn't explicitly mention integrating the work, to find the change in internal energy. He had mentioned measuring the change in volume and using the first law and might have pointed at a $pdV$ on the board.
Corinne commented that one way to handle the concern about having given a group a harder example is to acknowledge it, which he had. One can say “this is a harder example” and then ask the question of the class as a whole about the situation rather than focusing on that group further. Say, for example, “the temperature does go up, does the entropy also increase?” and generate that conversation with the class. Another thing to do, when a particular example is hard is tell the students, if your group solves your first example quickly, then go on to this other one (the one that happens to be hard) and that way get the strong groups thinking about the hard example.
David noted that all of the 3-D ones depend upon having already covered the standard thermodynamic quantities, pressure, volume, etc. The internal energy ones require the first law of thermodynamics and an understanding of heat and work and how to measure them, which had been discussed previously during day 1 or the math interlude.
### Explicit Articulation of the Purpose of the "Name the Experiment" Activity
After talking about a thought experiment for the last partial derivative, David explicitly discussed why he had the students do what they had just finished doing. Such explicit discussion of one’s teaching strategies can help students see the broader picture of their experiences in learning, particularly in a difficult context such as thermodynamics.
${\bf [00:55:19.20]}$
${\bf Roundy:}$ By the way, the reason I do this Name the Experiment thing, they're going to come back again, later, maybe not tomorrow, but we will have more of this kind of activity using harder derivatives.
He noted that one of the challenging things in thermodynamics is that a problem will be posed and there will be lots of derivatives in the solution and then the lecturer, or the text, will say something like “Oh that’s just isothermal compressibility.”
${\bf [00:56:04.08]}$
${\bf Roundy:}$ Then you say, “it's all beautiful” but “why? why did all that stuff happen?” and the reason all that stuff happened is because you're trying to convert something like that (gestures at final group's derivative) that is hard to measure, or something like this fixed volume thing, where you say, “Fixed volume? That's really hard to measure experimentally” and you might say “I’d like to, right, instead measure” something else easy to measure, and figure out what this thing I'm curious about is.” and so, in order to do that, I'm going to have you work on thinking about how you would actually measure some of these derivatives.
With ten minutes left in the class, David turned to clarifying and extending the concept of heat capacity. The previous day they had done a heat capacity experiment at constant pressure, $C_{p}$ . He contrasted that with heat capacity at constant volume, $C_{v}$. Constant volume means that no work ($pdV$) is being done. From the First Law of Thermodynamics one can say that whatever energy is put into heating, all that energy is going into changing the internal energy so $C_{v}$ is just $dU$ by $dT$ at constant volume. One might think that $C_{p}$ - the heat capacity at constant pressure - would just be $dU$ by $dT$ at constant pressure, but that is not the case. At constant pressure, there is some work being done, so the amount of energy transferred by heating is not equal to the amount of the change in internal energy because there is work being done by the system. Then he discussed a generalized form that could be used in thinking about heat capacity in a variety of cases, including rubber bands.
### Notes
$1.$ This interpretative narrative is based upon a video of the class session and discussions with the instructor, David Roundy, the director of the Physics Paradigms Program, Corinne Manogue, and Emily van Zee, a science education researcher. Amanda Abbott transcribed the video. In writing the narrative, Emily drew upon her research in the tradition of ethnography of communication (Hymes, 1972; Philipsen & Coutu, 2004; van Zee & Minstrell, 1997a,b), a discipline that studies cultures through the language phenomena observed. This interpretative narrative presents an example of students growing into participants in the culture of “thinking like a physicist,” in particular in using the ‘language’ of partial derivatives.
### References
Hymes, D. (1972). Models for the interaction of language and social life. In J. Gumperz & D. Hymes (Eds.), Directions in sociolinguistics: The ethnography of communication (pp. 35-71). New York: Holt, Rinehart & Winston.
Philipsen, G. & Coutu, L. (2004). The Ethnography of Speaking. In K. L. Fitch & R. E. Sanders (Eds.), Handbook of language and social interaction (pp.l 355-380. Mahwah, NJ: Lawrence Erlbaum.
van Zee, E. H. & Minstrell, J. (1997a). Reflective discourse: Developing shared understandings in a high school physics classroom. International Journal of Science Education, 19, 209-228.
van Zee, E. H. & Minstrell, J. (1997b). Using questioning to guide student thinking. The Journal of the Learning Sciences, 6, 229-271.
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Publications | {"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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.8586775064468384, "perplexity": 504.0491631096034}, "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-2020-10/segments/1581875144429.5/warc/CC-MAIN-20200219214816-20200220004816-00394.warc.gz"} |
https://wiseupwinnipeg.nationbuilder.com/engineering_speed_limits | ## Engineering Speed Limits
The 85th Percentile Engineering Standard for Setting Speed Limits...
(Excerpt from “Speed Zoning Information A Case of Majority Rule" Institute of Transportation Engineers)
Generally, traffic laws that reflect the behaviour of the majority of motorists are found to be successful, while laws that arbitrarily restrict the majority of motorists encourage violations, lack public support and usually fail to bring about desirable changes in driving behaviour. This is particularly true when it comes to establishing speed limits.
Speed limits are based on several fundamental concepts deeply rooted within the system of government and law:
1. driving behaviour is an extension of social attitude and the majority of drivers respond in a safe and reasonable manner as demonstrated by consistently favourable driving records;
2. the normally careful and competent actions of a reasonable person should be considered appropriate;
3. laws are established for the protection of the public and the regulation of unreasonable behaviour on the part of individuals; and
4. laws cannot be effectively enforced without the consent and voluntary compliance of the public majority.
One important objective in setting a speed limit is to inform drivers of a reasonable and safe maximum speed under normal driving conditions (Clear and proper signage/what is MUCTD?).
When less than ideal conditions exist, a driver must adjust vehicle speed accordingly as required by provisions of the Manitoba Highway Traffic Act.
It is a long accepted North American practice to recommend and establish speed limits based on the 85th percentile speed, being the speed at or below which 85% of motorists travel, in conjunction with a detailed engineering analysis of other factors such as collision information. (Known as the 85th percentile rule, or 85th percentile engineering standard)
Circumstances such as curves on the road, visibility restrictions, pedestrian and parking activity, and adjacent land uses (e.g., schools, shopping centres, etc.) are factors that determine the speed at which the vast majority of motorists elect to operate their vehicle.
A speed limit established on such a basis is also referred to as a “credible speed limit” in that the speed limit matches the image that is inspired by the roadway environment and the traffic operating circumstances encountered.
Features of the driving environment that are relevant to a ”credible speed limit” include the roadway width, the number of lanes, lane lining and marking, the presence of adjacent buildings, as well as trees, utility poles and furniture in the boulevard. Long, straight, wide sections of roadways with a smooth surface in an open clear road environment tend to lend themselves to a higher operating speed than is the case where such features are not present.
(Surprisingly absent from that description is the whim of politicians)
Establishing speed limits in this manner has proven to be effective in that it accommodates traffic in a safe and orderly way and enables the Police to focus their enforcement resources toward the 15% of drivers who operate at excessive speeds. (That is, excessively too high, or excessively too low. Ironically, enforcement usually only occurs for higher speed, not the lower speed traveller, even though both are proven to be dangerous to safer/more efficient traffic flow)
Such a criterion recognises that the majority of motorists operate their vehicle in a reasonable and prudent manner with due consideration for conditions encountered, including activity into and out of intersecting public streets and approaches as well as the presence of pedestrians and bicyclists on or near the roadway.
By setting speed limits using the 85th percentile speed, the range of speeds is lessened, providing a more uniform flow of traffic. Studies have shown that:
• more collisions occur when the speeds of vehicles are varied with extremely high or low speeds encountered [1];
• setting speed limits lower than that considered reasonable to the majority of drivers encourages disrespect of speed limits in general;
• posted speed limits which are set higher or lower than that dictated by roadway and traffic conditions are ignored by the majority of motorists; and that
• when speed limits are raised or lowered, there is very little impact on motorists’ actual speeds.
• 1“U.S. DOT Publication No. FHWA-RD-98-154”, 1998
Safe, “credible speed limits” can be expected to enhance motorists’ compliance to the speed limit, which in turn can result in a reduction in collisions than would otherwise be the case.
If a speed limit is not credible, motorists will be inclined to elect to drive at a speed that they perceive to be realistic. If speed limits are perceived as being incredible too frequently, it will challenge the public’s trust in the speed limit system generally. A speed limit can be incredible because the speed limit is either perceived as being too low or as being too high.
The net effect of incredible speed limits is that motorists will increasingly disregard that and consequently with frequent incredible limits, motorists will begin more and more disregarding road rules and regulations.
If it is noticed that drivers are disregarding speed and other regulations, it is often a symptom of the application of improper traffic engineering.
This is a map of the roads that were in a proposal to increase speeds just over 4+ years ago. To this day I don't know of any of these proposals being implemented. These were based on the appropriate speed per the 85 percentile which the city claims it uses to set speed limits. | {"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": 0.886854350566864, "perplexity": 1875.3675968195937}, "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/1526794867416.82/warc/CC-MAIN-20180526092847-20180526112847-00136.warc.gz"} |
https://infoscience.epfl.ch/record/189349 | ## Ultrafast Relaxation Dynamics of Osmium-Polypyridine Complexes in Solution
We present steady-state absorption and emission spectroscopy and femtosecond broadband photoluminescence up-conversion spectroscopy studies of the electronic relaxation of Os(dmbp)(3) (Os1) and Os(bpy)(2)(dpp) (Os2) in ethanol, where dmbp is 4,4'-dimethyl-2,2'-biypridine, bpy is 2,2'-biypridine, and dpp is 2,3-dipyridyl pyrazine. In both cases, the steady-state phosphorescence is due to the lowest (MLCT)-M-3 state, whose quantum yield we estimate to be <= 5.0 x 10(-3). For Os1, the steady-state phosphorescence lifetime is 25 ns. In both complexes, the photoluminescence excitation spectra map the absorption spectrum, pointing to an excitation wavelength-independent quantum yield. The ultrafast studies revealed a short-lived (<= 100 fs) fluorescence, which stems from the lowest singlet metal-to-ligand-charge-transfer ((MLCT)-M-1) state and decays by intersystem crossing to the manifold of (MLCT)-M-3 states. In addition, Os1 exhibits a 50 ps lived emission from an intermediate triplet state at an energy similar to 2000 cm(-1) above that of the long-lived (25 ns) phosphorescence. In Os2, the (MLCT)-M-1-(MLCT)-M-3 intersystem crossing is faster than that in Os1, and no emission from triplet states is observed other than the lowest one. These observations are attributed to a higher density of states or a smaller energy spacing between them compared with Os1. They highlight the importance of the energetics on the rate of intersystem crossing.
Published in:
Journal Of Physical Chemistry C, 117, 31, 15958-15966
Year:
2013
Publisher:
Washington, American Chemical Society
ISSN:
1932-7447
Laboratories: | {"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": 0.9406773447990417, "perplexity": 7632.4812197034735}, "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/1539583512679.76/warc/CC-MAIN-20181020080138-20181020101638-00399.warc.gz"} |
https://arxiv.org/search/advanced/gr-qc | # arXiv.org
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http://bladman7659.soup.io/post/641955922/Of-Course-These-Very-Strategies-Have-Proven | You are at the newest post.
### Of Course, These Very Strategies Have Proven Quite Effective In The Past, And Will Likely Continue To Work Well In The Future.
Number One and MOST important – Never, ever, under any circumstance borrow money make things easier by consolidating them and taking one single loan to pay off the total debt. Even if you have $500,000 right now, it is better the late night infomercials is called ‘lease optioning’. The next most ‘traditional’ method is to buy a fixer-upper, dollar bills for forty-five cents is likely to prove profitable even for mere mortals like us. Even if you have$ 500,000 right now, it is better does calculate the intrinsic value of the stocks he buys. Personal loans are classified as secured and unsecured loan common stock that historically has a steady or increasing dividends. There is something called investor eligibility that you need to meet for this form is what investors look at while using private money investing. | {"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": 0.32030925154685974, "perplexity": 1923.8406396058108}, "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/1539583512499.22/warc/CC-MAIN-20181019232929-20181020014429-00082.warc.gz"} |
https://math.stackexchange.com/questions/2562291/non-linear-assumed-form-in-galerkin-method | # Non-linear assumed form in Galerkin method
The wikipedia article on Method of mean weighted residuals has a section on choice of test functions, in particular it says about Galerkin method:
The Galerkin method, which uses the basis functions themselves as test functions or in the more general case of a nonlinear assumed form (where the nonlinearity is in the degrees of freedom) of the solution the Galerkin method uses the test functions: $$w_{i}={\frac {\partial u}{\partial a_{i}}}$$
I'm really interested in the "nonlinear assumed form". I would like to know what is known about Galerking when we assume nonlinear form of the trial function.
Unfortunately the wiki article gives only one reference to a book which I was unable to get hold of. Can anyone direct me to any other source where I can read more about this topic?
• Could it be that 'nonlinear assumed form' means that Galerkin function depend nonlinear on degree of freedom $a_i$? This wiki page is indeed very confusing. – daw Dec 21 '17 at 10:23
• I think that is exactly it, but I would like to know more about the consequences of this nonlinearity. All texts talk about Galerkin method where you assume that the unknown function is a linear combination of fixed basis functions. – tom Dec 21 '17 at 21:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8971607685089111, "perplexity": 293.29218252661695}, "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/1558232256184.17/warc/CC-MAIN-20190521002106-20190521024106-00014.warc.gz"} |
https://wikieducator.org/User:Vtaylor/CIS2_Open_Textbook_Reviews | # User:Vtaylor/CIS2 Open Textbook Reviews
## Summer 2010
I was taking a look at the online text books that they had for psychology, I'm currently not taking that course but I plan to in the fall. As I was looking at how wiki-books presents the
information of the online text book, I was impressed, they have it setup with chapter, sub
chapters, important terms and information in bold and sum highlighted with as link to find out
more about a term or certain section. I think this is a great idea because of outrageously
expensive text books that we have today, but this I can tell this is still in the making, I can't
wait till the have implemented this for more text books in subjects. In my real estate class they
don't the book for online and its costs 70 dollars, so it would be nice if they could get real
estate text books up for people taking the course like myself.For certain classes I would still
and dizzy.
• I must agree that I am not used to reading books online, it also gives me a headache. I do
envy the prices for the online textbooks..
• Many times when I am working on homework or just researching a product it comes to
workspace. Today we view a lot of information on the internet. We are all use to having the text
book on the desk and typing your essay or assignment on the computer in Word. However,
with ebooks the problem I find is you have to open a window for the ebook itself + the word and
having to switch back and forth becomes a pain and definitely causes headaches.
This book was about nanotechnology and how it can help in the fight against cancer.
Nanotechnology deals with very small robots being inserted into humans attacking and killing
cancer cells. This, theoretically, should help humans find ways to cure cancer and it sounds
like a great idea. The size of nanotechnology would be from 1 to 100 nanometers, which is very
small and hard to phathom how something so small will be able to help humans. I love
technology and what it has done with the medical field and I am a fan of stem cell research and
this would be a great text to teach because it just portrays how much computers have
advanced and the possibility of mitigating the affect of various illnesses.
This book is used by University of Minnesota. I could not find any specific professor who uses
this book other than ones at University of Minnesota. I would use this book in a classroom
because it contains good viewpoints and has detailed information about what the authors are
trying to say. I think eBooks if purchased can be easier to have because of being able to
search for certain topics via computers and not having to waste time flipping through many
pages to find what you want.
I was taking accounting class last quarter. As a metter of fact, I felt dizzy when i was reading
this open text book online. It only has four colours on all of the pages, it has no examples and
pictures. It's just like a handout with smaller font, and you can't even take notes on it. It's very
important for me to take notes on the text book because it helps me to relate each note and
the concept together. I usend such book once-it should not be called as a book, it's a handout
posted online by my math professer, Mr.Mosh. I printed his handout out anyways.
• I felt dizzy while I was reading the open book as well and mine was Open Shakespeare
,"Hamlet". I agree that it is hard to keep notes with the open books. I think they are very good
reading "tool" for some subjects, but not everything.
• I found this web-link very useful and I have bookmarked it in my “Computer Science”
bookmarked directory. I will use these books to help me with additional questions I have related
to certain computer science topics. The difference in cost, is that e-books like these are free,
the world wide web is full of information like these books on-line. I have had teachers reference
awesome teaching material for learning HTML. http://www.w3schools.com/ which I compared
to the ebook “Introduction To HTML”
I chose the topic of Education, specifically Reading. I liked the format and the user- friendly
book. Of course I would use it, as I am using an online book for this class. Online books are
easy to access, used and convenient, especially if you are traveling.
The cost is extremely important to college students. I believe that online text books are going
to be part of our future and one day make "real books" obsolete, especially with the movement
of a greener environment and the creation of new technology!
• Terrific! I think you are the first to use the online version of the CIS2 textbook. Many students
complain about the cost of the 'dead tree' version so the online option seems perfect for this
course.
• I'm also using online version of the book, not the paper one. Save some nature...
• The lowered cost of the online version was the reason I choose that over text. Also the fact
that our class is online just made sense to pick the online text.
hence selecting this book. The book is about how to communicate in the business world. For
example, how to write emails, memo's, letters etc in an office environment. This book looks like
it easy to read and is in color (which is a bonus as black and white can get boring). However, I
am definitely a type of person who prefers to read a textbook rather than an e-book as I feel it
is strenuous for the eyes. The difference in costs does play a huge role as being a student, you
are always looking for ways to save but e-book would not be the way to go for me. I have never had class where an instructor permitted open textbooks.
This would be a great textbook for a more advanced course in Network Security. I'm not sure if
I'd use an e-textbook. I understand that it is cheaper, who doesn't love something for half price.
I think that after staring at a computer screen at work all day, then staring at the screen for
homework, it gets to be too much for the eyes. I like the idea of highlighting and making notes
in the text of what I think is important. Also, I knew I'd be traveling for part of this class and that
computer/Internet access would be challenging. It worked better for me to have the text
When I was choosing classes to register (read Waitlist) for, there was another CIS class in
which the Instructor provided all class materials in digital format so the student didn't have to
purchase anything. I appreciated his attempt to be as green as possible.
• Most open textbooks are digital and free of charge. Some open textbooks are also available
in printed format at relatively low cost.
Did you decide against the other CIS course because there was no textbook?
How important is a print textbook?
• No, it wasn't a textbook issue, it was a wait-list issue. I was wait-listed for this class and the
other one. This was the one I was lucky enough to get a spot in. For me personally, a print
textbook is important. But then, I'm a girl who loves books, as Amazon and Barnes & Noble
can attest to.
I chose a biology textbook, as that relates to my major at UCSD. I actually really like this
method, but I would rather it be a companion to a print textbook. This is because using a
computer offers many shortcuts to reading, such as using Ctrl+F to find specific phrases.
Having a print textbook forces a student to read the material, instead of just finding the answer.
I would use a book like this as a supplemental resource. The cost in purchasing print
textbooks as compared to e-books is not that important to me. None of my instructors are
using these books. In fact, I had never even heard of these until this assignment.
The selected book is very similar to a former math class that I have taken. I believe the concept
of free books are excellent, however would pay the extra for printed version. The quality
compared to printed books are really not noticeable, to me. Personally, the online books do not
appeal to me, since I enjoy reading a book whenever, and wherever, and prefer turning to the
chosen pages rather than electronically.
At this time, I do not know anyone using the website to access textbooks.
This textbook to me is a very interesting textbook that teaches you about ethics in the
business world. As a business major, I have learned a lot of things, but my professors have
always told me that it is important to carry high ethical manners for credibility. This text not
only teaches you the fundamentals of ethics, but also it teaches how to use it in the real world
situations. The text shows stories of business that goes unethical and busts as well as
business that successfully deals out with ethics. To all business majors this is one text that
wouldn't hurt reading because we all have to agree that ethics is one important aspect in the
I found this open textbook to be very interesting. It really highlighted the theories that shape the
foundation of ethical behavior we have in business today. Being a profession in business and
commerce, I found the content to be very applicable to the mindsets of business leaders today.
So with that said, I would definitely consider using this open textbook online. It's convenient,
accessible from anywhere, and could become a great point of reference should I need to utilize
the texts.
The cost factor is definitely a big factor when considering a published textbook versus and it's
electronic form. Given the costs of textbooks today, having a more affordable option is always
welcome. However, I must say that for my own personal learning, I sometimes appreciate
having a textbook in front of me, in physical form. I find I can "mark up" my materials much
better than through the computer. So while cost is a factor, I think the ability to write notes,
highlight, etc. would be another determining condition.
Being out of school for a couple of years, I cannot remember any of my instructors using open
textbooks in their courses. However, I would not be surprised if more did today, given it's
convenience and cost savings.
This is a link to an open text book in C programing. I payed text book would offer more details,
greater description and more examples but this is a good reference for those that could not
afford to pay for a text book about the subject.
I chose the Open Textbook that taught how to use Excel 2007 because I actually had to use
Excel for a class last year. I think that the Open Textbook is very helpful and similar to an
actual book. I would not mind using it if my class required it. I would use the Open Textbook,
but if I were to buy a book, I would prefer to buy the book we carry around because it is easier
for me to read. There is also less distractions because when I'm on the computer, I tend to
multitask and get distracted easily. None of my instructors are using Open Textbooks right
now.
• I think that reading an Open textbook is hard also. Normally, while I am reading something on
the computer, I have hard times focusing on the "paper" for a long time because I get distracted
easy as well and I get a headache after looking at the computer for a long time. However, I
think that open books regarding Computer Science are better than written books because the
open Textbooks are giving us examples plus a "practice". Written books are good too but even
having a lot of pictures, I would still consider them as books with "dry information".
I really like this new idea of online text books or ebooks. The print verision is also great
because when it comes to the beginning of every quarter I have to spend $200 - 300 on text books! When I read in the syllabus that you can purchase a ebook of the required text book I was excited. Not only was this much cheaper but also easy to use. Currently, this is the only class i have had that offers a online book. This particular open text I found is a Microsoft Word 2007 training manual. I currently use the 2007 version and just reading the first 10 pages on this manual, I found new hotkeys and ways to help my word processing. On the downside to this open text is that it is in a .pdf format. If there was a way to represent/simulate a book with two pages on each side that would be great. This also depends on the screen size that the user has and if they can support more than one page of text on their screen. mixed This book is about business fundamentals. I've taken plenty of business classes and I would say it would be useful if those classes had a free textbook like this. However, I feel like it's not as indepth as some of the publishers hard copy textbooks. Being that it's free, I can't really blame the authors for that fact. I know that if these types of books were available for engineers and other science majors, it would be a huge plus. I've had many friends spend upwards of$200 for textbooks that they may only get to use one quarter, and then a new version comes
out and the value of theirs plummets.
• I don't like online books because I know that you can take notes in online book, and even
highlighted, but I feel different. It is not something that I'm used to and I like to read books. I
have not taken any class that requires open books.
"The Berlin Wall: 20 Years Later"
I think it is a great textbook that offers some first hand perspectives that would not be found in
a mass produced textbook. While it cannot offer the same amount of information or an as
in-depth look as larger textbooks, the price of it alone makes it a positive asset.
• For the open book ""The Berlin Wall: 20 Years Later", I really liked the fact that the book is
colorful and has pictures, as the same way an actual written book would have. This makes it
easier to read and follow. The price, I do agree, it is a positive asset.
Because I am an Accounting major, I decided to look at the free accounting text books. It
seems like a lot of the major issues are being touched upon, but the book doesn't really seem
to have any depth. A lot of accounting is really being able to complete problems, and there
seems to be very little of that in this book. I would much rather pay a \$200 for a book that was
written by professionals in the Accounting field who are recognized for their expertise. While
the books are very expensive and the companies tend to prey upon students, I think that the
barriers to entry of textbook writing is worth the price. I tend to buy my books used, so I never
pay the full price.
this book is pretty cool because it has the same business practices that my 100 dollar book
was trying to teach me when i took the class at de anza. something like this would be great if
a teacher or teachers could adapted this to their learning schedule because it could make
going to school for all students cheaper and we could charge for other things. like everyone
gets a laptop.
This is a programming book. The only reason why I picked this is that I am not a big fan of
ebooks, However I do like using them to search for material. So assuming that I can do the
basic coding and all that I am trying to learn is syntax, then I can see the allure of an ebook.
However if I'm not comfortable, I like having a physical book, so I can trace the code. I can write
my own notes in the margins. I like to write notes next to the actual text supplementing what I
don't understand, or I need to ask someone to explain something about a topic in the book.
This website has has a list of all of Shakespeare's plays, reviews from readers, and even links
to where you can read his plays online. I think this site would be very useful for teaching
literature and might be a valuable resource for teaching an english class.
It's not so much a textbook as a set of case studies for MIT's Sloan School of Management.
On the other hand, some of the case studies sounded really fascinating, like the one about
IBM. All the cases were formatted in PDF, ie Adobe Acrobat and therefore able to be
class/lecture makes life so much simpler, not the least of which is often the cost. At the same
time, you can't mark an eBook up with highlighters, fold pages over, underline words, or make
notes in the margins. So it depends on the subject matter to me. Some things just require a
physical book.
I took literature class the very first semester I had in DeAnza. Since a kid I love reading books
and "meeting" new characters. One of my favorite authors has always been William
Shakespeare. Most of his work has been turned into movies and had enormous success.
What I liked about the "Open Shakespeare" is that it gives me all kind of different information at
one place and just by "clicking" i can find what i really need. It is really hard to get all the
information at one place when it is an actual written book. However, I don't like the fact that
when I have to read, Hamlet for instance, I have to always keep my finger on the computer pad
and keep moving the page down. This irritates me and I am loosing the meaning of the context.
However, I did like the fact that I could read "Hamlet" in different formats.
I believe that 10 years from now, written books will be "antiques" and open books will be the
main tool for reading and studying.
love math. I have never used an open book in class but it is a good option. I do not personally
prefer them however because they need to be viewed by a computer which is usually inside.
I thought this book would be useful to me and relevant to my major (Business Economics with
an emphasis in accounting).
I use online textbooks all the time, this class included. I think that they are very useful and
usually much cheaper. However, that is only if you wish to keep your book. I usually buy mine
used hard copy and then sell them back to minimize costs. I have used online books before
and probably will use them again they are more useful in some ways (ctrl F) and lightweight.
• I think online textbooks have a place, but cannot completely replace physical textbooks. With
that said, I use both on a regular basis at school.
Through the Open Textbook site, I found an interesting article about anxiety. http://www.nimh.nih.gov/health/publications/anxiety-disorders/complete-index.shtml
I think it's important that everyone is educated in how anxiety works and affects people. A lot of
people have anxiety, and fail to recognize or realize that they even have a problem. Anxiety is
very detrimental to your health, and a lot of problems can arise with continued anxiety.
Learning about it is the first step to knowing how to deal with it, or identify it in yourself, or the
people you know.
This is not a good textbook at all. each chapter is about 1 page in content. My real textbook is
over 200 pages long. There is no way you can cover all of that content in such a short space.
The only time I would use an online textbook is when it is a PDF or online version of the actual
textbook. In that case, it becomes a good value for everyone involved. It is environmentally
friendly, cheap to produce, and easy to transport. I am currently using the online version of the
textbook for this course. As far as a free online textbook, I would not trust it since it comes
from a single author. There are not enough people to vet the information provided by the text
and there is no way to know if the author has some ulterior motive. I would rather use and
official textbook. | {"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": 0.27120789885520935, "perplexity": 1086.638218632146}, "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/1659882572077.62/warc/CC-MAIN-20220814204141-20220814234141-00636.warc.gz"} |
http://www.oalib.com/search?kw=Shengwu%20Xiong&searchField=authors | Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
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ISSN: 2333-9721
APC: Only $99 Submit 2019 ( 3 ) 2018 ( 73 ) 2017 ( 80 ) 2016 ( 65 ) Search Results: 1 - 10 of 6298 matches for " Shengwu Xiong " All listed articles are free for downloading (OA Articles) Page 1 /6298 Display every page 5 10 20 Item Advances in Multimedia , 2012, DOI: 10.1155/2012/343724 Abstract: Advances in Multimedia , 2012, DOI: 10.1155/2012/343724 Abstract: Video target tracking is a critical problem in the field of computer vision. Particle filters have been proven to be very useful in target tracking for nonlinear and non-Gaussian estimation problems. Although most existing algorithms are able to track targets well in controlled environments, it is often difficult to achieve automated and robust tracking of pedestrians in video sequences if there are various changes in target appearance or surrounding illumination. To surmount these difficulties, this paper presents multitarget tracking of pedestrians in video sequences based on particle filters. In order to improve the efficiency and accuracy of the detection, the algorithm firstly obtains target regions in training frames by combining the methods of background subtraction and Histogram of Oriented Gradient (HOG) and then establishes discriminative appearance model by generating patches and constructing codebooks using superpixel and Local Binary Pattern (LBP) features in those target regions. During the process of tracking, the algorithm uses the similarity between candidates and codebooks as observation likelihood function and processes severe occlusion condition to prevent drift and loss phenomenon caused by target occlusion. Experimental results demonstrate that our algorithm improves the tracking performance in complicated real scenarios. 1. Introduction Video target tracking is an important research field in computer vision for its wide range of application demands and prospects in many industries, such as military guidance, visual surveillance, visual navigation of robots, human-computer interaction and medical diagnosis [1–3], and so forth. The main task of target tracking is to track one or more mobile targets in video sequences so that the position, velocity, trajectory, and other parameters of the target can be obtained. Two main tasks needs to be completed by moving target tracking during the processing procedure: the first one is target detection and classification which detects the location of relevant targets in the image frames; the second one is the relevance of the target location of consecutive image frames, which identifies the target points in the image and determines their location coordinates, thus to determine the trajectory of the target as time changes. However, automated detection and tracking of pedestrians in video sequences is still a challenging task because of following reasons [4]. (1) Large intraclass variability which refers to various changes in appearance of pedestrians due to different poses, clothing, viewpoints, Computational Intelligence and Neuroscience , 2013, DOI: 10.1155/2013/369016 Abstract: Multiobjective evacuation routes optimization problem is defined to find out optimal evacuation routes for a group of evacuees under multiple evacuation objectives. For improving the evacuation efficiency, we abstracted the evacuation zone as a superposed potential field network (SPFN), and we presented SPFN-based ACO algorithm (SPFN-ACO) to solve this problem based on the proposed model. In Wuhan Sports Center case, we compared SPFN-ACO algorithm with HMERP-ACO algorithm and traditional ACO algorithm under three evacuation objectives, namely, total evacuation time, total evacuation route length, and cumulative congestion degree. The experimental results show that SPFN-ACO algorithm has a better performance while comparing with HMERP-ACO algorithm and traditional ACO algorithm for solving multi-objective evacuation routes optimization problem. 1. Introduction The evacuation planning in large-scale public area usually possesses two difficult points:(1)large scale: the large-scale public area has a complex flat structure. And it can hold thousands of people.(2)multisource and multisink: in evacuation process, the evacuees often start at different places in public area and run away from different exits. In a word, the evacuation planning in large-scale public area is a challenging problem. For solving this problem, researchers have put forward some effective methods. Shi et al. [1] used agent-based model to simulate and analyze evacuation process in large public building under fire conditions. Chen and Miller-Hooks [2] employed Benders decomposition to determine a set of evacuation routes and the assignment of evacuees to these routes for large building. Tayfur and Taaffe [3] utilized linear programming relaxation to model and solve a resource requirements and scheduling problem during hospital evacuations with the objective of minimizing cost within a prespecified evacuation completion time. Fang et al. [4] modeled evacuation process in a teaching building with multiexits, simulated it by cellular automata, and analyzed the multiexits choice phenomenon to find out the optimal exits choice combination for all evacuees. Usually, multiple macroscopic objectives are required to be considered in actual evacuation planning, and a set of nondominated plans are needed for decision making. Thus, evacuation planning problem could be transformed into multi-objective optimization problem. However, just a few researches, such as the literature [5–7], focused on that. Among these pieces of literature, the literature [7] successfully solved the multi-objective Computer Science , 2015, Abstract: In this paper, the linear complexity over$\mathbf{GF}(r)$of generalized cyclotomic quaternary sequences with period$2pq$is determined, where$ r $is an odd prime such that$r \ge 5$and$r\notin \lbrace p,q\rbrace$. The minimal value of the linear complexity is equal to$\tfrac{5pq+p+q+1}{4}$which is greater than the half of the period$2pq$. According to the Berlekamp-Massey algorithm, these sequences are viewed as enough good for the use in cryptography. We show also that if the character of the extension field$\mathbf{GF}(r^{m})$,$r$, is chosen so that$\bigl(\tfrac{r}{p}\bigr) = \bigl(\tfrac{r}{q}\bigr) = -1$,$r\nmid 3pq-1$, and$r\nmid 2pq-4\$, then the linear complexity can reach the maximal value equal to the length of the sequences.
Shengwu Zhu Journal of Mathematics Research , 2009, DOI: 10.5539/jmr.v1n1p8 Abstract: In this paper, we shall give a set R* and indicate its properties, and thus, some abnormal results, such as the limit number may be successor, the natural number may be transfinite, the infinite set can not be equipotent to its proper subset etc., will be obtained.
Intelligent Information Management (IIM) , 2010, DOI: 10.4236/iim.2010.25038 Abstract: In this paper, according to economics of real estate and macro-control theory, combine with the characteristics of the real estate market, macro-control of the real estate market is studied. After giving the dynamic model of three-dimensional nonlinear differential equations based on the total number of houses on the real estate business, the government’s averages housing investment funds and the standard price, systematically established the stability conditions of equilibrium point for this model. What’s more, through the use of extreme value analysis model, government funds have been invested in real estate business building devotion principles and the construction base of the real estate businessmen has also been estimated successfully. This provides the corresponding theoretical basis for government macro control policy-making.
PLOS ONE , 2013, DOI: 10.1371/journal.pone.0052384 Abstract: We have previously demonstrated that the CCR9/CCL25 signaling pathway plays an important role in drug resistance in human acute T-lymphocytic leukemia (T-ALL) by inducing activation of ERM protein with polarized distribution in T-ALL cell line MOLT4. However, the mechanism of action of the activated ERM protein in the drug resistance of MOLT4 cells induced by CCL25 remains uncharacterized. Here we investigated the mechanism of CCR9/CCL25-initiated drug resistance in CCR9-high-expressing T-ALL cells. Our results showed that 1) the function of P-gp was increased after treatment with CCL25; 2) P-gp colocalized and co-immunoprecipitated with p-ERM and F-actin in CCL25 treated cells; and 3) ERM-shRNA conferred drug sensitivity coincident with release of ERM interactions with P-gp and F-actin after treatment with CCL25. These data suggest it is pivotal that P-gp associate with the F-actin cytoskeleton through p-ERM in CCR9/CCL25 induced multidrug resistance of T-ALL cells. Strategies aimed at inhibiting P-gp-F-actin cytoskeleton association may be helpful in increasing the efficiency of therapies in T-ALL.
Chinese Science Bulletin , 1998, DOI: 10.1007/BF02883402 Abstract: For discrete time case a characterization of locally risk-minimizing strategies is given. Based on this characterization, it is evident that risk-minimizing strategies must be locally risk-minimizing.
Chinese Science Bulletin , 1999, DOI: 10.1007/BF02885857 Abstract: Fossil matoniaceous plants represented byPhlebopteris have been formerly described and illustrated from the Mesozoic of China mainly based on vegetative leaves, and no information is provided on the fine structure of fertile organs. Recent reinvestigation of the Hsiangchi Flora from the Lower Jurassic of West Hubei, China, obtained a rich collection of well preservedPhlebopteris specimens (both sterile and fertile). Studies onPhlebopteris polypodioides Brongniart using LM and SEM reveal not only the structures of sori and sporangia, but also the details ofin situ spores. Comparisons have been made between thein situ spores and other related fossil and extant matoniaceous spores, as well as the dispersed genera.
Wang Liejun Wu Shengwu Information Technology Journal , 2011, Abstract: In this study, we derive the Outage capacity region of Multiple-Input and Multiple-Output (MIMO) Broadcasting Channel (BC) under long-term average power constraint. Assuming channel state information known both at transmitter and receivers, we use the concept of equivalent channel and water-filling method to obtain the minimum required power for given rate and given channel state. In optimal power allocation strategy, we apply the ratio of reward to the minimum required power as criterion and give a threshold. Afterwards, we give a tail power allocation strategy, which together with the threshold algorithm guarantees the maximal reward of the long-term power.
Page 1 /6298 Display every page 5 10 20 Item | {"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": 0.45177173614501953, "perplexity": 4698.522792893139}, "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-2019-18/segments/1555578577686.60/warc/CC-MAIN-20190422175312-20190422201312-00551.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-14-trigonometric-graphs-identities-and-equations-prerequisite-skills-page-906/10 | ## Algebra 2 (1st Edition)
$\pi/3$
According to the unit circle, this value is found where $\theta = 60^o$ Converting to radians gives us $\pi/3$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9452029466629028, "perplexity": 1193.1888939678422}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027319724.97/warc/CC-MAIN-20190824041053-20190824063053-00386.warc.gz"} |
https://www.physicsforums.com/threads/spring-and-friction-assignment.173291/ | # Spring and friction assignment
1. Jun 8, 2007
### kar_04
another physics question..plz
Albertine sits ina reclining chair, in front of a large, compressed spring. The spring is compressed 5m from its equilibrium position, and a glass sits 19.8 m from her outstretched foot. Assuming that Albertine's mass is 60kg what is the coefficient of kinetic friction between the chair and the waxed floor?
U(kinetic) = F/N
N= 588
F=??
a=??
2. Jun 9, 2007
### chaoseverlasting
Assuming that she just touches the glass, $$0.5kx^2=\mu mgx$$ where x is 19.8 m, but you dont know k (spring const) which is reqd. | {"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": 0.842684268951416, "perplexity": 5282.386814287157}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719960.60/warc/CC-MAIN-20161020183839-00388-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://blog.nguyenvq.com/ | ## Compile R 3.2.2 on AIX 6.1
Here are my notes compiling 64-bit R 3.2.2 on AIX 6.1. As a pre-requisite, read the AIX notes from R-admin. Like the notes, I had GCC installed from here by our admin, along with many other pre-requisites. These were installed prior to compiling R. Note that you could grab newer versions of each package by going to http://www.oss4aix.org/download/RPMS/ (needed for R-dev).
## list of packages
http://www.oss4aix.org/download/RPMS/info/info-5.2-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/less/less-458-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/screen/screen-4.0.3-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/parallel/parallel-20140122-1.aix5.1.ppc.rpm
http://www.bullfreeware.com/download/bin/1464/readline-6.2-3.aix6.1.ppc.rpm
http://www.bullfreeware.com/download/bin/1465/readline-devel-6.2-3.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gmp/gmp-5.1.3-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gmp/gmp-devel-5.1.3-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/mpfr/mpfr-3.1.2-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/mpfr/mpfr-devel-3.1.2-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/libmpc/libmpc-1.0.2-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/libmpc/libmpc-devel-1.0.2-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gcc/gcc-4.8.2-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gcc/gcc-c++-4.8.2-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gcc/gcc-cpp-4.8.2-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gcc/gcc-gfortran-4.8.2-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gcc/libgcc-4.8.2-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gcc/libgomp-4.8.2-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gcc/libstdc++-4.8.2-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gcc/libstdc++-devel-4.8.2-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/libiconv/libiconv-1.14-2.aix5.1.ppc.rpm
## conversting unicode, ascii, etc; aix version is not compatible with
R
http://download.icu-project.org/files/icu4c/54.1/icu4c-54_1-AIX7_1-VA2.tgz
## dependency for unicode support; just need to extract to root /
http://www.oss4aix.org/download/RPMS/make/make-4.1-1.aix5.3.ppc.rpm ##
need the gnu version of make
## libm
https://www.ibm.com/developerworks/community/forums/html/topic?id=72e62875-0603-4d93-a9bf-9d80c6cdc6ea
http://www-01.ibm.com/support/docview.wss?uid=isg1fileset-1318926131
https://www.ibm.com/developerworks/java/jdk/aix/service.html ## jre
http://www.oss4aix.org/download/RPMS/libpng/libpng-1.6.12-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/libpng/libpng-devel-1.6.12-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/libjpeg/libjpeg-9a-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/libjpeg/libjpeg-devel-9a-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/icu/icu-gcc-4.8.1.1-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/icu/libicu-gcc-4.8.1.1-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/icu/libicu-gcc-devel-4.8.1.1-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/icu/libicu-gcc-doc-4.8.1.1-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gdb/gdb-7.8.1-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/expat/expat-2.1.0-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/curl/curl-7.27.0-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/curl/curl-devel-7.27.0-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/libxml2/libxml2-2.9.1-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/libxml2/libxml2-devel-2.9.1-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/ncurses/ncurses-5.9-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/ncurses/ncurses-devel-5.9-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gdbm/gdbm-1.11-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/gdbm/gdbm-devel-1.11-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/libtool/libtool-1.5.26-2.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/libtool/libtool-ltdl-1.5.26-2.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/libtool/libtool-ltdl-devel-1.5.26-2.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/zlib/zlib-1.2.8-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/zlib/zlib-devel-1.2.8-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/bzip2/bzip2-1.0.6-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/bzip2/bzip2-devel-1.0.6-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/pcre/pcre-8.37-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/pcre/pcre-devel-8.37-1.aix5.1.ppc.rpm
#### python
http://www.oss4aix.org/download/RPMS/python-setuptools/python-setuptools-0.6.24-1.aix5.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/python/python-2.7.5-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/python/python-devel-2.7.5-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/python/python-libs-2.7.5-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/python/python-test-2.7.5-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/python/python-tools-2.7.5-1.aix6.1.ppc.rpm
http://www.oss4aix.org/download/RPMS/python/tkinter-2.7.5-1.aix6.1.ppc.rpm
Add /opt/freeware/bin to PATH.
Now, download the R source tarball, extract, and cd.
Next, update src/main/dcf.c from the R-dev as a bug that causes readDCF to segfault when using install.packages; no need to do this for future versions of R. Then apply this patch from here to fix the following error:
gcc -maix64 -pthread -std=gnu99 -I../../../../include -DNDEBUG
-I../../../include -I../../../../src/include -DHAVE_CONFIG_H
-I../../../../src/main -I/usr/local/include -mminimal-toc -O2 -g
-mcpu=power6 -c gramRd.c -o gramRd.o
gcc -maix64 -pthread -std=gnu99 -shared -Wl,-brtl -Wl,-G -Wl,-bexpall
-Wl,-bnoentry -lc -L/usr/local/lib -o tools.so text.o init.o Rmd5.o
md5.o signals.o install.o getfmts.o http.o gramLatex.o gramRd.o -lm
-lintl
make[6]: Entering directory '/sas/data04/vinh/R-3.2.2/src/library/tools/src'
mkdir -p -- ../../../../library/tools/libs
make[6]: Leaving directory '/sas/data04/vinh/R-3.2.2/src/library/tools/src'
make[5]: Leaving directory '/sas/data04/vinh/R-3.2.2/src/library/tools/src'
make[4]: Leaving directory '/sas/data04/vinh/R-3.2.2/src/library/tools'
make[4]: Entering directory '/sas/data04/vinh/R-3.2.2/src/library/tools'
installing 'sysdata.rda'
Error: Line starting 'Package: tools ...' is malformed!
Execution halted
../../../share/make/basepkg.mk:150: recipe for target 'sysdata' failed
make[4]: *** [sysdata] Error 1
make[4]: Leaving directory '/sas/data04/vinh/R-3.2.2/src/library/tools'
Makefile:30: recipe for target 'all' failed
make[3]: *** [all] Error 2
make[3]: Leaving directory '/sas/data04/vinh/R-3.2.2/src/library/tools'
Makefile:36: recipe for target 'R' failed
make[2]: *** [R] Error 1
make[2]: Leaving directory '/sas/data04/vinh/R-3.2.2/src/library'
Makefile:28: recipe for target 'R' failed
make[1]: *** [R] Error 1
make[1]: Leaving directory '/sas/data04/vinh/R-3.2.2/src'
Makefile:59: recipe for target 'R' failed
make: *** [R] Error 1
Hopefully, this patch will make it to R-dev so that it is no longer needed for future versions of R.
export OBJECT_MODE=64
export CC="gcc -maix64 -pthread"
export CXX="g++ -maix64 -pthread"
export FC="gfortran -maix64 -pthread"
export F77="gfortran -maix64 -pthread"
export CFLAGS="-O2 -g -mcpu=power6"
export FFLAGS="-O2 -g -mcpu=power6"
export FCFLAGS="-O2 -g -mcpu=power6"
./configure --prefix=/path/to/opt ## custom location so I don't need root
make -j 16
make install
## add /path/to/opt/bin to PATH
The last step may complain about NEWS.pdf not found in a directory and a certain directory is not found in the destination. For the former, just do touch NEWS.pdf to where it’s supposed to be; for the latter, create the directory yourself.
## Automatically specify line break options with termstr as CRLF or LF in SAS when importing data
It could be annoying when dealing with data from multiple platforms: Windows uses the carriage return (CR) and line feed (LF) to indicate a new line, UNIX uses LF, and Mac uses CR. Most companies have SAS running on a UNIX/Linux server, and it’s hard to tell which characters indicate a new line without going to a terminal to inspect the file.
Here’s a sas macro that creates a filename handle that could be used in PROC IMPORT or a DATA step. It will automatically detect CRLF and if not, default to LF. This assumes SAS is running on a UNIX server with access to the head and awk commands.
%macro handle_crlf(file, handle_name, other_filename_options=) ;
/* if there is a carriage return at the end, then return 1 (stored in macro variable SYSRC) */
%sysexec head -n 1 "&file" | awk '/\r/ { exit(1) }' ; %if &SYSRC=1 %then %let termstr=crlf ; %else %let termstr=lf ; filename &handle_name "&file" termstr=&termstr &other_filename_options ; %mend ; /* %handle_crlf(file=/path/to/file.txt, handle_name=fh) ; proc import data=fh dbms=dlm replace outdata=d1 ; delimiter='|' ; run ; */ ## Repair line breaks within a field of a delimited file Sometimes some people generate delimited files with line break characters (carriage return and/or line feed) inside a field without quoting. I previously wrote about the case when the problematic fields are quoted. I also wrote about using non-ascii characters as field and new record indicators to avoid clashes. The following script reads in stdin and writes to stdout repaired lines by ensuring every output line has at least the number of delimiters (|) as the first/header line (call this the target number of delimiters) by continually concatenating lines (remove line breaks) until it reaches the point when concatenating the next line would yield more delimiters than the target number of delimiters. The script appears more complicated than it should be in order to address the case when there are more than one line breaks in a field (so don’t just concatenate one line but keep doing so) and the case when a line has more delimiters than the target number of delimiter (this could lead to an infinite loop if we restrict the number of delimiters to equal the target). #! /usr/bin/env python dlm='|' import sys from signal import signal, SIGPIPE, SIG_DFL # http://stackoverflow.com/questions/14207708/ioerror-errno-32-broken-pipe-python signal(SIGPIPE,SIG_DFL) ## no error when exiting a pipe like less line = sys.stdin.readline() n_dlm = line.count(dlm) line0 = line line_next = 'a' while line: if line.count(dlm) > n_dlm or line_next=='': sys.stdout.write(line0) line = line_next # line = sys.stdin.readline() if line.count(dlm) > n_dlm: ## line with more delimiters than target? line0 = line_next line_next = sys.stdin.readline() line = line.replace('\r', ' ').replace('\n', ' ') + line_next else: line0 = line line_next = sys.stdin.readline() line = line.replace('\r', ' ').replace('\n', ' ') + line_next ## Calculate the weighted Gini coefficient or AUC in R This post on Kaggle provides R code for calculating the Gini for assessing a prediction rule, and this post provides R code for the weighted version (think exposure for frequency and claim count for severity in non-life insurance modeling). Note that the weighted version is not well-defined when there are ties in the predictions and where the corresponding weights vary because different Lorentz curve (gains chart) could be drawn for different orderings of the observations; see this post for an explanation and some examples. Now, to explain the code. The calculation of the x values (variable random, the cumulative proportion of observations or weights) and y values (variable Lorentz, the cumulative proportion of the response, the good’s/1’s or positive values) are straightforward. To calculate the area between the Lorentz curve and the diagonal line, one could use the trapezoidal rule to calculate the area between the Lorentz curve and x-axis and then subtract the area of the lower triangle (1/2): \begin{align} Gini &= \sum_{i=1}^{n} (x_{i} – x_{i-1}) \left[\frac{L(x_{i}) + L(x_{i-1})}{2}\right] – \frac{1}{2} \ &= \frac{1}{2} \sum_{i=1}^{n} \left[ L(x_{i})x_{i} + L(x_{i-1})x_{i} – L(x_{i})x_{i-1} – L(x_{i-1})x_{i-1} \right] – \frac{1}{2} \ &= \frac{1}{2} \sum_{i=1}^{n} \left[ L(x_{i})x_{i} – L(x_{i-1})x_{i-1} \right] + \frac{1}{2} \sum_{i=1}^{n} \left[ L(x_{i-1})x_{i} – L(x_{i})x_{i-1} \right] – \frac{1}{2} \ &= \frac{1}{2} L(x_{n})x_{n} + \frac{1}{2} \sum_{i=1}^{n} \left[ L(x_{i-1})x_{i} – L(x_{i}) x_{i-1} \right] – \frac{1}{2} \ &= \frac{1}{2} \sum_{i=1}^{n} \left[ L(x_{i-1})x_{i} – L(x_{i}) x_{i-1} \right] \end{align} where the last equality comes from the fact that $$L(x_{n}) = x_{n} = 1$$ for the Lorentz curve/gains chart. The remaining summation thus corresponds to sum(dfLorentz[-1]*df$random[-n]) - sum(df$Lorentz[-n]*df\$random[-1]) inside the WeightedGini function since the $$i=1$$ term in the summation is 0 ($$x_i=0$$ and $$L(x_{0})=0$$ for the Lorentz curve), yielding $$n-1$$ terms in the code.
For the unweighted case, applying the trapezoidal rule on the area between the Lorentz curve and the diagonal line yields:
\begin{align} Gini &= \sum_{i=1}^{n} \frac{1}{n} \frac{\left[ L(x_{i}) – x_{i} \right] – \left[ L(x_{i-1}) – x_{i-1} \right] }{2} \ &= \frac{1}{2n} \sum_{i=1}^{n} \left[ L(x_{i}) – x_{i} \right] + \frac{1}{2n} \sum_{i=1}^{n} \left[ L(x_{i-1}) – x_{i-1} \right] \ &= \frac{1}{2n} \sum_{i=1}^{n} \left[ L(x_{i}) – x_{i} \right] + \frac{1}{2n} [L(x_{0}) – x_{0}] + \frac{1}{2n} \sum_{i=1}^{n-1} \left[ L(x_{i}) – x_{i} \right] \ &= \frac{1}{2n} \sum_{i=1}^{n} \left[ L(x_{i}) – x_{i} \right] + \frac{1}{2n} [L(x_{0}) – x_{0}] + \frac{1}{2n} \sum_{i=1}^{n-1} \left[ L(x_{i}) – x_{i} \right] + \frac{1}{2n} [L(x_{n}) – x_{n}] \ &= \frac{1}{2n} \sum_{i=1}^{n} \left[ L(x_{i}) – x_{i} \right] + \frac{1}{2n} [L(x_{0}) – x_{0}] + \frac{1}{2n} \sum_{i=1}^{n} \left[ L(x_{i}) – x_{i} \right] \ &= \frac{1}{n} \sum_{i=1}^{n} \left[ L(x_{i}) – x_{i} \right] \end{align}
where we repeatedly used the fact that $$L(x_{0}) = x_{0} = 0$$ and $$L(x_{n}) = x_{n} = 1$$ for a Lorentz curve and that $$1/n$$ is the width between points (change in cdf of the observations). The summation is what is returned by SumModelGini.
Note that both $$1/2$$ and $$1/n$$ are not multiplied to the sums in the weighted and unweighted functions since most people will use the normalized versions, in which case these factors just cancel.
## Make a printer wireless using a router with USB running OpenWRT
Many recent printers have wifi capability built in for wireless printing. Older printers or even some recent printers do not have this feature, but one could purchase a wireless adapter to turn the printer wireless. The adapters aren’t cheap, and a search for a cheap adapter led me to configuring the TP-Link WR-703N with OpenWRT as an affordable alternative (plug printer into router with a usb cable and print to router via a usb print server).
First, flash the router to OpenWRT by logging into the router at tplinklogin.net using the username/password admin/admin; follow this guide for pictures in navigating the default Chinese interface. Once flashed, the router will have wifi disabled and the ethernet port could be used to log onto the LAN network. Log into the router using a web browser at the destination 192.168.1.1. Set up wifi and turn the ethernet port to WAN by following these instructions; I changed my default router IP address to 192.168.94.1 to avoid clashing with my default “home” network when I plug it into my home network for internet access. Plug the current router into another router with internet access via ethernet. Then ssh into the TP-Link router on its network: root@192.168.94.1. Install the usb printer:
opkg update
opkg install p910nd kmod-usb-printer
Start the print server:
/etc/init.d/p910nd start
/etc/init.d/p910nd enable
Now, on a laptop or computer, connect to the same wifi network as this mini router and add a printer at the router’s ip with port 9100 after plugging a printer into the usb port. Install the necessary printer driver on the laptop or computer.
This setup creates a separate network for usb wireless printing. If we want to have the printer join an existing wifi network, then just set up the router as the first post I referenced.
## Optimized R and Python: standard BLAS vs. ATLAS vs. OpenBLAS vs. MKL
Revolution Analytics recently released Revolution Open R, a downstream version of R built using Intel’s Math Kernel Library (MKL). The post mentions that comparable improvements are observed on Mac OS X where the ATLAS blas library is used. A reader also expressed his hesitation in the Comments section for a lack of a comparison with ATLAS and OpenBLAS. This concept of using a different version of BLAS is documented in the R Administration manual, and has been compared in the past here and here. Now, as an avid R user, I should be using a more optimal version of R if it exists and is easy to obtain (install/compile), especially if the improvements are up to 40% as reported by the Domino Data Lab. I decided to follow the framework set out by this post to compare timings for the different versions of R on a t2.micro instance on Amazon EC2 running Ubuntu 14.04.
First, I install R and the various versions of BLAS and lapack and download the benchmark script:
sudo apt-get install libblas3gf libopenblas-base libatlas3gf-base liblapack3gf libopenblas-dev liblapack-dev libatlas-dev R-base R-base-dev
wget http://r.research.att.com/benchmarks/R-benchmark-25.R
echo "install.packages('SuppDists', dep=TRUE, repo='http://cran.stat.ucla.edu')" | sudo R --vanilla ## needed for R-benchmarks-25.R
One could switch which blas and lapack library are used via the following commands:
sudo update-alternatives --config libblas.so.3 ## select from 3 versions of blas: blas, atlas, openblas
sudo update-alternatives --config liblapack.so.3 ## select from 2 versions of lapack: lapack and atlas-lapack
Run R, issue Ctrl-z to send the process to the background, and see that the selected BLAS and lapack libraries are used by R by:
ps aux | grep R ## find the process id for R
lsof -p PROCESS_ID_JUST_FOUND | grep 'blas\|lapack'
Now run the benchmarks on different versions:
# selection: libblas + lapack
cat R-benchmark-25.R | time R --slave
...
171.71user 1.22system 2:53.01elapsed 99%CPU (0avgtext+0avgdata 425068maxresident)k
4960inputs+0outputs (32major+164552minor)pagefaults 0swaps
173.01
# selection: atlas + lapack
cat R-benchmark-25.R | time R --slave
...
69.05user 1.16system 1:10.27elapsed 99%CPU (0avgtext+0avgdata 432620maxresident)k
2824inputs+0outputs (15major+130664minor)pagefaults 0swaps
70.27
# selection: openblas + lapack
cat R-benchmark-25.R | time R --slave
...
70.69user 1.19system 1:11.93elapsed 99%CPU (0avgtext+0avgdata 429136maxresident)k
1592inputs+0outputs (6major+131181minor)pagefaults 0swaps
71.93
# selection: atlas + atlas-lapack
cat R-benchmark-25.R | time R --slave
...
68.02user 1.14system 1:09.21elapsed 99%CPU (0avgtext+0avgdata 432240maxresident)k
2904inputs+0outputs (12major+124761minor)pagefaults 0swaps
69.93
As can be seen, there’s about a 60% improvement using OpenBLAS or ATLAS over the standard libblas+lapack. What about MKL? Let’s test RRO:
sudo apt-get remove R-base R-base-dev
wget http://mran.revolutionanalytics.com/install/RRO-8.0-Beta-Ubuntu-14.04.x86_64.tar.gz
tar -xzf RRO-8.0-Beta-Ubuntu-14.04.x86_64.tar.gz
./install.sh
# check that it is using a different version of blas and lapack using lsof again
cat R-benchmark-25.R | time R --slave
...
51.19user 0.98system 0:52.24elapsed 99%CPU (0avgtext+0avgdata 417840maxresident)k
2208inputs+0outputs (11major+131128minor)pagefaults 0swaps
52.24
This is a 70% improvement over the standard libblas+lapack version, and a 25% improvement over the ATLAS/OpenBLAS version. This is quite a substantial improvement!
## Python
Although I don’t use Python much for data analysis (I use it as a general language for everything else), I wanted to repeat similar benchmarks for numpy and scipy as improvements have been documented. To do so, compile numpy and scipy from source and download some benchmark scripts.
sudo pip install numpy
less /usr/local/lib/python2.7/dist-packages/numpy/__config__.py ## openblas?
sudo pip install scipy
# test different blas
python
ps aux | grep python
lsof -p 20812 | grep 'blas\|lapack' ## change psid
wget https://gist.github.com/osdf/3842524/raw/df01f7fa9d849bec353d6ab03eae0c1ee68f1538/test_numpy.py
wget https://gist.github.com/osdf/3842524/raw/22e21f5d57a9526cbcd9981385504acdc7bdc788/test_scipy.py
One could switch blas and lapack like before. Results are as follows:
# selection: blas + lapack
time python test_numpy.py
FAST BLAS
version: 1.9.1
maxint: 9223372036854775807
dot: 0.214728403091 sec
real 0m1.253s
user 0m1.119s
sys 0m0.036s
time python test_scipy.py
cholesky: 0.166237211227 sec
svd: 3.56523122787 sec
real 0m19.183s
user 0m19.105s
sys 0m0.064s
# selection: atlas + lapack
time python test_numpy.py
FAST BLAS
version: 1.9.1
maxint: 9223372036854775807
dot: 0.211034584045 sec
real 0m1.132s
user 0m1.121s
sys 0m0.008s
time python test_scipy.py
cholesky: 0.0454761981964 sec
svd: 1.33822960854 sec
real 0m7.442s
user 0m7.346s
sys 0m0.084s
# selection: openblas + lapack
time python test_numpy.py
FAST BLAS
version: 1.9.1
maxint: 9223372036854775807
dot: 0.212402009964 sec
real 0m1.139s
user 0m1.130s
sys 0m0.004s
time python test_scipy.py
cholesky: 0.0431131839752 sec
svd: 1.09770617485 sec
real 0m6.227s
user 0m6.143s
sys 0m0.076s
# selection: atlas + atlas-lapack
time python test_numpy.py
FAST BLAS
version: 1.9.1
maxint: 9223372036854775807
dot: 0.217267608643 sec
real 0m1.162s
user 0m1.143s
sys 0m0.016s
time python test_scipy.py
cholesky: 0.0429849624634 sec
svd: 1.31666741371 sec
real 0m7.318s
user 0m7.213s
sys 0m0.092s
Here, if I only focus on the svd results, then OpenBLAS yields a 70% improvement and ATLAS yields a 63% improvement. What about MKL? Well, a readily available version costs money, so I wasn’t able to test.
## Conclusion
Here are my take-aways:
• Using different BLAS/LAPACK libraries is extremely easy on Ubuntu; no need to compile as you could install the libraries and select which version to use.
• Install and use RRO (MKL) when possible as it is the fastest.
• When the previous isn’t possible, use ATLAS or OpenBLAS. For example, we have AIX at work. Getting R installed on there is already a difficult task, so optimizing R is a low priority. However, if it’s possible to use OpenBLAS or ATLAS, use it (Note: MKL is irrelevant here as AIX uses POWER cpu).
• For Python, use OpenBLAS or ATLAS.
For those that want to compile R using MKL yourself, check this. For those that wants to do so for Python, check this.
Finally, some visualizations to summarize the findings:
# R results
timings <- c(173.01, 70.27, 71.93, 69.93, 52.24)
versions <- c('blas + lapack', 'atlas + lapack', 'openblas + lapack', 'atlas + atlas-lapack', 'MKL')
versions <- factor(versions, levels=versions)
d1 <- data.frame(timings, versions)
ggplot(data=d1, aes(x=versions, y=timings / max(timings))) +
geom_bar(stat='identity') +
geom_text(aes(x=versions, y=timings / max(timings), label=sprintf('%.f%%', timings / max(timings) * 100)), vjust=-.8) +
labs(title='R - R-benchmark-25.R')
ggsave('R_blas+atlas+openblas+mkl.png')
# Python results
timings <- c(3.57, 1.34, 1.10, 1.32)
versions <- c('blas + lapack', 'atlas + lapack', 'openblas + lapack', 'atlas + atlas-lapack')
versions <- factor(versions, levels=versions)
d1 <- data.frame(timings, versions)
ggplot(data=d1, aes(x=versions, y=timings / max(timings))) +
geom_bar(stat='identity') +
geom_text(aes(x=versions, y=timings / max(timings), label=sprintf('%.f%%', timings / max(timings) * 100)), vjust=-.8) +
labs(title='Python - test_scipy.py (SVD)')
ggsave('Python_blas+atlas+openblas.png')
## Change delimiter in a csv file and remove line breaks in fields
I wrote a script to convert delimiters in CSV files, eg, commas to pipes. I prefer pipe-delimited files because the the pipe-delimiter (|) will not clash data in the different fields 99.999% of the time. I also added the option to convert newline () and carriage return () characters in the data fields to spaces. This comes in handy when I use PROC IMPORT in SAS as line breaks cause it to choke.
Here’s my csvconvert.py script:
#! /usr/bin/env python
#### Command line arguments
import argparse
parser = argparse.ArgumentParser(description="Convert delimited file from one delimiter to another; defaults to converting CSV to pipe-delimited.")
parser.add_argument("--dlm-input", action="store", dest="dlm_in", default=",", required=False, help="delimiter of the input file; defaults to comma (,)", nargs='?', metavar="','")
parser.add_argument("--dlm-output", action="store", dest="dlm_out", default="|", required=False, help="delimiter of the output file; defaults to pipe (|)", nargs='?', metavar="'|'")
parser.add_argument("--remove-line-char", action="store_true", dest="remove_line_char", default=False, help="remove \\n and \\r characters in fields and replace with spaces")
parser.add_argument("--quote-char", action="store", dest="quote_char", default='"', required=False, help="quote character; defaults to double quote (\")", nargs='?', metavar="\"")
parser.add_argument("-i", "--input", action="store", dest="input", required=False, help="input file; if not specified, take from standard input.", nargs='?', metavar="file.csv")
parser.add_argument("-o", "--output", action="store", dest="output", required=False, help="output file; if not specified, write to standard output", nargs='?', metavar="file.pipe")
parser.add_argument("-v", "--verbose", action="store_true", dest="verbose", default=False, help="increase verbosity")
args = parser.parse_args()
# print args
# http://snipplr.com/view/45759/convert-csv-file-to-pipe-delineated-file/
import argparse
import csv
import sys
from signal import signal, SIGPIPE, SIG_DFL # http://stackoverflow.com/questions/14207708/ioerror-errno-32-broken-pipe-python
signal(SIGPIPE,SIG_DFL) ## no error when exiting a pipe like less
if args.input:
csv_reader = csv.reader(open(args.input, 'rb'), delimiter=args.dlm_in, quotechar=args.quote_char)
else:
csv_reader = csv.reader(sys.stdin, delimiter=args.dlm_in, quotechar=args.quote_char)
if args.output:
h_outfile = open(args.output, 'wb')
else:
h_outfile = sys.stdout
for row in csv_reader:
row = args.dlm_out.join(row)
if args.remove_line_char:
row = row.replace('\n', ' ').replace('\r', ' ')
h_outfile.write("%s\n" % (row))
h_outfile.flush()
# print row
Help description:
usage: csvconvert.py [-h] [--dlm-input [',']] [--dlm-output ['|']]
[--remove-line-char] [--quote-char ["]] [-i [file.csv]]
[-o [file.pipe]] [-v]
Convert delimited file from one delimiter to another; defaults to converting
CSV to pipe-delimited.
optional arguments:
-h, --help show this help message and exit
--dlm-input [','] delimiter of the input file; defaults to comma (,)
--dlm-output ['|'] delimiter of the output file; defaults to pipe (|)
--remove-line-char remove \n and \r characters in fields and replace with
spaces
--quote-char ["] quote character; defaults to double quote (")
-i [file.csv], --input [file.csv]
input file; if not specified, take from standard
input.
-o [file.pipe], --output [file.pipe]
output file; if not specified, write to standard
output
-v, --verbose increase verbosity
Usage:
cat myfile.csv | csvconvert.py --remove-line-char > myfile.pipe
## Issues with https proxy in Python via suds and urllib2
I recently had the need to access a SOAP API to obtain some data. SOAP works by posting an xml file to a site url in a format defined by the API’s schema. The API then returns data, also in a form of an xml file. Based on this post, I figured suds was the easiest way to utilize Python to access the API so I could sequentially (and hence, parallelize) query data repeatedly. suds did turn out to be relatively easy to use:
from suds.client import Client
url = 'http://www.ripedev.com/webservices/localtime.asmx?WSDL'
client = Client(url)
print client
client.service.LocalTimeByZipCode('90210')
This worked on my home network. At work, I had to utilize a proxy in order to access the outside world. Otherwise, I’d get a connection refuse message: urllib2.URLError: <urlopen error [Errno 111] Connection refused>. The modification to use a proxy was straightforward:
from suds.client import Client
proxy = {'http': 'proxy_username:proxy_password@proxy_server.com:port'}
url = 'http://www.ripedev.com/webservices/localtime.asmx?WSDL'
# client = Client(url)
client = Client(url, proxy=proxy)
print client
client.service.LocalTimeByZipCode('90210')
The previous examples were from a public SOAP API I found online. Now, the site I wanted to actually hit uses ssl for encryption (i.e., https site) and requires authentication. I thought the fix would be as simple as:
from suds.client import Client
proxy = {'https': 'proxy_username:proxy_password@proxy_server.com:port'}
url = 'https://some_server.com/path/to/soap_api?wsdl'
un = 'site_username'
pw = 'site_password'
# client = Client(url)
client = Client(url, proxy=proxy, username=un, password=pw)
print client
client.service.someFunction(args)
However, I got the error message: Exception: (404, u'/path/to/soap_api'). Very weird to me. Is it an authentication issue? Is it a proxy issue? If a proxy issue, how so, as my previous toy example worked. Tried the same site on my home network where there is no firewall, and things worked:
from suds.client import Client
url = 'https://some_server.com/path/to/soap_api?wsdl'
un = 'site_username'
pw = 'site_password'
# client = Client(url)
client = Client(url, username=un, password=pw)
print client
client.service.someFunction(args)
Conclusion? Must be a proxy issue with https. I used the following prior to calling suds to help with debugging:
import logging
logging.basicConfig(level=logging.INFO)
logging.getLogger('suds.client').setLevel(logging.DEBUG)
logging.getLogger('suds.transport').setLevel(logging.DEBUG)
logging.getLogger('suds.xsd.schema').setLevel(logging.DEBUG)
logging.getLogger('suds.wsdl').setLevel(logging.DEBUG)
My initial thoughts after some debugging: there must be something wrong with the proxy as the log shows python sending the request to the target url, but I get back a response that shows the path (minus the domain name) not found. What happened to the domain name? I notified the firewall team to look into this, as it appears the proxy is modifying something (url is not complete?). The firewall team investigated, and found that the proxy is returning a message that warns the ClientHello message is too large. This is one clue. The log also shows that the user was never authenticated and that the ssl handshake was never completed. My thought: still a proxy issue, as the python code works at home. However, the proxy team was able to access the https SOAP API through the proxy using the SOA Client plugin for Firefox. Now that convinced me that something else may be the culprit.
Googled for help, and thought this would be helpful.
import urllib2
import urllib
import httplib
import socket
class ProxyHTTPConnection(httplib.HTTPConnection):
_ports = {'http' : 80, 'https' : 443}
def request(self, method, url, body=None, headers={}):
#request is called before connect, so can interpret url and get
#real host/port to be used to make CONNECT request to proxy
proto, rest = urllib.splittype(url)
if proto is None:
raise ValueError, "unknown URL type: %s" % url
#get host
host, rest = urllib.splithost(rest)
#try to get port
host, port = urllib.splitport(host)
#if port is not defined try to get from proto
if port is None:
try:
port = self._ports[proto]
except KeyError:
raise ValueError, "unknown protocol for: %s" % url
self._real_host = host
self._real_port = port
httplib.HTTPConnection.request(self, method, url, body, headers)
def connect(self):
httplib.HTTPConnection.connect(self)
#send proxy CONNECT request
self.send("CONNECT %s:%d HTTP/1.0\r\n\r\n" % (self._real_host, self._real_port))
#expect a HTTP/1.0 200 Connection established
response = self.response_class(self.sock, strict=self.strict, method=self._method)
(version, code, message) = response._read_status()
#probably here we can handle auth requests...
if code != 200:
#proxy returned and error, abort connection, and raise exception
self.close()
raise socket.error, "Proxy connection failed: %d %s" % (code, message.strip())
#eat up header block from proxy....
while True:
#should not use directly fp probablu
line = response.fp.readline()
if line == '\r\n': break
class ProxyHTTPSConnection(ProxyHTTPConnection):
default_port = 443
def __init__(self, host, port = None, key_file = None, cert_file = None, strict = None, timeout=0): # vinh added timeout
ProxyHTTPConnection.__init__(self, host, port)
self.key_file = key_file
self.cert_file = cert_file
def connect(self):
ProxyHTTPConnection.connect(self)
#make the sock ssl-aware
ssl = socket.ssl(self.sock, self.key_file, self.cert_file)
self.sock = httplib.FakeSocket(self.sock, ssl)
class ConnectHTTPHandler(urllib2.HTTPHandler):
def do_open(self, http_class, req):
return urllib2.HTTPHandler.do_open(self, ProxyHTTPConnection, req)
class ConnectHTTPSHandler(urllib2.HTTPSHandler):
def do_open(self, http_class, req):
return urllib2.HTTPSHandler.do_open(self, ProxyHTTPSConnection, req)
from suds.client import Client
# from httpsproxy import ConnectHTTPSHandler, ConnectHTTPHandler ## these are code from above classes
import urllib2, urllib
from suds.transport.http import HttpTransport
opener = urllib2.build_opener(ConnectHTTPHandler, ConnectHTTPSHandler)
urllib2.install_opener(opener)
t = HttpTransport()
t.urlopener = opener
url = 'https://some_server.com/path/to/soap_api?wsdl'
proxy = {'https': 'proxy_username:proxy_password@proxy_server.com:port'}
un = 'site_username'
pw = 'site_password'
client = Client(url=url, transport=t, proxy=proxy, username=un, password=pw)
client = Client(url=url, transport=t, proxy=proxy, username=un, password=pw, location='https://some_server.com/path/to/soap_api?wsdl') ## some site suggests specifying location
This too did not work. Continued to google, and found that lot’s of people are having issues with https and proxy. I knew suds depended on urllib2, so googled about that as well, and people too had issues with urllib2 in terms of https and proxy. I then decided to investigate using urllib2 to contact the https url through a proxy:
## http://stackoverflow.com/questions/5227333/xml-soap-post-error-what-am-i-doing-wrong
## http://stackoverflow.com/questions/34079/how-to-specify-an-authenticated-proxy-for-a-python-http-connect
### at home this works
import urllib2
url = 'https://some_server.com/path/to/soap_api?wsdl'
password_mgr = urllib2.HTTPPasswordMgrWithDefaultRealm()
password_mgr.add_password(None,
uri=url,
user='site_username',
passwd='site_password')
auth_handler = urllib2.HTTPBasicAuthHandler(password_mgr)
opener = urllib2.build_opener(auth_handler)
urllib2.install_opener(opener)
page = urllib2.urlopen(url)
page.read()
### work network, does not work:
url = 'https://some_server.com/path/to/soap_api?wsdl'
proxy = urllib2.ProxyHandler({'https':'proxy_username:proxy_password@proxy_server.com:port', 'http':'proxy_username:proxy_password@proxy_server.com:port'})
password_mgr = urllib2.HTTPPasswordMgrWithDefaultRealm()
password_mgr.add_password(None,
uri=url,
user='site_username',
passwd='site_password')
auth_handler = urllib2.HTTPBasicAuthHandler(password_mgr)
opener = urllib2.build_opener(proxy, auth_handler, urllib2.HTTPSHandler)
urllib2.install_opener(opener)
page = urllib2.urlopen(site)
### also tried re-doing above, but with the custom handler as defined in the previous code chunk (http://code.activestate.com/recipes/456195/) running first (run the list of classes)
No luck. I re-read this post that I ran into before, and really agreed that urllib2 is severely flawed, especially when using https proxy. At the end of the page, the author suggested using the requests package. Tried it out, and I was able to connect using the https proxy:
import requests
import xmltodict
p1 = 'http://proxy_username:proxy_password@proxy_server.com:port'
p2 = 'https://proxy_username:proxy_password@proxy_server.com:port'
proxy = {'http': p1, 'https':p2}
site = 'https://some_server.com/path/to/soap_api?wsdl'
r = requests.get(site, proxies=proxy, auth=('site_username', 'site_password'))
r.text ## works
soap_xml_in = """<?xml version="1.0" encoding="UTF-8"?>
...
"""
headers = {'SOAPAction': u'""', 'Content-Type': 'text/xml; charset=utf-8', 'Content-type': 'text/xml; charset=utf-8', 'Soapaction': u'""'}
soap_xml_out = requests.post(site, data=soap_xml_in, headers=headers, proxies=proxy, auth=('site_username', 'site_password')).text
My learnings?
• suds is great for accessing SOAP, just not when you have to access an https site through a firewall.
• urllib2 is severely flawed. Things only work in very standard situations.
• requests package is very powerful and just works. Even though I have to deal with actual xml files as opposed to leveraging suds‘ pythonic structures, the xmltodict package helps to translate the xml file into dictionaries that only adds marginal effort to extract out relevant data.
NOTE: I had to install libuuid-devel in cygwin64 because I was getting an installation error.
## Use Android as a wireless mouse and keyboard for a Linux machine
I just discovered the app Home Remote Control that allows me to use my Android device as a wireless keyboard/mouse to a Linux computer over ssh. I just had to install ethtool on the Linux machine to get things to work.
## Upgrading Ubuntu 12.04 to 14.04 breaks encrypted LVM
My laptop runs Ubuntu and is fully encrypted (since version 10.04). Upgrade from 10.04 to 12.04 was smooth in the sense that my system booted fine, asking for the passphrase to unlock the LVM. However, when I upgraded from 12.04 to 14.04, things broke and my laptop no longer booted properly as the LVM never got encrypted. I had to do the following to get my laptop working again (after many rounds of trial and error):
• Boot a live usb Ubuntu session, de-crypted the LVM, and chroot’ed to run as the original OS
• Finish the upgrade session via apt-get update && apt-get upgrade
• It appears Ubuntu 14.04 installed some new package (did not write name down) that manages LVM or disks somehow (based on googling the error message). I removed this package.
• Saw lvm issues, so installed the package lvm2
• I made sure both dm-crypt and lvm2 were installed, and were accessible in initramfs, as cryptsetup was removed from initramfs since version 13.10. Had to do something with the following CRYPTSETUP issue.
• Based on this post, I modified various files, but things still did not boot properly. I believe what finally fixed it was explicitly pointing to the LVM by /dev/sda5 in the GRUB_CMDLINE_LINUX line in /etc/default/grub.
The following is summary of these files for me. /etc/crypttab:
# <target name> <source device> <key file> <options>
# sdb5_crypt UUID=731a44c4-4655-4f2b-ae1a-2e3e6a14f2ef none luks
sdb5_crypt UUID=731a44c4-4655-4f2b-ae1a-2e3e6a14f2ef none luks,retry=1,lvm=vg01
/etc/initramfs-tools/conf.d/cryptroot:
## vinh created http://www.joh.fi/posts/2014/03/18/install-ubuntu-1310-on-top-of-encrypted-lvm/
# CRYPTROOT=target=sdb5_crypt,source=/dev/disk/by-uuid/f1ba5a54-ac7e-419d-8762-43da3274aba4
CRYPTOPTS=target=sdb5_crypt,source=UUID=f1ba5a54-ac7e-419d-8762-43da3274aba4,lvm=vg01
Then run update-initramfs -k all -c in order to update the initramfs images.
Have this line in /etc/default/grub:
#GRUB_CMDLINE_LINUX="cryptopts=target=sdb5_crypt,source=/dev/disk/by-uuid/f1ba5a54-ac7e-419d-8762-43da3274aba4,lvm=vg01"
#GRUB_CMDLINE_LINUX="cryptopts=target=sdb5_crypt,source=UUID=f1ba5a54-ac7e-419d-8762-43da3274aba4,lvm=vg01"
GRUB_CMDLINE_LINUX="cryptopts=target=sdb5_crypt,source=/dev/sda5,lvm=vg01"
Run update-grub.
Again, I think the key is the source definition in the previous line. I kept trying to refer to it by uuid but that did not work. | {"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": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3535507023334503, "perplexity": 15701.040186031432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398447860.26/warc/CC-MAIN-20151124205407-00344-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.atmos-meas-tech.net/11/1377/2018/ | Journal cover Journal topic
Atmospheric Measurement Techniques An interactive open-access journal of the European Geosciences Union
Journal topic
Atmos. Meas. Tech., 11, 1377-1384, 2018
https://doi.org/10.5194/amt-11-1377-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
Atmos. Meas. Tech., 11, 1377-1384, 2018
https://doi.org/10.5194/amt-11-1377-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
Research article 08 Mar 2018
Research article | 08 Mar 2018
# Raindrop fall velocities from an optical array probe and 2-D video disdrometer
Raindrop fall velocities from an optical array probe and 2-D video disdrometer
Viswanathan Bringi1, Merhala Thurai1, and Darrel Baumgardner2 Viswanathan Bringi et al.
• 1Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, Colorado, USA
• 2Droplet Measurements Technologies, Longmont, Colorado, USA
Abstract
Back to toptop
We report on fall speed measurements of raindrops in light-to-heavy rain events from two climatically different regimes (Greeley, Colorado, and Huntsville, Alabama) using the high-resolution (50 µm) Meteorological Particle Spectrometer (MPS) and a third-generation (170 µm resolution) 2-D video disdrometer (2DVD). To mitigate wind effects, especially for the small drops, both instruments were installed within a 2∕3-scale Double Fence Intercomparison Reference (DFIR) enclosure. Two cases involved light-to-moderate wind speeds/gusts while the third case was a tornadic supercell and several squall lines that passed over the site with high wind speeds/gusts. As a proxy for turbulent intensity, maximum wind speeds from 10 m height at the instrumented site recorded every 3 s were differenced with the 5 min average wind speeds and then squared. The fall speeds vs. size from 0.1 to 2 and >0.7mm were derived from the MPS and the 2DVD, respectively. Consistency of fall speeds from the two instruments in the overlap region (0.7–2 mm) gave confidence in the data quality and processing methodologies. Our results indicate that under low turbulence, the mean fall speeds agree well with fits to the terminal velocity measured in the laboratory by Gunn and Kinzer from 100 µm up to precipitation sizes. The histograms of fall speeds for 0.5, 0.7, 1 and 1.5 mm sizes were examined in detail under the same conditions. The histogram shapes for the 1 and 1.5 mm sizes were symmetric and in good agreement between the two instruments with no evidence of skewness or of sub- or super-terminal fall speeds. The histograms of the smaller 0.5 and 0.7 mm drops from MPS, while generally symmetric, showed that occasional occurrences of sub- and super-terminal fall speeds could not be ruled out. In the supercell case, the very strong gusts and inferred high turbulence intensity caused a significant broadening of the fall speed distributions with negative skewness (for drops of 1.3, 2 and 3 mm). The mean fall speeds were also found to decrease nearly linearly with increasing turbulent intensity attaining values about 25–30 % less than the terminal velocity of Gunn–Kinzer, i.e., sub-terminal fall speeds.
1 Introduction
Back to toptop
Knowledge of the terminal fall speed of raindrops as a function of size is important in modeling collisional breakup and coalescence processes (e.g., List et al., 1987), in the radar-based estimation of rain rate, in retrieval of drop size distribution using Doppler spectra at vertical incidence (e.g., Sekhon and Srivastava, 1971) and in soil erosion studies (e.g., Rosewell, 1986). In these and other applications it is generally accepted that there is a unique fall speed ascribed to drops of a given mass or diameter and that it equals the terminal speed with adjustment for pressure (e.g., Beard, 1976). The terminal velocity measurements of Gunn and Kinzer (1949) under calm laboratory conditions and fits to their data (e.g., Atlas et al., 1973; Foote and du Toit, 1969; Beard and Pruppacher, 1969) are still considered the standard against which measurements using more modern optical instruments in natural rain are compared (Löffler-Mang and Joss, 2000; Barthazy et al., 2004; Schönhuber et al., 2008; Testik and Rahman, 2016; Yu et al., 2016). More recently, the broadening and skewness of the fall speed distributions of a given size (3 mm) in one intense rain event were attributed to mixed-mode amplitude oscillations (Thurai et al., 2013). Super- and sub-terminal fall speeds in intense rain shafts have been detected and attributed, respectively, to drop breakup fragments (sizes <0.5mm) and high wind/gusts (sizes 1–2 mm) (Montero-Martinez et al., 2009; Larsen et al., 2014; Montero-Martinez and Garcia-Garcia, 2016). Thus, there is some evidence that raindrops may not fall at their terminal velocity except under calm conditions and that the concept of a fall speed distribution for a drop of given mass (or, diameter) might need to be considered, which is the topic of this paper. The implications are rather profound, especially for numerical modeling of collision-coalescence and breakup processes, which are important for shaping the drop size distribution.
The fall speeds and concentration of small drops (<1mm) in natural rain are difficult to measure accurately given the poor resolution (>170µm) of most optical disdrometers and/or sensitivity issues. While cloud imaging probes (with high resolution 25–50 µm) on aircraft have been used for many years, they generally cannot measure the fall speeds. A relatively new instrument, the Meteorological Particle Spectrometer (MPS), is a droplet imaging probe built by Droplet Measurements Technologies (DMT, Inc.) under contract from the US Weather Service and specifically designed for drizzle as small as 50 µm and raindrops up to 3 mm. This instrument in conjunction with a lower-resolution 2-D video disdrometer (2DVD; Schoenhuber et al., 2008) is used in this paper to measure fall speed distributions in natural rain.
This paper briefly describes the instruments used, presents fall speed measurements from two sites under relatively low wind conditions and one case from an unusual tornadic supercell with high winds and gusts, and ends with a brief discussion and summary of the results.
2 Instrumentation and measurements
Back to toptop
The principal instruments used in this study are the MPS and third-generation 2DVD, both located within a 2∕3-scale Double Fence Intercomparison Reference (DFIR; Rasmussen et al., 2012) wind shield. As reported in Notaros et al. (2016), the 2∕3-scale DFIR was effective in reducing the ambient wind speeds by nearly a factor of 2–3 based on data from outside and inside the fence. The flow field in and around the DFIR has been simulated by Theriault et al. (2015) assuming steady ambient winds. They found that depending on the wind direction relative to the octagonal fence, weak vertical motions could be generated above the sensor areas. For 5 m s−1 speeds, the motions could range between 0.4 (down draft) and 0.2 m s−1 (up draft).
The instrument setup was the same for the two sites (Greeley, Colorado, and Huntsville, Alabama). Huntsville has a very different climate from Greeley, and its altitude is 212 m m.s.l. as compared with 1.4 km m.s.l. for Greeley. According to the Köppen–Trewartha climate classification system (Trewartha and Horn, 1980), this labels Greeley as a semiarid-type climate, whereas Huntsville is a humid subtropical-type climate (Belda et al., 2014).
The MPS is an optical array probe (OAP) that uses the technique introduced by Knollenberg (1970, 1976, 1981) and measures drop diameter in the range from 0.05 to 3.1 mm. A 64-element photodiode array is illuminated with a 660 nm collimated laser beam. Droplets passing through the laser cast a shadow on the array and the decrease in light intensity on the diodes is monitored with the signal processing electronics. A two-dimensional image is captured by recording the light level of each diode during the period that the array is shadowed. The fall velocity is derived using two methods. One uses the same approach as described by Montero-Martinez et al. (2009), in which the fall velocity is calculated from the product of the true air speed clock and ratio of the image height to width. Note that “width” is the horizontal dimension parallel to the array and “height” is along the vertical. The second method computes the fall velocity from the maximum horizontal dimension (spherical drop shape assumption) divided by the amount of time that the image is on the array, a time measured with a 2 MHz clock. In order to be comparable to the results of Montero-Martinez et al. (2009), their approach is implemented here for sizes >250µm. The fall velocity of smaller, slower-moving droplets is measured using the second technique.
The limitations and uncertainties associated with OAP measurements have been well documented (Korolev et al., 1991, 1998; Baumgardner et al., 2016). There are a number of potential artifacts that arise when making measurements with optical array probes (Baumgardner et al., 2016): droplet breakup on the probe tips that form satellite droplets, multiple droplets imaged simultaneously and out-of-focus drops whose images are usually larger than the actual drop (Korolev, 2007). The measured images have been analyzed to remove satellite droplets whose interarrival times are usually too short to be natural drops; multiple drops are detected by shape analysis and removed, and out-of-focus drops are detected and size corrected using the technique described by (Korolev, 2007). The sizing and fall speed errors primarily depend on the digitization error (±25µm). The fall speed accuracy according to the manufacturer (DMT) is <10 % for 0.25 mm and <1 % for sizes greater than 1 mm, limited primarily by the accuracy in droplet sizing.
The third-generation 2DVD is described in detail by Schoenhuber et al. (2007, 2008) and its accuracy of size and fall speed measurement has been well documented (e.g., Thurai et al., 2007, 2009; Huang et al., 2008; Bernauer et al., 2015). Considering the horizontal pixel resolution of 170 µm and other factors (such as “mismatched” drops), the effective sizing range is D>0.7mm. To clarify the mismatched drop problem: it is very difficult to match a drop detected in the top light-beam plane of the 2DVD to the corresponding drop in the bottom plane for tiny drops resulting in erroneous fall speeds. The fall velocity accuracy is determined primarily by the accuracy of calibrating the distance between the two orthogonal light “sheets” or planes and is <5 % for fall velocity <10m s−1. In our application, we utilize the MPS for measurement of small drops with D<1.2mm. The measurements from the MPS are compared with those from the 2DVD in the overlap region of D≈0.7–2.0 mm to ensure consistency of observations. The only fall velocity threshold used for the 2DVD is the lower limit set at 0.5 m s−1 in accordance with the manufacturer guidelines for rain measurements.
## 2.1 Fall speeds from Greeley, Colorado
We first consider a long duration (around 20 h) rain episode on 17 April 2015 which consisted of a wide variety of rain types/rates (mostly light stratiform <8mm h−1) as described in Table 2 of Thurai et al. (2017). Two wind sensors at a height of 1 m were available to measure the winds outside and inside the DFIR. Average wind speeds were, respectively, <1.5m s−1 inside the DFIR and <4m s−1 outside with light gusts. These wind sensors were specific to the winter experiment described in Notaros et al. (2016) and were unavailable for the rain measurement campaign after May 2015.
Figure 1a shows the fall speeds vs. D from the 2DVD (shown as contoured frequency of occurrence), along with mean and ±1σ standard deviation from the MPS. Also shown is the fit of Foote and du Toit (1969) (henceforth FT fit) to the terminal fall speed measurements of Gunn and Kinzer (1949) at sea level and after applying altitude corrections (Beard, 1976) for the elevation of 1.4 km m.s.l. for Greeley. Panels b and c show the histogram of fall speeds for diameter intervals (0.5±0.1) and (1±0.1mm) and (0.7±0.1) and (1.5±0.1mm), respectively. Panel a demonstrates the excellent “visual” agreement between the two instruments in the overlap size range (0.7–2 mm), which is quantified in Table 1. However, the altitude-adjusted FT fit is slightly higher than the measured values as shown in Table 1. Notable in Fig. 1a is the remarkable agreement in mean fall speeds between the FT fit and the MPS for D<0.5mm down to near the lower limit of the instrument (0.1 mm). Few measurements have been reported of fall speeds in this size range.
Figure 1(a) Fall velocity vs. diameter (D). The contoured frequency of occurrence from 2DVD data is shown in color (log scale). The mean fall velocity and ±1σ standard deviation bars are from MPS. The dark dashed line is from the fit to the laboratory data of Gunn and Kinzer (1949) and the purple line is the same except corrected for the altitude of Greeley, CO (1.4 km m.s.l.). (b) Relative frequency histograms of fall velocity for the 0.5±0.1mm and 1±0.1mm bins. (c) As in (b) but for the 0.7±0.1mm and 1.5±0.1mm bins.
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Table 1Expected fall velocities for various diameter intervals (bin width of 0.2 mm) from Foote and du Toit (1969) with altitude adjustment and the measured mean fall velocities with ±1σ (standard deviation).
The histograms in Fig. 1b and c show good agreement between 2DVD and MPS for 1 and 1.5 mm drop sizes, respectively, with respect to the mode, symmetry, spectral width and lack of skewness in the distributions. For the 1 mm size histogram, the mean is 3.8 m s−1 while the spectral width or standard deviation from MPS data is 0.6 m s−1. The corresponding coefficient of variation (ratio of standard deviation to mean) is 15.7 %. The finite bin width used (0.9–1.1 mm) causes a corresponding fall speed “spread” of around 0.6 m s−1, which is clearly a significant contributor to the measured coefficient of variation. Similar comments apply to the fall speed histogram for the 1.5 mm size shown in Fig. 1c. The definition of sub- or super-terminal fall speeds by Montero-Martinez et al. (2009) is based on fall speeds that are, respectively, less than 0.7 times the mean value or greater than 1.3 times the mean value (i.e., exceeding 30 % threshold on either side of the mean terminal fall speed). From examining the 1 mm size fall speed histogram there is negligible evidence of occurrences with fall speeds <2.66m s−1 (sub) or >4.94m s−1 (super). Similar comment also applies for the 1.5 mm size based on the corresponding histogram.
The histogram from MPS for the 0.5 mm sizes shows positive skewness with mean of 1.8 m s−1, spectral width of 0.65 m s−1 and corresponding coefficient of variation nearly doubling to 35 % (relative to the 1 mm size histogram). The finite bin width (0.4–0.6 mm) causes a corresponding fall speed “spread” of 0.4 m s−1, which contributes to the measured coefficient of variation. Nevertheless, it is not possible to rule out the low frequency of occurrence of sub- or super-terminal fall speeds that is less than 1.26 m s−1 or exceeding 2.34 m s−1, respectively, based on our data. Examination of the MPS-based fall speed histogram for the 0.7 mm size indicates negative skewness. As with the 0.5 mm drops it is not possible to rule out the occurrences of fall speeds <1.8m s−1 or >3.4m s−1, i.e., sub- or super-terminal fall speeds.
## 2.2 Fall speeds from Huntsville, Alabama
The first Huntsville event occurred on 11 April 2016 and consisted of precipitation associated with the mesoscale vortex of a developing squall line that moved across northern Alabama between 18:00 and 23:00 UTC and produced over 25 mm of rainfall in the Huntsville area. Figure 2a shows the ambient 10 m height wind speeds (3 s and 5 min averaged) recorded at the site. Maximum speeds were less than 5 m s−1 and wind gusts were light. As no direct in situ measurement of turbulence was available, we use the approach by Garrett and Yuter (2014), who estimate the difference between the maximum wind speed, or gust, which was sampled every 3 s, and the average wind speed derived from successive 5 min intervals. The estimated turbulent intensity is proportional to $E=\left(\text{gusts}-\text{average wind}{\right)}^{\mathrm{2}}/\mathrm{2}$. Figure 2b shows the E values, which were small (maximum E<0.4m2 s−2) and indicative of low turbulence. Also shown in Fig. 2b is the 2DVD-based time series of rainfall rate (R) averaged over 3 min; the maximum R is around 10 mm h−1.
Figure 2(a) The 3 s raw and 5 min averaged wind speeds at 10 m height. (b) Turbulent intensity estimates E and 3 min averaged R.
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Figure 3a shows the fall velocity vs. D comparison between the two instruments while panels b and c show the histograms for the 0.5 and 1 mm and 0.7 and 1.5 mm sizes, respectively. Similar to the Greeley event, the mean fall speed agreement between both instruments in the overlap region is excellent (see Table 1) and consistent with the FT fit to the Gunn–Kinzer laboratory data. As in Fig. 1a, the MPS data in Fig. 3a are in excellent agreement with FT fit for sizes <0.5mm.
Figure 3(a) As in Fig. 1a except for 11 April 2016 event. The dashed line is fit to Gunn–Kinzer at sea level. (b, c) As in Fig. 1b and c but for 11 April 2016 event.
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The 0.5 and 1 mm histogram shapes in Fig. 3b are quite similar to the Greeley case shown in Fig. 1b. The mean and SDs from the MPS data for the 0.5 and 1 mm bins are, respectively, [2±0.62] and [3.88±0.44] m s−1. The values for the 0.7 and 1.5 mm bins are, respectively, [2.6±0.6] and [5.4±0.4] m s−1. There is negligible evidence of sub- or super-terminal fall speed occurrences based on the 1 and 1.5 mm histograms. The comments made earlier with respect to Fig. 1b and c of the Greeley event for the 0.5 and 0.7 mm histograms are also applicable here; i.e., we cannot rule out the occasional occurrences of sub- or super-terminal fall speeds based on our data.
The second case considered is from 30 November 2016 wherein a supercell passed over the instrumented site from 03:00 to 03:30 UTC, producing about 15 min later a long-lived EF-2 tornado. Strong winds were recorded at the site, with 5 min averaged speeds reaching 10–12 m s−1 between 03:20 and 03:30 and E values in the range of 7–8 m2 s−2, indicating strong turbulence (Fig. 4a and b). The rain rates peaked at 70 mm h−1 during this time (Fig. 4b). About 3 h later several squall-line-type storm cells passed over the site from 07:00 to 09:00 UTC, again with strong winds but considerably lower E values 2–4 m2 s−2 and maximum R of 80 mm h−1. After 10:00 UTC the E values were much smaller (<0.5m2 s−2), indicating calm conditions. The peak R is also smaller at 30 mm h−1 at 10:00 UTC.
Figure 4c–e show the mean and ±1σ of the fall speeds from the 2DVD for the 1.3, 2 and 3 mm drop sizes, respectively. The MPS data are not shown here since during this event it was located outside the DFIR on its turntable and we did not want to confuse the wind effects between the two instruments. It is clear from Fig. 4c that during the supercell passage (03:00–03:30 UTC) the mean fall speed for 1.3 mm drops decreases (from 5 to 3.5 m s−1) and the standard deviation increases (from 0.5 to 1.5 m s−1). The histogram shapes also show increasing negative skewness (not shown). The same trend can be seen for the subsequent squall-line rain cell passage from 07:00 to 09:00 UTC. Similar trends are noted in Fig. 4d and less so in Fig. 4e.
Figure 4(a) As in Fig. 2a except for 30 November 2016 event. (b) As in Fig. 2b. (c) Mean and ±1σ SD of fall speeds from 2DVD for 1.3±0.1mm sizes. (d, e) As in (c) but for 2±0.1 and 3±0.1mm sizes, respectively.
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To expand on this observed correlation, Fig. 5 shows scatterplots of the mean fall speed and standard deviation vs. E for the 1.3 mm drops (panels a and b), while panels c and d and e and f show the same but for the 2 and 3 mm drops, respectively. The mean fall speed decreases with increasing E nearly linearly for E>1m2 s−2 but less so for the 3 mm size drops (Stout et al., 1995). This decrease relative to Gunn–Kinzer terminal fall speeds is termed as “sub-terminal” and our data are in general agreement with Montero-Martinez and Garcia-Garcia (2016), who found an increase in the numbers of sub-terminal drops with sizes between 1 and 2 mm under windy conditions using a 2-D precipitation probe with resolution of 200 µm (similar to 2DVD) but without a wind fence. The standard deviation of fall speeds (σf) vs. E is shown in panels 5b, d and f. When E>1m2 s−2, the σf is nearly constant at 1.5 m s−1 for both 1.3 and 2 mm drop sizes and constant at 1 m s−1 for the 3 mm size. For E<1, the σf is more variable and essentially uncorrelated with E. From the discussion related to Figs. 1b and c and 3b and c, σf values exceeding approximately 0.5 m s−1 can be attributed to physical, not instrumental or finite bin width effects (see also Table 1). Thus, the fall speed distributions are considerably broadened when E>1m2 s−2 due to increasing turbulence levels which is again consistent with the findings of Montero-Martinez and Garcia-Garcia (2016) as well as those of Garett and Yuter (2014). The latter observations, however, were of graupel fall speeds in winter precipitation using a multiangle snowflake camera (Garrett et al., 2012).
Figure 5(a, b) Mean fall speed and SD, respectively, vs. E for 1.3 mm sizes. (c, d) Same but for 2 mm sizes. (e, f) Same but for 3 mm.
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3 Discussion and conclusions
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We have reported on raindrop fall speed distributions using a high-resolution (50 µm) droplet spectrometer (MPS) collocated with moderate-resolution (170 µm) 2DVD (with both instruments inside a DFIR wind shield) to cover the entire size range (from 0.1 mm onwards) expected in natural rain. Turbulence intensity (E) was derived from wind/gust data at 10 m height following Garrett and Yuter (2014). For low turbulent intensities (E<0.4m2 s−2), in the overlap region of the two instruments (0.7–2 mm), the mean fall speeds were in excellent agreement with each other for both the Greeley, CO, and Huntsville, AL, sites, giving high confidence in the quality of the measurements. For D<0.5mm and down to 0.1 mm, the mean fall speeds from MPS from both sites were in remarkable agreement with FT fit to the laboratory data of Gunn and Kinzer (1949). In the overlap region, the mean fall speeds from the two instruments were in excellent agreement with the FT fit for the Huntsville site (no altitude adjustment required) and good agreement for the Greeley site (after adjustment for altitude of 1.4 km). For D>2mm, the mean fall speeds from 2DVD were in excellent agreement with the FT fit at both sites.
Our histograms of fall speeds for 1 and 1.5 mm sizes under low turbulence intensity conditions (E<0.4m2 s−2) from both MPS and 2DVD were in good agreement and did not show any evidence of either sub- or super-terminal speeds; instead, the histograms were symmetric with mean close to the Gunn–Kinzer terminal velocity with no significant broadening over that ascribed to instrument and/or finite bin width effects. (Note: sub-terminal implies fall speeds <0.7 times the terminal fall speed whereas super-terminal implies >1.3 times terminal value; Montero-Martinez et al., 2009.) However, for the 0.5 and 0.7 mm sizes, from the histogram of fall speeds using the MPS under the same conditions occasional occurrences of both sub- and super-terminal fall speeds, after accounting for instrumental and finite bin width effects, cannot be ruled out.
The only comparable earlier study is by Montero-Martinez et al. (2009) who used collocated 2-D cloud and precipitation probes (2D-C, 2D-P) but restricted their data to calm wind conditions. Their main conclusion was that the distribution of the ratio of the measured fall speed to the terminal fall speed for 0.44 mm size, while having a mode at 1 m s−1 was strongly positively skewed with tails extending to 5 m s−1 especially at high rain rates. In our data for the 0.5 and 0.7 mm sizes shown in Figs. 1b and c and 3b and c, no such strong positive skewness was observed in the fall speed histograms, and the corresponding ratio of MPS-measured fall speeds to terminal values does not exceed 1.5 to 2.
Another study by Larsen et al. (2014) appears to confirm the ubiquitous existence of super-terminal fall speeds for sizes <1mm using different instruments, one of which was a 2DVD similar to the one used in this study. However, it is well known that mismatched drops cause erroneous fall speed estimates from 2DVD for drops <0.5mm (Schoenhuber et al., 2008; Appendix in Huang et al., 2010; Bernauer et al., 2015). It is not clear whether Larsen et al. (2014) accounted for this problem in their analysis. In addition, their 2DVD was not located within a DFIR-like wind shield.
In a later study using only the 2D-P probe, Montero-Martinez and Garcia-Garcia (2016) found sub-terminal fall speeds and broadened distributions under windy conditions for 1–2 mm sizes in general agreement with our results using the 2DVD. Stout et al. (1995) simulated the motion of drops subject to nonlinear drag in isotropic turbulence and determined that there would be a significant reduction of the average drop settling velocity (relative to terminal velocity) of greater that 35 % for drops around 2 mm size when the ratio of root mean square (rms) velocity fluctuations (due to turbulence) relative to drop terminal velocity is around 0.8. Whereas we did not have a direct measure of the rms velocity fluctuations, the proxy for turbulence intensity (E) related to wind gusts during supercell passage (very large E around 7 m2 s−2) and two squall-line passages (moderate E between 2 and 5 m2 s−2) clearly showed a significant reduction in mean fall speeds of 25–30 % relative to terminal speed for 1.3 and 2 mm sizes (and less so for 3 mm drops), with significant broadening of the fall speed distributions relative to calm conditions by nearly a factor of 1.5 to 2.
While our dataset is limited to three events they cover a wide range of rain rates, wind conditions and two different climatologies. One caveat is that the response of the DFIR wind shield to ambient winds in terms of producing subtle vertical air motions near the sensor area is yet to be evaluated as future work. Analysis of further events with direct measurement of turbulent intensity, for example using a 3-D sonic anemometer at the height of the sensor, would be needed to generalize our findings.
Data availability
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Data availability.
Competing interests
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Competing interests.
Viswanathan Bringi and Merhala Thurai declare they have no conflict of interest. Darrel Baumgardner is employed by Droplet Measurements Technologies, Inc. (Longmont, Colorado, USA), who manufacture the Meteorological Particle Spectrometer used in this study.
Acknowledgements
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Acknowledgements.
Two of the authors (Viswanathan N. Bringi and Merhala Thurai) acknowledge support from the US National Science Foundation via grant AGS-1431127. The assistance of Patrick Gatlin of NASA/MSFC is gratefully acknowledged. Kevin Knupp and Carter Hulsey of the University of Alabama in Huntsville processed the wind data.
Edited by: Gianfranco Vulpiani
Reviewed by: Hidde Leijnse and four anonymous referees
References
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Short summary
Raindrop fall velocities are important for rain rate estimation, soil erosion studies and in numerical modelling of rain formation in clouds. The assumption that the fall velocity is uniquely related to drop size is made inherently based on laboratory measurements under still air conditions from nearly 68 years ago. There have been very few measurements of drop fall speeds in natural rain under both still and turbulent wind conditions. We report on fall speed measurements in natural rain shafts.
Raindrop fall velocities are important for rain rate estimation, soil erosion studies and in...
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Stefan Monnier committed Jul 14, 2002 1 ;;; tex-mode.el --- TeX, LaTeX, and SliTeX mode commands -*- coding: utf-8 -*- Eric S. Raymond committed May 30, 1992 2 Stefan Monnier committed Mar 18, 2004 3 ;; Copyright (C) 1985,86,89,92,94,95,96,97,98,1999,2002,03,2004 Richard M. Stallman committed Apr 13, 1997 4 ;; Free Software Foundation, Inc. Eric S. Raymond committed Jul 22, 1992 5 Richard M. Stallman committed Dec 22, 1994 6 ;; Maintainer: FSF Eric S. Raymond committed Jul 17, 1992 7 ;; Keywords: tex Eric S. Raymond committed Jul 16, 1992 8 Richard M. Stallman committed Jul 04, 1992 9 ;; Contributions over the years by William F. Schelter, Dick King, Richard M. Stallman committed Dec 22, 1994 10 ;; Stephen Gildea, Michael Prange, Jacob Gore, and Edward M. Reingold. Richard M. Stallman committed Jul 04, 1992 11 root committed Aug 28, 1990 12 13 14 15 ;; This file is part of GNU Emacs. ;; GNU Emacs is free software; you can redistribute it and/or modify ;; it under the terms of the GNU General Public License as published by Eric S. Raymond committed Jul 16, 1992 16 ;; the Free Software Foundation; either version 2, or (at your option) root committed Aug 28, 1990 17 18 19 20 21 22 23 24 ;; any later version. ;; GNU Emacs 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 General Public License for more details. ;; You should have received a copy of the GNU General Public License Richard M. Stallman committed Jan 26, 1996 25 26 27 ;; along with GNU Emacs; see the file COPYING. If not, write to the ;; Free Software Foundation, Inc., 59 Temple Place - Suite 330, ;; Boston, MA 02111-1307, USA. root committed Aug 28, 1990 28 Pavel Janík committed Jul 15, 2001 29 30 ;;; Commentary: Eric S. Raymond committed Jul 16, 1992 31 32 ;;; Code: Richard M. Stallman committed Feb 07, 1999 33 34 35 ;; Pacify the byte-compiler (eval-when-compile (require 'compare-w) Stefan Monnier committed Oct 15, 2000 36 (require 'cl) Richard M. Stallman committed Feb 07, 1999 37 38 (require 'skeleton)) Edward M. Reingold committed Apr 01, 1994 39 (require 'shell) Eric S. Raymond committed Apr 25, 1993 40 (require 'compile) Richard M. Stallman committed Jul 04, 1992 41 Richard M. Stallman committed Apr 13, 1997 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 (defgroup tex-file nil "TeX files and directories" :prefix "tex-" :group 'tex) (defgroup tex-run nil "Running external commands from TeX mode" :prefix "tex-" :group 'tex) (defgroup tex-view nil "Viewing and printing TeX files" :prefix "tex-" :group 'tex) Jim Blandy committed Aug 12, 1992 57 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 58 59 60 61 62 (defcustom tex-shell-file-name nil "*If non-nil, the shell file name to run in the subshell used to run TeX." :type '(choice (const :tag "None" nil) string) :group 'tex-run) Richard M. Stallman committed Jul 04, 1992 63 Jim Blandy committed Aug 12, 1992 64 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 65 (defcustom tex-directory "." Richard M. Stallman committed Aug 25, 1996 66 "*Directory in which temporary files are written. Richard M. Stallman committed May 02, 1994 67 You can make this /tmp' if your TEXINPUTS has no relative directories in it Richard M. Stallman committed Jul 04, 1992 68 and you don't try to apply \$tex-region] or \\[tex-buffer] when there are Richard M. Stallman committed Apr 13, 1997 69 70 71 \\input' commands with relative directories." :type 'directory :group 'tex-file) root committed Aug 28, 1990 72 Richard M. Stallman committed Dec 29, 1996 73 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 74 (defcustom tex-first-line-header-regexp nil Richard M. Stallman committed Dec 29, 1996 75 76 77 "Regexp for matching a first line which tex-region' should include. If this is non-nil, it should be a regular expression string; if it matches the first line of the file, Richard M. Stallman committed Apr 13, 1997 78 79 80 81 tex-region' always includes the first line in the TeX run." :type '(choice (const :tag "None" nil) regexp) :group 'tex-file) Richard M. Stallman committed Dec 29, 1996 82 Richard M. Stallman committed Aug 25, 1996 83 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 84 (defcustom tex-main-file nil Richard M. Stallman committed Aug 25, 1996 85 "*The main TeX source file which includes this buffer's file. Richard M. Stallman committed Jul 29, 1998 86 87 The command tex-file' runs TeX on the file specified by tex-main-file' if the variable is non-nil." Richard M. Stallman committed Apr 13, 1997 88 89 90 :type '(choice (const :tag "None" nil) file) :group 'tex-file) Richard M. Stallman committed Aug 25, 1996 91 Jim Blandy committed Aug 12, 1992 92 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 93 94 95 96 (defcustom tex-offer-save t "*If non-nil, ask about saving modified buffers before \\[tex-file] is run." :type 'boolean :group 'tex-file) root committed Aug 28, 1990 97 Jim Blandy committed Aug 12, 1992 98 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 99 (defcustom tex-run-command "tex" root committed Aug 28, 1990 100 "*Command used to run TeX subjob. Richard M. Stallman committed Feb 16, 1999 101 102 TeX Mode sets tex-command' to this string. See the documentation of that variable." Richard M. Stallman committed Apr 13, 1997 103 104 :type 'string :group 'tex-run) root committed Aug 28, 1990 105 Jim Blandy committed Aug 12, 1992 106 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 107 (defcustom latex-run-command "latex" root committed Aug 28, 1990 108 "*Command used to run LaTeX subjob. Richard M. Stallman committed Feb 16, 1999 109 110 LaTeX Mode sets tex-command' to this string. See the documentation of that variable." Richard M. Stallman committed Apr 13, 1997 111 112 :type 'string :group 'tex-run) Richard M. Stallman committed Jul 04, 1992 113 Richard M. Stallman committed Feb 07, 1999 114 115 116 ;;;###autoload (defcustom slitex-run-command "slitex" "*Command used to run SliTeX subjob. Richard M. Stallman committed Feb 16, 1999 117 118 SliTeX Mode sets tex-command' to this string. See the documentation of that variable." Richard M. Stallman committed Feb 07, 1999 119 120 121 :type 'string :group 'tex-run) Richard M. Stallman committed Feb 16, 1999 122 ;;;###autoload Stefan Monnier committed Jul 14, 2002 123 (defcustom tex-start-options "" Richard M. Stallman committed Apr 26, 2002 124 "*TeX options to use when starting TeX. Stefan Monnier committed Jul 14, 2002 125 126 127 128 These immediately precede the commands in tex-start-commands' and the input file name, with no separating space and are not shell-quoted. If nil, TeX runs with no options. See the documentation of tex-command'." :type 'string Richard M. Stallman committed Feb 07, 1999 129 :group 'tex-run Richard M. Stallman committed Apr 26, 2002 130 131 132 133 134 :version "21.4") ;;;###autoload (defcustom tex-start-commands "\\nonstopmode\\input" "*TeX commands to use when starting TeX. Stefan Monnier committed Jul 14, 2002 135 136 They are shell-quoted and precede the input file name, with a separating space. If nil, no commands are used. See the documentation of tex-command'." Richard M. Stallman committed Apr 26, 2002 137 138 139 140 141 142 :type '(radio (const :tag "Interactive $$nil$$" nil) (const :tag "Nonstop $$\"\\nonstopmode\\input\"$$" "\\nonstopmode\\input") (string :tag "String at your choice")) :group 'tex-run :version "21.4") Richard M. Stallman committed Feb 07, 1999 143 Stefan Monnier committed Nov 08, 2002 144 (defvar latex-standard-block-names Stefan Monnier committed Nov 10, 2000 145 146 147 148 149 150 151 152 153 '("abstract" "array" "center" "description" "displaymath" "document" "enumerate" "eqnarray" "eqnarray*" "equation" "figure" "figure*" "flushleft" "flushright" "itemize" "letter" "list" "minipage" "picture" "quotation" "quote" "slide" "sloppypar" "tabbing" "table" "table*" "tabular" "tabular*" "thebibliography" "theindex*" "titlepage" "trivlist" "verbatim" "verbatim*" "verse" "math") Richard M. Stallman committed Jul 04, 1992 154 155 "Standard LaTeX block names.") Jim Blandy committed Aug 12, 1992 156 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 157 (defcustom latex-block-names nil Richard M. Stallman committed Jul 04, 1992 158 "*User defined LaTeX block names. Stefan Monnier committed Nov 08, 2002 159 Combined with latex-standard-block-names' for minibuffer completion." Richard M. Stallman committed Apr 13, 1997 160 161 :type '(repeat string) :group 'tex-run) root committed Aug 28, 1990 162 Jim Blandy committed Aug 12, 1992 163 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 164 (defcustom tex-bibtex-command "bibtex" Richard M. Stallman committed Jul 04, 1992 165 "*Command used by tex-bibtex-file' to gather bibliographic data. Richard M. Stallman committed May 02, 1994 166 If this string contains an asterisk (*'), that is replaced by the file name; Richard M. Stallman committed Apr 13, 1997 167 168 169 otherwise, the file name, preceded by blank, is added at the end." :type 'string :group 'tex-run) root committed Aug 28, 1990 170 Jim Blandy committed Aug 12, 1992 171 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 172 (defcustom tex-dvi-print-command "lpr -d" Richard M. Stallman committed Jul 04, 1992 173 "*Command used by \\[tex-print] to print a .dvi file. Richard M. Stallman committed May 02, 1994 174 If this string contains an asterisk (*'), that is replaced by the file name; Richard M. Stallman committed Apr 13, 1997 175 176 177 otherwise, the file name, preceded by blank, is added at the end." :type 'string :group 'tex-view) Richard M. Stallman committed Jul 04, 1992 178 Jim Blandy committed Aug 12, 1992 179 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 180 (defcustom tex-alt-dvi-print-command "lpr -d" Richard M. Stallman committed Jul 04, 1992 181 "*Command used by \\[tex-print] with a prefix arg to print a .dvi file. Richard M. Stallman committed May 02, 1994 182 183 If this string contains an asterisk (*'), that is replaced by the file name; otherwise, the file name, preceded by blank, is added at the end. Richard M. Stallman committed Jul 04, 1992 184 Richard M. Stallman committed May 02, 1994 185 186 If two printers are not enough of a choice, you can set the variable tex-alt-dvi-print-command' to an expression that asks what you want; Richard M. Stallman committed Jul 04, 1992 187 188 189 190 191 192 for example, (setq tex-alt-dvi-print-command '(format \"lpr -P%s\" (read-string \"Use printer: \"))) would tell \\[tex-print] with a prefix argument to ask you which printer to Richard M. Stallman committed Apr 13, 1997 193 194 195 196 use." :type '(choice (string :tag "Command") (sexp :tag "Expression")) :group 'tex-view) root committed Aug 28, 1990 197 Jim Blandy committed Aug 12, 1992 198 ;;;###autoload Dave Love committed Aug 05, 2003 199 (defcustom tex-dvi-view-command '(if (eq window-system 'x) "xdvi" "dvi2tty * | cat -s") Richard M. Stallman committed May 02, 1994 200 "*Command used by \\[tex-view] to display a .dvi' file. Richard M. Stallman committed Nov 06, 2002 201 If it is a string, that specifies the command directly. Richard M. Stallman committed May 02, 1994 202 If this string contains an asterisk (*'), that is replaced by the file name; Richard M. Stallman committed Nov 06, 2002 203 otherwise, the file name, preceded by a space, is added at the end. Richard M. Stallman committed Jul 04, 1992 204 Richard M. Stallman committed Nov 06, 2002 205 206 If the value is a form, it is evaluated to get the command to use." :type '(choice (const nil) string sexp) Richard M. Stallman committed Apr 13, 1997 207 :group 'tex-view) root committed Aug 28, 1990 208 Jim Blandy committed Aug 12, 1992 209 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 210 (defcustom tex-show-queue-command "lpq" Richard M. Stallman committed Jul 04, 1992 211 "*Command used by \\[tex-show-print-queue] to show the print queue. Richard M. Stallman committed Apr 13, 1997 212 213 214 Should show the queue(s) that \\[tex-print] puts jobs on." :type 'string :group 'tex-view) root committed Aug 28, 1990 215 Jim Blandy committed Aug 12, 1992 216 ;;;###autoload Gerd Moellmann committed Nov 01, 1999 217 (defcustom tex-default-mode 'latex-mode root committed Aug 28, 1990 218 219 220 "*Mode to enter for a new file that might be either TeX or LaTeX. This variable is used when it can't be determined whether the file is plain TeX or LaTeX or what because the file contains no commands. Richard M. Stallman committed Apr 13, 1997 221 222 223 Normally set to either plain-tex-mode' or latex-mode'." :type 'function :group 'tex) root committed Aug 28, 1990 224 Jim Blandy committed Aug 12, 1992 225 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 226 227 228 (defcustom tex-open-quote "" "*String inserted by typing \\[tex-insert-quote] to open a quotation." :type 'string Stefan Monnier committed Jul 14, 2002 229 :options '("" "\"<" "\"" "<<" "«") Richard M. Stallman committed Apr 13, 1997 230 :group 'tex) root committed Aug 28, 1990 231 Jim Blandy committed Aug 12, 1992 232 ;;;###autoload Richard M. Stallman committed Apr 13, 1997 233 234 235 (defcustom tex-close-quote "''" "*String inserted by typing \\[tex-insert-quote] to close a quotation." :type 'string Stefan Monnier committed Jul 14, 2002 236 :options '("''" "\">" "\"'" ">>" "»") Richard M. Stallman committed Apr 13, 1997 237 :group 'tex) root committed Aug 28, 1990 238 Richard M. Stallman committed Jul 04, 1992 239 240 241 (defvar tex-last-temp-file nil "Latest temporary file generated by \\[tex-region] and \\[tex-buffer]. Deleted when the \\[tex-region] or \\[tex-buffer] is next run, or when the Richard M. Stallman committed May 02, 1994 242 tex shell terminates.") Richard M. Stallman committed Jul 04, 1992 243 Stefan Monnier committed Jul 14, 2002 244 (defvar tex-command "tex" Richard M. Stallman committed Feb 16, 1999 245 "*Command to run TeX. Stefan Monnier committed Jul 14, 2002 246 If this string contains an asterisk $$*'$$, that is replaced by the file name; Richard M. Stallman committed Apr 26, 2002 247 248 249 otherwise the value of tex-start-options', the $$shell-quoted$$ value of tex-start-commands', and the file name are added at the end with blanks as separators. Richard M. Stallman committed Feb 16, 1999 250 251 252 253 In TeX, LaTeX, and SliTeX Mode this variable becomes buffer local. In these modes, use \\[set-variable] if you want to change it for the current buffer.") root committed Aug 28, 1990 254 255 256 257 258 (defvar tex-trailer nil "String appended after the end of a region sent to TeX by \\[tex-region].") (defvar tex-start-of-header nil Richard M. Stallman committed Oct 19, 1995 259 "Regular expression used by \\[tex-region] to find start of file's header.") root committed Aug 28, 1990 260 261 (defvar tex-end-of-header nil Richard M. Stallman committed Oct 19, 1995 262 "Regular expression used by \\[tex-region] to find end of file's header.") root committed Aug 28, 1990 263 264 265 (defvar tex-shell-cd-command "cd" "Command to give to shell running TeX to change directory. Richard M. Stallman committed May 02, 1994 266 The value of tex-directory' is appended to this, separated by a space.") root committed Aug 28, 1990 267 268 269 270 271 272 273 274 275 276 277 278 (defvar tex-zap-file nil "Temporary file name used for text being sent as input to TeX. Should be a simple file name with no extension or directory specification.") (defvar tex-last-buffer-texed nil "Buffer which was last TeXed.") (defvar tex-print-file nil "File name that \\[tex-print] prints. Set by \\[tex-region], \\[tex-buffer], and \\[tex-file].") Stefan Monnier committed Jun 08, 2003 279 280 281 282 283 284 285 286 287 288 (defvar tex-mode-syntax-table (let ((st (make-syntax-table))) (modify-syntax-entry ?% "<" st) (modify-syntax-entry ?\n ">" st) (modify-syntax-entry ?\f ">" st) (modify-syntax-entry ?\C-@ "w" st) (modify-syntax-entry ?' "w" st) (modify-syntax-entry ?@ "_" st) (modify-syntax-entry ?* "_" st) (modify-syntax-entry ?\t " " st) Stefan Monnier committed May 10, 2003 289 290 ;; ~ is printed by TeX as a space, but it's semantics in the syntax ;; of TeX is not whitespace' (i.e. it's just like \hspace{foo}). Stefan Monnier committed Jun 08, 2003 291 292 293 294 295 296 297 298 (modify-syntax-entry ?~ "." st) (modify-syntax-entry ? "" st) (modify-syntax-entry ?\\ "/" st) (modify-syntax-entry ?\" "." st) (modify-syntax-entry ?& "." st) (modify-syntax-entry ?_ "." st) (modify-syntax-entry ?^ "." st) st) root committed Aug 28, 1990 299 "Syntax table used while in TeX mode.") Stefan Monnier committed Oct 01, 2000 300 301 302 303 ;;;; ;;;; Imenu support ;;;; root committed Aug 28, 1990 304 Stefan Monnier committed Sep 29, 2000 305 (defcustom latex-imenu-indent-string ". " Richard M. Stallman committed Jun 23, 1997 306 307 308 309 310 "*String to add repeated in front of nested sectional units for Imenu. An alternative value is \" . \", if you use a font with a narrow period." :type 'string :group 'tex) Stefan Monnier committed Sep 29, 2000 311 312 313 314 315 316 (defvar latex-section-alist '(("part" . 0) ("chapter" . 1) ("section" . 2) ("subsection" . 3) ("subsubsection" . 4) ("paragraph" . 5) ("subparagraph" . 6))) Stefan Monnier committed Oct 01, 2000 317 318 319 320 321 (defvar latex-metasection-list '("documentstyle" "documentclass" "begin{document}" "end{document}" "appendix" "frontmatter" "mainmatter" "backmatter")) Richard M. Stallman committed May 15, 1996 322 (defun latex-imenu-create-index () Stefan Monnier committed Oct 01, 2000 323 324 325 326 327 328 329 "Generate an alist for imenu from a LaTeX buffer." (let ((section-regexp (concat "\\\\" (regexp-opt (mapcar 'car latex-section-alist) t) "\\*?[ \t]*{")) (metasection-regexp (concat "\\\\" (regexp-opt latex-metasection-list t))) i0 menu case-fold-search) Richard M. Stallman committed Jun 23, 1997 330 331 332 333 334 335 336 337 338 339 340 341 (save-excursion ;; Find the top-most level in this file but don't allow it to be ;; any deeper than "section" (which is top-level in an article). (goto-char (point-min)) (if (search-forward-regexp "\\\\part\\*?[ \t]*{" nil t) (setq i0 0) (if (search-forward-regexp "\\\\chapter\\*?[ \t]*{" nil t) (setq i0 1) (setq i0 2))) ;; Look for chapters and sections. (goto-char (point-min)) Stefan Monnier committed Oct 01, 2000 342 (while (search-forward-regexp section-regexp nil t) Richard M. Stallman committed Jun 23, 1997 343 344 345 346 347 (let ((start (match-beginning 0)) (here (point)) (i (cdr (assoc (buffer-substring-no-properties (match-beginning 1) (match-end 1)) Stefan Monnier committed Sep 29, 2000 348 latex-section-alist)))) Richard M. Stallman committed Jun 23, 1997 349 350 351 352 353 354 (backward-char 1) (condition-case err (progn ;; Using sexps allows some use of matching {...} inside ;; titles. (forward-sexp 1) Stefan Monnier committed Sep 29, 2000 355 356 357 358 359 360 361 362 (push (cons (concat (apply 'concat (make-list (max 0 (- i i0)) latex-imenu-indent-string)) (buffer-substring-no-properties here (1- (point)))) start) menu)) Richard M. Stallman committed Jun 23, 1997 363 364 365 366 367 368 (error nil)))) ;; Look for included material. (goto-char (point-min)) (while (search-forward-regexp "\\\\\$$include\\|input\\|verbatiminput\\|bibliography\$$\ Stefan Monnier committed Oct 01, 2000 369 \[ \t]*{\$$[^}\n]+\$$}" Richard M. Stallman committed Jun 23, 1997 370 nil t) Stefan Monnier committed Oct 01, 2000 371 372 373 374 375 376 377 378 (push (cons (concat "<<" (buffer-substring-no-properties (match-beginning 2) (match-end 2)) (if (= (char-after (match-beginning 1)) ?b) ".bbl" ".tex")) (match-beginning 0)) menu)) Richard M. Stallman committed Jun 23, 1997 379 380 381 ;; Look for \frontmatter, \mainmatter, \backmatter, and \appendix. (goto-char (point-min)) Stefan Monnier committed Oct 01, 2000 382 383 (while (search-forward-regexp metasection-regexp nil t) (push (cons "--" (match-beginning 0)) menu)) Richard M. Stallman committed Jun 23, 1997 384 385 386 ;; Sort in increasing buffer position order. (sort menu (function (lambda (a b) (< (cdr a) (cdr b)))))))) Stefan Monnier committed Oct 01, 2000 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 ;;;; ;;;; Outline support ;;;; (defvar latex-outline-regexp (concat "\\\\" (regexp-opt (append latex-metasection-list (mapcar 'car latex-section-alist)) t))) (defun latex-outline-level () (if (looking-at latex-outline-regexp) (1+ (or (cdr (assoc (match-string 1) latex-section-alist)) -1)) 1000)) ;;;; ;;;; Font-Lock support ;;;; ;(defvar tex-font-lock-keywords ; ;; Regexps updated with help from Ulrik Dickow . ; '(("\\\\\$$begin\\|end\\|newcommand\$${\$$[a-zA-Z0-9\\*]+\$$}" ; 2 font-lock-function-name-face) ; ("\\\\\$$cite\\|label\\|pageref\\|ref\$${\$$[^} \t\n]+\$$}" ; 2 font-lock-constant-face) ; ;; It seems a bit dubious to use bold' and italic' faces since we might ; ;; not be able to display those fonts. ; ("{\\\\bf\$$[^}]+\$$}" 1 'bold keep) ; ("{\\\\\$$em\\|it\\|sl\$$\$$[^}]+\$$}" 2 'italic keep) ; ("\\\\\$$[a-zA-Z@]+\\|.\$$" . font-lock-keyword-face) ; ("^[ \t\n]*\\\\def[\\\\@]\$$\\w+\$$" 1 font-lock-function-name-face keep)) ; ;; Rewritten and extended for LaTeX2e by Ulrik Dickow . ; '(("\\\\\$$begin\\|end\\|newcommand\$${\$$[a-zA-Z0-9\\*]+\$$}" ; 2 font-lock-function-name-face) ; ("\\\\\$$cite\\|label\\|pageref\\|ref\$${\$$[^} \t\n]+\$$}" ; 2 font-lock-constant-face) ; ("^[ \t]*\\\\def\\\\\$$\\(\\w\\|@\$$+\\)" 1 font-lock-function-name-face) ; "\\\\\$$[a-zA-Z@]+\\|.\$$" ; ;; It seems a bit dubious to use bold' and italic' faces since we might ; ;; not be able to display those fonts. ; ;; LaTeX2e: \emph{This is emphasized}. ; ("\\\\emph{\$$[^}]+\$$}" 1 'italic keep) ; ;; LaTeX2e: \textbf{This is bold}, \textit{...}, \textsl{...} ; ("\\\\text\$$\\(bf\$$\\|it\\|sl\\){\$$[^}]+\$$}" ; 3 (if (match-beginning 2) 'bold 'italic) keep) ; ;; Old-style bf/em/it/sl. Stop at \\' and un-escaped &', for tables. ; ("\\\\\$$\\(bf\$$\\|em\\|it\\|sl\\)\\>\$$\\([^}&\$\\|\\\$^\$\$$+\\)" ; 3 (if (match-beginning 2) 'bold 'italic) keep)) ;; Rewritten with the help of Alexandra Bac . (defconst tex-font-lock-keywords-1 (eval-when-compile (let* (;; Names of commands whose arg should be fontified as heading, etc. (headings (regexp-opt '("title" "begin" "end" "chapter" "part" "section" "subsection" "subsubsection" "paragraph" "subparagraph" "subsubparagraph" Stefan Monnier committed Nov 08, 2002 444 445 446 "newcommand" "renewcommand" "providecommand" "newenvironment" "renewenvironment" "newtheorem" "renewtheorem") Stefan Monnier committed Oct 01, 2000 447 448 449 450 451 452 453 454 t)) (variables (regexp-opt '("newcounter" "newcounter*" "setcounter" "addtocounter" "setlength" "addtolength" "settowidth") t)) (includes (regexp-opt '("input" "include" "includeonly" "bibliography" "epsfig" "psfig" "epsf" "nofiles" "usepackage" Stefan Monnier committed May 25, 2001 455 "documentstyle" "documentclass" "verbatiminput" Stefan Monnier committed Oct 01, 2000 456 457 458 459 "includegraphics" "includegraphics*") t)) ;; Miscellany. (slash "\\\\") Stefan Monnier committed May 25, 2001 460 461 462 463 (opt " *\$$\$[^]]*\$ *\$$*") ;; This would allow highlighting \newcommand\CMD but requires ;; adapting subgroup numbers below. ;; (arg "\$$?:{\\(\\(?:[^{}\\]+\\|\\\\.\\|{[^}]*}\$$+\\)\\|\\\$a-z*]+\\)")) Stefan Monnier committed Nov 22, 2000 464 (arg "{\$$\\(?:[^{}\$+\\|\\\\.\\|{[^}]*}\$$+\\)")) Stefan Monnier committed Oct 01, 2000 465 (list Stefan Monnier committed Jul 14, 2002 466 467 468 469 470 471 ;; display $$math$$ ;; We only mark the match between $$and$$ because the $$delimiters ;; themselves have already been marked (along with ..) by syntactic ;; fontification. Also this is done at the very beginning so as to ;; interact with the other keywords in the same way as ... does. (list "\\\\\$$[^]+\$$\\\\" 1 'tex-math-face) Stefan Monnier committed Oct 01, 2000 472 473 ;; Heading args. (list (concat slash headings "\\*?" opt arg) Stefan Monnier committed Nov 14, 2000 474 475 476 477 478 479 480 481 482 483 ;; If ARG ends up matching too much (if the {} don't match, f.ex) ;; jit-lock will do funny things: when updating the buffer ;; the re-highlighting is only done locally so it will just ;; match the local line, but defer-contextually will ;; match more lines at a time, so ARG will end up matching ;; a lot more, which might suddenly include a comment ;; so you get things highlighted bold when you type them ;; but they get turned back to normal a little while later ;; because "there's already a face there". ;; Using keep' works around this un-intuitive behavior as well Stefan Monnier committed May 25, 2001 484 485 ;; as improves the behavior in the very rare case where you do ;; have a comment in ARG. Stefan Monnier committed Nov 14, 2000 486 3 'font-lock-function-name-face 'keep) Stefan Monnier committed Nov 08, 2002 487 488 (list (concat slash "\$$?:provide\\|\\(?:re\$$?new\\)command\\** *\$$\\\$A-Za-z@]+\$$") 1 'font-lock-function-name-face 'keep) Stefan Monnier committed Oct 01, 2000 489 ;; Variable args. Stefan Monnier committed May 25, 2001 490 (list (concat slash variables " *" arg) 2 'font-lock-variable-name-face) Stefan Monnier committed Oct 01, 2000 491 492 493 ;; Include args. (list (concat slash includes opt arg) 3 'font-lock-builtin-face) ;; Definitions. I think. Stefan Monnier committed May 25, 2001 494 '("^[ \t]*\\\\def *\\\\\$$\\(\\w\\|@\$$+\\)" Stefan Monnier committed Nov 14, 2000 495 1 font-lock-function-name-face)))) Stefan Monnier committed Oct 01, 2000 496 497 498 499 500 501 502 "Subdued expressions to highlight in TeX modes.") (defconst tex-font-lock-keywords-2 (append tex-font-lock-keywords-1 (eval-when-compile (let* (;; ;; Names of commands whose arg should be fontified with fonts. Stefan Monnier committed Nov 22, 2000 503 (bold (regexp-opt '("textbf" "textsc" "textup" Stefan Monnier committed Oct 01, 2000 504 "boldsymbol" "pmb") t)) Stefan Monnier committed Nov 22, 2000 505 (italic (regexp-opt '("textit" "textsl" "emph") t)) Stefan Monnier committed Apr 14, 2003 506 507 ;; FIXME: unimplemented yet. ;; (type (regexp-opt '("texttt" "textmd" "textrm" "textsf") t)) Stefan Monnier committed Oct 01, 2000 508 509 510 511 ;; ;; Names of commands whose arg should be fontified as a citation. (citations (regexp-opt '("label" "ref" "pageref" "vref" "eqref" Stefan Monnier committed May 25, 2001 512 "cite" "nocite" "index" "glossary" "bibitem" Stefan Monnier committed Nov 22, 2000 513 514 515 ;; These are text, rather than citations. ;; "caption" "footnote" "footnotemark" "footnotetext" ) Stefan Monnier committed Oct 01, 2000 516 517 518 t)) ;; ;; Names of commands that should be fontified. Andreas Schwab committed Feb 15, 2002 519 520 521 522 523 524 (specials-1 (regexp-opt '("\\" "\\*") t)) ;; "-" (specials-2 (regexp-opt '("linebreak" "nolinebreak" "pagebreak" "nopagebreak" "newline" "newpage" "clearpage" "cleardoublepage" "displaybreak" "allowdisplaybreaks" "enlargethispage") t)) Stefan Monnier committed Oct 01, 2000 525 526 527 528 (general "\$$[a-zA-Z@]+\\**\\|[^ \t\n]\$$") ;; ;; Miscellany. (slash "\\\\") Stefan Monnier committed May 25, 2001 529 (opt " *\$$\\[[^]]*\$ *\$$*") Stefan Monnier committed Nov 08, 2002 530 (args "\$$\\(?:[^{}&\\]+\\|\\\\.\\|{[^}]*}\$$+\\)") Stefan Monnier committed Nov 22, 2000 531 (arg "{\$$\\(?:[^{}\\]+\\|\\\\.\\|{[^}]*}\$$+\\)")) Stefan Monnier committed Oct 01, 2000 532 533 534 535 536 (list ;; ;; Citation args. (list (concat slash citations opt arg) 3 'font-lock-constant-face) ;; Stefan Monnier committed May 25, 2001 537 ;; Text between quotes ''. Stefan Monnier committed Jul 14, 2002 538 (cons (concat (regexp-opt ("" "\"<" "\"" "<<" "«") t) Sam Steingold committed Jul 01, 2002 539 "[^'\">{]+" ;a bit pessimistic Stefan Monnier committed Jul 14, 2002 540 (regexp-opt ("''" "\">" "\"'" ">>" "»") t)) Stefan Monnier committed May 25, 2001 541 542 'font-lock-string-face) ;; Stefan Monnier committed Oct 01, 2000 543 ;; Command names, special and general. Andreas Schwab committed Feb 15, 2002 544 545 546 (cons (concat slash specials-1) 'font-lock-warning-face) (list (concat "\$$" slash specials-2 "\$$\$$[^a-zA-Z@]\\|\\'\$$") 1 'font-lock-warning-face) Stefan Monnier committed Oct 01, 2000 547 548 549 550 (concat slash general) ;; ;; Font environments. It seems a bit dubious to use bold' etc. faces ;; since we might not be able to display those fonts. Stefan Monnier committed May 25, 2001 551 552 (list (concat slash bold " *" arg) 2 '(quote bold) 'append) (list (concat slash italic " *" arg) 2 '(quote italic) 'append) Stefan Monnier committed Nov 22, 2000 553 ;; (list (concat slash type arg) 2 '(quote bold-italic) 'append) Stefan Monnier committed Oct 01, 2000 554 555 ;; ;; Old-style bf/em/it/sl. Stop at \\' and un-escaped &', for tables. Stefan Monnier committed Nov 08, 2002 556 557 558 559 560 (list (concat "\\\\\$$em\\|it\\|sl\$$\\>" args) 2 '(quote italic) 'append) ;; This is separate from the previous one because of cases like ;; {\em foo {\bf bar} bla} where both match. (list (concat "\\\\bf\\>" args) 1 '(quote bold) 'append))))) Stefan Monnier committed Oct 01, 2000 561 562 "Gaudy expressions to highlight in TeX modes.") Stefan Monnier committed Nov 08, 2002 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 (defun tex-font-lock-suscript (pos) (unless (or (memq (get-text-property pos 'face) '(font-lock-constant-face font-lock-builtin-face font-lock-comment-face tex-verbatim-face)) ;; Check for backslash quoting (let ((odd nil) (pos pos)) (while (eq (char-before pos) ?\\) (setq pos (1- pos) odd (not odd))) odd)) (if (eq (char-after pos) ?_) '(face subscript display (raise -0.3)) '(face superscript display (raise +0.3))))) (defconst tex-font-lock-keywords-3 (append tex-font-lock-keywords-2 (eval-when-compile (let ((general "\$$[a-zA-Z@]+\\|[^ \t\n]\$$") (slash "\\\\") ;; This is not the same regexp as before: it has a +' removed. ;; The + makes the matching faster in the above cases (where we can ;; exit as soon as the match fails) but would make this matching ;; degenerate to nasty complexity (because we try to match the ;; closing brace, which forces trying all matching combinations). (arg "{\$$?:[^{}\\]\\|\\\\.\\|{[^}]*}\$$*")) ((,(concat "[_^] *\$$[^\n\\{}]\\|" slash general "\\|" arg "}\$$") (1 (tex-font-lock-suscript (match-beginning 0)) append)))))) "Experimental expressions to highlight in TeX modes.") Stefan Monnier committed Oct 01, 2000 593 594 (defvar tex-font-lock-keywords tex-font-lock-keywords-1 "Default expressions to highlight in TeX modes.") Karl Heuer committed Jul 28, 1995 595 Stefan Monnier committed Nov 08, 2002 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 (defvar tex-verbatim-environments '("verbatim" "verbatim*")) (defvar tex-font-lock-syntactic-keywords (let ((verbs (regexp-opt tex-verbatim-environments t))) ((,(concat "^\\\\begin *{" verbs "}.*\$$\n\$$") 2 "|") (,(concat "^\\\\end *{" verbs "}\$$.?\$$") 2 (unless (<= (match-beginning 0) (point-min)) (put-text-property (1- (match-beginning 0)) (match-beginning 0) 'syntax-table (string-to-syntax "|")) "<")) ;; ("^\$$\\\\\$$begin *{comment}" 1 "< b") ;; ("^\\\\end *{comment}.*\$$\n\$$" 1 "> b") ("\\\\verb\\**\$$[^a-z@*]\$$" 1 "\"")))) (defun tex-font-lock-unfontify-region (beg end) (font-lock-default-unfontify-region beg end) (while (< beg end) (let ((next (next-single-property-change beg 'display nil end)) (prop (get-text-property beg 'display))) (if (and (eq (car-safe prop) 'raise) (member (car-safe (cdr prop)) '(-0.3 +0.3)) (null (cddr prop))) (put-text-property beg next 'display nil)) (setq beg next)))) (defface superscript '((t :height 0.8)) ;; :raise 0.3 "Face used for superscripts.") (defface subscript '((t :height 0.8)) ;; :raise -0.3 "Face used for subscripts.") Stefan Monnier committed Nov 14, 2000 628 629 630 631 632 (defface tex-math-face '((t :inherit font-lock-string-face)) "Face used to highlight TeX math expressions.") (defvar tex-math-face 'tex-math-face) Stefan Monnier committed Nov 08, 2002 633 634 635 636 637 (defface tex-verbatim-face ;; '((t :inherit font-lock-string-face)) '((t :family "courier")) "Face used to highlight TeX verbatim environments.") (defvar tex-verbatim-face 'tex-verbatim-face) Stefan Monnier committed Nov 14, 2000 638 639 640 ;; Use string syntax but math face for .... (defun tex-font-lock-syntactic-face-function (state) Stefan Monnier committed Nov 08, 2002 641 642 643 644 645 646 647 648 649 650 651 652 653 654 (let ((char (nth 3 state))) (cond ((not char) font-lock-comment-face) ((eq char ?) tex-math-face) (t (when (char-valid-p char) ;; This is a \verb?...? construct. Let's find the end and mark it. (save-excursion (skip-chars-forward (string ?^ char)) ;; Use end' ? (when (eq (char-syntax (preceding-char)) ?/) (put-text-property (1- (point)) (point) 'syntax-table '(1))) (unless (eobp) (put-text-property (point) (1+ (point)) 'syntax-table '(7))))) tex-verbatim-face)))) Stefan Monnier committed Nov 14, 2000 655 Stefan Monnier committed Oct 01, 2000 656 root committed Aug 28, 1990 657 (defun tex-define-common-keys (keymap) Richard M. Stallman committed May 02, 1994 658 "Define the keys that we want defined both in TeX mode and in the TeX shell." root committed Aug 28, 1990 659 660 661 662 663 (define-key keymap "\C-c\C-k" 'tex-kill-job) (define-key keymap "\C-c\C-l" 'tex-recenter-output-buffer) (define-key keymap "\C-c\C-q" 'tex-show-print-queue) (define-key keymap "\C-c\C-p" 'tex-print) (define-key keymap "\C-c\C-v" 'tex-view) Richard M. Stallman committed Oct 27, 1993 664 665 666 (define-key keymap [menu-bar tex] (cons "TeX" (make-sparse-keymap "TeX"))) Stefan Monnier committed Nov 14, 2000 667 668 (define-key keymap [menu-bar tex tex-kill-job] '(menu-item "Tex Kill" tex-kill-job :enable (tex-shell-running))) Richard M. Stallman committed Nov 06, 1993 669 (define-key keymap [menu-bar tex tex-recenter-output-buffer] Stefan Monnier committed Nov 14, 2000 670 671 '(menu-item "Tex Recenter" tex-recenter-output-buffer :enable (get-buffer "*tex-shell*"))) Richard M. Stallman committed Oct 27, 1993 672 673 (define-key keymap [menu-bar tex tex-show-print-queue] '("Show Print Queue" . tex-show-print-queue)) Richard M. Stallman committed Nov 06, 1993 674 (define-key keymap [menu-bar tex tex-alt-print] Stefan Monnier committed Nov 14, 2000 675 676 677 678 679 680 681 682 683 '(menu-item "Tex Print (alt printer)" tex-alt-print :enable (stringp tex-print-file))) (define-key keymap [menu-bar tex tex-print] '(menu-item "Tex Print" tex-print :enable (stringp tex-print-file))) (define-key keymap [menu-bar tex tex-view] '(menu-item "Tex View" tex-view :enable (stringp tex-print-file)))) (defvar tex-mode-map (let ((map (make-sparse-keymap))) Stefan Monnier committed May 25, 2001 684 (set-keymap-parent map text-mode-map) Stefan Monnier committed Nov 14, 2000 685 686 687 688 689 690 691 692 693 694 695 696 697 (tex-define-common-keys map) (define-key map "\"" 'tex-insert-quote) (define-key map "(" 'skeleton-pair-insert-maybe) (define-key map "{" 'skeleton-pair-insert-maybe) (define-key map "[" 'skeleton-pair-insert-maybe) (define-key map "" 'skeleton-pair-insert-maybe) (define-key map "\n" 'tex-terminate-paragraph) (define-key map "\M-\r" 'latex-insert-item) (define-key map "\C-c}" 'up-list) (define-key map "\C-c{" 'tex-insert-braces) (define-key map "\C-c\C-r" 'tex-region) (define-key map "\C-c\C-b" 'tex-buffer) (define-key map "\C-c\C-f" 'tex-file) Stefan Monnier committed May 15, 2003 698 (define-key map "\C-c\C-c" 'tex-compile) Stefan Monnier committed Nov 14, 2000 699 (define-key map "\C-c\C-i" 'tex-bibtex-file) Stefan Monnier committed Nov 08, 2002 700 701 (define-key map "\C-c\C-o" 'latex-insert-block) (define-key map "\C-c\C-e" 'latex-close-block) Stefan Monnier committed Nov 14, 2000 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 (define-key map "\C-c\C-u" 'tex-goto-last-unclosed-latex-block) (define-key map "\C-c\C-m" 'tex-feed-input) (define-key map [(control return)] 'tex-feed-input) (define-key map [menu-bar tex tex-bibtex-file] '("BibTeX File" . tex-bibtex-file)) (define-key map [menu-bar tex tex-validate-region] '(menu-item "Validate Region" tex-validate-region :enable mark-active)) (define-key map [menu-bar tex tex-validate-buffer] '("Validate Buffer" . tex-validate-buffer)) (define-key map [menu-bar tex tex-region] '(menu-item "TeX Region" tex-region :enable mark-active)) (define-key map [menu-bar tex tex-buffer] '("TeX Buffer" . tex-buffer)) (define-key map [menu-bar tex tex-file] '("TeX File" . tex-file)) map) Stefan Monnier committed May 25, 2001 717 718 719 720 721 "Keymap shared by TeX modes.") (defvar latex-mode-map (let ((map (make-sparse-keymap))) (set-keymap-parent map tex-mode-map) Stefan Monnier committed Apr 14, 2003 722 (define-key map "\C-c\C-s" 'latex-split-block) Stefan Monnier committed May 25, 2001 723 724 725 726 727 728 729 730 map) "Keymap for latex-mode'. See also tex-mode-map'.") (defvar plain-tex-mode-map (let ((map (make-sparse-keymap))) (set-keymap-parent map tex-mode-map) map) "Keymap for plain-tex-mode'. See also tex-mode-map'.") Richard M. Stallman committed Nov 06, 1993 731 Stefan Monnier committed Sep 29, 2000 732 733 734 735 736 (defvar tex-shell-map (let ((m (make-sparse-keymap))) (set-keymap-parent m shell-mode-map) (tex-define-common-keys m) m) Richard M. Stallman committed May 02, 1994 737 "Keymap for the TeX shell. Edward M. Reingold committed Dec 21, 1994 738 Inherits shell-mode-map' with a few additions.") root committed Aug 28, 1990 739 Richard M. Stallman committed Jan 26, 1996 740 741 742 743 744 745 746 747 748 749 750 751 752 (defvar tex-face-alist '((bold . "{\\bf ") (italic . "{\\it ") (bold-italic . "{\\bi ") ; hypothetical (underline . "\\underline{") (default . "{\\rm ")) "Alist of face and TeX font name for facemenu.") (defvar tex-latex-face-alist ((italic . "{\\em ") ,@tex-face-alist) "Alist of face and LaTeX font name for facemenu.") Stefan Monnier committed May 25, 2001 753 754 ;; This would be a lot simpler if we just used a regexp search, ;; but then it would be too slow. Stefan Monnier committed Apr 14, 2003 755 (defun tex-guess-mode () Stefan Monnier committed Sep 29, 2000 756 (let ((mode tex-default-mode) slash comment) root committed Aug 28, 1990 757 758 759 760 761 762 763 (save-excursion (goto-char (point-min)) (while (and (setq slash (search-forward "\\" nil t)) (setq comment (let ((search-end (point))) (save-excursion (beginning-of-line) (search-forward "%" search-end t)))))) Stefan Monnier committed Nov 14, 2000 764 765 766 767 768 769 (when (and slash (not comment)) (setq mode (if (looking-at (eval-when-compile (concat (regexp-opt '("documentstyle" "documentclass" Stefan Monnier committed Jul 14, 2002 770 "begin" "subsection" "section" Stefan Monnier committed May 10, 2003 771 772 "part" "chapter" "newcommand" "renewcommand") 'words) Stefan Monnier committed Nov 14, 2000 773 774 775 776 777 778 "\\|NeedsTeXFormat{LaTeX"))) (if (looking-at "document\$$style\\|class\$$\$$\$.*\$\$$?{slides}") 'slitex-mode 'latex-mode) 'plain-tex-mode)))) Stefan Monnier committed Sep 29, 2000 779 (funcall mode))) Eric S. Raymond committed Apr 28, 1993 780 Stefan Monnier committed Apr 14, 2003 781 782 783 784 785 786 ;; tex-mode' plays two roles: it's the parent of several sub-modes ;; but it's also the function that chooses between those submodes. ;; To tell the difference between those two cases where the function ;; might be called, we check delay-mode-hooks'. (define-derived-mode tex-mode text-mode "generic-TeX" (tex-common-initialization)) Stefan Monnier committed May 10, 2003 787 788 789 790 791 792 793 794 795 ;; We now move the function and define it again. This gives a warning ;; in the byte-compiler :-( but it's difficult to avoid because ;; define-derived-mode' will necessarily define the function once ;; and we need to define it a second time for autoload' to get the ;; proper docstring. (defalias 'tex-mode-internal (symbol-function 'tex-mode)) ;;;###autoload (defun tex-mode () "Major mode for editing files of input for TeX, LaTeX, or SliTeX. Stefan Monnier committed Apr 14, 2003 796 797 798 799 800 Tries to determine (by looking at the beginning of the file) whether this file is for plain TeX, LaTeX, or SliTeX and calls plain-tex-mode', latex-mode', or slitex-mode', respectively. If it cannot be determined, such as if there are no commands in the file, the value of tex-default-mode' says which mode to use." Stefan Monnier committed May 10, 2003 801 802 803 804 805 (interactive) (if delay-mode-hooks ;; We're called from one of the children already. (tex-mode-internal) (tex-guess-mode))) Stefan Monnier committed Apr 14, 2003 806 Jim Blandy committed May 13, 1991 807 ;;;###autoload Eric S. Raymond committed Apr 23, 1993 808 (defalias 'TeX-mode 'tex-mode) Jim Blandy committed May 13, 1991 809 ;;;###autoload Richard M. Stallman committed Feb 16, 1999 810 811 (defalias 'plain-TeX-mode 'plain-tex-mode) ;;;###autoload Eric S. Raymond committed Apr 23, 1993 812 (defalias 'LaTeX-mode 'latex-mode) root committed Aug 28, 1990 813 Roland McGrath committed May 09, 1991 814 ;;;###autoload Stefan Monnier committed Apr 14, 2003 815 (define-derived-mode plain-tex-mode tex-mode "TeX" root committed Aug 28, 1990 816 817 818 819 820 821 822 823 824 825 826 827 828 "Major mode for editing files of input for plain TeX. Makes and } display the characters they match. Makes \" insert when it seems to be the beginning of a quotation, and '' when it appears to be the end; it inserts \" only after a \\. Use \\[tex-region] to run TeX on the current region, plus a \"header\" copied from the top of the file (containing macro definitions, etc.), running TeX under a special subshell. \\[tex-buffer] does the whole buffer. \\[tex-file] saves the buffer and then processes the file. \\[tex-print] prints the .dvi file made by any of these. \\[tex-view] previews the .dvi file made by any of these. \\[tex-bibtex-file] runs bibtex on the file of the current buffer. Markus Rost committed Jan 29, 1999 829 Use \\[tex-validate-buffer] to check buffer for paragraphs containing root committed Aug 28, 1990 830 831 832 mismatched 's or braces. Special commands: Stefan Monnier committed May 25, 2001 833 \\{plain-tex-mode-map} root committed Aug 28, 1990 834 835 836 837 838 839 840 841 842 Mode variables: tex-run-command Command string used by \\[tex-region] or \\[tex-buffer]. tex-directory Directory in which to create temporary files for TeX jobs run by \\[tex-region] or \\[tex-buffer]. tex-dvi-print-command Command string used by \\[tex-print] to print a .dvi file. Richard M. Stallman committed Jul 04, 1992 843 844 845 tex-alt-dvi-print-command Alternative command string used by \\[tex-print] (when given a prefix argument) to print a .dvi file. root committed Aug 28, 1990 846 847 848 849 850 851 tex-dvi-view-command Command string used by \\[tex-view] to preview a .dvi file. tex-show-queue-command Command string used by \\[tex-show-print-queue] to show the print queue that \\[tex-print] put your job on. Richard M. Stallman committed May 02, 1994 852 853 854 Entering Plain-tex mode runs the hook text-mode-hook', then the hook tex-mode-hook', and finally the hook plain-tex-mode-hook'. When the special subshell is initiated, the hook tex-shell-hook' is run." Stefan Monnier committed Apr 14, 2003 855 856 857 858 (set (make-local-variable 'tex-command) tex-run-command) (set (make-local-variable 'tex-start-of-header) "%\\*\\*start of header") (set (make-local-variable 'tex-end-of-header) "%\\*\\*end of header") (set (make-local-variable 'tex-trailer) "\\bye\n")) root committed Aug 28, 1990 859 Roland McGrath committed May 09, 1991 860 ;;;###autoload Stefan Monnier committed Apr 14, 2003 861 (define-derived-mode latex-mode tex-mode "LaTeX" root committed Aug 28, 1990 862 863 864 865 866 867 868 869 870 871 872 873 874 "Major mode for editing files of input for LaTeX. Makes and } display the characters they match. Makes \" insert when it seems to be the beginning of a quotation, and '' when it appears to be the end; it inserts \" only after a \\. Use \\[tex-region] to run LaTeX on the current region, plus the preamble copied from the top of the file (containing \\documentstyle, etc.), running LaTeX under a special subshell. \\[tex-buffer] does the whole buffer. \\[tex-file] saves the buffer and then processes the file. \\[tex-print] prints the .dvi file made by any of these. \\[tex-view] previews the .dvi file made by any of these. \\[tex-bibtex-file] runs bibtex on the file of the current buffer. Markus Rost committed Jan 29, 1999 875 Use \\[tex-validate-buffer] to check buffer for paragraphs containing root committed Aug 28, 1990 876 877 878 mismatched 's or braces. Special commands: Stefan Monnier committed May 25, 2001 879 \\{latex-mode-map} root committed Aug 28, 1990 880 881 882 883 884 885 886 887 888 Mode variables: latex-run-command Command string used by \\[tex-region] or \\[tex-buffer]. tex-directory Directory in which to create temporary files for LaTeX jobs run by \\[tex-region] or \\[tex-buffer]. tex-dvi-print-command Command string used by \\[tex-print] to print a .dvi file. Richard M. Stallman committed Jul 04, 1992 889 890 891 tex-alt-dvi-print-command Alternative command string used by \\[tex-print] (when given a prefix argument) to print a .dvi file. root committed Aug 28, 1990 892 893 894 895 896 897 tex-dvi-view-command Command string used by \\[tex-view] to preview a .dvi file. tex-show-queue-command Command string used by \\[tex-show-print-queue] to show the print queue that \\[tex-print] put your job on. Richard M. Stallman committed Feb 07, 1999 898 Entering Latex mode runs the hook text-mode-hook', then Richard M. Stallman committed May 02, 1994 899 900 tex-mode-hook', and finally latex-mode-hook'. When the special subshell is initiated, tex-shell-hook' is run." Stefan Monnier committed Apr 14, 2003 901 902 903 904 (set (make-local-variable 'tex-command) latex-run-command) (set (make-local-variable 'tex-start-of-header) "\\\\document\$$style\\|class\$$") (set (make-local-variable 'tex-end-of-header) "\\\\begin\\s-*{document}") Stefan Monnier committed Nov 27, 2003 905 (set (make-local-variable 'tex-trailer) "\\end{document}\n") Richard M. Stallman committed Jan 03, 1994 906 907 908 ;; A line containing just$$ is treated as a paragraph separator. ;; A line starting with $$starts a paragraph, ;; but does not separate paragraphs if it has more stuff on it. Stefan Monnier committed Sep 29, 2000 909 (setq paragraph-start Stefan Monnier committed Apr 14, 2003 910 (concat "[ \t]*\$$\\\\\\|" Stefan Monnier committed Sep 29, 2000 911 912 913 914 915 916 917 918 "\\\\[][]\\|" "\\\\" (regexp-opt (append (mapcar 'car latex-section-alist) '("begin" "label" "end" "item" "bibitem" "newline" "noindent" "newpage" "footnote" "marginpar" "parbox" "caption")) t) "\\>\\|\\\\[a-z]*" (regexp-opt '("space" "skip" "page") t) Stefan Monnier committed Nov 10, 2000 919 "\\>\$$")) Stefan Monnier committed Sep 29, 2000 920 (setq paragraph-separate Stefan Monnier committed Nov 10, 2000 921 (concat "[\f%]\\|[ \t]*\$$\\|" Stefan Monnier committed Sep 29, 2000 922 923 924 925 926 927 928 "\\\\[][]\\|" "\\\\" (regexp-opt (append (mapcar 'car latex-section-alist) '("begin" "label" "end" )) t) "\\>\\|\\\\\\(" (regexp-opt '("item" "bibitem" "newline" "noindent" "newpage" "footnote" "marginpar" "parbox" "caption")) Stefan Monnier committed Nov 10, 2000 929 930 "\\|\\\\\\|[a-z]*\\(space\\|skip\\|page[a-z]*\$$" "\\>\\)[ \t]*\$$\\|%\$$\\)")) Stefan Monnier committed Sep 29, 2000 931 932 933 (set (make-local-variable 'imenu-create-index-function) 'latex-imenu-create-index) (set (make-local-variable 'tex-face-alist) tex-latex-face-alist) Stefan Monnier committed Jul 14, 2002 934 (add-hook 'fill-nobreak-predicate 'latex-fill-nobreak-predicate nil t) Stefan Monnier committed Oct 15, 2000 935 (set (make-local-variable 'indent-line-function) 'latex-indent) Stefan Monnier committed Nov 10, 2000 936 (set (make-local-variable 'fill-indent-according-to-mode) t) Stefan Monnier committed Sep 29, 2000 937 938 (set (make-local-variable 'outline-regexp) latex-outline-regexp) (set (make-local-variable 'outline-level) 'latex-outline-level) Stefan Monnier committed Oct 01, 2000 939 (set (make-local-variable 'forward-sexp-function) 'latex-forward-sexp) Stefan Monnier committed Apr 14, 2003 940 (set (make-local-variable 'skeleton-end-hook) nil)) root committed Aug 28, 1990 941 Brian Fox committed Sep 21, 1993 942 ;;;###autoload Stefan Monnier committed Sep 29, 2000 943 (define-derived-mode slitex-mode latex-mode "SliTeX" root committed Aug 28, 1990 944 945 946 947 948 949 950 951 952 953 954 955 956 "Major mode for editing files of input for SliTeX. Makes and } display the characters they match. Makes \" insert when it seems to be the beginning of a quotation, and '' when it appears to be the end; it inserts \" only after a \\. Use \\[tex-region] to run SliTeX on the current region, plus the preamble copied from the top of the file (containing \\documentstyle, etc.), running SliTeX under a special subshell. \\[tex-buffer] does the whole buffer. \\[tex-file] saves the buffer and then processes the file. \\[tex-print] prints the .dvi file made by any of these. \\[tex-view] previews the .dvi file made by any of these. \\[tex-bibtex-file] runs bibtex on the file of the current buffer. Markus Rost committed Jan 29, 1999 957 Use \\[tex-validate-buffer] to check buffer for paragraphs containing root committed Aug 28, 1990 958 959 960 mismatched 's or braces. Special commands: Stefan Monnier committed May 25, 2001 961 \\{slitex-mode-map} root committed Aug 28, 1990 962 963 964 965 966 967 968 969 970 Mode variables: slitex-run-command Command string used by \\[tex-region] or \\[tex-buffer]. tex-directory Directory in which to create temporary files for SliTeX jobs run by \\[tex-region] or \\[tex-buffer]. tex-dvi-print-command Command string used by \\[tex-print] to print a .dvi file. Richard M. Stallman committed Jul 04, 1992 971 972 973 tex-alt-dvi-print-command Alternative command string used by \\[tex-print] (when given a prefix argument) to print a .dvi file. root committed Aug 28, 1990 974 975 976 977 978 979 tex-dvi-view-command Command string used by \\[tex-view] to preview a .dvi file. tex-show-queue-command Command string used by \\[tex-show-print-queue] to show the print queue that \\[tex-print] put your job on. Richard M. Stallman committed May 02, 1994 980 981 982 983 Entering SliTeX mode runs the hook text-mode-hook', then the hook tex-mode-hook', then the hook latex-mode-hook', and finally the hook slitex-mode-hook'. When the special subshell is initiated, the hook tex-shell-hook' is run." root committed Aug 28, 1990 984 (setq tex-command slitex-run-command) Stefan Monnier committed Sep 29, 2000 985 (setq tex-start-of-header "\\\\documentstyle{slides}\\|\\\\documentclass{slides}")) root committed Aug 28, 1990 986 987 (defun tex-common-initialization () Richard M. Stallman committed Jul 04, 1997 988 ;; Regexp isearch should accept newline and formfeed as whitespace. Stefan Monnier committed Sep 29, 2000 989 (set (make-local-variable 'search-whitespace-regexp) "[ \t\r\n\f]+") Richard M. Stallman committed Dec 23, 1993 990 ;; A line containing just$$ is treated as a paragraph separator. Stefan Monnier committed Sep 29, 2000 991 992 (set (make-local-variable 'paragraph-start) "[ \t]*$\\|[\f\\\\%]\\|[ \t]*\\$\\$") Richard M. Stallman committed Jan 03, 1994 993 994 ;; A line starting with$$starts a paragraph, ;; but does not separate paragraphs if it has more stuff on it. Stefan Monnier committed Sep 29, 2000 995 996 997 998 999 (set (make-local-variable 'paragraph-separate) "[ \t]*$\\|[\f\\\\%]\\|[ \t]*\\$\\$[ \t]*$") (set (make-local-variable 'comment-start) "%") (set (make-local-variable 'comment-add) 1) (set (make-local-variable 'comment-start-skip) Stefan Monnier committed Nov 02, 2002 1000 "\$$\\(^\\|[^\\\n]\$$\$$\\\\\\\\\$$*\\)\$$%+ *\$$") Stefan Monnier committed Sep 29, 2000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 (set (make-local-variable 'parse-sexp-ignore-comments) t) (set (make-local-variable 'compare-windows-whitespace) 'tex-categorize-whitespace) (set (make-local-variable 'facemenu-add-face-function) (lambda (face end) (let ((face-text (cdr (assq face tex-face-alist)))) (if face-text face-text (error "Face %s not configured for %s mode" face mode-name))))) (set (make-local-variable 'facemenu-end-add-face) "}") (set (make-local-variable 'facemenu-remove-face-function) t) (set (make-local-variable 'font-lock-defaults) Stefan Monnier committed Nov 08, 2002 1013 1014 '((tex-font-lock-keywords tex-font-lock-keywords-1 tex-font-lock-keywords-2 tex-font-lock-keywords-3) Stefan Monnier committed Sep 29, 2000 1015 1016 nil nil ((?$ . "\"")) nil ;; Who ever uses that anyway ??? Stefan Monnier committed Nov 14, 2000 1017 1018 (font-lock-mark-block-function . mark-paragraph) (font-lock-syntactic-face-function Stefan Monnier committed Nov 08, 2002 1019 1020 1021 1022 1023 1024 . tex-font-lock-syntactic-face-function) (font-lock-unfontify-region-function . tex-font-lock-unfontify-region) (font-lock-syntactic-keywords . tex-font-lock-syntactic-keywords) (parse-sexp-lookup-properties . t))) Stefan Monnier committed Jul 14, 2002 1025 1026 ;; TABs in verbatim environments don't do what you think. (set (make-local-variable 'indent-tabs-mode) nil) Stefan Monnier committed Nov 08, 2002 1027 ;; Other vars that should be buffer-local. root committed Aug 28, 1990 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 (make-local-variable 'tex-command) (make-local-variable 'tex-start-of-header) (make-local-variable 'tex-end-of-header) (make-local-variable 'tex-trailer)) (defun tex-categorize-whitespace (backward-limit) ;; compare-windows-whitespace is set to this. ;; This is basically a finite-state machine. ;; Returns a symbol telling how TeX would treat ;; the whitespace we are looking at: null, space, or par. (let ((category 'null) (not-finished t)) (skip-chars-backward " \t\n\f" backward-limit) (while not-finished (cond ((looking-at "[ \t]+") (goto-char (match-end 0)) Richard M. Stallman committed Jul 24, 1997 1044 (if (eq category 'null) root committed Aug 28, 1990 1045 1046 (setq category 'space))) ((looking-at "\n") Richard M. Stallman committed Jul 24, 1997 1047 (cond ((eq category 'newline) root committed Aug 28, 1990 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 (setq category 'par) (setq not-finished nil)) (t (setq category 'newline) ;a strictly internal state (goto-char (match-end 0))))) ((looking-at "\f+") (setq category 'par) (setq not-finished nil)) (t (setq not-finished nil)))) (skip-chars-forward " \t\n\f") | {"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": 2, "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": 0.9653778076171875, "perplexity": 21560.09846838856}, "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/1674764494936.89/warc/CC-MAIN-20230127033656-20230127063656-00633.warc.gz"} |
http://mathhelpforum.com/geometry/55551-geometry-circle.html | # Math Help - Geometry of a Circle
1. ## Geometry of a Circle
1) ABCDEF is a hexagon inscribed in a circle. What is x + y + z?
2) In this figure, find the value of angle ABC + angle ADC and figure out if ABCD a cyclic quadrilateral. Therefore, find angle BDC
3a) In this figure, AOB is the diameter. Prove that points OAMP are concylic.
3b) Find two angles equal to angle OPA.
Thanks for any help.
2. Hello, BG5965!
1) $ABCD{E}F$ is a hexagon inscribed in a circle. .What is $x + y + z$ ?
An inscribed angle is measured by one-half its intercepted arc.
We have: . $\begin{array}{ccc}x &=& \frac{1}{2}(BC + CD + DE + EF) \\ \\[-4mm]
y &=& \frac{1}{2}(AB + DE + EF + FA) \\ \\[-4mm]
z &=& \frac{1}{2}(FA + AB + BC + CD) \end{array}$
Add: . $x+y+z\;=\;\tfrac{1}{2}(2AB + 2BC + 2CD + 2DE + 2EF + 2FA)$
. . . . $x+y+z \;= \;AB + BC + CD + DE + EF + FA$
. . . . $x+y+z \;=\;360^o$
3. Hello again, BG5965!
I have 3(b) . . .
3a) In this figure, $AOB$ is the diameter.
Prove that points OAMP are concylic.
3b) Find two angles equal to $\angle OPA.$
I assume that $O$ is the center of the circle.
$OA = OP$ . . . both are radii.
Hence, $\Delta AOP$ is isosceles.
. . Therefore: . $\angle PAO \:=\:\angle OPA$
$\angle APB = 90^o$ . . . It is inscribed in a semicircle.
Right triangles $APB$ and $MOB$ share angle $PBA$
Then: . $\Delta APB \sim \Delta MOB$
. . Therefore: . $\angle BMO \:=\:\angle PAB \:=\:\angle OPA$ | {"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": 18, "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": 0.887427806854248, "perplexity": 1365.9293730739848}, "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/1438042990112.50/warc/CC-MAIN-20150728002310-00009-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://labs.tib.eu/arxiv/?author=M.C.S.%20Williams | • ### The Extreme Energy Events HECR array: status and perspectives(1703.06358)
March 18, 2017 astro-ph.IM, astro-ph.HE
The Extreme Energy Events Project is a synchronous sparse array of 52 tracking detectors for studying High Energy Cosmic Rays (HECR) and Cosmic Rays-related phenomena. The observatory is also meant to address Long Distance Correlation (LDC) phenomena: the network is deployed over a broad area covering 10 degrees in latitude and 11 in longitude. An overview of a set of preliminary results is given, extending from the study of local muon flux dependance on solar activity to the investigation of the upward-going component of muon flux traversing the EEE stations; from the search for anisotropies at the sub-TeV scale to the hints for observations of km-scale Extensive Air Shower (EAS).
• ### INFN What Next: Ultra-relativistic Heavy-Ion Collisions(1602.04120)
Feb. 12, 2016 nucl-ex, nucl-th
This document was prepared by the community that is active in Italy, within INFN (Istituto Nazionale di Fisica Nucleare), in the field of ultra-relativistic heavy-ion collisions. The experimental study of the phase diagram of strongly-interacting matter and of the Quark-Gluon Plasma (QGP) deconfined state will proceed, in the next 10-15 years, along two directions: the high-energy regime at RHIC and at the LHC, and the low-energy regime at FAIR, NICA, SPS and RHIC. The Italian community is strongly involved in the present and future programme of the ALICE experiment, the upgrade of which will open, in the 2020s, a new phase of high-precision characterisation of the QGP properties at the LHC. As a complement of this main activity, there is a growing interest in a possible future experiment at the SPS, which would target the search for the onset of deconfinement using dimuon measurements. On a longer timescale, the community looks with interest at the ongoing studies and discussions on a possible fixed-target programme using the LHC ion beams and on the Future Circular Collider.
• The ALICE Collaboration has measured inclusive J/psi production in pp collisions at a center of mass energy sqrt(s)=2.76 TeV at the LHC. The results presented in this Letter refer to the rapidity ranges |y|<0.9 and 2.5<y<4 and have been obtained by measuring the electron and muon pair decay channels, respectively. The integrated luminosities for the two channels are L^e_int=1.1 nb^-1 and L^mu_int=19.9 nb^-1, and the corresponding signal statistics are N_J/psi^e+e-=59 +/- 14 and N_J/psi^mu+mu-=1364 +/- 53. We present dsigma_J/psi/dy for the two rapidity regions under study and, for the forward-y range, d^2sigma_J/psi/dydp_t in the transverse momentum domain 0<p_t<8 GeV/c. The results are compared with previously published results at sqrt(s)=7 TeV and with theoretical calculations.
• ### Multiplicity Studies and Effective Energy in ALICE at the LHC(0709.1664)
Sept. 11, 2007 hep-ph
In this work we explore the possibility to perform effective energy'' studies in very high energy collisions at the CERN Large Hadron Collider (LHC). In particular, we focus on the possibility to measure in $pp$ collisions the average charged multiplicity as a function of the effective energy with the ALICE experiment, using its capability to measure the energy of the leading baryons with the Zero Degree Calorimeters. Analyses of this kind have been done at lower centre--of--mass energies and have shown that, once the appropriate kinematic variables are chosen, particle production is characterized by universal properties: no matter the nature of the interacting particles, the final states have identical features. Assuming that this universality picture can be extended to {\it ion--ion} collisions, as suggested by recent results from RHIC experiments, a novel approach based on the scaling hypothesis for limiting fragmentation has been used to derive the expected charged event multiplicity in $AA$ interactions at LHC. This leads to scenarios where the multiplicity is significantly lower compared to most of the predictions from the models currently used to describe high energy $AA$ collisions. A mean charged multiplicity of about 1000-2000 per rapidity unit (at $\eta \sim 0$) is expected for the most central $Pb-Pb$ collisions at $\sqrt{s_{NN}} = 5.5 TeV$. | {"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": 0.9468097686767578, "perplexity": 1630.9305950321393}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370506959.34/warc/CC-MAIN-20200402111815-20200402141815-00504.warc.gz"} |
https://en.wikisource.org/wiki/Page:Popular_Science_Monthly_Volume_6.djvu/51 | # Page:Popular Science Monthly Volume 6.djvu/51
Letters on the Proper Sphere of Government, (Occupied several years as a Railroad Engineer.) 1842 Planned Social Statics, 1846 Social Statics published, 1850 Theory of Population, ${\displaystyle {\begin{matrix}{\Big \}}\end{matrix}}}$ 1852 The Development Hypothesis, Philosophy of Style, Over-Legislation, ${\displaystyle {\begin{matrix}{\big \}}\end{matrix}}}$ 1853 The Universal Postulate, Manners and Fashion, ${\displaystyle {\begin{matrix}{\bigg \}}\end{matrix}}}$ 1854 The Genesis of Science, The Art of Education, Evolution first conceived as Universal, Principles of Psychology, (Breakdown of eighteen months.) 1855 Progress, its Law and Cause, ${\displaystyle {\begin{matrix}{\bigg \}}\end{matrix}}}$ 1857 Origin and Function of Music, Transcendental Physiology, Representative Government, State Tamperings with Money and Banks, ${\displaystyle {\begin{matrix}{\Bigg \}}\end{matrix}}}$ 1858 Moral Education, The Nebular Hypothesis, Archetype and Homologies of the Vertebrate Skeleton, Evolution first conceived as the Basis of a System of Philosophy, The Laws of Organic Form, ${\displaystyle {\begin{matrix}{\Bigg \}}\end{matrix}}}$ 1859 Physical Education, What Knowledge is of most Worth, Illogical Geology, Prospectus of the System of Philosophy drawn up, The Emotions and the Will, ${\displaystyle {\begin{matrix}{\Bigg \}}\end{matrix}}}$ 1860 The Social Organism, The Physiology of Laughter, Parliamentary Reforms, Prison Ethics, Prospectus of the Philosophical System published, First Principles, 1862 Classification of the Sciences, 1864 Principles of Biology, 1867 Principles of Psychology, The Study of Sociology, ${\displaystyle {\begin{matrix}{\big \}}\end{matrix}}}$ 1873 Descriptive Sociology, Principles of Sociology, Part I. 1874 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "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": 0.3974401652812958, "perplexity": 12530.35238793835}, "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-2017-17/segments/1492917122739.53/warc/CC-MAIN-20170423031202-00230-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://aux.planetmath.org/pollardsrhoalgorithm | # Pollard's $\rho$ algorithm
## Primary tabs
Synonym:
Pollard $\rho$ algorithm, Pollard's rho algorithm, Pollard rho algorithm
Major Section:
Reference
Type of Math Object:
Algorithm
## Mathematics Subject Classification
### Pollard's rho in Mathematica
Mathematica may be able to quickly factor F8, but it doesn't do it by Pollard's Rho algorithm. It uses ECM (ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/rpb161t...).
### Re: Pollard's rho in Mathematica
> Mathematica may be able to quickly factor F8, but it doesn't do it by Pollard's Rho algorithm. It uses ECM (ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/rpb161t...).
There is a way to make Mathematica show you the steps it takes to calculate something. I think it was in Theodore Glynn's book. Of course I could be wrong in assuming that it uses Pollard's rho. The program would have no nostalgia for how the number was first factored, it would simply choose the method most suitable for the number according to the way it was programmed.
I followed the link you provide but I'm having a hard time extracting the PostScript file from the archive. After that I'll have to figure out how to make my computer make me a PDF.
### Re: Pollard's rho in Mathematica
According to the Mathematica documentation (http://tinyurl.com/33pa2r), it currently uses:
"FactorInteger switches between trial division, Pollard p-1, Pollard rho, elliptic curve and quadratic sieve algorithms."
F8 factorization will take too long on both Pollard tests; ECM in the other hand finds it almost instantaneously, so I doubt QS has any chance of being used.
PS: the paper is just about the ECM, not about the claim on Mathematica's factoring algorithm; you can extract the .gz with 7-zip, and open the .ps with GSView/Ghostview.
### Re: Pollard's rho in Mathematica
> "FactorInteger switches between trial division, Pollard p-1, Pollard rho, elliptic curve and quadratic sieve algorithms."
That is exactly what the Mathematica 4.2 built-in help says. I thought about making my own Mathematica implementation of Pollard's rho, but I'm not that good a programmer. (I actually tried using Combinatorica::Partitions to find Goldbach representations of even numbers! It wasn't until the second draft of that that I switched to just subtracting odd numbers in order and then testing for primality).
I'll try 7-zip, but if there are no legal bugaboos, I'd much rather view the PostScript as a PDF in Adobe Reader.
### PS-PDF woe
>> I'll try 7-zip, but if there are no legal bugaboos, I'd much rather >> view the PostScript as a PDF in Adobe Reader.
Yeah, isn't is puzzeling that the founders of Adobe invented both PostScript and PDF but somehow their free PDF product doesn't read PostScript!
I've looked into the PDF, PS codec and the truth is they are both remarkably similar (though PDF does have compression and hypertext linking). After all, anything that can print to a laser printer must have in its code a PS codec, so why not have a full-fledged reader for PS!!
I'm with you PrimeFan, it is annoying. (...someday we can all be on Linux and this wont be a problem. :)
### Re: PS-PDF woe
But James, download GSView and Ghostscript from
Or I'm missing something?
perucho
### Re: PS-PDF woe
Yes, I know, I have had to use this when on Windows machines, but my lament is that PostScript is unusable to people who have a generic software setups. Ghost is a great product but I think Adobe reader should work with PostScript as well (as should Word for that matter.)
### Re: PS-PDF woe
I second the notion...it just isn't right...plus there are all these strange compatibility issues that I've encountered when including eps graphics in LaTeX documents generated using METAPOST, where FIRST I have to convert the file to ps, then use PS2PDF or whatever, and it's just annoying. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 33, "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": 0.5744217038154602, "perplexity": 2612.5027237063437}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806426.72/warc/CC-MAIN-20171121204652-20171121224652-00499.warc.gz"} |
https://math.stackexchange.com/questions/257955/irreducibles-are-prime-in-a-ufd | # Irreducibles are prime in a UFD
Any irreducible element of a factorial ring $D$ is a prime element of $D$.
Proof. Let $p$ be an arbitrary irreducible element of $D$. Thus $p$ is a non-unit. If $ab \in (p)\smallsetminus\{0\}$, then $ab = cp$ with $c \in D$. We write $a,\,b,\,c$ as products of irreducibles: $$\displaystyle a \;=\; p_1\cdots p_l, \quad b \;=\; q_1\cdots q_m, \quad c \;=\; r_1\cdots r_n.$$ Here, one of those first two products may be empty, i.e., it may be a unit. We have $$\displaystyle p_1\cdots p_l\,q_1\cdots q_m \;=\; r_1\cdots r_n\,p\tag{1}$$
Due to the uniqueness of prime factorization, every factor $r_k$ is an associate of certain of the $l+m$ irreducibles on the left hand side of $(1)$. Accordingly, $p$ has to be an associate of one of the $p_i$'s or $q_j$'s. It means that either $a \in (p)$ or $b \in (p)$. Thus, $(p)$ is a prime ideal of $D$, and its generator must be a prime element.
It may be too simple, but why $a \in (p)$ instead of $p_1 \in (p)$? Is it because $p$ has to be an associate of one of the $p_i$'s or $q_j$'s? Let's say $p_2$ is an associate of $p$. So, $p_2=pw$, $w\in R$. Since $a=p_1p_2\cdots p_l$ then $a=p_1pwp_3\cdots p_l$ and $a=p(p_1p_3\cdots p_lw)$, $p_1p_3\cdots p_lw \in R$ so $a$ is divisible by $p$ hence $a\in (p)$?
• Your reasoning looks correct to me. – Zach L. Dec 13 '12 at 15:12
• Yes, more simply $\rm\: p\mid p_i\mid a,\:$ i.e. it follows by transitivity of "divides" (or "contains", if expressed using ideals) – Bill Dubuque Dec 13 '12 at 15:32
The proof given above is probably the standard one to show that a factorial domain is an AP-domain. But there is another proof using the following application: for an irreducible $p\in D$, let's define $e_p\colon D\setminus \{0\}\rightarrow \Bbb{N}$ given by $a\mapsto e_p(a)=\#$ of times that $p$ or its associates appear in the irreducible factorization of $a$.
We notice that because $D$ is factorial the application given above it's well defined. Moreover, if $a\in D^{\times}$, then $e_p(a)=0$ for every irreducible $p$, and if $a\in D\setminus{D^{\times}_0}$, then $e_p(a)=0$ iff $p\not\mid a$. Equivalently, $e_p(a)>0$ iff $p\mid a$.
We have the following:
Lemma: Let $D$ be a factorial domain and $a,b\in D\setminus \{0\}$. Then $$e_p(ab)=e_p(a)+e_p(b)$$ for every irreducible $p\in D$.
Proof: As $D$ is factorial, we can write $a=p^{e_p(a)}\ldots$ and $b=p^{e_p(b)}\ldots$, then $$ab=(p^{e_p(a)}\ldots)(p^{e_p(b)}\ldots)=p^{e_p(a)+e_p(b)}\ldots$$ Hence, $e_p(ab)=e_p(a)+e_p(b)$.
Now we're going to prove that if $D$ is factorial, then $D$ is an AP-domain. Let $p$ be an irreducible element in $D$ and let $a,b\in D$ such that $p\mid ab$. If $ab=0$, then $a=0$ or $b=0$, so in this case clearly $p\mid a$ or $p\mid b$. If $ab\neq 0$, since $p\mid ab$ we have $e_p(ab)>0$, so by the above lemma we find $$e_p(ab)=e_p(a)+e_p(b)>0.$$ Therefore we deduce that necessarily $e_p(a)>0$ or $e_p(b)>0$, i.e., $p\mid a$ or $p\mid b$. Hence, $p$ is prime.
As a remark, this kind of ideas applied to $\Bbb{Z}$ can be found in the first pages of the book "A Classical Introduction to Modern Number Theory" by K. Ireland and M. Rosen.
• It is important to know different perspectives. – W.Leywon Jun 17 '17 at 3:38
• @W.Leywon that's true. – Xam Jun 25 '17 at 14:59
It's trivial to show that primes are irreducible. So, assume that $$a$$ is an irreducible in a UFD (Unique Factorization Domain) $$R$$ and that $$a \mid bc$$ in $$R$$. We must show that $$a \mid b$$ or $$a \mid c$$. Since $$a\mid bc$$, there is an element $$d$$ in $$R$$ such that $$bc=ad$$. Now replace $$b,c$$ and $$d$$ by their factorizations as a product of irreducibles and use uniqueness.
• what if $d$ is non invertible? – user370967 Nov 25 '17 at 16:43
• The uniqueness of the factorization implies that. There is no need to use invertibility to cancel out the elements,just simply compare the factors of both sides of the equation,they shall be equal,one-to-one. – W.Leywon Nov 30 '17 at 8:59 | {"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": 12, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.982725977897644, "perplexity": 104.52310412684204}, "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/1620243988775.25/warc/CC-MAIN-20210507060253-20210507090253-00135.warc.gz"} |
https://www.physicsforums.com/threads/del-1-r-r-r-2.367226/ | # Homework Help: Del(1/r) = -R/r^2
1. Jan 5, 2010
### boardbox
1. The problem statement, all variables and given/known data
let R be the separation vector from (a,b,c) to (x,y,z) and r be the magnitude of R.
Show that: del(1/r) = -R/r^2
2. Relevant equations
del is the gradient operator
3. The attempt at a solution
The problem is that I keep getting a 3/2 power in the denominator when I calculate the left hand side.
r = sqrt((x-a)^2 + (y-b)^2 + (z-c)^2)
1/r = ((x-a)^2 + (y-b)^2 + (z-c)^2)^-0.5
del(1/r) = -R/[(x-a)^2 + (y-b)^2 + (z-c)^2]^(3/2)
Last edited: Jan 5, 2010
2. Jan 5, 2010
3. Jan 5, 2010
### boardbox
I see, you're right. I missed the hat on R in my text. Thanks. | {"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": 0.9733166694641113, "perplexity": 3365.2490623154154}, "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-2018-39/segments/1537267160754.91/warc/CC-MAIN-20180924205029-20180924225429-00194.warc.gz"} |
https://www.physicsforums.com/threads/surface-area-and-electric-field-strength.770145/ | # Surface Area and electric field strength
1. Sep 10, 2014
### Samson4
This is my first post but I have frequented the forum for a little while now. I tried to figure things out myself and often times I am lead here by google. So my question is this:
How is the electric field of an object altered when the surface area of the object is altered?
Example: Take a perfectly smooth sphere and charge it to 100 volts. Then take the same sphere and etch it to increase it's surface area 100 times. By that I mean similar to etching aluminium with hcl in capacitor production. When comparing the electric fields, are they different in anyway?
My first thought was that by increasing the surface area; therefore capacitance, you would have more electric field lines on the altered sphere. Since those must terminate on a conductor or at infinity, it would have a stronger electric field. I was thinking with the idea that every charge carrier gets an electric field line. I don't think that is correct.
Now I think the answer is that the electric fields are the same. However, the altered sphere would feel a stronger force if placed in an external electric field. Instead of individual charges getting field lines, I think it's more accurate to say that the electric field is a measure of total surface charge pressure.
Can you guys shed some light on this or point me to something that will help me understand?
Last edited: Sep 10, 2014
2. Sep 10, 2014
### Staff: Mentor
Yes, at any distance from the object. As long at the object is spherically symmetrical, there is some distance away at which its electrical field will be indistinguishable from that of a point charge. (If this were not true, we would find $F=Cq_1q_2/r^2$ to be much less useful).
No. Although you increase the surface area by making the object rough or spiky, you also decrease the average charge density per unit area. Grovel through the surface integral and you'll end up with the same net force, as long as the applied electrical field is constant in the neighborhood of the object.
If you haven't already done so, check out Gauss's Theorem (google will find it). It's not quite directly applicable, but it is a very powerful tool for making the shape of the distribution of charge within a volume irrelevant outside that volume.
3. Sep 10, 2014
### Samson4
Sorry, I tried to be very specific with my question. I forgot to mention that the new sphere would also be charged to 100volts. So it would have 100 times the charges on it's surface. You still answer that question even though I didn't explain it enough.
To really clarify, I'd like to give another example. We have two spheres, a and b. A is smaller than b; but a has higher surface area than b. A is placed inside b and charged to 100volts. They are insulated from each other. Gaussian theorem states that the outer sphere would create an equal charge to the inner sphere. How does sphere b's electric field compare to a's. Are they still the same and if so, what happens to the charge from all the extra carriers needed to bring sphere A to 100v because of it's higher capacitance. | {"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": 0.9156547784805298, "perplexity": 304.04439886366316}, "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/1521257647299.37/warc/CC-MAIN-20180320052712-20180320072712-00445.warc.gz"} |
http://mathhelpforum.com/math-topics/161141-physics-circular-motion.html | # Math Help - physics- circular motion
1. ## physics- circular motion
calculate the orbital height about the earth of a geostationay satellite.
the following data will be required:
earth's mass 6.0x10^24 kg
g.s G=6.67x10^-11
i know the time is going to be 24 hours but i have no idea what equations to use....
thanks.
2. Use the fact that the grativational pull provides the centripetal force.
$F_c = F_g$
Let m be the mass of the satellite,
r the distance from the Earth's centre,
omega the angular velocity of the satellite,
M the mass of the Earth.
$mr\omega ^2 = \dfrac{GMm}{r^2}$
Simplify;
$r^3\omega ^2 = GM$
Now, fill in the values that you know.
$r^3\left(\dfrac{2\pi}{24\times3600}\right)^2 = (6.67\times10^{-11})(6\times10 ^{24})$
You should be able to get the value of r, the distance of the satellite from the centre of the Earth and then, the height above the ground.
Of course, there are assumptions made here to make it easier.
Post the answer that you get!
Answer I got: 3.59 x 10^7 m
3. thank you very much!
i got 3.6x10^7 m.
4. Use your mouse to select my whole post and you'll see the spoiler I hid in there. That's the right 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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9242527484893799, "perplexity": 1199.8646650089045}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931007797.72/warc/CC-MAIN-20141125155647-00019-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/non-static-and-isotropic-solution-for-einstein-field-eq.897068/ | # A Non static and isotropic solution for Einstein Field Eq
Tags:
1. Dec 13, 2016
Hello dear friends, today's question is:
In a non static and spherically simetric solution for Einstein field equation, will i get a non diagonal term on Ricci tensor ? A R[r][/t] term ?
I'm getting it, but not sure if it is right.
Thanks.
2. Dec 13, 2016
### PAllen
Note that spherically symmetric is not the same as isotropic. Your title says one, your post says another. A spherically symmetric solution is anisotropic.
Isoptropic, non-static solutions include all the FLRW solutions except where cosmological constant exactly balances the expansion.
The spherically symmetric non-static vacuum solution is unique (without cosmological constant, at least). It is the interior of an eternal Schwarzschild BH. The Ricci tensor is identically zero because it is vacuum. For non-vacuum solutions, you will, indeed get an (r,t) Ricci term (given normal meaning of such coordinates). This should be the only non diagonal term [ noting (r,t) obviously = (t,r) term].
3. Dec 14, 2016
### Staff: Mentor
This is not quite correct. FRW spacetime is spherically symmetric and also isotropic. The key is that FRW spacetime is spherically symmetric about every point. A spacetime like Schwarzschild spacetime, which is not isotropic, is not.
More technically: "spherically symmetric" means there is a 3-parameter family of spacelike Killing vector fields with the commutation relations of SO(3). But that does not preclude there being more than one such family. In FRW spacetime, there is an infinite number of such families (one centered on each spatial point--and to be really technical, there is such an infinite family in each spacelike hypersurface, instead of just one as there is in Schwarzschild spacetime).
4. Dec 14, 2016
### Staff: Mentor
Can you post your actual math?
5. Dec 14, 2016
### PAllen
Is there some reason you want this? For non-vacuum case, the existence of this term is well known.
6. Dec 14, 2016
### Staff: Mentor
I'm interested in how the "non-static" assumption is formulated.
7. Dec 14, 2016
### PAllen
You just use a metric ansatz with angular coordinates, and two unknown functions of r and t. Different formulations of the ansatz lead to different styles of coordinates. In all such set ups, you find the Ricci tensor is diagonal except for the r,t components. One reference for this is chapter 7, section 3, of Synge's book.
8. Dec 14, 2016
### Staff: Mentor
Are you referring to equations (70) and (71) of Synge? The first is a general ansatz with three unknown functions, and the second gives different specializations that determine one of the functions in terms of the other two.
9. Dec 14, 2016
### vanhees71
Yes, that's the way Birkhoff's theorem is proven, i.e., you make a ansatz for a spherically symmetric solution of the free Einstein equation. Then you'll get out that in fact the metric components are necessarily time-independent, i.e., any spherically symmetric solution of the vacuum Einstein equation is static.
10. Dec 14, 2016
### PAllen
Yes. Or, as is done earlier in that chapter, simpler forms of the ansatz for different coordinate styles are given. Those equations (70, 71) use a most general form to derive universal results.
11. Dec 14, 2016
### PAllen
Note, Birkhoff really states the existence of an extra killing field. Only if it is time like do you then get staticity. Thus, the BH exterior is static, while the interior is not because dt is space like in the interior, so you have an axial extra killing field.
12. Dec 15, 2016
I'm coursing general relativity for the first time, first contact with these strange spaces.
Draft saved Draft deleted
Similar Discussions: Non static and isotropic solution for Einstein Field Eq | {"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": 0.9322491884231567, "perplexity": 1285.0660927546414}, "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-2017-51/segments/1512948592972.60/warc/CC-MAIN-20171217035328-20171217061328-00110.warc.gz"} |
https://www.physicsforums.com/threads/heisenberg-vs-schrodinger-picture.417553/ | # Heisenberg vs Schrodinger Picture
1. Jul 21, 2010
### nateHI
Please tell me where my understanding of the Heisenberg and/or the Schrodinger picture falls apart.
-Schrodinger says the state vector of a system changes with time according to a unitary operator that doesn't change with time.
-Hesienberg says the state vector of a system doesn't change with time but the operator acting on that state vector has time dependence.
-In the heisenberg equation of motion http://en.wikipedia.org/wiki/Heisenberg_picture , the second term (partial A with respect to t) dissapears (A doesn't change with time) when the Hamiltonian is autonomous.
- But if the Hamiltonian is autonomous and we are in the Heisenberg picture then A(t)=A(0) and the system will not evolve with time because the operator A is the only time dependent variable.
-However, in the schrodinger picture, the phase of a stationary state can evolve with time.
Conclusion:
-Therefore, the two pictures do not seem entirely equal.
2. Jul 21, 2010
### olgranpappy
No. A(t) does not equal A(0) in the Heisenberg picture in general.
In general,
$$A(t)=e^{i H t}A(0)e^{-i H t}$$
A(t) only equals A(0), for all t, if A(0) commutes with the Hamiltonian.
3. Jul 22, 2010
### tom.stoer
Forget about explicit time dependence for a moment (no time dependent force or something like that.
You have an operator H, an evolution operatur U(t) = exp(-iHt) and state vector |state>
Now you can calculate
|state, t> = U(t) |state>
and look at
A |state, t>;
this is the Schrödinger picture for some operator A.
Or you can look at
U*(t) A |state, t> = U*(t) A U(t) |state> = A(t) |state>
with A(t) = U*(t) A U(t);
this is the Heisenberg picture for A(t).
It's just playing around with A.
4. Jul 22, 2010
### xepma
First of all, I don't really see why you draw the conclusion that A(t) = A(0).. This is certainly not true if A does not commute with the Hamiltonian while not having some explicit time dependence.
Anyways, it's best to look at the one thing that matters in quantum mechanics: the amplitudes. Written in the Schroedinger picture they look like:
$$\langle \psi(t)| A | \psi(t)\rangle$$
where the time evolution of the state is dictated by some unitary evolution operator $$U(t,t_0)$$ acting on some reference state $$| \psi(t_0)\rangle$$, so
$$| \psi(t)\rangle = U(t,t_0) | \psi(t_0)\rangle$$
In the Heisenberg picture the time evolution is switched, such that it is located within the operators. This simply amounts to letting the unitary evolution operators act on the operators instead (I'm dropping the t_0 label on the reference state).
$$\langle \psi(t)| A | \psi(t)\rangle = \langle \psi|U^\dag(t,t_0) AU(t,t_0) | \psi\rangle = \langle \psi|A_H(t) | \psi\rangle$$
where $$A_H(t) = U^\dag(t,t_0) AU(t,t_0)$$ is now the Heisenberg representation of the operator A. Strictly speaking you should always attach a subscript to each operator and state to label in which picture they are in.
Now the moral of the story is that the amplitude, which is usually denoted as $$\langle A \rangle$$, is the same in both pictures. This is pretty clear from the above since I just moved around the time evolution operators. If for instance the states only have some phase factor time dependence in the Schroedinger picture (and $$A_H(t) = A(0)$$ like you mentioned), then this still doesn't matter for a quantum amplitude -- the phases just cancel.
Now, in the following notation (which is neither the Schroeding or Heisenberg picture)
$$\langle \psi|U^\dag(t,t_0) AU(t,t_0) | \psi\rangle$$
all time dependence is put explicitly into the time evolution operators U. This is, for me, the clearest way to look at both quantum mechanics and QFT. You start with some state $$| \psi\rangle$$ at some reference time t0. You let the system evolve until it hits A at time t. Then you evolve backwards in time, back to the reference state at t0 again. And this manipulation will give you the quantum amplitude $$\langle A(t)\rangle$$. It turns out to be unpractical to work with this notation directly -- the operator U(t,t_0) is problematic -- and that is why you switch to either the Schroedinger or Heisenberg picture, depending on what type of problem you are dealing with.
( It will get 'worse' by the way. In the Dirac picture (also interaction picture) the time evolution is split! Both the operators and the states have their own time evolution operator -- both evolving with respect to a different part of the Hamiltonian. This comes down to writing U(t,t') as a product of two unitary operators, U_0 and U_I -- one then acts on the states, and the other on the operators. )
Last edited: Jul 22, 2010
5. Jul 22, 2010
### nateHI
OK I get it now, just because A doesn't change with time (partial of A with respect to t = 0) doesn't meant that A(t) doesn't change with time because the time dependence for A(t) is actually in U. | {"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": 0.9247428178787231, "perplexity": 596.7484923065915}, "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-2019-04/segments/1547583659944.3/warc/CC-MAIN-20190118070121-20190118092121-00350.warc.gz"} |
https://noa.gwlb.de/receive/cop_mods_00050947 | # The long-term trend and production sensitivity change in the US ozone pollution from observations and model simulations
We investigated the ozone pollution trend and its sensitivity to key precursors from 1990 to 2015 in the United States using long-term EPA Air Quality System (AQS) observations and mesoscale simulations. The modeling system, a coupled regional climate–air quality model (CWRF-CMAQ; Climate-Weather Research Forecast and the Community Multiscale Air Quality), captured well the summer surface ozone pollution during the past decades, having a mean slope of linear regression with AQS observations of inline-formula∼0.75. While the AQS network has limited spatial coverage and measures only a few key chemical species, CWRF-CMAQ provides comprehensive simulations to enable a more rigorous study of the change in ozone pollution and chemical sensitivity. Analysis of seasonal variations and diurnal cycle of ozone observations showed that peak ozone concentrations in the summer afternoon decreased ubiquitously across the United States, up to 0.5 ppbv yrinline-formula−1 in major non-attainment areas such as Los Angeles, while concentrations at certain hours such as the early morning and late afternoon increased slightly. Consistent with the AQS observations, CMAQ simulated a similar decreasing trend of peak ozone concentrations in the afternoon, up to 0.4 ppbv yrinline-formula−1, and increasing ozone trends in the early morning and late afternoon. A monotonically decreasing trend (up to 0.5 ppbv yrinline-formula−1) in the odd oxygen (inline-formula $M5inlinescrollmathmlchem{\mathrm{normal O}}_{x}={\mathrm{normal O}}_{normal 3}+{\mathrm{normal NO}}_{normal 2}$ 69pt13ptsvg-formulamathimgc13b7ba31a73eaac2dac9773cc4bcd0b acp-20-3191-2020-ie00001.svg69pt13ptacp-20-3191-2020-ie00001.png ) concentrations are simulated by CMAQ at all daytime hours. This result suggests that the increased ozone in the early morning and late afternoon was likely caused by reduced NO–inline-formulaO3 titration, driven by continuous anthropogenic inline-formulaNOx emission reductions in the past decades. Furthermore, the CMAQ simulations revealed a shift in chemical regimes of ozone photochemical production. From 1990 to 2015, surface ozone production in some metropolitan areas, such as Baltimore, has transited from a VOC-sensitive environment (inline-formula>50 % probability) to a inline-formulaNOx-sensitive regime. Our results demonstrated that the long-term CWRF-CMAQ simulations can provide detailed information of the ozone chemistry evolution under a changing climate and may partially explain the US ozone pollution responses to regional and national regulations.
### Zitieren
Zitierform:
He, Hao / Liang, Xin-Zhong / Sun, Chao / et al: The long-term trend and production sensitivity change in the US ozone pollution from observations and model simulations. 2020. Copernicus Publications.
### Zugriffsstatistik
Gesamt:
Volltextzugriffe: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.6575198769569397, "perplexity": 4912.85312710825}, "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/1656103640328.37/warc/CC-MAIN-20220629150145-20220629180145-00756.warc.gz"} |
https://chemistry.stackexchange.com/questions/133879/why-is-na-reduced-instead-of-h-in-the-electrolysis-of-dilute-naclaq-with-mer | # Why is Na+ reduced instead of H+ in the electrolysis of dilute NaCl(aq) with Mercury cathode?
In a class of electrolysis, my instructor told me that Hg forms Na-Hg in the electrolysis of dilute NaCl aqueous solution. For this reason, sodium cations are reduced in the cathode instead of hydronium ions. However, he did not give me any reason for how creating an amalgam affects the reduction of the metal or, why Hg creates amalgam in the first place. So, I would like to know why Hg creates amalgam whereas other metals like Pt don't form them ? And also how this affects the reduction of Na.
• It has to something with the hydrogen overpotential of Hg electrode. – M. Farooq May 17 at 23:21
• Well,what do you mean by "hydrogen overpotential" ?And what has it got to do with Hg? – Habib May 17 at 23:31
• Umm, proper question would be rather why Na is reduced with Hg electrode and is this even really the case. While Na might perhaps be reduced also with other electrodes, it obviously won't become part of amalgamate, as only solutions in Hg are called like that. – Mithoron May 18 at 0:13
• Please see “Castner–Kellner process” in wikipedia. Definitely sodium can be electrolyzed from brine. The reason it works is the overpotential for hydrogen reduction at a mercury cathode. In other words, there is a kinetic hindrance to hydrogen reduction and sodium forms an alloy with mercury, so back-reaction with the aqueous solution is impeded. Traditionally, mercury alloys with other metals are called amalgams. Liquid gallium, for example, also readily forms alloys with many metals, but gallium has not been known since ancient times. – Ed V May 18 at 0:24
• In the case of sodium ion reduction at the liquid mercury cathode, the sodium atoms get “dissolved” in the liquid mercury. So they move around like in a ordinary solution. But this keeps them from contacting the brine, so they do not get to react with the brine. Those at the interface of brine and mercury can so react, producing hydrogen gas and NaOH. For platinum, for example, the sodium would, at best, form a plating: the atoms could not get sheltered, as it were, by getting into the interior of the platinum. So it is a matter of getting the sodium away from the brine and mercury does it. – Ed V May 18 at 0:56
The mental conflict occurs because two processes could theoretically happen (electrodeposition of sodium, or electrodeposition of hydrogen), and the theoretically unfavorable one occurs - and not only occurs, but is so favorable that it was used commercially to produce NaOH as EdV noted.
So why is the theoretically favored process not favored in this instance? Hydrogen evolution is a multi-step process, and if you wave your hands a lot, you can call it overpotential and be off to other projects.
When (or if) sodium is deposited (on mercury), it can dissolve and be removed from contact with water. So this process is a slam-dunk if you push with a high enough voltage.
But the hydrogen evolution should take place at a lower voltage. Why not? If/when hydrogen is deposited, its first appearance is atomic: one H atom sticks somewhere on the mercury surface; the second step is deposition of another H atom somewhere else. The next step, for the two atoms to form a molecule, requires some mobility on the part of the hydrogen and the mercury surface. At this point, conventional wisdom suggests that the molecule could dissolve into the solution, but eventually the aqueous solution becomes saturated. Then the first molecule must meet another molecule, and another, to begin the process of forming a bubble which will eventually rise to the surface of the aqueous solution. Somewhere in there is an impediment to the formation of bubbles, and it may involve mobility of atoms or molecules, or attachment of hydrogen to mercury, but the process is difficult enough to allow the voltage to be raised high enough to allow deposition of sodium to proceed.
And then we say that mercury has a high overpotential for deposition of hydrogen.
Now suppose we do the same, but with gallium. Sodium should deposit and become dissolved as with mercury, if the voltage is high enough. (Sodium alloys with gallium; amalgamation is a chemical term that should be reserved for mercury alloys.) But the hydrogen overpotential on gallium is less than on mercury (Ref 1), and gallium and hydrogen are deposited together from a solution of $$Ga^{+3}$$ (Ref 2). So, in the case of liquid gallium, it seems that hydrogen would be more easily deposited than sodium (unless, perhaps you used a very high voltage, which would override the thermodynamic preference because diffusion couldn't keep up with the current demands).
If you heat metallic sodium with let’s say gold / silver / mercury / platinum etc.. it will form an amalgam.
I have tried this with a tiny bit of gold some years ago.
I did heat Na in an inert argon atmosphere to several hundred degrees in an quartz tube and drop the gold inside.
It did take only fractions of a second till the amalgam was formed.
The result is a green looking amalgam when it is hot - when it cools down it is a brown solid substance.
Given the fact that gold is a pretty inert substance, I guess that sodium will do the same with Pt, Rh, Ir, Ru, Os, Ag, Hg ...
I guess during electrolysis you have metallic Na formed. I don’t mean junks of Na but Na atoms and some of them may live long enough to amalgamate with the Hg.
• Not every metal amalgamates. See chemistry.stackexchange.com/q/117633/79678. – Ed V Jun 23 at 15:02
• The question was not about Na amalgam nor amalgams in general, but specifically why it or metallic Na is formed at all during NaCl electrolysis with Hg as the cathode. And yes, other metal than mercury obviously cannot form amalgams as they are not mercury. – Poutnik Jun 23 at 15:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6395993232727051, "perplexity": 1850.9547877667947}, "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/1600400211096.40/warc/CC-MAIN-20200923144247-20200923174247-00235.warc.gz"} |