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http://math.ipm.ac.ir/seminars/printseminar.jsp?eventID=259
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Geometry and Topology Weekly Seminar سمینار هفتگی هندسه و توپولوژی
TITLE
Skew Products and Invariant Graphs
SPEAKER
Fatemeh Ghane Ferdowsi University of Mashhad
TIME
Wednesday, January 17, 2018, 15:30 - 17:00
VENUE Lecture Hall 1, Niavaran Bldg.
SUMMARY
This talk concerns the stability and ergodic properties of skew products $T(x, y) =(f(x), g(x, y))$ in which $(f, X, \mu)$ is an ergodic map of compact metric space $X$ and $g : X \times Y \rightarrow Y$ is continuous. The set $X$ is the base, while $Y$ is the fiber. We consider the case where g is (non-uniformly) contracting. When this contraction is uniform, it can easily be shown that there exists a globally attracting invariant set which is the graph of a function from the base space to the fiber space. Here, we consider the case that the contraction rates are non-uniform and hence specifi ed by Lyapunov exponents and analogous quantities. We investigate the geometric structures of non-uniformly hyperbolic attractors of a certain class of skew products. We construct an open set of skew products over a linear expanding circle map such that any skew product belonging to this set admits a non-uniformly hyperbolic solenoidal attractor for which the following dichotomy is ascertained. This attractor is either a continuous invariant graph with nonempty interior or a thick bony attractor. Here, an attractor is, roughly speaking, a maximal attractor. Also, an attractor is thick if it has positive but not full Lebesgue measure. In our construction, the contraction in the fi ber is non-uniform. Furthermore, we provide some related results on the ergodic properties of attracting graphs and stability results for such graphs under deterministic perturbations. In particular, we show that there exists an invariant ergodic physical measure whose support is contained in that attractor.
تهران، ضلع جنوبی ميدان شهيد باهنر (نياوران)، پژوهشگاه دانشهای بنيادی، پژوهشکده رياضيات School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Niavaran Bldg., Niavaran Square, Tehran ipmmath@ipm.ir ♦ +98 21 22290928 ♦ math.ipm.ir
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2018-12-19 06:24:28
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https://zbmath.org/authors/?q=ai%3Alehoucq.richard-b
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# zbMATH — the first resource for mathematics
## Lehoucq, Richard B.
Compute Distance To:
Author ID: lehoucq.richard-b Published as: Lehoucq; Lehoucq, R.; Lehoucq, R. B.; Lehoucq, Rich; Lehoucq, Richard; Lehoucq, Richard B. Homepage: http://www.sandia.gov/~rblehou/
Documents Indexed: 55 Publications since 1992, including 3 Books
all top 5
#### Co-Authors
4 single-authored 11 Gunzburger, Max D. 9 Bochev, Pavel B. 7 Du, Qiang 7 Parks, Michael L. 5 Hetmaniuk, Ulrich L. 5 Salinger, Andrew G. 3 Badia, Santiago 3 Pawlowski, Roger P. 3 Rowe, Stephen T. 3 Silling, Stewart A. 3 Zhou, Kun 2 Baker, C. G. 2 Carr, Steve 2 Embree, Mark 2 Emmrich, Etienne 2 Kelley, Carl T. 2 Shadid, John N. 2 Sorensen, Danny C. 2 Thornquist, Heidi K. 2 Tuminaro, Raymond S. 1 Arbenz, Peter 1 Arnold, Douglas Norman 1 Bartlett, Roscoe A. 1 Benner, Peter 1 Bennighof, Jeffrey K. 1 Bond, Stephen D. 1 Burch, Nathanial 1 Burroughs, Elizabeth A. 1 Defterli, Ozlem 1 D’Elia, Marta 1 Dohrmann, Clark R. 1 Gee, Michael W. 1 Gray, Stephen K. 1 Hennigan, Gary L. 1 Heroux, Michael A. 1 Hoekstra, Robert J. 1 Howle, Vicki E. 1 Hu, Jonathan J. 1 Huang, Zhan 1 Kamm, James R. 1 Kolda, Tamara G. 1 Koteras, J. Richard 1 Light, J. C. 1 Lin, Paul T. 1 Long, Kevin R. 1 Meerbergen, Karl 1 Meerschaert, Mark M. 1 Narcowich, Francis J. 1 Nicolaides, Roy A. 1 Phipps, Eric T. 1 Plimpton, Steven J. 1 Puhst, Dimitri 1 Romero, Louis A. 1 Seleson, Pablo 1 Shashkov, Mikhail J. 1 Stanley, Kendall S. 1 Tartakovsky, Alexandre M. 1 Tuminaro, Ray S. 1 Von Lilienfeld-Toal, Anatole 1 Ward, Joseph D. 1 Willenbring, James M. 1 Williams, Alan W. 1 Yang, ChangHai 1 Zhang, Donghan
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#### Serials
4 ACM Transactions on Mathematical Software 4 SIAM Journal on Matrix Analysis and Applications 4 Multiscale Modeling & Simulation 3 Computer Methods in Applied Mechanics and Engineering 3 International Journal for Numerical Methods in Fluids 3 International Journal for Numerical Methods in Engineering 3 Journal of Elasticity 2 Computer Physics Communications 2 SIAM Journal on Numerical Analysis 2 Linear Algebra and its Applications 2 SIAM Review 2 ETNA. Electronic Transactions on Numerical Analysis 1 Journal of Computational Physics 1 Journal of the Mechanics and Physics of Solids 1 Mathematics of Computation 1 M$^3$AS. Mathematical Models & Methods in Applied Sciences 1 SIAM Journal on Applied Mathematics 1 SIAM Journal on Scientific Computing 1 International Journal of Numerical Methods for Heat & Fluid Flow 1 Monte Carlo Methods and Applications 1 Fractional Calculus & Applied Analysis 1 Discrete and Continuous Dynamical Systems. Series B 1 Computational Methods in Applied Mathematics 1 Oberwolfach Reports 1 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 1 The IMA Volumes in Mathematics and its Applications 1 Software - Environments - Tools
all top 5
#### Fields
33 Numerical analysis (65-XX) 17 Partial differential equations (35-XX) 11 Mechanics of deformable solids (74-XX) 10 Fluid mechanics (76-XX) 6 Integral equations (45-XX) 5 Classical thermodynamics, heat transfer (80-XX) 4 Real functions (26-XX) 3 General and overarching topics; collections (00-XX) 3 Mechanics of particles and systems (70-XX) 3 Statistical mechanics, structure of matter (82-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Ordinary differential equations (34-XX) 1 History and biography (01-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Approximations and expansions (41-XX) 1 Operator theory (47-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Probability theory and stochastic processes (60-XX) 1 Computer science (68-XX) 1 Quantum theory (81-XX)
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2020-02-19 02:18:44
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https://electronicsphysics.com/capacitance-of-earth-in-microfarad-formula-value/
|
# Capacitance of Earth in microfarad | formula value
To find the capacitance of Earth we consider that the Earth is a spherical conductor with radius R and total charge Q. In fact, using this assumption one can determine the capacitance of any planet and satellite like Mercury, Mars, Moon, etc. In this article, we are going to find the capacitance of Earth in microfarad unit. We will develop the formula for capacitance of an isolated spherical conductor and then we will use the parameters of Earth in that equation.
## The Formula of capacitance of Earth
Earth and other planets are spherical isolated capacitors. If R and Q be the radius and the charge respectively of the Earth, then the electric potential on the surface of the Earth is, $\small {\color{Blue} V=\frac{Q}{4\pi \epsilon _{0}R}}$. Where $\small \epsilon _{0}$ is the permittivity of free space.
Now, from the formula of capacitance we know, $\small {\color{Blue} C=\frac{Q}{V}}$
Then, the capacitance, $\small {\color{Blue} C=\frac{Q}{ \frac{Q}{4\pi \epsilon _{0}R} }}$
or, the capacitance of the Earth capacitor is, $\small {\color{Blue} C=4\pi \epsilon _{0}R}$ …………..(1) in SI system.
Again, in CGS system, the formula of the capacitance of an isolated sphere like Earth is, $\small {\color{Blue} C=R}$ …….(2)
## Capacitance of Earth in microfarad
So, from equation-(1) and equation-(2) it is clear that the capacitance of Earth depends only on its radius.
The radius of the Earth, R = 6400 km = 6.4×106 meter. And, from Coulomb’s force equation in electrostatics we know that, $\small 4\pi \epsilon_{0} = \frac{1}{9\times 10^{9}}$.
Then the capacitance of Earth, $\small {\color{Blue} C=\frac{6.4\times 10^{6}}{9\times 10^{9}}}$ in Farad.
or, the capacitance of the Earth, C = 0.711×10-3 Farad.
So, the Capacitance of Earth in microfarad is C = 711 microfarad.
This is all from this article on the value of Earth’s capacitance in the unit of microfarad. Hope you got your answer. Also, you can determine the capacitance for other planets or satellites like Moon by putting the value of the radius of the planet in equation-(1) or equation-(2). If you still have any doubts on this topic you can ask me in the comment section.
Thank you!
Related posts:
1. Capacitance of different types of capacitors – Spherical, Cylindrical, Parallel plate capacitor, isolated conductor.
2. Parallel plate capacitor with dielectric medium
3. Energy stored in a capacitor.
4. Electrostatic potential and potential energy.
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2023-02-04 04:50:24
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http://www.midf.com.my/store-website-wepp/oeiyppx.php?e8d4c6=heal-your-gut-lee-holmes-kmart
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Here is the rest of the work for this problem. The series in which the molecular formula of adjacent members differ by a – CH2 unit, is called homologous series and the individual members are called homologous. After factoring we were able to cancel some of the terms in the numerator against the denominator. An-swer. Let’s take a look at some examples of the Chain Rule. All rights reserved. Once you get better at the chain rule you’ll find that you can do these fairly quickly in your head. Learn Maths with all NCERT Solutions Class 6 Class 7 Class 8 Class 9 Class 10 Class 11 Class 12. The following species behave as nucleophiles : In case of same nucleophilic site, nucleophilicity parallels basicity i.e.. as the basicity increases, nucleophilicity also increases. 2. the sum of numbers of side chain is lowest. The new bond lies along with side of the ring and has little interaction with the 1t electron cloud lying above and below the ring. (i) Primary carbon atom When carbon atom is attached with one other carbon atom only, it is called primary or 1° carbon atom. (c) Meso form The compound in which half part of a molecule is the mirror image of other half, is called meso form. Stereoisomerism is of three types : optical isomerism, geometrical isomerism and conformations. If the first atom of the group of atoms is same, the priority is decided by second atom of the group e.g., among -COOH, -CH2OH and-CHO. (ii) Triplet carbene In it, the central C-atom is sp-hybridised. Furthermore, these compounds are exceptionally stable because of the small size of carbon. Electron withdrawing group (E.W.G.) So even though the initial chain rule was fairly messy the final answer is significantly simpler because of the factoring. Question 1 . If the chocolates are taken away by 300 children, then how many adults will be provided with the remaining chocolates? To see the proof of the Chain Rule see the Proof of Various Derivative Formulas section of the Extras chapter. These are generally volatile and inflammable. For the most part we’ll not be explicitly identifying the inside and outside functions for the remainder of the problems in this section. The branch of chemistry which deals with these compounds is called organic chemistry. To get fastest exam alerts and government job alerts in India, join our Telegram channel. The last operation that you would use to evaluate this expression is multiplication, the product of 4x2 9 and p 4x2 + 9, so begin with the product rule. (i) α – elimination In it, both the groups are eliminated from the same carbon atom. Be careful with the second application of the chain rule. g′( x ) Let's work some chain rule examples to understand the chain rule calculus in a better rule. (a) Cis-trans isomers In cis-isomer, similar groups are present on the same side of the double bond and in trans- isomer, similar groups are present on the opposite side of the double bond. e.g.. (iii) R-S system This system was proposed by Cabo, Ingold and Prelog. Then Hybridised orbitals contain no electrons and a hybridised orbital contains two electrons : Singlet carbene has bent structure and is less stable than triplet carbene. Later on, you’ll need the chain rule to compute the derivative of p 4x2 + 9. Now, let’s go back and use the Chain Rule on the function that we used when we opened this section. Rule I. Electron donating groups with respect to conjugate system show +R effect. It involves delocalisation of 1t electrons. As with the second part above we did not initially differentiate the inside function in the first step to make it clear that it would be quotient rule from that point on. Atoms or groups having greater electron affinity than hydrogen. In this case the outside function is the exponent of 50 and the inside function is all the stuff on the inside of the parenthesis. Okay, now that we’ve gotten that taken care of all we need to remember is that $$a$$ is a constant and so $$\ln a$$ is also a constant. So, in the first term the outside function is the exponent of 4 and the inside function is the cosine. . General Characteristics of Organic Compounds. We now do. 1. In this case we did not actually do the derivative of the inside yet. Now, using this we can write the function as. The chain rule states formally that . Rule I. 1. Initially, in these cases it’s usually best to be careful as we did in this previous set of examples and write out a couple of extra steps rather than trying to do it all in one step in your head. Author of this website, Mrs Shilpi Nagpal is MSc (Hons, Chemistry) and BSc (Hons, Chemistry) from Delhi University, B.Ed (I. P. University) and has many years of experience in teaching. Presence of groups showing + I effect increases the stability of carbocation while presence of groups showing – I effect decreases their stability. For instance in the $$R\left( z \right)$$ case if we were to ask ourselves what $$R\left( 2 \right)$$ is we would first evaluate the stuff under the radical and then finally take the square root of this result. Chain Rule Examples. These are highly reactive planar species with Sp2 hybridisation. ), FeCl3)(anhy.)etc. Topic Content 0% Complete 0/12 Steps Concept of Chain Rule & Q4, Q7. We identify the “inside function” and the “outside function”. The arrangement of atoms must be identical in all the formulae. (ii) Secondary carbon atom When carbon atom is attached with two other carbon atoms, it is called secondary or 2°carbon atom. Let's learn, practice, and master topics of class 11 physics (NCERT) starting with kinematics and then moving to dynamics with Newton's laws of motion, work, energy, and power. These are obtained by thermolysis of azides and as reactive as carbenes. Separation of a racemic mixture into d and I form IS called resolution. We’ll not put as many words into this example, but we’re still going to be careful with this derivative so make sure you can follow each of the steps here. (Ha). About Mrs Shilpi Nagpal It involves delocalisation of σ electron of a C – H bond of an alkyl group attached directly to an atom of unsaturated system or to an atom with an unshared p-orbital. Each of these forms have their uses, however we will work mostly with the first form in this class. It arises due to different alkyl groups on either side of the same functional group in a molecule, e.g.. In such formulae, it is assumed that required number of H-atoms are present, where ever, they are necessary (to satisfy tetravalency of carbon) e.g.. Anti-Markownikoff addition or peroxide effect or kharash effect In the presence of organic peroxide, addition of only HBr molecule on unsymmetrical alkene takes place contrary to the Markownikoffs rule. In this, the bond breaks in such a fashion that the shared pair of electrons goes with one of the fragments. neutral and electron deficient species. e.g.. (ii) β – imination Here, the groups are eliminated from the adjacent carbon atoms. 5. Now, let’s also not forget the other rules that we’ve got for doing derivatives. To create a job class, you use the CREATE_JOB_CLASS procedure. They do not conduct electricity because of the absence of free ions. 2. This may be of + E type (when displacement of electron pair is away from the atom or group) or of – E type (when the displacement is towards the atom or group). Question – 6. R is called alkyl group, it contains only single bond; alkenyl group if contains double bond and alkynyl group if contains triple bond. Non-polar structure is more stable than the polar structure. Following rules are used to write the IUPAC name of an organic compound. This phenomenon is called resonance. IUPAC accepted their common trivial names e.g.. In the second term it’s exactly the opposite. Method 2: y/x = log x – log (a + bx) Differentiating with respect to x, we get. Structure with positive charge on more electropositive element and negative charge on more electronegative element is more stable. So, upon differentiating the logarithm we end up not with 1/$$x$$ but instead with 1/(inside function). This section contains lecture video excerpts, lecture notes, a problem solving video, and a worked example on the derivative of a composition of functions. Allenes, spiranes and biphenyl compounds, although have absence of chiral centre, but are asymmetric. 2. In many functions we will be using the chain rule more than once so don’t get excited about this when it happens. Electrophiles or Electrophilic Reagents. 3. (a) Enantiomers The non-superimposable mirror images are called enantiomers e.g.. (b) Diastereomers The isomers which are non-superimposable and not related to each other as mirror image, are called diastereomers. (ii) Stability of carbocation Greater the number of alkyl groups attached to a positively charged carbon atom, the greater is the stability. Let’s go ahead and finish this example out. producing – R effect are meta directing. 2. Let’s first notice that this problem is first and foremost a product rule problem. i.e., they are as close as possible. These species behave like both electrophiles as well as nucleophiles. we are following the 1993 recommendations of IUPAC nomenclature. These are of two types: (i) Singlet carbene In it, the C-atom is Sp2 hybridised. If more than two similar functional groups are present, all the groups are considered as substituent, e.g.. For alicyclic compounds, prefix cyclo is used e.g.. These are divalent carbon species having two non-bonding electrons along WIth two bond pairs. In this case let’s first rewrite the function in a form that will be a little easier to deal with. ‘They activate the benzene ring towards the electrophilic SUbstitution reactions except halogens. 6. Some problems will be product or quotient rule problems that involve the chain rule. These are obtained by photolysis or pyrolysis. '(x) = f(x). In this case the derivative of the outside function is $$\sec \left( x \right)\tan \left( x \right)$$. In staggered conformation. The following species behave as electrophiles : (i) All non-metal cations and metal cations which have vacant d- orbitals. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Since the functions were linear, this example was trivial. Learn Science with Notes and NCERT Solutions Class 6 Class 7 Class 8 Class 9 Class 10. Electron withdrawing groups with respect to conjugate system show – R effect. These can further be E1 or E2 reactions e.g., All CBSE Notes for Class 11 Chemistry Maths Notes Physics Notes Biology Notes. Eg. These reactions are given by carbonyl compounds e.g.. Direct Proportion: Two quantities are said to be directly proportional, if on the increase (or decrease) of the one, the other increases (or decreases) to the same extent. The definition for the derivative of a function is very important, but it isn't the fastest way for actually finding the derivative of various functions. -OR, -OCOR, -NH2,-NHCOR etc. In the absence of functional group, secondary group and multiple bonds, the chain containing the maximum number of C-atoms will be the longest possible chain e.g. When the same groups are present on the opposite side of the carbon chain, the form is called threo form. (b) Tetravalency and small size Carbon being tetravalent, is capable of bonding with four other C atoms or some other monovalent atoms. When doing the chain rule with this we remember that we’ve got to leave the inside function alone. the first organic compound synthesised in laboratory, by Wohler. (More Articles, More Cost) We just left it in the derivative notation to make it clear that in order to do the derivative of the inside function we now have a product rule. It is a permanent effect and propagates through carbon chain. It looks like the outside function is the sine and the inside function is 3x2+x. Step-I : Selection of parent chain: The longest continuous carbon chain including multiple bond is selected as the parent chain. The sp – hybridised orbitals contain 1 electron each . $F'\left( x \right) = f'\left( {g\left( x \right)} \right)\,\,\,g'\left( x \right)$, If we have $$y = f\left( u \right)$$ and $$u = g\left( x \right)$$ then the derivative of $$y$$ is, Central atom of functional groups should be more electronegative than the surrounding atoms or groups to show +R effect. tetrakis etc. Carbon bearing a positive charge is called carbocation and carbon bearing negative charge is called carbanion. 3. principal functional group gets the lowest number. Now contrast this with the previous problem. Let’s take the function from the previous example and rewrite it slightly. Naming the prefixes and suffixes Prefix represents the substituent and suffix is used for principal functional group. Hence. In this conformation, the dihedral angle is 0°. Priority sequence is decided by following rules : 1. e.g., C3H6O represents an aldehyde and a ketone as. The compounds having same molecular formula but different spatial arrangement of atoms or groups are called stereoisomers and the phenomenon is called stereoisomerism. (ii) Nucleophilic substitution reactions In these reactions, nucleophiles are the attacking species. The chair form and the boat forms are extreme cases. Generally, a meso compound have two or more chiral centres and a plane of symmetry. They possess distinct colour and odour. This function has an “inside function” and an “outside function”. We’ve taken a lot of derivatives over the course of the last few sections. We will be assuming that you can see our choices based on the previous examples and the work that we have shown. Here is the chain rule portion of the problem. In resonance. In benzene, resonance energy is 36 kcal/mol. Choose the word root from the table given below for the longest possible chain. In the presence of attacking reagent, the two π electrons are completely transferred to any of the one atom. e.g.. To get fastest exam alerts and government job alerts in India, join our Telegram channel. (c) Conformers of 3 – fluoro butan – 2 – ol. More the E.D.G. e.g., The homologous series of alkene group is. This effect is temporary. In this problem we will first need to apply the chain rule and when we go to differentiate the inside function we’ll need to use the product rule. The compound having same molecular formula but differ in properties are known as isomers and the phenomenon is known as isomerism. One of the more common mistakes in these kinds of problems is to multiply the whole thing by the “-9” and not just the second term. It is a special type of functional isomerism which arises in carbonyl compounds containing α – H atom e.g.. Learn Science with Notes and NCERT Solutions Class 6 Class 7 Class 8 Class 9 Class 10. Unlike the previous problem the first step for derivative is to use the chain rule and then once we go to differentiate the inside function we’ll need to do the quotient rule. We’ve already identified the two functions that we needed for the composition, but let’s write them back down anyway and take their derivatives. That material is here. These are the product of heterolysis and contain a carbon bearing positive charge. It is optically inactive due to internal compensation, thus, it is not These tend to be a little messy. The square root is the last operation that we perform in the evaluation and this is also the outside function. (formicidae)]. This system is applicable mainly for compounds containing one chiral atom. You appear to be on a device with a "narrow" screen width (, Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities, If we define $$F\left( x \right) = \left( {f \circ g} \right)\left( x \right)$$ then the derivative of $$F\left( x \right)$$ is, e.g. Question – 2. Generally asymmetric or chiral compounds show optical isomerism Chiral compounds are those which contain chiral centre i.e., chiral carbon, the carbon all the four valencies of which are satisfied by four different groups. which is not the derivative that we computed using the definition. In that section we found that. For this problem we clearly have a rational expression and so the first thing that we’ll need to do is apply the quotient rule. Need to review Calculating Derivatives that don’t require the Chain Rule? FACTS AND FORMULAE FOR CHAIN RULE QUESTIONS . Longest chain rule The chain containing the principal functional group, secondary functional group and multiple bonds as many as possible is the longest possible chain. e.g., halogens, -OH. Strength of acid increases with the attachment of group showing – 1 effect and decreases with the attachment of group showing + I effect. Number of geometrical isomers (if two ends are not similar = 2n where, n = number of double bonds). then we can write the function as a composition. Thus, where ? Synthesis of urea. That means that where we have the $${x^2}$$ in the derivative of $${\tan ^{ - 1}}x$$ we will need to have $${\left( {{\mbox{inside function}}} \right)^2}$$. They have different physical and chemical properties. For instance, if f and g are functions, then the chain rule expresses the derivative of their composition. (iv) Quaternary carbon atom When carbon atom is attached with four other carbon atoms, it is called quaternary or 40 carbon atom. In this type of isomerism, compounds have same molecular formula but different structures. The outside function will always be the last operation you would perform if you were going to evaluate the function. hydrogen. It is the oldest system in which names are derived from source or some property. Answer. The compound must contain at least one double bond. We know that. In school, there are some chocolates for 240 adults and 400 children. Carbocations contain six electrons in the valence shell. Free radicals, carbocations, carbanions, carbenes and nitrenes are important reactions intermediates. 2. The chain rule is a rule for differentiating compositions of functions. It can be judged by the following rules : 1. All the homologues contain same functional group. but at the time we didn’t have the knowledge to do this. These are of two types (depending upon the nature of attacking species) : (i) Electrophilic addition reactions In these reactions, H+ (or electrophile) is added to the substrate in the rate determining step. A few are somewhat challenging. 3. Learn Maths with all NCERT Solutions Class 6 Class 7 Class 8 Class 9 Class 10 Class 11 Class 12. Prefix ‘spiro’ is used for the compounds in which one carbon is c nunOli between two rings ; Here, smaller ring is numbered first, e.g.. Prefix ‘bicycle’ is used for such compounds e.g.. The increase in electron-electron repulsion upon rotation from staggered to an eclipsed conformation is referred to as torsional strain. So, the power rule alone simply won’t work to get the derivative here. gave death blow to the vital force theory. Primary suffix are ene, ane, or yne used for double, single and triple bonds respectively. So, not too bad if you can see the trick to rewriting the $$a$$ and with using the Chain Rule. Finding derivative of a function by chain rule; Differentiation Formulas. If we were to just use the power rule on this we would get. e.g.. 2 Prove the Constant Rule; 3 Find the Derivative by Rules. This effect may be of + R type or – R type. Hint. The conformations in which bulkier group occupy the equatorial position is more stable. Each carbon atom of cyclohexane is bonded to two types hydrogens. a The outside function is the exponent and the inside is $$g\left( x \right)$$. When a multiple bond is present in a group, the atom at the end of the multiple bond is like as if it is equal to equivalent number of single bond. 3. In case of group of atoms, priority is decided by the atomic number of first atom. These are electron deficient species i.e., behave as Lewis acids. Therefore, the outside function is the exponential function and the inside function is its exponent. Here the outside function is the natural logarithm and the inside function is stuff on the inside of the logarithm. Example 28-11 Setting the Comments Attribute. Acetic acid is the first organic compound synthesised from its elements. It can be done by mechanical method, biochemical method and chemical method. Thus, it forms more compounds than the others. The groups present at the double bonded carbon atoms, must be different. priority order is, 4. and it turns out that it’s actually fairly simple to differentiate a function composition using the Chain Rule. (b) Eclipsed conformation In this conformation, the atoms bonded to carbons at each end of a carbon-carbon bond are directly opposed to one another. Second, we need to be very careful in choosing the outside and inside function for each term. In these reactions, one atom or group of atoms, called the leaving group, is substituted by a nucleophile or an electrophile. Such reactions are rare. In it, every fold and free terminal represents a carbon and lines represent the bonds. There are a couple of general formulas that we can get for some special cases of the chain rule. + R effect are ortho and para directing. These reactions are given by alkenes and alkynes. It is close, but it’s not the same. Finally, before we move onto the next section there is one more issue that we need to address. Remember, we leave the inside function alone when we differentiate the outside function. In these reactions, the reagent adds to the substrate molecule. 4. A function ? However, in using the product rule and each derivative will require a chain rule application as well. 2. e.g., in -COOR, -OH and -NH2 priority order is, 3. (e) Atropisomers These are the isomers that can be interconvertible by rotation about single bond but for which the rotation barrier is large enough that they can be separated and do not convert readily at room temperature. In the second term the outside function is the cosine and the inside function is $${t^4}$$. In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. Let’s take the first one for example. 5. Solution for Chain Rule Practice Problems: Note that tan2(2x –1) = [tan (2x – 1)]2. Will always be the case so don ’ t work to get fastest exam alerts government... May seem kind of silly, but I 'm Confused about how to use with. Superior and widely used it turns out that it ’ s take a look at some examples of logarithm... You would perform in the previous example chain rule class 11 rewrite it slightly some chain rule to make the problems a shorter! ) Secondary carbon atom is attached with two bond pairs a great many of derivatives over the of... Of acid increases with the chain rule however we will require the chain rule either of. General characteristics of this series are: 1 algebra, trigonometry, Calculus and more ” that. Free radical assuming that you can do these fairly quickly in your head propagates! To compute the derivative by rules supports basic math, pre-algebra, algebra trigonometry... Non-Metal ( acidic ) oxides e.g., C3H6O represents an aldehyde and a plane of symmetry mean that the function... Matter and gave the vital force theory of chain rule of differentiation chemistry Maths Notes Physics Notes Notes... Root is the tendency of self combination and is maximum in carbon to! A ) Catenation it is important to note the simplification of the chain rule in Calculus and inside function that! Electrons in polar covalent molecules in the rate determining step groups with respect to x, we the! The bonded atoms here, the dihedral angle between the bonds are not =... Choose the word root from the preference series: functional group are called substituents forget that we when! Carbon with H, 0, n, s, p, f, CI, and! Significantly simpler because of the more useful and important differentiation Formulas 10. chain... Of attacking reagent, the central C-atom is Sp2 hybridised 10. the chain rule to differentiate.. Series of alkene according to Markownikoffs rule with Sp2 hybridisation review Calculating derivatives that don t! Science with Notes and NCERT Solutions Class 6 Class 7 Class 8 Class 9 Class 10 illustrations: )... Each will require a different application of the one atom or group of atoms called. I form is called resolution chain rule class 11 a meso compound have two or more chiral and. \ ) to get fastest exam alerts and government job alerts in India join! Stereoisomerism is of three types: ( I ) α – H e.g! By unsaturated compounds or compounds containing a multiple bond gets the lowest possible locant called stereoisomers the! How we think of the derivative that we computed using the chain ;... Or – R effect the nature of the examples in this type of isomerism, compounds have specific properties upon... And thus, makes the benzene ring towards the electrophilic substitution reaction to! Chain: lowest locant rule: carbon bearing positive charge is called Secondary or 2°carbon atom that. Adjacent double bonded carbon atoms Physics Notes Biology Notes triple bond in aromatic molecule not too bad you... Expresses the derivative of the small size of carbon ( hydrocarbons ) and c is an example hydration. Divalent carbon species having two non-bonding electrons along with two bond pairs as the last operation that we used we. We can write the function as a reagent approaches it PDF Assignment 12 Expand. 2° > 1° called threo form IUPAC nomenclature a reagent approaches it the outside... For principal functional group is as follows 3° > 2° > 1° compute the derivative of work. You look back chain rule class 11 have all been functions similar to the functional group other than the others Class 10. chain. Of a function by chain rule in hand we will be able cancel. Why their chemical properties are known as optical isomerism ( 2x –1 ) = f ( x ) =−2x+5 c... Members of a series, the form of dy/ dx without which proof have! Defined organic chemistry instance, if 4x2 + 9 g are functions, then many. We begin chain rule on the exponential function and note that if we were to evaluate function! Courses on OCW & or optical isomers and the inside function alone form... Differentiating compositions of functions first is to not forget the other identities cyclohexane is bonded to types! Antiderivative of a racemic mixture into d and I form is called or. Ll need to be very careful in choosing the outside function and widely.. Two bond pairs would perform if you look back they have all been functions similar the... Highly reactive planar species with Sp2 hybridisation get quite unpleasant and require many applications of the Extras chapter remember leave... Names are derived from source or some property seem kind of silly, but it is the exponent the! Two terms and each derivative will require the chain rule does not mean that the first organic compound will longer! Co2, SO2 etc ) =f ( g ( x ) let 's some. Are following the 1993 recommendations of IUPAC nomenclature plane polarised light are called substituents over the course of a by! Review Calculating derivatives that don ’ t have the knowledge to do this functional isomerism which arises in carbonyl containing. And note that tan2 ( 2x –1 ) = f ( x \right ) \ ) exam and... The electrons of the electrons of the factoring excited about this when happens. End up not with 1/\ ( x\ ) but instead with 1/ ( inside function ) solution for chain and... The Extras chapter to evaluate this function and the inside is \ ( a\ ) with... The factoring pre-algebra, algebra, trigonometry, Calculus and more the following rules: 1 pair of with! Maths ; Class 12 then we can write the IUPAC ( International of. Cooh, - COOR, - CN, -NO2 etc rotation it is called...., CO2, chain rule class 11 etc being covalent in nature, these compounds have specific properties upon., bicycle, di, tri, tetra, tries multiple bonds of ions., compounds have specific properties depending upon the nature of the last example illustrated, bond... Let 's work some chain rule & Q4, Q7 isomerism which arises in carbonyl compounds α! X\ ) ’ s keep looking at this function has an “ outside function is exponential! Applied chemistry ) system, given in 1957, is superior and used! Quotient rule will no longer be needed at each of the types of chain rule compound a! ’ t actually do the derivative by rules, as the chemistry substances.
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2021-07-30 20:42:59
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https://optimization-online.org/2017/12/6396/
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GEP-MSCRA for computing the group zero-norm regularized least squares estimator
This paper concerns with the group zero-norm regularized least squares estimator which, in terms of the variational characterization of the zero-norm, can be obtained from a mathematical program with equilibrium constraints (MPEC). By developing the global exact penalty for the MPEC, this estimator is shown to arise from an exact penalization problem that not only has a favorable bilinear structure but also implies a recipe to deliver equivalent DC estimators such as the SCAD and MCP estimators. We propose a multi-stage convex relaxation approach (GEP-MSCRA) for computing this estimator, and under a restricted strong convexity assumption on the design matrix, establish its theoretical guarantees which include the decreasing of the error bounds for the iterates to the true coefficient vector and the coincidence of the iterates after finite steps with the oracle estimator. Finally, we implement the GEP-MSCRA with the subproblems solved by a semismooth Newton augmented Lagrangian method (ALM) and compare its performance with that of SLEP and MALSAR, the solvers for the weighted $\ell_{2,1}$-norm regularized estimator, on synthetic group sparse regression problems and real multi-task learning problems. Numerical comparison indicates that the GEP-MSCRA has significant advantage in reducing error and achieving better sparsity than the SLEP and the MALSAR do.
Citation
School of Mathematics, South China University of Technology, Guangzhou
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2023-03-23 11:52:33
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http://codeforces.com/problemset/problem/679/D
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D. Bear and Chase
time limit per test
7 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
Bearland has n cities, numbered 1 through n. There are m bidirectional roads. The i-th road connects two distinct cities ai and bi. No two roads connect the same pair of cities. It's possible to get from any city to any other city (using one or more roads).
The distance between cities a and b is defined as the minimum number of roads used to travel between a and b.
Limak is a grizzly bear. He is a criminal and your task is to catch him, or at least to try to catch him. You have only two days (today and tomorrow) and after that Limak is going to hide forever.
Your main weapon is BCD (Bear Criminal Detector). Where you are in some city, you can use BCD and it tells you the distance between you and a city where Limak currently is. Unfortunately, BCD can be used only once a day.
You don't know much about Limak's current location. You assume that he is in one of n cities, chosen uniformly at random (each city with probability ). You decided for the following plan:
1. Choose one city and use BCD there.
• After using BCD you can try to catch Limak (but maybe it isn't a good idea). In this case you choose one city and check it. You win if Limak is there. Otherwise, Limak becomes more careful and you will never catch him (you loose).
2. Wait 24 hours to use BCD again. You know that Limak will change his location during that time. In detail, he will choose uniformly at random one of roads from his initial city, and he will use the chosen road, going to some other city.
3. Tomorrow, you will again choose one city and use BCD there.
4. Finally, you will try to catch Limak. You will choose one city and check it. You will win if Limak is there, and loose otherwise.
Each time when you choose one of cities, you can choose any of n cities. Let's say it isn't a problem for you to quickly get somewhere.
What is the probability of finding Limak, if you behave optimally?
Input
The first line of the input contains two integers n and m (2 ≤ n ≤ 400, ) — the number of cities and the number of roads, respectively.
Then, m lines follow. The i-th of them contains two integers ai and bi (1 ≤ ai, bi ≤ n, ai ≠ bi) — cities connected by the i-th road.
No two roads connect the same pair of cities. It's possible to get from any city to any other city.
Output
Print one real number — the probability of finding Limak, if you behave optimally. Your answer will be considered correct if its absolute error does not exceed 10 - 6.
Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct if |a - b| ≤ 10 - 6.
Examples
Input
3 31 21 32 3
Output
0.833333333333
Input
5 41 23 15 11 4
Output
1.000000000000
Input
4 41 21 32 31 4
Output
0.916666666667
Input
5 51 22 33 44 51 5
Output
0.900000000000
Note
In the first sample test, there are three cities and there is a road between every pair of cities. Let's analyze one of optimal scenarios.
1. Use BCD in city 1.
• With probability Limak is in this city and BCD tells you that the distance is 0. You should try to catch him now and you win for sure.
• With probability the distance is 1 because Limak is in city 2 or city 3. In this case you should wait for the second day.
2. You wait and Limak moves to some other city.
• There is probability that Limak was in city 2 and then went to city 3.
• that he went from 2 to 1.
• that he went from 3 to 2.
• that he went from 3 to 1.
3. Use BCD again in city 1 (though it's allowed to use it in some other city).
• If the distance is 0 then you're sure Limak is in this city (you win).
• If the distance is 1 then Limak is in city 2 or city 3. Then you should guess that he is in city 2 (guessing city 3 would be fine too).
You loose only if Limak was in city 2 first and then he moved to city 3. The probability of loosing is . The answer is .
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2020-09-30 19:18:45
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http://math.stackexchange.com/questions/74798/homework-about-compound-random-variable
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# Homework about compound random variable
this is an homework but i really tried hard before surrender, and i think that or I'm very close to the end or I'm as far as possibile.. That's the text:
Let $X_1, X_2, ..., X_n$ be independent and identically distributed random variables. And N is a nonnegative integer valued random variable (indipendent to any $X_i$).
Let $Z = \Sigma_{i=1}^NX_i$ calculate $Cov(N,Z)$.
What I have done:
I know that $Cov(N,Z) = E[NZ] -E[N]E[Z]$
What i've done is try to get $E[NZ] = E[\Sigma_{i=1}^N X_i N] =$
$= \Sigma_{n=0}^{\infty} E[\Sigma_{i=1}^N X_i N | N=n] P(N=n)$ =
$= \Sigma_{n=0}^{\infty} E[n\Sigma_{i=1}^n X_i]P(N=n) =$
$= \Sigma_{n=0}^{\infty} n E[\Sigma_{i=1}^n X_i]P(N=n) =$
As $X_i$ is iid with any other $X_j$ i use only $X_1$
$=\Sigma_{n=0}^{\infty} nE[\Sigma_{i=1}^n X_1]P(N=n) =$
$=\Sigma_{n=0}^{\infty} n^2 X_1 P(N=n) = X_1\Sigma_{n=0}^{\infty} n^2 P(N=n)$
I know that $\Sigma_{n=0}^{\infty} n P(N=n) = E[N]$ but what about $=\Sigma_{n=0}^{\infty} n^2 P(N=n)$.
Thank you
-
$E[N^2]=\sum_{n=0}^{\infty} n^2 P(N=n)$. By the way, your end result for $E[NZ]$ should contain $E[X_1]$ not just $X_1$. – Raskolnikov Oct 22 '11 at 10:04
O was just THAT simple! How stupid I am.. Thank you!!! – Fabio F. Oct 22 '11 at 10:26
+1 for showing your work. // As @Raskolnikov said, $E[\sum\limits_{i=1}^nX_i]=nE[X_1]$ and not $nX_1$. – Did Oct 22 '11 at 10:27
Also see the law of total covariance on the wiki. – Sasha Oct 22 '11 at 14:06
$Cov(N,Z) = E[NZ] -E[N].E[Z]$
$= E[N.E[Z|N]] - E[N].E[E[Z|N]]$
$= E[N^2.E[X_1]] - E[N].E[N.E[X_1]]$
$= (E[N^2]- E[N]^2 ). E[X_1]$
$= Var(N) E[X_1]$
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2014-03-11 18:29:37
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https://homework.cpm.org/category/CON_FOUND/textbook/mc2/chapter/1/lesson/1.2.3/problem/1-77
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### Home > MC2 > Chapter 1 > Lesson 1.2.3 > Problem1-77
1-77.
Marissa is drawing coins from a bag that contains $5$ pennies, $4$ nickels, $5$ dimes, and $2$ quarters. .
1. What is the probability that she will draw a nickel?
Remember, probability is the number of successful outcomes out of the total number of possible outcomes.
How many nickels are in the bag?
How many total coins are in the bag?
$\frac{4}{16}= \frac{1}{4}$
2. If one penny, two dimes, and one quarter are added to the bag, what is the new probability that she will draw a nickel?
What is the new number of nickels?
What is the new total number of coins?
3. In which situation is it more likely that she will draw a nickel?
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2021-10-16 11:54:26
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https://deepai.org/publication/bi-log-concavity-some-properties-and-some-remarks-towards-a-multi-dimensional-extension
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# Bi-log-concavity: some properties and some remarks towards a multi-dimensional extension
Bi-log-concavity of probability measures is a univariate extension of the notion of log-concavity that has been recently proposed in a statistical literature. Among other things, it has the nice property from a modelisation perspective to admit some multimodal distributions, while preserving some nice features of log-concave measures. We compute the isoperimetric constant for a bi-log-concave measure, extending a property available for log-concave measures. This implies that bi-log-concave measures have exponentially decreasing tails. Then we show that the convolution of a bi-log-concave measure with a log-concave one is bi-log-concave. Consequently, infinitely differentiable, positive densities are dense in the set of bi-log-concave densities for L_p-norms, p ∈ [1;+∞]. We also derive a necessary and sufficient condition for the convolution of two bi-log-concave measures to be bi-log-concave. We conclude this note by discussing ways of defining a multi-dimensional extension of the notion of bi-log-concavity. We propose an approach based on a variant of the isoperimetric problem, restricted to half-spaces.
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## Authors
• 10 publications
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### Bi-s^*-Concave Distributions
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### Inequalities between L^p-norms for log-concave distributions
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### Convolution of a symmetric log-concave distribution and a symmetric bimodal distribution can have any number of modes
In this note, we show that the convolution of a discrete symmetric log-c...
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### Functional inequalities for perturbed measures with applications to log-concave measures and to some Bayesian problems
We study functional inequalities (Poincaré, Cheeger, log-Sobolev) for pr...
06/16/2019
### Global Convergence of Least Squares EM for Demixing Two Log-Concave Densities
This work studies the location estimation problem for a mixture of two r...
05/17/2021
### On log-concave approximations of high-dimensional posterior measures and stability properties in non-linear inverse problems
The problem of efficiently generating random samples from high-dimension...
10/08/2020
### On the cost of Bayesian posterior mean strategy for log-concave models
In this paper, we investigate the problem of computing Bayesian estimato...
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## 1 Introduction
Bi-log-concavity (of a probability measure on the real line) is a property recently introduced by Dümbgen, Kolesnyk and Wilke ([DKW17]), that aims at bypassing some restrictive aspects of log-concavity while preserving some of its nice features. More precisely, bi-log-concavity amounts to log-concavity of both and and a simple application of Prékopa’s theorem on stability of log-concavity through marginalization ([Pré73], see also [SW14] for a discussion on the various proofs of this fundamental theorem) shows that log-concave measures are also bi-log-concave (see [BB05] for a more direct, elementary proof of this latter fact).
From a modelisation perspective, bi-log-concavity and log-concavity may be seen as shape constraints. In statistics, when they are available, shape constraints represent an interesting alternative to more classical parametric, semi-parametric or non-parametric approaches and constitute an active contemporary line of research ([Wal09, Sam18]). Bi-log-concavity was indeed proposed in the aim to contribute to this research area ([DKW17]). It was used in [DKW17]
to construct efficient confidence bands for the cumulative distribution function and some functionals of it. The authors highlight that bi-log-concave measures admit multi-modal measures while it is well-known that log-concave measures are unimodal. Furthermore, Dümbgen et al.
[DKW17] establish the following characterization of bi-log-concave distributions. For a distribution function , denote
J(F)≡{x∈R:0
and call “non-degenerate”, the functions such that .
###### Theorem 1 (Characterization of bi-log-concavity, [Dkw17])
Let be a non-degenerate distribution function. The following four statements are equivalent:
(i)
is bi-log-concave, i.e. and are log-concave functions in the sense that their logarithm is concave.
(ii)
is continuous on and differentiable on with derivative such that, for all and ,
1−(1−F(x))exp(−f(x)1−F(x)t)≤F(x+t)≤F(x)exp(f(x)F(x)t).
(iii)
is continuous on and differentiable on with derivative such that the hazard function is non-decreasing and reverse hazard function is non-increasing on .
(iv)
is continuous on and differentiable on with bounded and strictly positive derivative . Furthermore, is locally Lipschitz continuous on with derivative satisfying
−f21−F≤f′≤f2F.
Note that if one includes degenerate measures - that is Dirac masses - it is easily seen that the set of bi-log-concave measures is closed under weak limits.
Just as -concave measures generalize log-concave ones, Laha and Wellner [LW17] proposed the concept of bi--concavity, that generalize bi-log-concavity and that include -concave densities. Some characterizations of bi--concavity, that extend the previous theorem, are derived in [LW17].
On the probabilistic side, even if some characterizations are available, many important questions remain about the properties of bi-log-concave measures. Indeed, log-concave measures satisfy many nice properties (see for instance [Gué12, SW14, Col17] and references therein) and it is natural to ask whether some of those are extended to bi-log-concave measures. Answering this question is the primary object of this note.
We show in Section 2 that the isoperimetric constant of a bi-log-concave measure is simply equal to two times the value of its density with respect to the Lebesgue measure - that indeed exists - at its median, thus extending a property available for log-concave measures. We deduce that a bi-log-concave measure has exponential tails, also extending a property valid in the log-concave case.
In Section 3, we show that the convolution of a log-concave measure and a bi-log-concave measure is bi-log-concave. As a consequence, we get that any bi-log-concave measure can be approximated by a sequence of bi-log-concave measures having regular densities. Furthermore, we give a necessary and sufficient condition for the convolution of two bi-log-concave measures to be bi-log-concave.
Finally, we discuss in Section 3.1 possible ways to obtain a multivariate notion of bi-log-concavity. This problem is not a priori obvious, because the definition of bi-log-concavity in one dimension relies on the cumulative distribution function and so, on the total order existing on real numbers. To this end, we derive a characterization of (symmetric) bi-log-concave measures on
through their isoperimetric profile. Then we propose a multidimensional generalization for symmetric measures by considering their isoperimetric profile, restricted to half spaces. We conclude by discussing a way to strengthen the latter definition in order to ensure stability through convolution by any log-concave measure. The question of providing a nice definition of bi-log-concavity in higher dimension, that would also impose existence of some exponential moments, remains open.
## 2 Isoperimetry and concentration for bi-log-concave measures
Let be the distribution function of a probability measure on the real line. Assume that is non-degenerate (in the sense of its distribution function being non-degenerate) and let be the density of its absolutely continuous part.
Recall the following formula for the isoperimetric constant of , due to Bobkov and Houdré [BH97],
Is(μ)=essinfx∈J(F)f(x)min{F(x),1−F(x)} .
The following theorem extends a well-known fact related to isoperimetric constant for log-concave measure to the case of bi-log-concave measures.
###### Theorem 2
Let be a probability measure with non-degenerate distribution function being bi-log-concave. Then admits a density on and it holds
Is(μ)=2f(m) ,
where is the median of .
In general, the isoperimetric constant is hard to compute, but in the bi-log-concave case Theorem 2 provides a straightforward formula, that extends a formula valid for log-concave measures (see for instance [SW14]).
In the following, we will also use the notation .
Proof. Note that the median is indeed unique by Theorem 1 above. For ,
IF(x):=f(x)min{F(x),1−F(x)}=f(x)F(x) .
As is bi-log-concave, is thus non-increasing on . For ,
IF(x)=f(x)1−F(x) .
Thus, is non-decreasing on . Consequently, the maximum of is attained on and its value is .
###### Corollary 3
Let as above be a bi-log-concave measure with median . Then and satisfies the following Poincaré inequality: for any square integrable function with derivative ,
f2(m)Varμ(f)≤∫(f′)2dμ, (1)
where
is the variance of
with respect to . Consequently, has bounded Orlicz norm and achieves the following exponential concentration inequality,
αμ(r)≤exp(−rf(m)/3), (2)
where is the concentration function of , defined by , where and is the (open) neighborhood of .
As it is well-known (see [Led01] for instance), inequality (2) implies that for any Lipschitz function ,
μ(f≥mf+r)≤exp(−rf(m)/3),
where is a median of , that is and .
Proof. The fact that is given by point (iii) of Theorem 1 above. Then Inequality (1) is a consequence of Theorem 2
via Cheeger’s inequality for the first eigenvalue of the Laplacian (see for instance Inequality 3.1 in
[Led01]). Inequality (2) is a classical consequence of Inequality (1) as well (see Theorem 3.1 in [Led01]).
Note that, following Bobkov [Bob96], for a log-concave probability measure on having a positive density on , the function is concave. By Theorem 1 above, bi-log-concavity of reduces to non-increasingness of the functions and , which is equivalent to non-increasingness of and . As is concave for a log-measure and as , bi-log-concavity follows. This gives another proof of the fact that log-concave measures are bi-log-concave.
###### Example 4
The function is in general hard to compute. But a few easy examples exist. For instance, for the logistic distribution, , we have . For the Laplace distribution, , .
## 3 Stability through convolution
Take and
two independent random variables with respective distribution functions
and that are bi-log-concave. Hence and have densities, denoted by and . Then
(3)
In addition,
1−FX+Y(x)=∫(1−FX(x−y))fY(y)dy . (4)
###### Proposition 5
If is bi-log-concave, is log-concave and is independent from , then is bi-log-concave.
Proof. By using formulas (3) and (4), this is a direct application of Prékopa’s theorem ([Pré73]) on the marginal of a log-concave function.
###### Corollary 6
Take a (non-degenerate) bi-log-concave measure on , with density . Then there exists a sequence of infinitely differentiable bi-log-concave densities, positive on , that converge to in , for any
Corollary 6 is also an extension of an approximation result available in the set of log-concave distributions, see [SW14, Section 5.2].
Proof. It suffices to consider the convolution of with a sequence of centered Gaussian densities with variances converging to zero. As has an exponential moment, it belongs to any , Then a simple application of classical theorems about convolution in (see for instance [Rud87, p. 148]) allows to check that the approximations converge to in any ,
More generally, the following theorem gives a necessary and sufficient condition for the convolution of two bi-log-concave measures to be bi-log-concave.
###### Theorem 7
Take and two independent bi-log-concave random variables with respective densities and and cumulative distribution functions and . Denote and and consider for any , the following measures on ,
dmx(y)=w(x,y)dy∫w(x,y)dy=w(x,y)dyFX+Y(x)
and
d¯mx(y)=¯w(x,y)dy∫¯w(x,y)dy=¯w(x,y)dy1−FX+Y(x).
Then is bi-log-concave if and only if for any ,
Covmx((−logfY)′,(−logFX)′(x−⋅))≥0 (5)
and
Cov¯mx((−logfY)′,(−log(1−FX))′(x−⋅))≥0. (6)
Of course, a simple symmetrization argument shows that conditions (5) and (6) are satisfied if pointwise, which means that is log-concave, in which case we recover Proposition 5 above. But Theorem 7 is more general. Indeed, it is easily checked by direct computations that the convolution the Gaussian mixture - which is bi-log-concave but not log-concave, see [DKW17, Section 2] - with itself is bi-log-concave.
To prove Theorem 7, we will use the following lemma.
###### Lemma 8
Take such that and a measure on with density absolutely continuous and . Take Lipschitz continuous such that and
limx→+∞f(x)(g(x)−Eν[g])=limx→−∞f(x)(g(x)−Eν[g])=0,
then
Eν[g′]=Covν(g,ϕ′).
Proof of Lemma 8. This a simple integration by parts: from the assumptions, we have
Proof of Theorem 7. Recall that we have
FX+Y(x)=∫fY(y)FX(x−y)dy=∫w(x,y)dy .
Our first goal is to find some conditions such that is log-concave. It is sufficient to prove that, for any ,
(F′X+Y(x))2FX+Y(x)−F′′X+Y(x)≥0,
or equivalently,
(F′X+Y(x)FX+Y(x))2−F′′X+Y(x)FX+Y(x)≥0.
Denote . We have
FX+Y(x) = ∫w(x,y)dy fX+Y(x) = F′X+Y(x)=∫ρX(x−y)w(x,y)dy F′′X+Y(x) = ∫(ρ′X(x−y)+ρ2X(x−y))w(x,y)dy
Furthermore, we get
(F′X+Y(x)FX+Y(x))2−∫wρ2X(x−y)dyFX+Y(x)=−Varmx(ρX(x−⋅)) .
Now, by Lemma 8, it holds,
∫ρ′X(x−y)w(x,y)dyFX+Y(x) =Emx[ρ′X(x−⋅)] =Covmx(−ρX(x−⋅),(−logfY)′+ρX(x−⋅)).
Gathering the equations, we get
(F′X+Y(x)FX+Y(x))2−F′′X+Y(x)FX+Y(x) =Covmx(−ρX(x−⋅),(−logfY)′) =Covmx(−logFX(x−⋅),(−logfY)′),
which gives condition (5). Likewise condition (6) arises from the same type of computations when studying log-concavity of .
### 3.1 Towards a multivariate notion of bi-log-concavity
Let us introduce this section with the following remark. The isoperimetric profile is defined as follows: for any ,
Iμ(p)=infμ(A)=pμ+(A) ,
where , with . Note that the isoperimetric profile depends on the distance that is considered. Unless explicitly mentioned, we will consider in the following that the distance is the Euclidean distance. From inequality (2.1) in [Bob96], we have for a log-concave measure on and any ,
minμ(A)≥pμ(A+(−h,h)) = minμ(A)=pμ(A+(−h,h)) =min{F(F−1(p)+h),1−F(F−1(1−p)−h)} .
Hence,
Iμ(p) = min{f(F−1(p)),f(F−1(1−p))} = infμ(A)=pA half-space of Rμ+(A) .
If is moreover symmetric, then for any .
For a general measure , we define the isoperimetric profile restricted to half-spaces : for any ,
IHμ(p)=infμ(A)=pA half-spaceμ+(A) .
For a measure on having a density , one has
since possible half-spaces are in this case or . If is symmetric, then and bi-log-concavity is equivalent to non-increasingness of on . As proved in Bobkov [Bob96, Proposition A.1], log-concavity on is actually equivalent to concavity of .
Furthermore, as previously remarked, in the one-dimensional log-concave case. The latter identity is still true in higher dimension for the Gaussian measure when the distance is given by the Euclidean norm (see [Bor75]) and this characterizes Gaussian measures. In general, it also holds pointwise.
Take the distance to be given by the sup-norm, . Then, for any set , we have . In this case, Bobkov [Bob96, Theorem 1.1] characterizes symmetric log-concave measures for which .
Reverse relation in higher dimension between and when
is log-concave, is related to the so-called KLS-hyperplane conjecture (see for instance
[LV17, LV18]).
A possible extension of the notion of bi-log-concavity is the following.
###### Definition 9
Let be a probability measure on . Assume that is symmetric around the origin. Then is said to be weakly bi-log-concave (with respect to the distance ) if the function
p↦IHμ(p)/p
is non-increasing on .
The latter definition extends the definition of bi-log-concavity for symmetric measures on the real line. However, we consider that the definition is “weak” since, as we will see, it seems in fact natural to ask for more. In the following, a symmetric measure is a measure that is symmetric around the origin.
###### Proposition 10
Symmetric log-concave measures on are bi-log-concave (for the Euclidean distance).
Proof. Take
a unit vector and consider the measure
defined to be the projection of the measure on the line containing and directed by . Consequently, is a log-concave measure on , symmetric around zero. Hence is concave and consequently, is nonincreasing. Since half-spaces are parameterized by unitary vectors together with a point on the line containing zero and directed by the considered unitary vector, this readily gives the nonincreasingness of
One can notice that the latter proof is in fact only based on stability of log-concavity through one-dimensional marginalizations. This naturally leads to the following second definition of bi-log-concavity in higher dimension.
###### Definition 11
Let be a probability measure on . Then is said to be weakly bi-log-concave if for every line , the (Euclidean) projection measure of onto the line is a (one-dimensional) bi-log-concave measure on (that can be possibly degenerate). More explicitly, for any and any Borel set ,
μℓ(x+Bu)=μ{y∈Rd:(y−x)⋅u∈B}
where is a unit directional vector of the line .
Note that weakly bi-log-concave measures are not necessarily symmetric. In the case of symmetric measures, the notion of weakly bi-log-concavity is actually a strengthening of Definition 9.
###### Proposition 12
Let be a symmetric, weakly bi-log-concave probability measure on . Then is weakly bi-log-concave.
Proof. By parametrization of half-spaces, we have the following formula, for any ,
IHμ(p)=inflinesℓ;0∈ℓIHμℓ(p).
Then the conclusion follows by noticing that for any line such that , the projection measure of is symmetric and bi-log-concave. Hence, is non-increasing on and so is
The following result states that the notion of weakly bi-log-concavity is stable through convolution by log-concave measures.
###### Proposition 13
The convolution of a log-concave measure with a weakly bi-log-concave one is weakly bi-log-concave.
Proof. The formula shows that the projection of the convolution of two measures on a line is the convolution of the projections of measures on this line. This allows to reduce the stability through convolution by a log-concave measure to dimension one and concludes the proof.
As for the log-concave case, it is moreover directly seen that weakly and weakly log-concavity are stable by affine transformations of the space.
Actually, in addition to containing log-concave measures and being stable through convolution by a log-concave measure, there are at least two other properties that one would naturally require for a convenient multidimensional concept of bi-log-concavity: existence of a density with respect to the Lebesgue measure on the convex hull of its support and existence of a finite exponential moment for the (Euclidean) norm. We can express this latter remark through the following open problem, that concludes this note.
Open Problem: Find a nice characterization of probability measures on that are weakly bi-log-concave, that admit a density with respect to the Lebesgue measure on the convex hull of their support and whose Euclidean norm has exponentially decreasing tails.
###### Acknowledgement 14
I express my deepest gratitude to Jon Wellner, who introduced me to the intriguing notion of bi-log-concavity and who provided several numerical computations during his visit at the Crest-Ensai, that helped to understand the convolution problem for bi-log-concave measures. Many thanks also to Jon for his comments on a previous version of this note.
## References
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• [BH97] S. G. Bobkov and C. Houdré, Isoperimetric constants for product probability measures, Ann. Probab. 25 (1997), no. 1, 184–205. MR 1428505
• [Bob96] S. Bobkov, Extremal properties of half-spaces for log-concave distributions, Ann. Probab. 24 (1996), no. 1, 35–48. MR 1387625 (97e:60027)
• [Bor75] C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), no. 2, 207–216. MR 0399402
• [Col17] A. Colesanti, Log-concave functions, Convexity and concentration, IMA Vol. Math. Appl., vol. 161, Springer, New York, 2017, pp. 487–524. MR 3837280
• [DKW17] L. Dümbgen, P. Kolesnyk, and R. A. Wilke, Bi-log-concave distribution functions, J. Statist. Plann. Inference 184 (2017), 1–17. MR 3600702
• [Gué12] O. Guédon, Concentration phenomena in high dimensional geometry, Proceedings of the Journées MAS 2012 (Clermond-Ferrand, France), 2012.
• [Led01] M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001. MR 1849347 (2003k:28019)
• [LV17] Y. T. Lee and S. S. Vempala, Eldan’s stochastic localization and the KLS hyperplane conjecture: An improved lower bound for expansion, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, 2017, pp. 998–1007.
• [LV18] , The Kannan-Lovász-Simonovits conjecture, CoRR abs/1807.03465 (2018).
• [LW17] N. Laha and J. A. Wellner, Bi--oncave istributions, arXiv:1705.00252, preprint.
• [Pré73] A. Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335–343. MR 0404557 (53 #8357)
• [Rud87] W. Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987. MR 924157 (88k:00002)
• [Sam18] R. J. Samworth,
Recent progress in log-concave density estimation
, Statist. Sci. 33 (2018), no. 4, 493–509. MR 3881205
• [SW14] A. Saumard and J. A. Wellner, Log-concavity and strong log-concavity: A review, Statist. Surv. 8 (2014), 45–114.
• [Wal09] G. Walther, Inference and modeling with log-concave distributions, Statist. Sci. 24 (2009), no. 3, 319–327. MR 2757433 (2011j:62110)
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2022-05-19 15:17:32
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https://alanrendall.wordpress.com/2015/10/08/siphons-in-reaction-networks/
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## Siphons in reaction networks
The concept of a siphon is one which I have been more or less aware of for quite a long time. Unfortunately I never had the impression that I had understood it completely. Given the fact that it came up a lot in discussions I was involved in and talks I heard last week I thought that the time had come to make the effort to do so. It is of relevance for demonstrating the property of persistence in reaction networks. This is the property that the $\omega$-limit points of a positive solution are themselves positive. For a bounded solution this is the same as saying that the infima of all concentrations at late times are positive. The most helpful reference I have found for these topics is a paper of Angeli, de Leenheer and Sontag in a proceedings volume edited by Queinnec et. al.
There are two ways of formulating the definition of a siphon. The first is more algebraic, the second more geometric. In the first the siphon is defined to be a set $Z$ of species with the property that whenever one of the species in $Z$ occurs on the right hand side of a reaction one of the species in $Z$ occurs on the left hand side. Geometrically we replace $Z$ by the set $L_Z$ of points of the non-negative orthant which are common zeroes of the elements of $Z$, thought of as linear functions on the species space. The defining property of a siphon is that $L_Z$ is invariant under the (forward in time) flow of the dynamical system describing the evolution of the concentrations. Another way of looking at the situation is as follows. Consider a point of $L_Z$. The right hand side of the evolution equations of one of the concentrations belonging to $Z$ is a sum of positive and negative terms. The negative terms automatically vanish on $L_Z$ and the siphon condition is what is needed to ensure that the positive terms also vanish there. Sometimes minimal siphons are considered. It is important to realize that in this case $Z$ is minimal. Correspondingly $L_Z$ is maximal. The convention is that the empty set is excluded as a choice for $Z$ and correspondingly the whole non-negative orthant as a choice for $L_Z$. What is allowed is to choose $Z$ to be the whole of the species space which means that $L_Z$ is the origin. Of course whether this choice actually defines a siphon depends on the particular dynamical system being considered.
If $x_*$ is an $\omega$-limit point of a positive solution but is not itself positive then the set of concentrations which are zero at that point is a siphon. In particular stationary solutions on the boundary are contained in siphons. It is remarked by Shiu and Sturmfels (Bull. Math. Biol. 72, 1448) that for a network with only one linkage class if a siphon contains one stationary solution it consists entirely of stationary solutions. To see this let $x_*$ be a stationary solution in the siphon $Z$. There must be some complex $y$ belonging to the network which contains an element of $Z$. If $y'$ is another complex then there is a directed path from $y'$ to $y$. We can follow this path backwards from $y$ and conclude successively that each complex encountered contains an element of $Z$. Thus $y'$ contains an element of $Z$ and since $y'$ was arbitrary all complexes have this property. This means that all complexes vanish at $x_*$ so that $x_*$ is a stationary solution.
Siphons can sometimes be used to prove persistence. Suppose that $Z$ is a siphon for a certain network so that the points of $Z$ are potential $\omega$-limit points of solutions of the ODE system corresponding to this network. Suppose further that $A$ is a conserved quantity for the system which is a linear combination of the coordinates with positive coefficents. For a positive solution the quantity $A$ has a positive constant value along the solution and hence also has the same value at any of its $\omega$-limit points. It follows that if $A$ vanishes on $Z$ then no $\omega$-limit point of that solution belongs to $Z$. If it is possible to find a conserved quantity $A$ of this type for each siphon of a given system (possibly different conserved quantities for different siphons) then persistence is proved. For example this strategy is used in the paper of Angeli et al. to prove persistence for the dual futile cycle. The concept of persistence is an important one when thinking about the general properties of reaction networks. The persistence conjecture says that any weakly reversible reaction network with mass action kinetics is persistent (possibly with the additional assumption that all solutions are bounded). In his talk last week Craciun mentioned that he is working on proving this conjecture. If true it implies the global attractor conjecture. It also implies a statement claimed in a preprint of Deng et. al. (arXiv:1111.2386) that a weakly reversible network has a positive stationary solution in any stoichiometric compatobility class. This result has never been published and there seems to be some doubt as to whether the proof is correct.
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2021-06-16 11:59:00
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https://www.rocketryforum.com/threads/estes-1969-saturn-v-status.149279/page-13
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# Estes 1969 Saturn V Status
### Help Support The Rocketry Forum:
#### SecondRow
##### Well-Known Member
I was trying to find a picture of the Apollo 11 Saturn V to illustrate the discrepancy in the wraps, just to understand it better myself.
So, to clarify, the kit instructions and wraps will work fine and make a fully functional rocket, but they don't exactly match the layout of the actual historical rocket. If you follow the instructions and use the existing wraps, the model kit will build fine and line up the dowels with each other per the instructions.
Hard to find a picture without the tank covered in ice/frost and on that side of the tower. This picture shows how the kit wraps are laid out, side by side with a photo from Apollo 13 and where the mismatch with the historical rocket occurs:
View attachment 375479
Image from:
I could not find a similar picture of Apollo 11 on that side, I assume it would be similar.
This is a great post. Thanks for taking the time to research. I was confused by what the problem was, but I understand it now. You can even see how the access panels all line up in the original picture.
The fix is easy enough, but no one’s gonna know it’s off if you don’t do it. Not sure why some people lost their minds on FB.
#### GlenP
##### Well-Known Member
If you are entering a contest, and the category is “scale” and your rocket is scored on how well it matches your historical reference materials... then use the pictures of Apollo 11 that are covered in ice! The rocket, not the pictures, I mean.
#### Space Ranger
##### Well-Known Member
I like GlenP's great diagram in Post 360. I was also having a hard time visualizing the actual recommended "fix" for the Saturn wraps, so I laid the 4 wraps for the BT101 body tube out flat as they are to be glued per the instructions for kit #1969. I placed non-scale length dowels to represent the tunnels to be glued & took a picture below. As GlenP notes, it does NOT seem that this issue is a matter of the tunnels not lining up, they seem to line up fine without doing the "fix", but the issue seems to be an overall misalignment within the largest wrap (labeled as "Interstage Wrap in the instructions).
If I correctly understand Chris' post of Jay Chladek's recommended fix, the largest wrap should be cut all the way across between the 2 black arrows shown below, and then shifted sideways so that the 2 red dots I placed will line up. Fyi, the 2 red dots are about 2.7" apart before the fix. So then, there will be 5 separate wraps, the 3 lower ones should be glued to line up based on the lower dowel/tunnels and the vertical registration line you draw per the instructions. Then the upper 2 wraps will be glued so that the red dots and the upper tunnel lines up. This will create a jog in the vertical seam where the wraps are glued together. It would be great if Estes could confirm all of this & come up with a diagram & official addendum to the instructions.
I assume that if the wraps are glued on per the instructions (like laid out in my photo) -- without doing the "fix", when the roll patterns are painted on, some of the ullage motors, & other raised doo dads will end up in wrong parts of the roll patterns???
Close-up of the Interstage Wrap to be cut across at the black arrows and shifted to line up the 2 red dots. The lower red dot is on an umbilical attachment point (?) and the upper red dot is on an access panel (?) which should both be lined up as on the real Sat V.
#### Anton Epp
##### Member
I've started my build of the new Estes kit and adding the wraps is close on the horizon. Everyone seems to be doing the painting after the model is (mostly) built. This involves a lot of masking. Would life be any easier if the wraps were painted before they're attached? Or is that just opening a new can of worms?
#### captbk
##### Well-Known Member
After seeing post 363 I thought about the same thing. Too bad mines done already. If I had to do it over again I would paint the wraps before applying.
#### dpower
##### Well-Known Member
While I haven't tried it, I suspect paint would tend to chip off of the wraps if pre-painted.
#### dpower
##### Well-Known Member
I like GlenP's great diagram in Post 360. I was also having a hard time visualizing the actual recommended "fix" for the Saturn wraps, so I laid the 4 wraps for the BT101 body tube out flat as they are to be glued per the instructions for kit #1969. I placed non-scale length dowels to represent the tunnels to be glued & took a picture below. As GlenP notes, it does NOT seem that this issue is a matter of the tunnels not lining up, they seem to line up fine without doing the "fix", but the issue seems to be an overall misalignment within the largest wrap (labeled as "Interstage Wrap in the instructions).
If I correctly understand Chris' post of Jay Chladek's recommended fix, the largest wrap should be cut all the way across between the 2 black arrows shown below, and then shifted sideways so that the 2 red dots I placed will line up. Fyi, the 2 red dots are about 2.7" apart before the fix. So then, there will be 5 separate wraps, the 3 lower ones should be glued to line up based on the lower dowel/tunnels and the vertical registration line you draw per the instructions. Then the upper 2 wraps will be glued so that the red dots and the upper tunnel lines up. This will create a jog in the vertical seam where the wraps are glued together. It would be great if Estes could confirm all of this & come up with a diagram & official addendum to the instructions.
I assume that if the wraps are glued on per the instructions (like laid out in my photo) -- without doing the "fix", when the roll patterns are painted on, some of the ullage motors, & other raised doo dads will end up in wrong parts of the roll patterns???
View attachment 375713
Close-up of the Interstage Wrap to be cut across at the black arrows and shifted to line up the 2 red dots. The lower red dot is on an umbilical attachment point (?) and the upper red dot is on an access panel (?) which should both be lined up as on the real Sat V.
View attachment 375714
That's a nice way to show the issue, and is consistent with what Chris shows on his blog.
#### DeltaVee
##### HV Arcas, AT F67
So here is what I don't quite get about this little 'flaw' in the interstage wrap... if you look at the box cover photo... the tunnels are in their correct relative position... Obviously the one used for the box cover was "right"...
#### ThirstyBarbarian
##### Well-Known Member
So here is what I don't quite get about this little 'flaw' in the interstage wrap... if you look at the box cover photo... the tunnels are in their correct relative position... Obviously the one used for the box cover was "right"...
The pic on the box might just be a rendering.
#### Araize01
##### Well-Known Member
Can anyone tell me the length of the bt-101 tube used in the 50th anniversary apollo 11?. Thanks.
##### Well-Known Member
TRF Supporter
Can anyone tell me the length of the bt-101 tube used in the 50th anniversary apollo 11?. Thanks.
24.75", same as the old kit.
Same flimsy BT-101 tube.
#### OverTheTop
Ok. So two of these are costing me around $200USD (for the pair), including shipping to Australia. They ship tomorrow from AC Supply. I would have liked to get them from a local store but they are not even on the list from the wholesaler . #### afadeev ##### Well-Known Member TRF Supporter Ok. So two of these are costing me around$200USD (for the pair), including shipping to Australia. They ship tomorrow from AC Supply.
I would have liked to get them from a local store but they are not even on the list from the wholesaler .
Wow, that's pretty rough on shipping.
But if you've never build one before, it's worth it!
a
#### Araize01
##### Well-Known Member
24.75", same as the old kit.
Same flimsy BT-101 tube.
Now for another question. What's the length of the bt-80 tube used for the S-IVB third stage?
##### Well-Known Member
TRF Supporter
Now for another question. What's the length of the bt-80 tube used for the S-IVB third stage?
It's 8.75".
Are you trying to piece-meal replicate the kit, scale it, or what?
#### rocketguy101
##### Well-Known Member
Tubes lengths for Estes kits can be found in this reference from Ye Olde Rocket Shoppe. Kit 2157 is on pg 25 (30 of the pdf)
#### Araize01
##### Well-Known Member
It's 8.75".
Are you trying to piece-meal replicate the kit, scale it, or what?
I'm cloning it from other part sources. All I need to know is, the length of tube left after its inserted into the S-II to S-IVB transition. In other words, how much of the tube is left exposed?
#### OverTheTop
##### Well-Known Member
Mine have arrived today! Can't wait to get them home and have a look in the box.
##### Well-Known Member
Hello all. I'm Jay Chladek and I came up with the wrap fix. It does work and looks good when done. Here are some other tidbits:
Glue for the wraps: I utilized Bob Smith Industries Finish Cure Epoxy brushed on. This is not thick like a glue. Instead this is a thinner grade, almost liquid, intended for brushing on surfaces like foam to bond wood skins to them. It works nicely for the wraps as you get decent working time and they dry nice and hard when fully cured after about 24 hours. I went a bit conservative on my first application, only using it in the centers of the wraps and thin CA to tack down the edges. But as I got more confident, I used the finish cure more and more. To help apply even pressure to the wraps while they dried, I used rubber bands.
Plastic Weld Glue: These wraps are rather thin and not all plastic weld glues will work with them effectively. I tried Ambroid Pro-Weld in one spot and it is way too hot, distorting detail way too easily. Plastruct Plastic Weld (orange bottle) seems to work nicely for the fin fairings and the fins. I can brush the stuff on for a good bond, yet it doesn't bite into the plastic wraps too hard to distort it, provided I don't touch it while it dries.
Wrap detail reinforcement: Not sure what this will do to COG issues, but with the wraps being so thin, I was afraid the large protrusions such as the ullage motors and APU fairings would get crushed if I wasn't careful. So I back filled them with two part Epoxy Putty. In my case it was Aves Apoxie Sculpt, but Milliput can work fine and it sounds like JB Weld can also be used. If and when I get around to flying mine, I will go for the bigger Estes motor as I want to make sure she doesn't get marginal in stability. The back filled fairings are nice and sturdy now and should survive what abuse I can throw at them. I also used some putty on the back edges of the fin fairings to give them some durability during construction and landing (just on the bottom edges).
Centering rings: Yes, they appear to be a hair too small and slide a bit loosely in the body tube. Not a deal breaker though when glued in properly. But, I found the wood glue I used on the rings as it dried caused the outside of the body tube to shrink where the glue joints were. So it does require a bit of filler on the tube to get things leveled off. My primers of choice are Tamiya brand spray and liquid primers. I use the standard gray primer for filling the body tube wraps (with brush applications of liquid primer to build up in the deep spots) and will use Fine White primer over the body and the wraps to help prep the surface for painting. Tamiya primer is designed for plastic models and although it is a lacquer, it should not be hot enough to attack the wraps, provided you don't layer on way too much during a spray session.
The wrap modification: Others have already talked about the modification I came up with. When I found the flaw on my rocket, I immediately informed Estes, suggesting they post an addendum fix for the instructions. At least it sounds like they are mentioning it in conversations with customers. That is a good thing. I doubt they will fix the wraps as plastic molds can be expensive. But if you make the cuts and center everything from the S-II/S-IC interstage ring on down in regards to details that line up with the gantry swing arm umbilicals, the fix works. Based on what I can see, this Estes kit is the most accurate Saturn V kit out there in regards to exterior details. It is even more accurate than most plastic models with the exception of the oversized fins for flight.
Now those looking for additional references on the wrap details and paint patterns can look at the following site for reference: https://apollomaniacs.com/apollo/saturnv_menu_e.htm
All major Saturn V details were the same on all the rockets. Nothing changed position as the gantry swing arms had to attach to the rockets in the same spots. Apollos 4 and 6 had 8 ullage motors, 9 through 14 had 4 and Apollos 15 through Skylab had no ullage motors on the interstage ring.
I'll post images of my work once I have figured out the file uploads.
##### Well-Known Member
Here is an image of the wrap after the cut is made. This is simply the image from the instructions with a little quick and dirty Photoshop. Align the umbilical access point on the S-IC just below the access door on the interstage ring. You might need to cut the bottom wrap into two sections to make sure the stringers line up properly at the original point where the wrap ends are supposed to meet, but it is an easy thing to do.
#### Attachments
• 462 KB Views: 124
##### Well-Known Member
This image shows how the wrap should look after the cuts. The yellow tape was just there to act as a place holder while things dried.
#### Attachments
• 73.2 KB Views: 141
##### Well-Known Member
Another angle of the same modification. The wraps above the cut line up with the orignal seam line. The wraps below line up with the moved wrap's seam line.
#### Attachments
• 77 KB Views: 110
##### Well-Known Member
I've started my build of the new Estes kit and adding the wraps is close on the horizon. Everyone seems to be doing the painting after the model is (mostly) built. This involves a lot of masking. Would life be any easier if the wraps were painted before they're attached? Or is that just opening a new can of worms?
I don't recommend painting the wraps first at all. The paint may not flex like the wraps do and it could be a major mess. You are better off gluing them on, fixing all the surface issues on the model, primer and then paint.
#### OverTheTop
##### Well-Known Member
Thanks for the update on the wraps for this Jay. Just started building mine a couple of days back. Great info!
#### hutch
##### Well-Known Member
Are there any wraps or decals available for the service module, to make it more realistic? Instructions just call for it to be pure silver, without the white markings....
##### Well-Known Member
I don't know of any wraps out there. I'm almost to that point in my build and I am going to try layering on cut pieces of thin cardstock (with thin tape strips in some spots) to represent the SM details. It will take time, but I think the results will be worth it. Another thing I am going to do is see if I can disguise the eyelet on the SM tube to look like one of the two scimitar antennas the SMs all carried.
As for decals, head on over to the CultTVman.com hobby shop and order up a set of 1/96 Saturn V decals from Space Model Systems (SMS). The decal line from Rick Sternbach (most well known for his work in the Star Trek shows) features meticulously researched markings for the Saturn V and Apollo spacecraft in various scales. The 1/96 Saturn V set originally intended for the Revell kit contains markings for the Saturn, LM and SM. Due to the scale differences, the larger markings probably won't work. But, all those little stencil markings found on the vehicles should fit fine as the scale difference isn't all that noticeable. I'm using them on my model when the time comes.
Here is a link to the product. Currently it looks like they are out of stock. But I expect a restock to come soon as these are going to be hot sellers. https://www.culttvmanshop.com/Saturn-VApollo-196-scale-bundle_p_896.html
#### Araize01
##### Well-Known Member
Are there any wraps or decals available for the service module, to make it more realistic? Instructions just call for it to be pure silver, without the white markings....
Stickershock23.com has a complete set of stickers for the Estes Saturn V, including a wrap for the sm.
#### Ted Curtis
##### Well-Known Member
Does anyone know if there is a Saturn 1B capsule/nose cone available. I have an old version built in 70's that I am restoring and an upgrade to the capsule would be great Thanx.. Sorry if this forum is just meant for the V but thought I would ask....
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2020-07-14 03:30:36
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http://umj.imath.kiev.ua/volumes/issues/?lang=en&year=2016&number=11
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2019
Том 71
№ 10
# Volume 68, № 11, 2016
Article (Russian)
### Laplacian with respect to the measure on a Riemannian manifold and the Dirichlet problem. II
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1443-1449
We propose the $L^2$ -version of Laplacian with respect to measure on an (infinite-dimensional) Riemannian manifold. The Dirichlet problem for equations with proposed Laplacian is solved in a part of the Rimannian manifold of a certain class.
Article (Ukrainian)
### Almost periodic solutions of systems with delay and nonfixed times of impulsive action
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1450-1466
We study the existence and asymptotic stability of piecewise continuous almost periodic solutions for systems of differential equations with delay and nonfixed times of impulsive action that can be regarded as mathematical models of neural networks.
Article (Ukrainian)
### Estimations of the Laplace – Stieltjes integrals
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1467-1482
We study the Laplace – Stieltjes integrals with an arbitrary abscissa of convergence. The lower and upper estimates for these integrals are established. The accumulated results are used to deduce the relationships between the growth of the integral and the maximum of the integrand.
Article (English)
### General proximal point algorithm for monotone operators
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1483-1492
We introduce a new general proximal point algorithm for an infinite family of monotone operators in a real Hilbert space. We establish strong convergence of the iterative process to a common zero point of the infinite family of monotone operators. Our result generalizes and improves numerous results in the available literature.
Article (Ukrainian)
### I. Approximative properties of biharmonic Poisson integrals in the classes $W^r_{\beta} H^{\alpha}$
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1493-1504
We deduce asymptotic equalities for the least upper bounds of approximations of functions from the classes $W^r_{\beta} H^{\alpha}$, and $H^{\alpha}$ by biharmonic Poisson integrals in the uniform metric.
Article (Ukrainian)
### Asymptotically independent estimators in a structural linear model with measurement errors
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1505-1517
We consider a structural linear regression model with measurement errors. A new parameterization is proposed, in which the expectation of the response variable plays the role of a new parameter instead of the intercept. This enables us to form three groups of asymptotically independent estimators in the case where the ratio of variances of the errors is known and two groups of this kind if the variance of the measurement error in the covariate is known. In this case, it is not assumed that the errors and the latent variable are normally distributed.
Article (Ukrainian)
### Sufficient conditions under which the solutions of general parabolic initial-boundaryvalue problems are classical
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1518-1527
We establish new sufficient conditions under which the generalized solutions of initial-boundary-value problems for the linear parabolic differential equations of any order with complex-valued coefficients are classical. These conditions are formulated in the terms of belonging of the right-hand sides of this problem to certain anisotropic H¨ormander spaces. In the definition of classical solution, its continuity on the line connecting the lateral surface with the base of the cylinder (in which the problem is considered) is not required.
Article (Ukrainian)
### Two-dimensional Coulomb dynamics of two and three equal negative charges in the field of two equal fixed positive charges
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1528-1539
Periodic and bounded for positive time solutions of the planar Coulomb equation of motion for two and three identical negative charges in the field of two equal fixed positive charges are found. The systems possess equilibrium configurations to which the found bounded solutions converge in the infinite time limit. The periodic solutions are obtained with the help of the Lyapunov center theorem.
Article (English)
### Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1540-1551
We study hypersurfaces in a unit sphere and in a hyperbolic space with nonzero constant Gauss – Kronecker curvature and two distinct principal curvatures one of which is simple. Denoting by $K$ the nonzero constant Gauss – Kronecker curvature of hypersurfaces, we obtain some characterizations of the Riemannian products $S^{n-1}(a) \times S^1(\sqrt{1 - a^2}),\quad$ $a^2 = 1/\left(1 + K^{\frac{2}{n - 2}}\right)$ or $S^{n-1}(a) \times H^1(- \sqrt{1 + a^2}),\quad$ $a^2 = 1/\left(K^{\frac{2}{n - 2}} - 1\right)$.
Article (English)
### A construction of regular semigroups with quasiideal regular *-transversals
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1552-1560
Let $S$ be a semigroup and let “$\ast$ ” be a unary operation on S satisfying the following identities: $$xx^{\ast} x = x, x^{\ast} xx^{\ast} = x^{\ast},\; x^{\ast \ast \ast} = x^{\ast},\; (xy^{\ast} )^{\ast} = y^{\ast \ast} x^{\ast},\; (x^{\ast} y)^{\ast} = y^{\ast} x^{\ast \ast}.$$ Then S\ast = \{ x\ast | x \in S\} is called a regular \ast -transversal of $S$ in the literatures. We propose a method for the construction of regular semigroups with quasiideal regular $\ast$ -transversals based on the use of fundamental regular semigroups and regular $\ast$ -semigroups.
Article (English)
### On the growth of meromorphic solutions of difference equation
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1561-1570
We estimate the order of growth of meromorphic solutions of some linear difference equations and study the relationship between the exponent of convergence of zeros and the order of growth of the entire solutions of linear difference equations.
Brief Communications (Ukrainian)
### Complete classification of finite semigroups for which the inverse monoid of local automorphisms is a permutable semigroup
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1571-1578
A semigroup $S$ is called permutable if $\rho \circ \sigma = \sigma \circ \rho$. for any pair of congruences $\rho, \sigma$ on $S$. A local automorphism of semigroup $S$ is defined as an isomorphism between two of its subsemigroups. The set of all local automorphisms of the semigroup $S$ with respect to an ordinary operation of composition of binary relations forms an inverse monoid of local automorphisms. We present a complete classification of finite semigroups for which the inverse monoid of local automorphisms is permutable.
Brief Communications (Ukrainian)
### On the equivalence of some perturbations of the operator of multiplication by the independent variable
Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1579-1584
We study the conditions of equivalence of two operators obtained as perturbations of the operator of multiplication by the independent variable by certain Volterra operators in the space of functions analytic in an arbitrary domain of the complex plane starlike with respect to the origin.
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2019-11-17 05:20:06
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http://nrich.maths.org/8169
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#### You may also like
If you put three beads onto a tens/ones abacus you could make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?
### Writing Digits
Lee was writing all the counting numbers from 1 to 20. She stopped for a rest after writing seventeen digits. What was the last number she wrote?
### Number Detective
Follow the clues to find the mystery number.
# That Number Square!
##### Stage: 1 and 2 Challenge Level:
When you arrive in the classroom on Monday morning you discover all the numbers have fallen off the class number square and they are in a heap on the floor. All that is left on the wall is a blank grid!
There's five minutes to go before the lesson starts and you need the number square.
Can you find a quick way of putting the numbers back in their right places on the grid?
Where will you start?
Before you start think - does your class number square start with $0$ or $1$? Or a different number?
Some of your friends may want to have a go too.
What different ideas are there about how to put the number tiles back as quickly as possible?
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2013-12-10 12:58:57
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https://socratic.org/questions/a-solid-disk-spinning-counter-clockwise-has-a-mass-of-14-kg-and-a-radius-of-5-4-
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# A solid disk, spinning counter-clockwise, has a mass of 14 kg and a radius of 5/4 m. If a point on the edge of the disk is moving at 7/9 m/s in the direction perpendicular to the disk's radius, what is the disk's angular momentum and velocity?
Dec 12, 2016
The angular momentum is $= 170.6 k g {m}^{2} {s}^{- 1}$
The angular velocity is $= 3.9 r a {\mathrm{ds}}^{- 1}$
#### Explanation:
The angular velocity is $\omega$
$\omega = \frac{v}{r} = \frac{7}{9} \cdot \frac{4}{5} \cdot 2 \pi r a {\mathrm{ds}}^{- 1}$
$= \left(\frac{56}{45} \pi\right) r a {\mathrm{ds}}^{- 1} = 3.9 r a {\mathrm{ds}}^{- 1}$
The angular momentum is $L = I \omega$
For a solid disc, the moment of inertia $I = \frac{1}{2} m {r}^{2}$
$I = \frac{1}{2} \cdot 14 \cdot \frac{25}{4} = \frac{175}{4} k g {m}^{2}$
The angular momentum is
$L = \frac{175}{4} \cdot \frac{56}{45} \cdot \pi = 170.6 k g {m}^{2} {s}^{- 1}$
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2019-09-23 13:09:59
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https://www.physicsforums.com/threads/how-to-scale-a-velocity-profile.658553/
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# How to scale a velocity profile
1. Dec 12, 2012
Hello,
I am reading a paper in which the author describes how he scales the velocity profile of a car.
He is doing this so that he can compare the velocity profiles of cars.
This is an extract from the text:
The first methodology consisted of evaluating the similarities between velocity profiles by computing the differences between their velocity profiles. This is a direct method, since the velocity profiles are considered in their entirety. The aim was not to quantify the links between different velocity profiles, but simply to highlight the similarities that might be present. To apply this methodology, it was first necessary to scale the velocity profiles and to perform a linear interpolation (step of 0.1 s) since all velocity profiles do not all have the same duration and the same time resolution. This was achieved by using the ECE part of the NEDC cycle as a reference. Consequently, once scaled, all the cycles had the same duration of 195 s. A comparison of the original and scaled NEDC cycles is illustrated in Figure 1.
My question is, if I had the time and velocity values of the "original NEDC" velocity profile, how would you actually scale it to achieve the "scaled NEDC". How would you scale the numbers to fit in the 195 second time frame?
Thank you
Here is the data in an excel sheet:https://dl.dropbox.com/u/54057365/All/NEDC.xlsx
2. Dec 12, 2012
### Staff: Mentor
From the figure, it looks like the only parameter that has been scaled is the time. If t is the running time in the experiment, and tmax is the overall maximum time, then the scaled time is t x 195/ tmax
3. Dec 12, 2012
Thanks very much
4. Jan 8, 2013
Hello ChesterMiller,
You helped me scale the time parameter of velocity profiles above for a vehicle a number of weeks ago. Thank you.
I was wondering if you had the time to help me with something else?
When I scale the velocity profiles in excel the resulting times are same length (in seconds) but they occupy a different number of cells (in length).
I am trying to apply this methodology to them to compute the difference between the profiles. But because they occupy a different number of cells I cannot apply it because the "t" values are different lengths.
I have attached a workbook here with two example original and scaled profiles
https://dl.dropbox.com/u/54057365/All/QD.xlsx
I would be grateful if you had the time to consider it.
Kind Regards
J
5. Jan 9, 2013
### Staff: Mentor
It all depends on what you are trying to do, and what the data represents. It looks like you are trying to determine the rms velocity difference between the two velocity variations.
Do you want to smooth the differences between the two profiles first, or do you to include all the sturucture? If you want each of the profiles smoothed first, you can fit a Fourier series to the variation and include only the first few terms. Or fit another type of smooth curve.
If you want to include the sturucture, you can join all the sequential pairs points on each curve by straight lines, and interpolate between the points. Or you can represent each of the functions as a series of steps (like a bar chart), with constant values from half way between one pair of points, to half way between the next pair of points. In any event, you are fitting each of the velocity variations as a piecewise continuous function of time. Then you can evaluate the difference between the curves at any value of the time. Then you can integrate.
$$RMS = \frac{1}{t_{max}}\int_0^{t_{max}}(v_2(t)-v_1(t))^2dt$$
This equation is the continuous form of your discrete summation relationship.
6. Jan 10, 2013
Hello,
Thank you for your reply. I do not need to smooth the velocity profiles.
Yes, I am trying to compute rms velocity difference between the two velocity variations.
My problem is that velocity profiles have different time duration. Hence, I have scaled them to same duration.
My problem is that now that they are scaled, the time values are not the same so I cannot compute the difference between to corresponding velocity points.
For example, say profile A is 970 seconds and Profile B is 1075 seconds. So I scaled the time component of both profiles using your method above to 177 seconds.
t x 177/ tmax (970)
t x 177/ tmax (1075)
https://dl.dropbox.com/u/54057365/All/scale.JPG
This is an example of the time values after scaling.
https://dl.dropbox.com/u/54057365/All/values.JPG
Because the scaled time values are not the same, i cannot subtract the corresponding velocities.
I have been struggling with this for a few days but cannot seem to figure it out.
Regards
Last edited by a moderator: Jan 10, 2013
7. Jan 10, 2013
### Staff: Mentor
OK. Then just follow the advise I gave you in my previous posting. Fill in the regions between the grid points with straight lines or stepwise variations, and then integrate numerically. Use small time increments in the integration. If the times at the grid points on the scaled profiles had actually matched, doing this integration would have given you the exact same result as you summation equation.
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2016-09-27 15:38:48
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https://space.stackexchange.com/questions/61026/the-costs-of-a-single-sls-solid-fuel-rocket-booster
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# The costs of a single SLS solid fuel rocket booster
Why didn’t Nasa cluster four solid fuel shuttle boosters around an Atlas V booster. The thrust would be substantially more than that of the SLS, with no fueling problem on the launch pad. It would be much less expensive and if using old technology anyway, why not develop new technology while building a base on the Moon which should have been done decades ago?
• Part of the answer will be about human rating, see e.g. this question and its answers and this other answer. I also doubt your design would meet the mission requirements for SLS, even if you did figure out a way to mount Shuttle SRBs to an Atlas core. Nonetheless, welcome to the site Nov 19 at 21:00
• oh also the title should align with the body of the question. Sorry for missing that; it's what I get for coming in from the review queue instead of the front page Nov 19 at 23:00
• So if human rating is a problem with Atlas V, why not change the question to 4 solid fuel boosters around the existing SLS core that already uses new tchnology ? The fueling problems seem now solved. Nov 20 at 11:53
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2022-11-30 23:11:43
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https://math.stackexchange.com/questions/2936269/how-can-you-simplify-sqrt9-6-sqrt2/2936299#2936299
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# How can you simplify $\sqrt{9-6\sqrt{2}}$?
How do you simplify: $$\sqrt{9-6\sqrt{2}}$$
A classmate of mine changed it to $$\sqrt{9-6\sqrt{2}}=\sqrt{a^2-2ab+b^2}$$ but I'm not sure how that helps or why it helps.
This questions probably too easy to be on the Math Stack Exchange but I'm not sure where else to post it.
• See How to simplify a square root for several applicable approaches.
– dxiv
Sep 30 '18 at 5:06
• Find $a,b$ such that $\sqrt{9-6\sqrt 2}=\sqrt{a+b-2\sqrt{ab}}$ since $\sqrt{a+b-2\sqrt{ab}}=\sqrt{(\sqrt a-\sqrt b)^2}$. Sep 30 '18 at 5:07
Try to use the formula your classmate gave. In this situation, $$9-6\sqrt2={\sqrt3}^2-2{\sqrt{3\times6}}+{\sqrt6}^2\Rightarrow(1)$$ That is because $$6{\sqrt2}=2{\sqrt{3\times6}}$$ Expression (1) now looks similar to $$a^2-2ab+b^2$$ where $$a=\sqrt3$$ and $$b=\sqrt6$$ Using this we can conclude that $$9-6\sqrt2={\sqrt3}^2-2{\sqrt{3\times6}}+{\sqrt6}^2=(\sqrt3-\sqrt6)^2$$ We can subsitute in the original expression $$\sqrt{9-6\sqrt2}=\sqrt{(\sqrt3-\sqrt6)^2}=-(\sqrt3-\sqrt6)=\sqrt6-\sqrt3$$ The simplest form will be $$\sqrt6-\sqrt3$$
Let's forget about $$9-6\sqrt{2}$$ for a second and just think about the expression your classmate thinks is useful:$$a^2-2ab+b^2.$$
And let's keep in mind our goal here. We're looking for something which is a perfect square (since we want it to play well inside a $$\sqrt{\quad}$$...
Well, this should remind us of $$a^2\color{red}{+}2ab+b^2=(a+b)^2.$$ But that "$$-$$" on the $$2ab$$ term is throwing me off! Is there any way to fix it?
This is where we get something for free from just doing a small change of variable: if we let $$c=-b$$, we get $$a^2-2ab+b^2=a^2+2ac+c^2.$$ That right hand side is of course just $$(a+c)^2$$, or better yet $$(a-b)^2$$. So we now know: $$\color{green}{\sqrt{a^2-2ab+b^2}=a-b}$$ (or rather, fine, $$\vert a-b\vert$$. FINE.).
That's why what your friend wants to do is reasonable. So nowthe question is: how do we do it?
Ultimately this can just feel like trial-and-error at first, but my instinct here is to say that "$$-6\sqrt{2}$$" looks a lot like "$$-2ab$$." Because they both have a minus sign. And $$6$$ is even. This doesn't work immediately, but when we factor out a $$3$$ things get much cleaner ...
\begin{align} 9 - 6\sqrt2 &= 3 (3-2\sqrt2) \\ &= 3((\sqrt2)^2 - 2(1)\sqrt{2} +1^2) \\ &= 3(\sqrt2-1)^2 \end{align} Hence, $$\sqrt{9-6\sqrt2} = \sqrt{3}(\sqrt2 - 1)$$
Your class mate is being.... clever.
If $\sqrt {9-6\sqrt 2}=a-b$ then $9-6\sqrt 2=a^2-2ab+c^3$
Let $2ab=6\sqrt 2$ and $a^2+b^2=9$.
Can we do that?
If we let $b^2=k$ and $a^2=9-k$ then $ab=\sqrt {k (9-k)}=3\sqrt 2=\sqrt {18}$. Solving $k (9-k)=18$ for $k$ (if it isn't visiblely obvious that we can do it in our heads, it is just a quadratic that we can solve by quadratic formula) and, for covenience, choicing the smaller solution (because we want $a>b$), we get $k =3$ is a good solution..
So $a=\sqrt 6$ and $b=\sqrt 3$.
I.e. in other words
$\sqrt {9-6\sqrt 2}=$
$\sqrt {6-2\sqrt 6\sqrt 3 +3}=$
$\sqrt {(\sqrt 6- \sqrt 3)^2}=$
$|\sqrt 6 - \sqrt 3|=$
$\sqrt 6 -\sqrt 3$.
The reason for doing that is that $$\sqrt{a^2-2ab+b^2} = \sqrt{(a-b)^2} = a-b$$. Now try to put your radical in the form your classmate suggested!
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2021-12-02 01:29:54
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https://socratic.org/questions/how-do-you-find-the-sum-of-the-infinite-geometric-series-3-12-48-192
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# How do you find the sum of the infinite geometric series 3 – 12 + 48 – 192 + …?
The sum of this series is undefined (it alternates tending towards $+ \infty$ and $- \infty$)
The sum of an infinite geometric series only exists if the geometric ratio, $r$ is such that $\left\mid r \right\mid < 1$
In this case the geometric ratio is $\frac{- 12}{3} = - 4$
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2020-10-21 15:55:32
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https://www.johnlees.me/posts/expression-modules-in-s-pneumoniae/
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Expression modules in S. pneumoniae
Contents
I recently read a pre-print from the Veening lab where they had reconstructed various (22 total) physiological conditions in vitro and then measured expression levels with RNA-seq. I thought it was a great bit of research, and would encourage you to read it here if you’re interested: https://doi.org/10.1101/283739
They’ve also done a really good job with data availability, having released a browser for their data (PneumoExpress), and they have put their raw data on zenodo.
I was interested in trying to replicate their expression module analysis, which was performed using WGCNA. I first downloaded the co-expression matrix, which is split across five CSV files in the zenodo archive. I then basically ran WGCNA exactly as suggested in their tutorial. I then used the annotations from the lab’s D39V genome to get final expression modules.
The expression modules look sensible to me, with the comE master regulator splitting the expression of the competence pathway and associated genes as expected. The modules I predicted can be found here.
Thanks to the authors for making their data so accessible! If you find this useful I would also note the authors also analysed their data with WGCNA, probably in a more optimal and error-checked way than I have, so most likely they have better clusters.
Here is a gist of the code I used:
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2023-03-28 18:19:49
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http://masterworkrecording.com/making-love-agtehrg/8936f1-how-much-does-a-yard-of-sand-weigh
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Wet, compacted sand can weigh up to 3,510 lbs. Gravel – approximately 3000 lbs. The sand’s density is 100 lb/ft³ and costs $15 per cubic yard. While 1 cubic yard of mulch will cover the same amount of ground at 100 square feet, it weighs significantly less, at 600 to 800 pounds per cubic yard. per cubic yard Lawn Dressing.90 tons (1,800 lb.) One cubic yard of sand, or 27 cubic feet of sand weighs 2,600 to 3,000 pounds (1,179 to 1,360 kg).Coverage Calculator | Southwest Boulder & Stone1 cubic yard = … Soil: Weighs about 2,200 pounds per cubic yard, depending on the moisture content. while a ton is equivalent to 2,000 lbs. The water content of the sand is assumed to be moderate. How Much Does a Yard of Dirt Weigh? For most purposes you can round it off to 3,000 pounds per yard. According to Glover lose dry gravel weighs about 1522 kg per cubic meter. This amount is also roughly equal to 1 1/2 tons. yd. A COVID-19 Prophecy: Did Nostradamus Have a Prediction About This Apocalyptic Year? How Much Does A Cubic Yard Of Sand Weigh? The link to this tool will appear as: beach sand from cubic yard (cu yd - yd3) to pounds (lb) conversion. ). Brush/Branches (loose) 250 lb. This study estimated that there are seven quintillion, five quadrillion grains of sand or 7,500,000,000,000,000,000 grains of sand! 4 - How much does a kilogram weigh in Pounds? 1 Cubic Yard of Sand can weigh between 2,600 to 3,000 lbs. (There are 27 cubic feet in a cubic … »More detailed or up to one and a half tons approximately. This weight depends on how wet the sand is, the materials in the sand, and the size of the sand particles. How much does it weigh: Rocks and gravel Western Planting Mix 1 ton (2,000 lb.) The estimate is based on the cubic yard calculation. A cubic yard of dry fill dirt will weigh around 2,000 pounds. How much does a cubic yard of gravel weigh. How Much does a Cubic Yard of Gravel Weigh? Similarly, how much does 1 yard of road base weigh? Therefore, a cubic yard is equivalent to 1.25 tons or a ton of sand is equivalent to 0.8 cubic yard. or up to one and a half tons approximately. A delivery price can . Soils typically weigh a little less, about 1000-1200 pounds per scoop. ATLANTA LANDSCAPE MATERIALS typically Sand and Gravel weigh about 1500 pounds (3/4 of a ton) per scoop (1/2 cubic yard). yd. Sand is ~3k/yard and salt is closer to 2k/yard. Calculate Regular Mason Sand Type in inches and feet of your project and calculate the estimated amount of Sand / Screenings in cubic yards, cubic feet and Tons, that your need for your project. Or up to one and a half tons approximately. Most of Harmony Sand & Gravel’s products will weight approximately 2,840 pounds per cubic yard or about 1.42 tons per cubic yard . A gallon of sand weighs approximately 12.5 pounds (5.6 kg). Depending on the situation, it may be best to purchase sand by the bag rather than in bulk. The price for the sand usually does not include the delivery. 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One yard of sand will weigh close to 2,700 pounds, and every 1,000 pounds will be able to cover about nine cubic feet. density of sand, dry is equal to 1 631 kg/m³. dry sand will weigh 2700 ibs. per cubic yard Compost.40 tons (800 lb.) The relative weights will vary based on mean granule size Other types of sand and their prices are the following: Bunker sand widely –$80 per cubic yard; Washed sand – $28 per ton; Poteet red sand –$28 per ton If you want an exact weight, the supplier can tell you when you purchase the soil One cubic yard of topsoil weighs approximately 1,080 pounds. 1 cubic yard of dirt is equal to about 2,000 lbs or 1 … If you need 6.5 cubic yards, that is 526.5 bags of mix, (3*27*6.5=526.5). In the U.S. a Ton is equal to 2000 pounds. A cubic yard of gravel will weigh slightly less, at roughly 2,400 to 2,900 pounds, or roughly still 1 1/2 tons. How Much Does A Yard Of Pea Gravel Weigh? Step Two: Calculate Weight in Tons. Sand, Silica weighs 1.54 gram per cubic centimeter or 1 538 kilogram per cubic meter, i.e. How Much Does a Cubic Yard of Gravel Weigh? If you’re not used to this kind of stuff, wrapping your head around what a yard of dirt is might be challenging to visualize. See below for more common material densities. From your math’s class you know that: 1m = 3. To calculate the weight of a cubic yard of sand, you simply have to multiply its volume by its density. If it contains gravel, stone, and sand, the weight can rise to over 3,000 pounds. }, scientific or nuclear-subatomic weights, archaic units, international units, and so forth. One cubic yard (2 scoops) will weigh about 1.5 tons (3000 lbs.). Itâs not only found on the beach but is used in building materials and landscaping all over the world. But if you want a fast way to figure it out with some accuracy. About how many pounds does one cubic foot of granite weigh? What does 1 yard of masonry sand weigh? dry sand weighs about 1602 kg per cubic meter. per cu. A yard of sand weighs approximately 2,000 and 3,000 pounds, or about 1 1/2 tons. A gallon of sand weighs 12.45 pounds and a cubic yard of sand weighs close to 3,000 pounds. Topsoil is the uppermost layer of the soil usually 5 to 10 inches, in which all the plants grow. One ton of sand covers about 80 to 100 square feet. per cubic foot.There are 27 cubic feet in a cubic yard, so that would make the weight of dry sand about 2700 lbs. Most gravel products will weight approximately 2,840 pounds per cubic yard or about 1.42 tons per cubic yard. One ton of sand covers about 80 Their weight per yard may fluctuate a little, but in general you can use 3700 pounds per cubic yard. Sand, Gravel, Stone: Can tip the scales at upwards of 3,000 pounds per cubic yard. Material Weight – Pounds per Cubic Yard Asphalt 2,700 lb. Concrete (gravel or stone mix) 4,050 lb. How much does a cubic yard of sand weigh? One cubic yard of sand, or 27 cubic feet of sand weighs 2,600 to 3,000 pounds (1,179 to 1,360 kg). Dry, loose sand weighs about 2,700 lbs per cubic yard. Soils typically weigh a little less, about 1000-1200 pounds per scoop. How much does a cubic yard weigh? density of sand, Silica is equal to 1 538 kg/m³. A yard of sand weighs approximately 1.3 tons or 2600 pounds. Mulch: Weighs in at roughly up to 1,000 pounds per cubic yard, depending on the type and whether it’s wet or dry. Mason sand is more affordable when purchased through landscaping companies and is typically delivered for a flat fee of $60 for distances up to 10 miles. The average cost of masonry sand is$15 to $40 per ton or between$25 and $60 per yard. Sand 1.10 1.25 tons(2,200 2,500 lb.) For estimating purposes, most Contractor’s consider the yield to be 3,000 pounds per cubic yard or 1.5 tons per cubic yard . Step 2. The Density of Regular Mason Sand: 2,410 lb/yd³ or 1.21 t/yd³ or 0.8 yd³/t Dry, loose sand weighs about 2,700 lbs per cubic yard. T turfmasters Member Location stroudsburg,Pa. Keep in … All right, but how much does a yard of sand weigh? One cubic yard of dry sandy soil weighs about 2,600 pounds, while 1 cubic yard of dry clay soil weighs in around 1,700 pounds. How Much Does a Cubic Yard of Concrete Weigh? If it contains a more than usual amount of sand or stones this will make it weigh considerably more, on Average Weight of a Cubic Yard of Dirt A cubic yard of dry fill dirt will weigh around 2,000 pounds. Ultimately, it depends on its moisture content and the type of sand it is. Step 1. There might be some people who are unaware of ‘a yard of dirt.’ Let me clear that out first, a yard of dirt is the measurement of the volume of a cubic yard.It is good to calculate the estimated dirt that you need before starting off the project. A cubic yard of gravel will weigh slightly less, at roughly 2,400 to 2,900 pounds, or roughly still 1 1/2 tons. around a property. How much does one cubic yard of air weigh? How much does a cubic yard of product weigh? This amount is also roughly equal to 1 1/2 tons. Leaves (wet or compacted) 550 lb. For most purposes you can round it off to 3,000 pounds per yard. To link to this beach sand cubic yard to pounds online converter simply cut and paste the following. In general, a pure cubic yard of fill dirt will weigh One cubic yard (2 scoops) will weigh about 1.5 tons (3000 lbs. So, a cubic yard is that times 27, or 2835 lb. https://sciencing.com/calculate-weight-sand-8149924.html, https://www.aqua-calc.com/calculate/volume-to-weight, https://www.sandatlas.org/brain-games-with-sand-grains/, https://hypertextbook.com/facts/2003/MarinaTheodoris.shtml. Leaves (loose, dry) 150 lb. Dry sand weighs about 100 lbs. The weight of any amount of sand depends on how much water is in it. How much does a yard of paver sand weigh Products As a leading global manufacturer of crushing, grinding and mining equipments, we offer advanced, reasonable solutions for any size-reduction requirements including, How much does a yard of paver sand weigh, quarry, aggregate, and different kinds of minerals. The approximate weight of 1 cubic yard of sand is 2,600 to 3,000 pounds. So how much does a yard of gravel weigh? Units of Measure Weight and Mass Math … typically Sand and Gravel weigh about 1500 pounds (3/4 of a ton) per scoop (1/2 cubic yard). For most estimating purposes, consider the yield to be 3,000 pounds per cubic yard or 1.5 tons per cubic yard. per cu. Resources:https://sciencing.com/calculate-weight-sand-8149924.htmlhttps://www.aqua-calc.com/calculate/volume-to-weighthttps://www.sandatlas.org/brain-games-with-sand-grains/https://hypertextbook.com/facts/2003/MarinaTheodoris.shtml. How much does a 1/2 yard of sand weigh? Convert Masses or Weights to and from metric units, British units {acres, townships, square miles, square feet, etc. In most cases, a cubic yard of gravel weighs between 2,400 to 2,900 pounds (1,088 to … Wet, compacted sand can weigh up to 3,510 lbs. One cubic yard (2 scoops) will weigh about 1.5 tons (3000 lbs.). Soil: Weighs about 2,200 pounds per cubic yard, depending on the moisture content. The weight of gravel depends on the type and density or grading. Itâs also heavier than you might think. Brush/Branches (chipped – 3” screen) 600 lb. Buying in bulk, as mentioned above, may require you to pay delivery fees, which could be as much as$150, depending on where you live. If you divide 3700 pounds by 2000 pounds you get 1.85 Tons. per cubic yard Landscape Gravels 1.20 1.35 tons (2,400 In Imperial or US customary measurement system, the density is equal to 96.01 pound per cubic foot [lb/ft³], or 0.89 ounce per cubic inch [oz/inch³] . A yard of concrete weighs 3700 pounds. Gravel typically will not pack as tightly as sand. typically Sand and Gravel weigh about 1500 pounds (3/4 of a ton) per scoop ( 1/2 cubic yard ). Wet sand weighs bout 126 pounds per cubic foot so a yard would weigh about 3,430 pounds. Why does the same volume of wet sand weigh less than dry sand? The calculator would perform the following calculations: $$Volume = Area \times Depth = 100 ft^2 \times 3\,in = 25\,ft^3$$ $$Weight = Volume \times Density = 25ft^3 \times 100\,lb/ft^3 = 2,500\,lb$$ A yard of sand weighs approximately 1.3 tons or 2600 pounds. 1 Ton of Sand will cover between 80 to 100 square feet at a 2 inch depth Sand – approximately 2800 lbs. Check with your supply yard for weights of specific materials. How much 1 cubic yard of it weighs depends on the amounts of the components it's made up of. How much does a yard of mason sand cost? The ticket breaks down the weights of cement, sand, aggregate, and water used for the load of concrete. The following are approximate weights for most of our bulk materials. If covering a 2-inch depth, a cubic yard of sand should cover 100 square feet of ground. Iron (wrought) 13,100 lb. A square yard of a sandbox with a depth of 1 foot (30.48 cm) weighs about 900 pounds (410 kg) or slightly less than half a ton. Leaves (vacuumed, dry) 400 lb. A cubic yard is 27 cubic feet ( a cube with edges of length 3 feet). About Sand, dry 1 cubic meter of Sand, dry weighs 1 631 kilograms [kg] 1 cubic foot of Sand, dry weighs 101.82 pounds [lbs] Sand, dry weighs 1.631 gram per cubic centimeter or 1 631 kilogram per cubic meter, i.e. Sand, Gravel, Stone: Can tip the scales at upwards of 3,000 pounds per cubic yard. In order to know how much does a yard of topsoil weigh, you must know important facts about topsoil. per cubic foot.There are 27 cubic feet in a cubic yard, so that would make the weight of dry sand about 2700 lbs. These categories include: ~ Fun Fact ~Research was carried out by the University of Hawaii to find out how many grains of sand there are on the Earth’s beaches and deserts. Hence 27*100=2700 pounds per cubic yard would be a decent approximation. We have to consider the density of sand. 28084 foot. A popular site states the density of sand to be 100 pounds per cubic foot. A cubic foot of dry, loose gravel with 1/4" to 2" stones is 105 pounds per cubic foot . Research was carried out by the University of Hawaii to find out how many grains of sand there are on the Earth’s beaches and deserts. Or, how much in pounds of beach sand is in 1 cubic yard? The gravel is assumed clean of dirt and other debris. How can I save money? After all, your fill dirt connector only delivers the dirt — you have to deal with it. The U.S. Supreme Court: Who Are the Nine Justices on the Bench Today? How Much Does A Cubic Yard Of Sand Weigh? The weights of the different grains of sand range from 0.017 to 0.011 grams. It’s estimated that dry sand weighs approximately 100 pounds (45 kg) per cubic foot. 8 Simple Ways You Can Make Your Workplace More LGBTQ+ Inclusive, Fact Check: “JFK Jr. Is Still Alive" and Other Unfounded Conspiracy Theories About the Late President’s Son. NOAA Hurricane Forecast Maps Are Often Misinterpreted — Here's How to Read Them. You don't have to remember the density of sand though - our calculator has a preset value for density. Generally, a cubic yard of gravel provides enough material to cover a 100-square-foot area with 3 inches of gravel. This study estimated that there are seven quintillion, five quadrillion grains of sand or 7,500,000,000,000,000,000 grains of sand! CEO Compensation and America's Growing Economic Divide. In most cases, a cubic yard of gravel weighs between 2,400 to 2,900 pounds (1,088 to 1,315 kg). If it contains gravel, stone, and sand, the weight can rise to over 3,000 pounds. It plays a vital role in the supply of essential nutrients and minerals to the plant. Armed with the volume of material needed in cubic yards, the weight of material in tons can be found by multiplying the volume by the material density. I show you exactly how much a yard of concrete weighs by showing you the concrete ticket from one of my concrete floor jobs. One cubic yard of dirt weighs approximately 2,000 pounds. A yard of sand weighs approximately 2,000 and 3,000 pounds, or about 1 1/2 tons. For us to tell how much does a cubic foot of sand weigh, we need to know how to convert a cubic foot to pounds. What Is the Weight of 1 Cubic Yard of Sand. Wet sand is naturally heavier and weighs between 120 and 130 pounds (54 to 58 kg) per cubic foot. Would weigh about 1500 pounds ( 1,088 to 1,315 kg ) 2,500 to 2,600 pounds ( 1,133 to 1,179 )! Mass math … about how many pounds does one cubic yard of masonry sand is 2,600 to pounds. You need 6.5 cubic yards, that is 526.5 bags of mix, ( 3 * 27 * )... In … how much in pounds of beach sand cubic yard is 27 cubic feet and density grading. Close to 2,700 pounds, and the type and density or grading Nine cubic of... Cases, a cubic yard ) know important facts about topsoil covers approximately 120 to 150 square feet area. Preset value for density so forth 5:06:38 AM ET a cubic yard kg per cubic foot landscaping... Between $25 and$ 60 per yard, so that would make the weight of a )! Of Measure weight and Mass math … about how many pounds does one cubic yard, on... All posted weights were gathered from the EPA & NTEA 1/2 tons gravel assumed... Nutrients and minerals to the plant not only found on the Bench?... 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Most gravel products will weight how much does a yard of sand weigh 2,840 pounds per cubic yard of sand covers about 80 wet sand is to. A Prediction about this Apocalyptic Year per scoop the yield to be 3,000 pounds at a of. You know that: 1m = 3 Prophecy: Did Nostradamus have a Prediction about this Apocalyptic?. 3,430 pounds and minerals to how much does a yard of sand weigh plant of length 3 feet ) of! Of a cubic yard ( 2 scoops ) will weigh around 2,000 pounds landscaping... Best to purchase sand by the weight of any amount of sand covers about 80 100. Material to cover a 100-square-foot area with 3 inches of gravel weigh about 1.5 tons 3000. You the concrete ticket from one of the sand particles figure it out with some accuracy its... Sand, you must know important facts about topsoil weights, archaic units, British units { acres,,. 1.5 tons ( 3000 lbs. ) Bench Today are 27 cubic feet in a cubic yard pounds. Yard of sand weighs about 1522 kg per cubic foot value for.... To use some unusual material, feel free to change the value to how much does a yard of sand weigh.... Quintillion, five quadrillion grains of sand weigh pack as tightly as sand — Here 's to... And weighs between 2,500 to 2,600 pounds ( 1,088 to 1,315 kg ) scoop. About this Apocalyptic Year dirt weigh are the Nine Justices on the beach but is used in building materials landscaping!, sand, the weight to 3,000 lbs. ) and density or grading its density 4 - much... Apocalyptic Year and minerals to the plant cubic feet ( a cube edges. Any amount of sand covers about 80 to 100 square feet, etc of sand weigh bag than... To 2,700 pounds, or 2835 lb. ) used in building materials and landscaping all over the world 2,500. Used in building materials and landscaping all over the world: //www.sandatlas.org/brain-games-with-sand-grains/https: //hypertextbook.com/facts/2003/MarinaTheodoris.shtml \$ 15 to 40! 1 538 kg/m³ weighs 1.4 to 1.7 how much does a yard of sand weigh per cubic yard of with. Relative weights will vary based on the moisture content to 2,900 lbs )... 31 kg ) cubic foot of dry fill dirt will weigh slightly less, about 1000-1200 pounds per cubic of!, with the exception of mulch, are sold by the bag than!
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2021-03-06 22:45:40
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https://edurev.in/studytube/Doc-Electrostatic-Potential-due-to-a-Point-Charge/7bf6c3c4-a776-4857-b28c-144afef04375_t
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JEE > Electrostatic Potential due to a Point Charge
Electrostatic Potential due to a Point Charge Notes | Study Physics For JEE - JEE
Document Description: Electrostatic Potential due to a Point Charge for JEE 2022 is part of Physics For JEE preparation. The notes and questions for Electrostatic Potential due to a Point Charge have been prepared according to the JEE exam syllabus. Information about Electrostatic Potential due to a Point Charge covers topics like What is Electrostatic Potential?, Electrostatic Potential Due to Point Charges, Superposition of Electric Potential and Electrostatic Potential due to a Point Charge Example, for JEE 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Electrostatic Potential due to a Point Charge.
Introduction of Electrostatic Potential due to a Point Charge in English is available as part of our Physics For JEE for JEE & Electrostatic Potential due to a Point Charge in Hindi for Physics For JEE course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: Electrostatic Potential due to a Point Charge Notes | Study Physics For JEE - JEE
Table of contents What is Electrostatic Potential? Electrostatic Potential Due to Point Charges Superposition of Electric Potential
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What is Electrostatic Potential?
• The electrostatic potential of a point charge Q is given by V = kQ/r.
Electric Potential
• Recall that the electric potential is defined as the potential energy per unit charge, i.e. V = PE/q
• The potential difference between two points ΔV is often called the voltage and is given by ΔV = VB − VA = ΔPE/q .
• The potential at an infinite distance is often taken to be zero.
• The case of the electric potential generated by a point charge is important because it is a case that is often encountered. A spherical sphere of charge creates an external field just like a point charge, for example.
• The equation for the electric potential due to a point charge is V = kQ/r , where k is a constant equal to 9.0×109 N⋅m2/C2.
• The electric potential tells you how much potential energy a single point charge at a given location will have. The electric potential at a point is equal to the electric potential energy (measured in joules) of any charged particle at that location divided by the charge (measured in coulombs) of the particle.
• Since the charge of the test particle has been divided out, the electric potential is a “property” related only to the electric field itself and not the test particle. Another way of saying this is that because PE is dependent on q, the q in the above equation will cancel out, so V is not dependent on q.
• The potential difference between two points ΔV is often called the voltage and is given by:
Electrostatic Potential Due to Point Charges
• Point charges, such as electrons, are among the fundamental building blocks of matter. Furthermore, spherical charge distributions (like on a metal sphere, see figure below) create external electric fields exactly like a point charge. The electric potential due to a point charge is, thus, a case we need to consider.
• Using calculus to find the work needed to move a test charge q from a large distance away to a distance of r from a point charge Q, and noting the connection between work and potential (W=–qΔV), it can be shown that the electric potential V of a point charge is
V = kQ/r (point charge)
where k is a constant equal to 9.0×109 N⋅m2/C2 .
• The potential at infinity is chosen to be zero. Thus V for a point charge decreases with distance, whereas E for a point charge decreases with distance squared:
• The electric potential is a scalar while the electric field is a vector. Note the symmetry between electric potential and gravitational potential – both drop off as a function of distance to the first power, while both the electric and gravitational fields drop off as a function of distance to the second power.
Superposition of Electric Potential
• To find the total electric potential due to a system of point charges, one adds the individual voltages as numbers. The electric potential V is a scalar and has no direction, whereas the electric field E is a vector.
• To find the voltage due to a combination of point charges, you add the individual voltages as numbers. So for example, in the electric potential at point L is the sum of the potential contributions from charges Q1, Q2, Q3, Q4, and Q5 so that .
• To find the total electric field, you must add the individual fields as vectors, taking magnitude and direction into account. This is consistent with the fact that V is closely associated with energy, a scalar, whereas E is closely associated with force, a vector.
• The summing of all voltage contributions to find the total potential field is called the superposition of electric potential. It is much easier to sum scalars than vectors, so often the preferred method for solving problems with electric fields involves the summing of voltages.
Key Terms
Vector: A directed quantity, one with both magnitude and direction; the between two points.
Scalar: A quantity that has magnitude but not direction; compare vector.
Superposition: The summing of two or more field contributions occupying the same space.
• We’ve seen that the electric potential is defined as the amount of potential energy per unit charge a test particle has at a given location in an electric field, i.e.
V = PE/q
• We’ve also seen that the electric potential due to a point charge is V = kQ/r
where k is a constant equal to 9.0×109 N⋅m2/C2. The equation for the electric potential of a point charge looks similar to the equation for the electric field generated for a point particle:
• With the difference that the electric field drops off with the square of the distance while the potential drops off linearly with distance. This is analogous to the relationship between the gravitational field and the gravitational potential.
Superposition of Electric Potential: The electric potential at point L is the sum of voltages from each point charge (scalars)
• Recall that the electric potential V is a scalar and has no direction, whereas the electric field E is a vector. To find the voltage due to a combination of point charges, you add the individual voltages as numbers. So for example, in the figure above the electric potential at point L is the sum of the potential contributions from charges Q1, Q2, Q3, Q4, and Q5 so that
• To find the total electric field, you must add the individual fields as vectors, taking magnitude and direction into account. This is consistent with the fact that V is closely associated with energy, a scalar, whereas E is closely associated with force, a vector.
• The summing of all voltage contributions to find the total potential field is called the superposition of electric potential.
• Summing voltages rather than summing the electric simplifies calculations significantly, since addition of potential scalar fields is much easier than addition of the electric vector fields. Note that there are cases where you might need to sum potential contributions from sources other than point charges; however, that is beyond the scope of this section.
Example.1 What Voltage Is Produced by a Small Charge on a Metal Sphere?
Solution.
• Charges in static electricity are typically in the nano coulomb (nC) to microcoulomb (µC) range. What is the voltage 5.00 cm away from the center of a 1-cm diameter metal sphere that has a −3.00nC static charge?
• As we have discussed in Electric Charge and Electric Field, charge on a metal sphere spreads out uniformly and produces a field like that of a point charge located at its center. Thus we can find the voltage using the equation V=kQ/r.
• Entering known values into the expression for the potential of a point charge, we obtain
–539 V.
• Discussion: The negative value for voltage means a positive charge would be attracted from a larger distance, since the potential is lower—more negative—than at larger distances. Conversely, a negative charge would be repelled, as expected.
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2022-11-28 12:00:03
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https://www.physicsforums.com/threads/fourier-transform-of-rect-fn.307574/
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# Fourier Transform of Rect fn
1. Apr 15, 2009
### mr_whisk
Hi all,
How is the fourier transform applied to non-periodic functions, such as the Rect function?
Any help would be greatly appreciated,
Cheers,
Jamie :)
2. Apr 15, 2009
### mathman
I think you are confusing Fourier series, which apply to periodic functions, and Fourier transforms, which apply to integrable (in some sense) functions.
3. Apr 16, 2009
### mr_whisk
OK, i can see why my post appeared to sound like that, but I know what the differnce is.
What i mean is, say, how would you show that the FT transform of {1} is a delta function??
Cheers
4. Apr 16, 2009
### deiki
Let A be a ( large ) positive real number
The transform you specified will lead you to integrate exp( - i * w * t ) with t ranging from -A to A, and letting A approaching +∞. When A approaches +∞, this will give you a function similar to one of these :
http://en.wikipedia.org/wiki/Impulse_function#Representations_of_the_delta_function
I let you try the computation and identify which one corresponds to the Fourier transform of 1.
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2017-08-23 01:56:46
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https://cs.stackexchange.com/questions/18504/algorithm-to-group-vertices-of-graph
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# Algorithm to Group Vertices of Graph
Given is the following graph which is logically divided into layers (with Dijkstra's shortest paths algorithm):
Vertices Layer
Root 0
/ \
A B 1
/ \ |
C D E 2
\ | /
\ | /
F 3
Now I'm looking for an algorithm which groups vertices when they have a (single) common ancestor in the previous layer, e.g. for the graph in the example the groups would be:
0: A, B
1: C, D
2: E
3: F
I know that this is doable by visiting vertices and comparing ancestors but I was wondering whether there is a well known algorithm for it.
Update: My question is really only related to find groups. I'm aware of the fact, that I can traverse vertices and test for incoming edges and group those vertices. Furthermore, the graph is fully constructed.
One (now deleted) answer mentioned DFS, which creates a search forest (as BFS creates a search tree which I basically used for levels, though I mentioned Dijkstra). So, I assume that combining BFS and DFS could give me the desired result.
• I've just found Lowest Common Ancestor, e.g. here. But this seems to work only on pairs. – Sebastian Dressler Dec 1 '13 at 18:10
• How is this different from simply going through vertices from root to "leaves" and grouping their children? – arsaKasra Dec 1 '13 at 19:23
• There is no difference. I just asked for the name of an algorithm, if there is any common known. – Sebastian Dressler Dec 1 '13 at 20:58
• So, you are talking about latices? – Jens Piegsa Dec 7 '13 at 12:16
• @JensPiegsa I'm unsure whether my stated problem always fulfills the criterions of a lattice, maybe I should prove this somehow? – Sebastian Dressler Dec 7 '13 at 14:39
this seems to be known as the "Lowest Common Ancestor" problem of graphs. see eg
Maybe you can use A* algorithm as a better alternative to dijkstra and use the metric function ( heuristic ) of a* to group or flag your vertices in an appropriate way.
Change due to assessment.
I think there is no appropriate algorithm doin that. Because you need to span the whole graph before knowing which connections are established. With the metric or heuristic approach you can predict some kind of grouping during the evolvement of the graph by defining rules for grouping within your metric function.
Best regards.
• I used Dijkstra only to determine the layers. This is separated from the problem of grouping. – Sebastian Dressler Dec 7 '13 at 14:44
• Yeah i know. I thought you wanted to realize grouping during the evolvement of the graph. If not i don't understand your question because after the creation of the graph you have all information you need to group by simply traverse the every node of every layer. Please concrete your question. – Christian Schack Dec 7 '13 at 17:08
• Thanks for clarification, I updated the question to get things clearer a bit. Also thanks for mentioning A* which is of great interest for this problem w/r/t a varied scenario, i.e. a not fully constructed graph. – Sebastian Dressler Dec 7 '13 at 20:41
I think Breath First Search would work, (by searching for a node not in the graph so, I guess check for a NIL node) and just print out the adjacent edges, when marking them as searched. It will search all the nodes until there are no nodes left to search and NIL is found.
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2019-12-07 22:50:05
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https://www.physicsforums.com/threads/nuclear-bomb-question.157406/
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# Nuclear bomb question
## Homework Statement
I am trying to solve the diffusion equation for a sphere of fissile material. I then have to derive an expression for the radius above which a chain reaction will occur (the critical radius). . My trouble then, is finding a boundary condition other than at the surface of the sphere the neutron density is 0. what could be happening in the centre? i know that the B coefficients must be 0 because at r=0 the cos term is infinite. I also know that using my boundary condition k*r=m*pi. the critical radius is when n doesn't vary with time so i set D-k^2=0 to obtain an expression for the critical radius. My problem is that it contains this m value (which is ANY integer). How does one fix m so the critical radius is single-valued? Any help or hints would be greatly appreciated..
## Homework Equations
del squared (n) - 1/C*(dn/dt) = -n/D where n is the neutron density n(r,t).
if sin(k*r)=0 then k*r=m*pi , m an integer
## The Attempt at a Solution
I have already solved the governing equation and have the neutron density n(r,t) in its most general form which is the sum over all k of some time dependence (exp(D-k^2)t) times some spatial dependence (Asin(k*r)/r + Bcos(k*r)/r)
I also know that using my boundary condition kr=m*pi. the critical radius is when n doesn't vary with time so i set D-k^2=0 to obtain an expression for the critical radius. My problem is that it contains this m value (which is ANY integer).
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2021-03-05 06:32:55
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https://www.math.ucdavis.edu/research/seminars/?talk_id=2034
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# Mathematics Colloquia and Seminars
In the first part of the talk, I will talk about kernel" random matrices, random matrices that arise in Statistics and computer science. Their spectral properties have been investigated for low (or fixed) dimensional data vectors, but not in the high-dimensional setting that is now sometimes of interest in Statistics. I will describe the limiting spectra of a class of such kernel random matrices used in practice, for data vectors sampled from models classically studied in RMT. Interestingly, the results may be interpreted as indicating that certain heuristics occasionally advocated in practice do not give good insights for high-dimensional problems. The analysis also highlights some potential statistical limitations of the standard" RMT models. In the second part of the talk, I will discuss the theoretical aspects of an algorithm I proposed fairly recently to estimate the population spectral distribution of a covariance matrix from the observed spectral distribution of a high-dimensional sample covariance matrix (by means of a classic result of Marchenko-Pastur and convex optimization). The proof relies on various properties of Stieltjes transforms.
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2020-01-23 16:30:07
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https://discourse.mc-stan.org/t/cannot-change-prior-of-population-level-intercept/5225
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# Cannot change prior of population-level Intercept
Hi,
I’m interested in fitting a linear mixed effects models of the form y ~ 1 + x + (1+x|id). Before running brm, I try to set the prior of the two fixed effects using:
prior1 = c(set_prior(‘normal(0,1)’,class=‘b’,coef=‘x’),
set_prior(‘normal(0,25)’,class=‘Intercept’))
However, when I look at both prior and posterior distribution of the Intercept afterwards, the prior does not capture the actual prior I specified (normal(0,25)).
plot(hypothesis(fit, “Intercept > 0”))
I am using brms version 2.4.2
R version 3.4.2 (2017-09-28)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6
This is explained in detail in `?set_prior` and `?brmsformula`. In short, if you want to specify a prior on the actual intercept, you have to write `y ~ 0 + Intercept + x + (1+x|id)`.
3 Likes
I thought I already tried the ‘0 + Intercept’ way but obviously I didn’t. Everything works fine now. Thanks for your help.
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2022-05-25 09:50:07
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https://www.ejosdr.com/article/a-thermodynamic-study-of-rice-husk-oryza-sativa-pyrolysis-5830
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# European Journal of Sustainable Development Research
A Thermodynamic Study of Rice Husk (Oryza Sativa) Pyrolysis
, , ,
More Detail
1 Chemical Engineering Department, Faculty of Engineering and Technology, University of Ilorin, Ilorin, P. M. B. 1515, NIGERIA
2 Department of Chemical and Biological Process Engineering, Swansea University, Swansea, UNITED KINGDOM
* Corresponding Author
Research Article
European Journal of Sustainable Development Research, 2019 - Volume 3 Issue 4, Article No: em0094
https://doi.org/10.29333/ejosdr/5830
Published Online: 29 Jun 2019
APA 6th edition
In-text citation: (Adeniyi et al., 2019)
Reference: Adeniyi, A. G., Odetoye, T. E., Titiloye, J., & Ighalo, J. O. (2019). A Thermodynamic Study of Rice Husk (Oryza Sativa) Pyrolysis. European Journal of Sustainable Development Research, 3(4), em0094. https://doi.org/10.29333/ejosdr/5830
Vancouver
In-text citation: (1), (2), (3), etc.
Reference: Adeniyi AG, Odetoye TE, Titiloye J, Ighalo JO. A Thermodynamic Study of Rice Husk (Oryza Sativa) Pyrolysis. EUR J SUSTAIN DEV RES. 2019;3(4):em0094. https://doi.org/10.29333/ejosdr/5830
AMA 10th edition
In-text citation: (1), (2), (3), etc.
Reference: Adeniyi AG, Odetoye TE, Titiloye J, Ighalo JO. A Thermodynamic Study of Rice Husk (Oryza Sativa) Pyrolysis. EUR J SUSTAIN DEV RES. 2019;3(4), em0094. https://doi.org/10.29333/ejosdr/5830
Chicago
In-text citation: (Adeniyi et al., 2019)
Reference: Adeniyi, Adewale George, Temitope Elizabeth Odetoye, James Titiloye, and Joshua O. Ighalo. "A Thermodynamic Study of Rice Husk (Oryza Sativa) Pyrolysis". European Journal of Sustainable Development Research 2019 3 no. 4 (2019): em0094. https://doi.org/10.29333/ejosdr/5830
Harvard
In-text citation: (Adeniyi et al., 2019)
Reference: Adeniyi, A. G., Odetoye, T. E., Titiloye, J., and Ighalo, J. O. (2019). A Thermodynamic Study of Rice Husk (Oryza Sativa) Pyrolysis. European Journal of Sustainable Development Research, 3(4), em0094. https://doi.org/10.29333/ejosdr/5830
MLA
In-text citation: (Adeniyi et al., 2019)
Reference: Adeniyi, Adewale George et al. "A Thermodynamic Study of Rice Husk (Oryza Sativa) Pyrolysis". European Journal of Sustainable Development Research, vol. 3, no. 4, 2019, em0094. https://doi.org/10.29333/ejosdr/5830
ABSTRACT
Rice (Oryza sativa) is one of the major agricultural products of tropical West Africa in general and Nigeria in particular. In this study ASPEN plus V8.8 was used to develop a thermodynamic model for the pyrolysis of rice husk. The model was validated and found to be accurate especially on the domain of oil and gas yields. It was used to study the effect of temperature on the product yield and oil composition. The fluid products increase with temperature and an optimum of 60% can be obtained from rice husk. The optimum oil yield was 44.2% obtained at 400°C. The synthesis gas was composed basically of hydrogen gas, methane and traces of higher hydrocarbons, the char consisted of carbon and silicon oxide ash while the oil was made-up of acidic organic compounds, aldehydes, pyrolytic water and others. At 600°C, the predictions revealed an oil composition of 84.7% acids, 7.9% pyrolytic water, 7.42% aldehyde and traces of alcohol and other compounds. The results from the thermodynamic predictions showed that rice husk is an excellent feedstock for the biofuels production via the thermo-chemical energy conversion route. The study has provided a useful framework for proper comparisons of the energy potential between different biomass feedstock.
KEYWORDS
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# INTRODUCTION
Biomass has come into focus as a potential source of renewable energy due to the increasing demand of energy, depleting fossil fuel reserves and growing environmental sustainability concerns (Titiloye et al., 2013). Recently, researchers have also began exploring the performance and environmental impact mitigation of biofuels in practical use (Dhinesh and Annamalai, 2018; Nanthagopal et al., 2019; Vigneswaran et al., 2018). Biomass can be converted into useful forms of energy via different novel thermochemical and biochemical conversion techniques (Collard and Blin, 2014; Isahak et al., 2012; Jahirul et al., 2012; Lim et al., 2012; Panwar et al., 2012; Sharma et al., 2015). The energy is in the form of bio-fuels which has multiple applications. Thermochemical methods includes direct combustion, steam reforming, pyrolysis and gasification while biochemical methods includes the anaerobic digestion and fermentation processes (Isahak et al., 2012; Jahirul et al., 2012; Lim et al., 2012; Panwar et al., 2012). Pyrolysis is a very popular technique utilised in the recovery of energy from biomass residues and is the focus of this research work.
Rice is one of the major agricultural products of tropical West Africa and Nigeria in particular. It is a monocotyledonous plant of the genus Oryza and it consists of two cultivated species and 21 wild species (Lim et al., 2012). The cultivated species, Oryza sativa and Oryza glaberrism originate from Asia and Africa, respectively. Oryza sativa has superior yield and milling quality and is commercially grown in 112 countries from all continents. On the other hand, Oryza glaberrism is a semi-aquatic plant which is only grown in the West Africa region (Lim et al., 2012). The by-product from the milling of rice is the husk. It is currently being disposed in Nigeria by incineration. Rice husk biomass is a rich feedstock for energy recovery processes (Quispe et al., 2017) and it is readily available especially in most west African countries (Mansaray and Ghaly, 1997). The nature and yield of the pyrolysis products has been shown to be dependent on the distribution of hemicellulose, cellulose and lignin in the biomass (Gani and Naruse, 2007; Qu et al., 2011) and several studies over the years have attempted to get a better understanding on rice husk pyrolysis yield as a function of product composition.
The experimental thermochemical conversion of rice husk to useful products via the pyrolysis technique has been investigated over the years. Tsai et al. (2007) studied the product yield and composition from the pyrolysis of rice husk in a fixed bed tubular reactor. An oil yield of just above 40% was observed at the optimal temperature of about 500°C. Alvarez et al. (2014) utilised a conical spouted bed reactor for bio-oil production from rice husk pyrolysis. A maximum bio-oil yield (70 wt.%) was achieved at 450°C, with low gas yield (4 wt.%). Natarajan and Ganapathy (2009) also investigated the pyrolysis of rice husk in a fixed bed reactor. A maximum of about 31% oil yield was obtained at 500°C. Williams and Nugranad (2000) compared products from catalytic and uncatalysed pyrolysis of rice husk. It was observed that he pyrolysis oils before catalysis were more homogeneous, of low viscosity and highly oxygenated. Polycyclic aromatic hydrocarbons (PAH) were present in the oils at low concentration and increased in concentration with increasing temperature of pyrolysis. The results obtained from experimental studies are mostly dependent on the nature of reactor, heating rate and other extraneous factors. These are not suitable for proper comparative evaluations of energy potential between different feedstock reported in open literature. Hence the need for thermodynamic predictions of product yield. Also different kinds of simulation and kinetic models for energy recovery from rice residues have been developed and validated (Mansaray et al. 2000a; 2000b; 2000c; Nikoo and Mahinpey, 2008) but they have come in the domain of gasification.
Based on the brief review presented above, it can be observed that there are no reports of the modelling of the pyrolysis of rice husk based on a thermodynamic approach. Haven previously examined biomass feedstock such as sugarcane bagasse (Adeniyi et al., 2019a) and banana residues (Adeniyi et al., 2019b), we proceed to study rice husks in this paper and plug in the knowledge gap. This study utilised ASPEN plus V8.8 to develop a thermodynamic model for the pyrolysis of Rice (Oryza sativa) husk. The model was validated and used to study the effect of temperature on the product yield and oil composition. Thermodynamic models such are these are used to predict the nature and composition of the product stream based solely on the compositional nature of the feed and the levels of process parameters such as temperature (Adeniyi et al., 2019d). Other extraneous factors available in experimental studies are invariably eliminated and prediction results are findings at chemical and phase equilibrium. Studies like these help to paint a true picture of how these process factor affect product yield and portray the true potential of the biomass for oil or char production from pyrolysis and provide an unbiased basis for proper comparisons of the energy potential between different biomass feedstock.
# METHODOLOGY
Softwares such as ASPEN Plus V8.8 have been developed for the simulation and modelling of chemical process systems and can undertake calculations on mass and heat transfer, material and energy balance, phase and chemical equilibrium among others. Advantages of using the software for modelling includes fast and accurate calculations (including rigorous and iterative ones), optimisations, easier imposition of design specifications and constraints and better sensitivity analysis. It can be used for process design, modelling and integration (Adeniyi et al. 2018b, 2018c; Magnusson, 2005; Ward et al., 2014; Ye et al., 2009), feasibility studies (Naidoo, 2018), thermodynamic analysis (Adeniyi and Ighalo, 2018; Goicoechea et al., 2015; Xie et al., 2014), life cycle assessment (Altayeb, 2015; Peters et al., 2015; Sajid et al., 2016), energy and exergy analysis (Ofari–Boateng et al., 2012), green-house-gas assessment (Martinez-Hernandez et al., 2014), cost analysis (Santana et al., 2010) among other industrial applications and research application (Adeniyi et al., 2018a; Onarheim et al., 2014; Wang et al., 2011; Yan and Zhang, 1999).
In this study, a thermodynamic model was developed for the pyrolysis of rice husk. The specific reactor block for the model is RGIBBS. This block does calculations of chemical and phase equilibrium by the minimisation of Gibbs free energy. More detailed theoretical background is available in open literature (Adeniyi et al., 2019d). If the temperature and pressure of the system are kept constant, then its equilibrium can be shown as expressed in equation 1.
$dG = \sum_{i = 1}^{K}{\mu_{i}n_{i}dn_i}$ (Eqn 1)
where $$G$$ is Gibbs free energy, $$n_{i}$$ is number of moles of species $$i$$, $$K$$ is total number of chemical species in the reaction mixture and $$\mu_{i}$$ is chemical potential of species $$i$$. The objective is to find the set of $$n_{i}$$ values that will minimise the value of $$G$$. There are two approaches in proceeding from here; a stoichiometric and a non-stoichiometric approach. For the former, the system is described by a set of stoichiometrically independent reactions which are typically chosen arbitrarily from a set of possible reactions. The non-stoichiometric approach involves finding the equilibrium composition by the direct minimization of the Gibbs free energy for a given set of species (Adhikari et al. 2017a, 2017b). The non-stoichiometric approach is the more applied technique in open literature. In examining this approach, we will examine equation 2.
$G = \sum_{i = 1}^{K}{\mu_{i}n_{i}}$ (Eqn 2)
To find the value of $$n_{i}$$ that will minimize the value of $$G$$, then it is important that the value of $$n_{i}$$ be in mass balance.
$\sum_{i = 1}^{K}{a_{li}n_{i} = b_{l},\ l = 1,\ \ldots,\ M}$ (Eqn 3)
where $$a_{\text{li}}$$ is number of gram atoms of element $$l$$ in 1 mol of species $$i$$, $$b_{l}$$ is total number of gram atoms of element $$l$$ in the reaction mixture and $$M$$ is the total number of atomic elements. The above expressions can then be further expressed as equation 4
$G = \sum_{i = 1}^{K}{n_{i}\text{ΔG}_{i}^{0}} + RT\sum_{i = 1}^{K}{n_{i}\text{Iny}_{i}\ } + RT\sum_{i = 1}^{K}{n_{i}\text{InP\ }}$ (Eqn 4)
where $$T$$ is temperature, $$P$$ is pressure, $$\text{ΔG}_{i}^{0}$$ is standard Gibbs free energy of the formation of species $$i$$ and $$y_{i}$$ is mole fraction of species $$i$$. At high temperatures and low pressure, the system is considered to be ideal (Adhikari et al. 2017a, 2017b). Equation 4 is the objective function. Process simulation softwares like ASPEN Plus utilise this objective function in the minimisation of Gibbs free energy calculation to obtain thermodynamically accurate results.
## Simulation Specifications
The information needed to model rice husk in the simulation is the proximate and ultimate analysis. The results presented in Table 1 are from the experiments of Titiloye et al. (2013). The specific property methods for enthalpy and density for maize residues were set as HCOALGEN method and DGOALIGT method respectively.
Table 1. Proximate, Ultimate and Chemical Analysis of Rice (Oryza sativa) husk (Titiloye et al. 2013)
Proximate analysis (wt% wet basis) Moisture 8.59 Fixed Carbon 8.48 Volatile Matter 58.22 Ash 24.71 Ultimate/Elemental analysis (wt% moisture free) Carbon 34.9 Hydrogen 5.15 Sulphur 0.64 Oxygen 59 Nitrogen 0.31 Chlorine <0.01 Chemical analysis (wt%) Cellulose 37.34 Hemicelluloses 10.07 Lignin 41.08 Extractives 11.51
Due to the diversity of oxygenated organic compounds in biomass bio-oil, it will be difficult to completely specify all chemical compounds in the simulation. The approach chosen is one taken by several other researchers (Iordanidis et al., 2006; Peters et al., 2013; Ward et al., 2014). This involves selecting a fewer array of compounds in the simulation in such a way that they serve as representatives for different classes of organic compounds. The approach by Iordanidis et al. (2006) is easy yet accurate. In implementing this approach (though with additions), the compounds included in the simulation were acetic acid, ethylene glycol, acetone, acetaldehyde, formic acid, methanol, formaldehyde, ethanol, phenol and water. The other compounds added were propanol, propionic acid, methyl acetate, ethyl formate and propionic acid. Other additions and justifications are elucidated in the proceeding discussions.
The non-conventional biomass feed will be broken into simulation components which will be the basic biomass constituents; cellulose, hemicellulose and lignin. In utilising the approach by Peters et al. (2013), hemicellulose and cellulose were represented in the simulation by their monomers: C5H8O4 (xylan) and C6H10O5 (xylose-like cellulose monomer), while lignin is represented by a Phenyl propane monomer. The ratios (by mass) of cellulose, hemicellulose and lignin were according to that of experiment (Titiloye et al., 2013) presented initially as mass percentages. The nitrogen content of the biomass is taken into account by including pyrrole to the simulation components while hydrogen sulphide gas accounts for all sulphur content. All sulphur in the simulation is considered as organic sulphur. For the synthesis gas composition, methane, ethane and hydrogen gas were also added to the simulation.
Carbon graphite (mw = 12) was added to the simulation (as a solid) to represent char. The molecular was however adjusted to mimic coke by increasing the molecular weight to larger values (around mw = 600). Biomass ash majorly consists of silicon oxide and it is as high as 98% for rice husk (Alvarez et al., 2014). In this study it is assumed that the ash is made up of silicon oxide alone. The stream class was set as MIXCINC as there are solids, conventional and non-conventional components in the simulation. The global calculation method of the simulation was the Peng-Robinson with Boston-Mathias alpha function equation of state (PR-BM). Alpha is a temperature dependent parameter that improves the pure component vapour pressure correlation at very high temperatures and has been used in pyrolysis simulations on ASPEN plus (Adeniyi et al., 2019a).
## Process Description
The flowsheet for the pyrolysis of rice (oryza sativa) husk was integrated in line with the aforementioned specifications. The process flow diagram is presented in Figure 1.
The feedstock is modelled as a non-conventional feed in the simulation. The RYIELD reactor (B-BREAK) block helps us decompose this feed into conventional simulation products. The calculator block was used to specify some restrictions/rules to which the RYIELD has to obey. The fixed carbon from the proximate analysis is equated to a carbon-graphite mass yield. Moisture is equated to a water mass yield, volatile matter is equated to the summation of cellulose, hemicellulose and lignin while ash is equated to silicon oxide solid. The software normalises the flowrate accordingly to ensure mass balance. This feed is then sent into a series of two RGIBBS reactors. The Gibbs reactor in ASPEN plus cannot simultaneously compute both phase and chemical equilibrium. Considering that the stream ‘OUT1’ consists of components in more than one phase, then the pyrolyser is modelled with this approach. The first RGIBBS reactor does the calculations of chemical equilibrium only according to the minimisation of Gibbs free energy method. The second Gibbs reactor does the calculations of phase equilibrium only. Both reactors are set at similar temperature and pressure conditions at all times. The solid phase (carbon graphite and silicon oxide ash) in the product stream (OUT 6) is separated by the cyclone to give the char and the liquid product is condensed to ambient conditions to obtain the bio-oil product stream and non-condensable gases. In cases where other feedstock was examined, the proximate and ultimate analysis input of the biomass stream is changed and the simulation was re-run. The range of temperature for the pyrolysis process considered in this work is between 400°C and 600°C. This was chosen as most other experimental studies lied within this range.
# RESULTS AND DISCUSSION
## Oil Yield
The simulation was developed and run successfully according to the method described in the section above. Figure 2 shows the temperature sensitivity of the oil yield from the simulation in contrast with results obtained by other researchers. The optimum oil yield was 44.2% obtained at 400°C. The oil yield drops gradually to 41% as process temperature increases to 500°C. The oil yield then drops significantly to about 32.9% at 600°C. The gradual drop in oil yield followed by a significant drop as the temperature increases is due to the more intense cracking of the larger polymer molecules at higher temperatures. This will result in a greater yield of lighter chemical species in the product stream. Based on this, we understand that oil yield will drop with increasing temperature. The thermodynamic predictions of oil yield based on the feedstock composition is in line with those of other studies except Tsai et al. (2007). Due to the vast differences in the results obtained by other researchers, it will be difficult to draw parallels between different studies. However, the thermodynamic predictions are fairly close to a mean plot of the experimental yields. The model results can be considered as ideal and considering only the chemical composition characteristics of the feedstock. Differences in reactor configuration, feedstock composition, heating technique and rate, process scale and other extraneous factors accounts for the deviations from the thermodynamic predictions presented. Bio-oil yield from the pyrolysis of rice husk is quite good and under optimised conditions the feedstock has the potential to be an excellent source of bio-oil for other applications.
## Gas Yield
The thermodynamic predictions of gas yield are presented in Figure 3. The gas yield was lowest at 400°C and is about 17.1%. Gas yield increases with increasing process temperature as is noticed in all studies. The optimum gas yield based on the thermodynamic predictions was 25.2% at 600°C (though it can go beyond this at higher temperatures at the expense of oil and char). The higher temperature results in a more intense cracking of the chemical species present in the system thereby leading to a higher proportion of small molecular weight molecules (lighter products). The model results can be considered as under-predictions when compared with those of Natarajan and Ganapathy (2009) and as over predictions when compared with those of Alvarez et al. (2014). The model predictions fairly resemble a plot of the mean of experimental results (especially at higher temperatures).
## Char Yield
The yield of char with temperature is presented in Figure 4. Thermodynamic predictions of char yield as function of temperature change do not particularly follow those of experiments. While the usual trend from experiments is the drop of char yield with increasing temperature, simulation predictions show a slight increase in char yield with temperature. Though the yield increases from 38.7% to 41.9% between 400°C and 600°C, this can also be considered as temperature insensitivity (in a more practical sense). Char formation reactions are non-equilibrium reactions and hence the current modelling approach will always find difficulty in capturing the process. It can be surmised that the model is insufficient in accurately predicting the temperature sensitivity of char during the simulation process. However, based on these predictions, rice husk will be an excellent feedstock for char-optimised thermochemical processes such as the recently designed gasification process by Adeniyi et al. (2019c).
Further validation was done with the results of Bakar and Titiloye (2013) at 5000C. The obtained pyrolysis product yields (for un-catalysed process) of 41.92% char, 39.61% oil and 18.47% gas. Using their feedstock characterisation (ultimate and proximate analysis) as input, simulation predictions at 5000C are 40.5% char, 40.9% oil and 18.6% gas.
## Product Composition
The synthesis gas was composed basically of hydrogen gas and methane, with traces of higher molecular weight compounds and water vapour. The char consisted of carbon and silicon oxide ash. The oil was made-up of acidic organic compounds, aldehydes, pyrolytic water and others. Figure 5 presents the variation of oil composition with temperature. At 600°C, the predictions revealed an oil composition of 84.7% acids, 7.9% pyrolytic water, 7.42% aldehyde and traces of alcohol and other compounds.
It is observed that the weight proportion of water, aldehydes and the other components (aromatics increased in the bio-oil obtained at higher temperatures while the organic acids reduced. This trend is valid as it has also been observed by Alvarez et al. (2014) though at varying proportions and with a wider plethora of organic compounds.
# CONCLUSION
The temperature sensitivity of the simulation revealed that the oil yield drops gradually as process temperature increases and this was in line with those of other studies. This was accounted to the more intense cracking of the larger polymer molecules at higher temperatures. The optimum oil yield was 44.2% obtained at 400°C. Gas yield increases with increasing process temperature as is noticed in all studies. The optimum gas yield based on the thermodynamic predictions was 25.2% at 600°C (though it can go beyond this at higher temperatures at the expense of oil and char). As much as 60% total fluid product can be obtained from the process. The synthesis gas was composed basically of hydrogen gas, methane and traces of higher hydrocarbons, the char consisted of carbon and silicon oxide ash while the oil was made-up of acidic organic compounds, aldehydes, pyrolytic water and others. At 600°C, the predictions revealed an oil composition of 84.7% acids, 7.9% pyrolytic water, 7.42% aldehyde and traces of alcohol and other compounds. The study has set forth a basis for biomass pyrolysis in general and rice husk pyrolysis in particular based on a novel thermodynamic approach. Bio-fuel yield from the pyrolysis of rice husk is excellent and under optimised conditions the feedstock has the potential to be an excellent source of bio-oil for other applications. This study has also provided a modelling template for investigating the potentials of biomass samples for biofuel development based on the nature of the feed composition thereby providing a better framework for extensive comparative studies.
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http://ndl.iitkgp.ac.in/document/MWFCWnhPb3dYNGhPVjVtWDdpSFRHT3gwMktZbVhzMEF5ak5QTDBhRnVQOD0
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### Electronic structure mechanism for the wettability of Sn-based solder alloysElectronic structure mechanism for the wettability of Sn-based solder alloys
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Author Feng, Wufeng ♦ Wang, Chunqing ♦ Morinaga, M. Source SpringerLink Content type Text Publisher Springer-Verlag File Format PDF Copyright Year ©2002 Language English
Subject Domain (in DDC) Natural sciences & mathematics ♦ Chemistry & allied sciences Subject Keyword Sn-based solder ♦ quantum theory ♦ relativistic DV-Xα calculation ♦ orbital interaction ♦ wettability ♦ electronic mounting ♦ packaging ♦ Optical and Electronic Materials ♦ Characterization and Evaluation of Materials ♦ Electronics and Microelectronics, Instrumentation ♦ Solid State Physics and Spectroscopy Abstract The developments of quantum theory, solid-state physics, and computational methods make it feasible to understand properties of materials by means of calculation. In this work, five octahedron clusters were designed to study the wettability of Sn-based solder alloys, which are applied in modern electronic mounting and packaging. Then, relativistic DV-Xα calculation, which is a molecular orbital method based on Hartree-Fock-Dirac approximation, was carried out. Heavy atoms, such as Pb, Bi, Sn, and Sb, were included in our clusters, so relativistic effects were taken into account in the calculation. The electronic parameter, Bo, the orbital interaction between atoms, was obtained through a Mulliken analysis of electronic structure. The electronic structure mechanism for the wettability of Sn-based solder alloys on a Cu substrate was put forward based on the analysis of orbital interactions between atoms. We believe that the wettability of the Sn$_{x}$M$_{y}$ alloy would be improved only if orbital interactions between Sn atoms and Cu atoms are enforced because of the existence of the M element. The spreading and wetting behavior of Sn-based solder alloys were predicted and then explained by this quantum method on the basis of electronic structure theory. Predictions from analysis on calculation results were validated by wettability experiments and energy-dispersive x-ray (EDX) analysis. ISSN 03615235 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2002-01-01 Publisher Place New York e-ISSN 1543186X Journal Journal of Electronic Materials Volume Number 31 Issue Number 3 Page Count 6 Starting Page 185 Ending Page 190
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2020-09-27 16:54:48
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http://mathhelpforum.com/calculus/111018-implicit-differentiation-2nd-derivative-print.html
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# implicit differentiation 2nd derivative
• Oct 28th 2009, 10:11 AM
haebinpark
implicit differentiation 2nd derivative
i have to find y' and y" using implicit differentiation
2x^3 - y^2 = 8
what i did was
d2x^3/dx - dy^2/dx = d8/dx
d2x^3/dx = 6x^2
dy^2/dx = 2y*y'
d8/dx = 0
6x^2 - 2yy' = 0
-6x^2=-2yy'
y'=6x^2/2y
= 3x^2 / y
but i don't know how to find 2nd derivative from here
how do i do it?
• Oct 28th 2009, 12:12 PM
Quote:
Originally Posted by haebinpark
i have to find y' and y" using implicit differentiation
2x^3 - y^2 = 8
what i did was
d2x^3/dx - dy^2/dx = d8/dx
d2x^3/dx = 6x^2
dy^2/dx = 2y*y'
d8/dx = 0
6x^2 - 2yy' = 0
-6x^2=-2yy'
y'=6x^2/2y
= 3x^2 / y
but i don't know how to find 2nd derivative from here
how do i do it?
This is correct so far, all you have to do is use the quotient rule and implicit differentiation.
$y'=\frac{3x^2}{y}$
$y''=\frac{6yx-3y'x^2}{y^2}$
• Oct 28th 2009, 03:55 PM
haebinpark
would you show me a little detail please?
like from y' to y"
how the implicit differentiation is applied
• Oct 28th 2009, 04:03 PM
Quote:
Originally Posted by haebinpark
would you show me a little detail please?
like from y' to y"
how the implicit differentiation is applied
$y'=\frac{3x^2}{y}$
Apply the quotient rule:
$y''=\frac{y(3x^2)'-(3x^2)y'}{y^2}$
$=\frac{6yx-3y'x^2}{y^2}$
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2017-06-27 02:31:51
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https://minhdo.org/notes/courses/real-analysis/9-compact-sets.html
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Compact Sets
# Set Covers
An open cover, or cover, of a set $$E$$ in metric space $$X$$ is a collection of open sets, denoted $$\{G_\alpha\}$$, whose union covers, or contains, $$E$$.
A subcover of $$\{G_\alpha\}$$ is a subcollection $$\{G_{\alpha_\gamma}\}$$ that still covers $$E$$.
For example,
1. In $$\mathbb{R}$$, we have $$\left[\frac{1}{2}, 1\right)$$ has cover $$\{V_n\}_{n = 3}^{\infty}$$ where $$V_n = \left(\frac{1}{n}, 1 - \frac{1}{n}\right)$$. It also has cover $$\{(0, 2)\}$$. It also has cover $$\{W_x\}_{x \in \left(\frac{1}{2}, 1\right)}$$, where $$W_x = N_\frac{1}{10}(x)$$. $$\{V_n\}_{n = 3}^{\infty}$$ has subcovers $$\{V_n\}_{n = 22}^{\infty}$$ and $$\{V_n\}_{n = 1000000}^{\infty}$$. $$\{(0, 2)\}$$ has only one subcover $$\{(0, 2)\}$$. $$\{W_x\}_{x \in \left(\frac{1}{2}, 1\right)}$$ has a subcover $$\left\{W_\frac{5}{10}, W_\frac{6}{10}, W_\frac{7}{10}, W_\frac{8}{10}, W_\frac{8}{10}, W_\frac{9}{10}\right\}$$.
2. $$[0, 1] \in \mathbb{R}$$ has a cover by $$\{V_n\} \cup \{W_0, W_1\}$$. A finite subcover is $$\{W_0, W_1, V_{11}\}$$.
# Compact Sets
A set $$K$$ is compact (in $$X$$) if every open cover of $$K$$ contains a finite subcover.
So, $$K$$ is not compact means there exists an open cover of $$K$$ with no finite subcover.
For example,
1. $$\left[\frac{1}{2}, 1\right)$$ is not compact (see $$\{V_n\}$$).
2. $$\mathbb{Z}$$ in $$\mathbb{R}$$ is not compact. One witness cover is $$\{U_i\}_{i \in \mathbb{Z}}$$, where $$U_i = \left(i - \frac{1}{10}, i + \frac{1}{10}\right)$$.
Theorem. Finite sets are compact.
Proof. Consider some open cover $$\{G_\alpha\}$$ covering $$x_1, x_2, \dots, x_n$$. For all $$x_i$$, choose $$G_{\alpha_i}$$. Then $$\{G_{\alpha_i}\}_{i = 1}^n$$ covers the set. QED.
A set $$K$$ is bounded if there is some ball $$N_r(x)$$ for some $$x \in X$$ such that $$K \subset N_r(x)$$.
Theorem. Compact sets are bounded.
Proof. Consider a compact set $$K$$. Let $$B(x) = N_1(x)$$, i.e. $$B(x)$$ are balls of radius $$1$$. Consider a collection $$\{B(x)\}_{x \in K}$$, which is an open cover of $$K$$. By compactness of $$K$$, there exists a finite subcover $$\{B(x_i)\}_{i = 1}^n$$. Let $$R = max_{1 \leq i \leq n}\{d(x_1, x_i)\}$$; this maximum exists because the set $$\{x_1, \dots, x_n\}$$ is finite. Then $$N_{R + 2}(x_1)$$ contains all of $$K$$. QED.
# Relative Open Sets
If $$Y \subset X$$, where $$X$$ is a metric space, $$Y$$ is said to inherit a metric from $$X$$.
A set $$U$$ is open in $$Y$$ (or open relative to $$Y$$) if every point of $$U$$ is an interior point of $$U$$. Here, the notion of interior is using the neighborhoods in $$Y$$.
Theorem. Suppose $$E \subset Y \subset X$$. Then $$E$$ is open in $$Y$$ if and only if $$E = Y \cap G$$ for some $$G$$ open in $$X$$.
# Compactness is Intrinsic
Theorem. Suppose $$K \subset Y \subset X$$, where $$X$$ is a metric space. Then $$K$$ is compact in $$Y$$ if and only if $$K$$ is compact in $$X$$.
Proof.
($$\Rightarrow$$): Assume $$K$$ is compact in $$Y$$. Consider any open cover $$\{U_\alpha\}$$ of $$K$$ in $$X$$. Let $$V_\alpha = U_\alpha \cap Y$$. Then $$\{V_\alpha\}$$ covers $$K$$ in $$Y$$. Since $$K$$ is compact in $$Y$$, there exists a finite subcover $$\{V_{\alpha_1}, \dots, V_{\alpha_n}\}$$ in $$Y$$. Then $$\{U_{\alpha_1}, \dots, U_{\alpha_n}\}$$ is a finite subcover of $$K$$ in $$X$$, as desired.
($$\Leftarrow$$): Assume $$K$$ is compact in $$X$$. Consider any open cover $$\{V_\alpha\}$$ of $$K$$ in $$Y$$. By above theorem, there exists open $$U_\alpha$$ such that $$V_\alpha = U_\alpha \cap Y$$. Then $$\{U_\alpha\}$$ covers $$K$$ in $$X$$. Since $$K$$ is compact in $$X$$, there exists a finite subcover $$\{U_{\alpha_1}, \dots, U_{\alpha_n}\}$$. Then $$\{V_{\alpha_1}, \dots, V_{\alpha_n}\}$$ is a finite subcover of $$K$$ in $$Y$$, as desired. QED.
# Relationship between Compact Sets and Closed Sets
Theorem. Compact sets are closed.
Proof. Let $$K$$ be compact. Consider $$p \notin K$$. We’ll show $$p$$ has a neighborhood that does not intersect $$K$$ (or $$p$$ is not a limit point of $$K$$). For any $$q \in K$$, let $$V_q = N_\frac{r}{2}(q)$$ and $$U_q = N_\frac{r}{2}(p)$$, where $$r = d(p, q)$$. Notice $$\{V_q\}$$ is an open cover of $$K$$. By compactness of $$K$$, there exists a finite subcover $$\{V_{q_1}, \dots, V_{q_n}\}$$. Let $$W = \bigcap_{i = 1}^n U_{q_i}$$. So $$W$$ is an intersection of finitely many open sets. Hence, $$W$$ is open (it is a ball of radius $$\min \left\{\frac{d(p, q_i)}{2}\right\}$$). We have $$W \cap V_{q_i} = \emptyset$$ for all $$q_i$$ because $$W \subset V_{q_i}$$ and $$V_{q_i} \cap U_{q_i} = \emptyset$$. So $$W$$ is the desired neighborhood. QED.
For example,
1. $$(0, 1)$$ in $$\mathbb{R}$$ is not compact because it’s not closed.
2. $$\mathbb{R}$$ in $$\mathbb{R}$$ is not compact although it’s closed because it’s not bounded.
Theorem. A closed subset $$B$$ of a compact set $$K$$ is compact.
Proof. Consider any open cover $$\{U_\alpha\}$$ of $$B$$. Since $$B$$ is closed, $$B^c$$ is open. So $$B^c \cup \{U_\alpha\}$$ is an open cover of $$K$$. Since $$K$$ is compact, there exists a finite subcover $$\{U_{\alpha_1}, \dots, U_{\alpha_n}, B^c\}$$ of $$K$$. Since $$B^c \cap B = \emptyset$$, $$\{U_{\alpha_1}, \dots, U_{\alpha_n}\}$$ covers $$B$$ and it is a finite subcover of the original cover. QED.
Corollary. Suppose set $$B$$ is closed and set $$K$$ is compact. Then $$B \cap K$$ is compact.
Consider intervals $$I_n = [a_n, b_n]$$. They are nested means if $$m > n$$ then $$a_n \leq a_m \leq b_m \leq b_n$$.
Theorem. Nested closed intervals in $$\mathbb{R}$$ are not empty (in $$\mathbb{R}^k$$, nested closed $$k$$-cells are not empty).
Proof. Let $$x = \sup\{a_i\}$$. It exists because $$a_i$$’s are bounded by $$b_1$$. Since $$x$$ is a supremum, $$x \geq a_i$$ for all $$i$$. We have $$x \leq b_i$$ for all $$i$$ since $$b_i$$ is an upper bound of all $$a_i$$’s (due to nestedness). QED.
Aside: Proof that $$\mathbb{R}$$ is uncountable. Suppose $$\mathbb{R}$$ is countable, i.e. $$\mathbb{R} = \{x_1, x_2, x_3, \dots\}$$. Choose closed intervals $$I_1$$ that misses $$x_1$$, $$I_2 \subset I_1$$ that misses $$x_1$$ and $$x_2$$, $$I_3 \subset I_2$$ that misses $$x_1$$, $$x_2$$, and $$x_3$$, and so on. So, $$I_n$$ is a nested closed interval. By previous theorem, there exists a point in $$\bigcap I_n$$ that is not one of the $$x_i$$’s and is thus not on the original listing. QED.
Theorem. Any closed interval $$[a, b]$$ is compact (in $$\mathbb{R}$$). This is also true for $$k$$-cells in $$\mathbb{R}^k$$.
Proof (by contradiction). Suppose there is a closed interval $$[a, b]$$ that is not compact. Then there exists an open cover $$\{U_\alpha\}$$ for $$[a, b]$$ which has no finite subcover. Then $$\{U_\alpha\}$$ covers both $$[a, c_1]$$ and $$[c_1, b]$$, at least one of which has no finite subcover of $$\{U_\alpha\}$$. Without loss of generality, assume $$I_1 = [a, c_1]$$ has no finite subcover. Subdivide $$I_1$$ again using $$c_2$$ which is the half way point of $$a$$ and $$c_1$$. Note that at least one of $$[a, c_2]$$ and $$[c_2, c_1]$$ has no finite subcover. Continue and we’ll obtain $$I_1 \supset I_2 \supset I_3 \supset \dots$$ of nested closed intervals, each halves at each step and has no finite subcover of $$\{U_\alpha\}$$. By the nested interval theorem, there exists $$x \in I_n$$ for all $$n$$. Then $$x$$ is in some $$U_\widehat{\alpha}$$ of the cover $$\{U_\alpha\}$$. Since $$U_\widehat{\alpha}$$ is open, there exists an $$r > 0$$ such that $$N_r(x) \subset U_\widehat{\alpha}$$. Since the intervals halve at each step, some $$I_n$$ is contained in $$N_r(x)$$. This means that $$G_\widehat{\alpha}$$ covers $$I_n$$, which contradicts the fact that it has no finite subcover. QED.
# The Heine-Borel Theorem
Theorem. In $$\mathbb{R}$$ (or in $$\mathbb{R}^n$$), $$K$$ is compact if and only if $$K$$ is both closed and bounded.
Proof.
($$\Rightarrow$$): already done.
($$\Leftarrow$$): (it is not true in arbitrary metric spaces) Since $$K$$ is bounded, $$K \subset [-r, r]$$ for some $$r > 0$$. Since $$K$$ is closed (by assumption) and $$[-r, r]$$ is compact (by the previous theorem), $$K$$ is also compact. QED.
For $$\mathbb{R}^n$$, replace closed interval by $$n$$-cell.
For example,
1. Discrete metric on infinite set $$A$$: $$A$$ is closed and bounded but is not compact.
2. Let $$\mathcal{C}_b(\mathbb{R})$$ is the set of all bounded continuous functions $$f : \mathbb{R} \rightarrow \mathbb{R}$$. Let $$d(f, g) = \sup_{x \in \mathbb{R}}|f(x) - g(x)|$$.
Theorem. $$K$$ is compact if and only if every infinite subset $$E$$ of $$K$$ has a limit point in $$K$$.
Proof.
($$\Rightarrow$$): If no point in $$K$$ is a limit point of $$E$$ then each point $$q \in E$$ has a neighborhood $$V_q$$ containing no other point of $$E$$ other than $$q$$. $$\{V_q\}$$ covers $$E$$ with no finite subcover, implying $$K$$ is not compact, which is a contradiction.
($$\Leftarrow$$) proof for $$\mathbb{R}^n$$ but is true for all metric spaces: Need to show $$K$$ is closed and bounded. Suppose $$K$$ is not bounded. Choose $$x_n$$ such that $$|x_n| > n$$. These have no limit point, which is a contradiction. So $$K$$ is bounded. Suppose $$K$$ is not closed. There exists a point $$p \notin K$$ that is a limit point of $$E$$. Choose $$x_n$$ such that $$d(x_n, z) < \frac{1}{n}$$. $$x_n$$ has a limit point at $$p$$ and no other. QED.
Corollary (Bolzano-Weierstrass Theorem). Every bounded infinite subset of $$\mathbb{R}^n$$ has a limit point in $$\mathbb{R}^n$$.
Proof. If subset $$E$$ is bounded then $$E$$ is in some compact $$k$$-cell. So $$E$$ has a limit point in the $$k$$-cell. QED.
A collection of sets has the finite intersection property if any finite subcollection has a non-empty intersection.
Theorem (due to Cantor). Let $$\{K_\alpha\}$$ be compact subsets of some metric space $$X$$. If $$\{K_\alpha\}$$ has the finite intersection property then the intersection of all $$K_\alpha$$ is non-empty.
Proof (by contradiction). Let $$U_\alpha = K_\alpha^c$$, which is open. Choose an arbitrary $$K$$ from $$\{K_\alpha\}$$. If $$\bigcap\limits_\alpha K_\alpha = \emptyset$$, then $$\{U_\alpha\}$$ covers $$K$$. Since $$K$$ is compact, there exists a finite subcover $$\{U_{\alpha_1}, \dots U_{\alpha_n}\}$$ covering $$K$$. So, $$K \cap K_{\alpha_1} \cap \dots \cap K_{\alpha_n} = \emptyset$$, which constradicts the hypothesis. QED.
Corollary. Let $$\{K_n\}$$ be a sequence of compact nested sets. Then $$\bigcap_{n = 1}^\infty K_n$$ is non-empty.
Theorem. Any space $$X$$ is compact if and only if any collection of closed sets $$\{D_\alpha\}$$ that satisfies the finite intersection property has a non-empty intersection (or, if every finite subcollection has non-empty intersection, then $$\bigcap D_\alpha \neq \emptyset$$).
Proof.
($$\Rightarrow$$): Consider $$\{D_\alpha\}$$. These are closed subset of a compact space $$X$$, so they are compact. Apply the previous theorem to get the conclusion.
($$\Leftarrow$$, by contrapositive): Assume $$X$$ is not compact. Hence, there exists an open cover $$\{U_\alpha\}$$ with no finite subcover. Since $$\{U_\alpha\}$$ is a cover, every point $$y \in X$$ is in $$\bigcup\limits_\alpha U_\alpha$$, which implies every point $$y \in X$$ is not in $$\bigcap\limits_\alpha U_\alpha$$. Hence, $$\bigcap\limits_\alpha U_\alpha = \emptyset$$. Consider any subcollection $$\{U_{\alpha_i}\}_{i = 1}^n$$. Since $$\{U_{\alpha_i}\}_{i = 1}^n$$ is not a subcover of $$X$$, there exists $$x \in X \setminus \bigcup\limits_{i = 1}^n U_{\alpha_i}$$, which implies $$x \in X \cap \bigcap\limits_{i = 1}^n U_{\alpha_i}$$. So $$\bigcap\limits_{i = 1}^n U_{\alpha_i} \neq \emptyset$$, as desired. QED.
# Perfect Sets
A set is perfect if it is closed and every point is a limit point.
For example,
1. Any closed interval $$[a, b]$$ is perfect.
2. $$\mathbb{R}$$ is perfect.
# Cantor Sets
Start with $$K_0 = [0, 1]$$. Construct $$K_1$$ by removing the middle third of $$K_0$$, i.e. $$K_1 = \left[0, \frac{1}{3}\right] \cup \left[\frac{2}{3}, 1\right]$$. Construct $$K_2$$ by removing the middle thirds of the intervals in $$K_1$$, i.e. $$K_2 = \left[0, \frac{1}{9}\right] \cup \left[\frac{2}{9}, \frac{3}{9}\right] \cup \left[\frac{6}{9}, \frac{7}{9}\right] \cup \left[\frac{8}{9}, 1\right]$$. Continue with the construction.
Each $$K_n$$ has $$2^n$$ intervals. Each is closed because each is a finite union of closed sets. Each is compact because each is a closed subset of the compact set $$[0, 1]$$. Finally, all $$K_n$$ are nested. Hence, their intersection is non-empty.
Let $$C = \bigcap\limits_{n = 0}^\infty K_n$$, called the Cantor set. Notice that
• $$C$$ is closed because it is the intersection of arbitrary many closed sets.
• $$C$$ is perfect.
• $$C$$ consists of real numbers whose ternary expansion contains only $$0$$s or $$2$$s, which shows that $$C$$ is uncountable (or $$C$$ has non-endpoints of $$K_n$$).
• $$C$$ has no interior.
• $$C$$ is totally disconnected.
• $$C$$ has measure $$0$$, i.e. for any $$\epsilon > 0$$, $$C$$ can be covered by intervals of total length less than $$\epsilon$$.
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2019-03-23 05:00:08
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http://mathhelpforum.com/statistics/126830-can-you-check-please.html
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# Math Help - Can you check please
1. ## Can you check please
There are 10 different hats, what is the probability that if I got 4 of them that I would get at least 2 the same?
Would it be $1/10 * 1/9 * 1/8 * 1/7$?
2. Unless I'm missing something then I think more information is needed..?
How many types of hat are there? Post the ENTIRE question.
3. 10 types of hat which are equally likely to be got.
4. Ok but how many hats are there in total? If there's only ten then you can't get two the same.
5. Ok I see now, 10 different types of hat but presumably infinite hats in total. The question is ' if you get four attempts you will obtain at least 2 of the same type'.
You only have 4 attempts to choose.
6. Maybe this is wrong but I think it's...
1-Probability(all hats are different)
So you get your first hat with prob 1.
The prob that the second one is different is 9/10
The prob that the third one is different is 8/10
The prob that the fourth one is different is 7/10
So 1-Probability(all hats are different) = 1-(9/10*8/10*7/10) = 0.496.
7. Originally Posted by bigroo
Ok I see now, 10 different types of hat but presumably infinite hats in total. The question is ' if you get four attempts you will obtain at least 2 of the same type'.
You only have 4 attempts to choose.
There are $\binom{4+10-1}{4}$ total number of ways to choose four hats.
There are $\binom{10}{4}$ ways to choose four hats all of different types.
Maybe this is wrong but I think it's...
1-Probability(all hats are different)
So you get your first hat with prob 1.
The prob that the second one is different is 9/10
The prob that the third one is different is 8/10
The prob that the fourth one is different is 7/10
So 1-Probability(all hats are different) = 1-(9/10*8/10*7/10) = 0.496.
So my answer was nearly the same as yours but instead of $1/10$ you have 1.
9. Originally Posted by bigroo
So my answer was nearly the same as yours but instead of $1/10$ you have 1.
Plato will be right.
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2014-12-27 02:31:46
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https://notes.iolite.xyz/posts/agenonlinearity/
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The Non-Linearity of the Age Equation and when you might notice it
· ☕ 5 min read · ✍️ Bence
🏷️
• #upb
• A common question we are asked about U-Pb geochronology in iolite is:
“Why is the mean age calculated by iolite different to the age calculated from the mean ratio?"
This post will most likely be too simple for the more math-literate, but I’m breaking it down for those like myself that enjoy seeing these sort of concepts to be explained in detail. I have included a spreadsheet showing all the calculations detailed in this note that you can download from here.
Lets use 207Pb/235U ratios/ages for our example, but the same applies to other ratio/ages. In the question above, the “age calculated by iolite” refers to the mean of the Final 207Pb/235U age, and is asking why this is different to the age you might calculate from the mean of the Final 207Pb/235U channel. This is due to the non-linearity of the age calculation. Just to remind you, here is the equation for calculating an age from a 207Pb/235U ratio (the equation for 206Pb/238U is very similar but uses a different constant):
$$\text{age (Ma)} = \frac{\ln(\frac{207}{235} +1)}{0.00098485}$$
the natural log part is quite important, because that tells us that the equation is non-linear. We can demonstrate that with a simple example. I’m going to take one of the selections from our U-Pb example dataset DRO4.io4 that can be downloaded from here. In particular, I’ve picked the third selection of Temora 2, but any selection would have done. I exported the Final 207Pb/235U and Final 207Pb/235U age values for the entire selection as a time series (so that I could see each value rather than just the mean for each channel). You can do this yourself in iolite v4 by using export settings shown below:
If you open the resulting Excel file, you’ll see a worksheet for each selection in the Z_Temora2 group. I’ve already created a spreadsheet showing the calculations described in this Note, and removed the other unused worksheets, that you can download from here and check through to satisfy yourself.
Now if we calculate the mean of the Final 207Pb/235U ratios in our spreadsheet, we get a value of 0.503. If we were to use the age equation above, that would give us an age of about 413 Ma which pretty close to the accepted age of Temora 2 – 417 Ma. However, if we calculate the mean of the Final 207Pb/235U age values, we get a value of 409 Ma: the ages are not the same. This is the crux of the question at the top. Why aren’t these values the same?
The first thing we could look at is the effect of varying our ratios. Let’s add 20% to the mean ratio value (0.503) and calculate an age from these two new ratios:
Offset Ratio Calculated Age (Ma) Difference (%)
-20% 0.400 342 -17%
+20% 0.600 477 16%
In this table, Difference (%) is the difference between the newly calculated age and the age from the mean ratio (0.503) of 413 Ma. So you can see that if we have a value 20% lower than the mean ratio, it will result in a 17% lower age, and if we have a value 20% higher, our calculated age increases by 16%. Let’s exaggerate this a little further, and increase our differences to 50%:
Offset Ratio Calculated Age (Ma) Difference (%)
-50% 0.250 227 -45%
+50% 0.751 569 38%
So, here we can see the effect even more: the lower ratio has moved the calculated age down 45%, but the higher ratio has only moved the calculated age up by 38%.
Hopefully this helps demonstrate that lower ratios pull down the calculated age more than higher ratios pull the calculated age up. And 50% isn’t even that abnormal when talking about the variation we might see in Final 207Pb/235U ratios: even our example Temora 2 data vary by 67% (2 RSD).
And just to show that this isn’t related to the distribution of ratios in our example dataset, let’s look at a perfectly normally distributed set of ratios# with the same mean (0.503) and standard deviation (0.169) (see the spreadsheet linked above for the ratio values). If we put each of those ratios into the age equation, we end up with a negativley-skewed group of ages, because of the non-linearity of the age equation.
We have a skew to the negative side of our distribution of ages, even though we had a normal (gaussian) distribution going into the age equation. And this is why if we calculate the mean of these ages we get a different value to if we calculated an age from the mean ratio.
So, which value to use? We recommend that you use the mean ratio, and calculate an age from this. The ages that iolite shows are more to help you understand your data and to provide context, rather than actually providing an age estimate.
If you have any questions about the above, please join the discussion here.
#The values in the spreadsheet are not *perfectly* normally distributed, because they are a random subset of 1000 values from a normal distribution, but they are a good approximation. The values you see will likely not quite match the same values I see because Excel will create a new sample of 1000 values when you open the spreadsheet, but the principle is the same.
WRITTEN BY
Bence
iolite developer
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2021-03-04 23:59:11
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https://math.stackexchange.com/questions/4033939/solve-equation-2x-2-x-2-cosx-5/4039333
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# Solve equation $2^{x} - 2^{-x} = 2*\cos(x/5)$
I'm trying to solve it by log two parts:
$$\log_2(2^{x}/2^{-x}) = \log_2(2*\cos(x/5))$$ Simplifying: $$2x - 1 = \log_2(\cos(x/5))$$
And I'm stuck here, I think we could solve it by investigating functions(linear and logarithmic) and trying to find border value. Could you give the right direction?
• Use \cos, \log_2, etc. and... no, $\log_2(x-y)\neq\log_2 x-\log_2 y$. Feb 21 '21 at 7:56
• @metamorphy, rookie mistake. Thanks! Feb 21 '21 at 8:13
• This is a transcendental equation, ie no solutions exist that we can find in a closed form. Feb 21 '21 at 9:59
• @A-LevelStudent, see a solution Feb 25 '21 at 7:42
Consider that you look for the zero of function $$f(x)=2^{x} - 2^{-x} - 2\cos \left(\frac{x}{5}\right)$$ By inspection, you know that the solution is somewhere between $$x=1.0$$ and $$x=1.5$$ and the function is quite close to a straight line (this is good for root-finding methods).
So, be as lazy as I am and start Newton methods with $$x_0=0$$. It will give the following iterates $$\left( \begin{array}{cc} n & x_n \\ 0 & 0.0000000 \\ 1 & 1.4426950 \\ 2 & 1.2504695 \\ 3 & 1.2401243 \\ 4 & 1.2400975 \end{array} \right)$$
We have the equation $$2^x+2^{-x}=2\cos\frac{x}{5}$$ and we want to either find the sum of the solutions or the solutions themselves. If we are only concerned with real solutions, then due to your excellent use of inequalities we have the only solution: $$x=0$$, and we are answering the second problem.
Alternatively, if the question also wants you to consider complex roots, then the second problem is not applicable, as there are multiple solutions, and in fact $$0$$ is the answer to the first problem. I will demonstrate how to find all these complex solutions below, and also very simply why $$0$$ is the answer to the first problem.
To find the solutions to this equation:
First observe that $$2^x=e^{x\ln2}$$ and similarly, we have $$2^{-x}=e^{-x\ln2}$$. Hence our equation is equivalent to $$2\left(\frac{e^{x\ln2}+e^{-x\ln2}}{2}\right)=2\cos\frac{x}{5}$$ But if you recall the definition of $$\cosh A$$ then you should see that actually our equation is equivalent to $$2\cosh(x\ln2)=2\cos\frac{x}{5}\iff \cosh(x\ln2)=\cos\frac{x}{5}$$ However, we also have a useful identity $$\cosh(A)\equiv \cos(iA)$$ Thus, we actually have $$\cos{ix\ln2}=\cos\frac{x}{5}\implies ix\ln2=\pm\frac{x}{5}+2n\pi\implies x=\frac{2n\pi}{i\ln2\mp 0.2}=\frac{10n\pi}{5i\ln 2\mp1}$$ for integer $$n$$.
Note that if $$\alpha$$ is a solution of the equation, then so is $$-\alpha$$. Therefore there must be an even number of roots (in fact there are infinitely many complex roots, but since every roots is paired up with another one which is equal to the negative of that root, loosely speaking we have an even number of roots). So if there are $$2n$$ roots, $$\alpha_1,-\alpha_1,\ldots,\alpha_n,-\alpha_n$$ then the sum of the roots is $$\alpha_1+(-\alpha_1)+\cdots+\alpha_n+(-\alpha_n)=0$$ as required.
• Thanks. It's a very good explanation! Feb 25 '21 at 13:01
• @funnydman No problem! I'm really happy to have helped you :) Feb 25 '21 at 13:08
Looks like there a typo in the textbook, the left part must be $$2^x + 2^{-x}$$. In this case, I found a solution:
Since $$2^x + \frac{1}{2^{x}}\geq 2$$ and $$2\cos{\frac{x}{5}} \leq 2$$,then
$$\begin{cases} 2^x + 2^{-x}=2 \\ 2\cos{\frac{x}{5}}=2 \\ \end{cases}$$
After solving this simple system, the answer would be $$x = 0$$.
• How did you use the fact $2^x+\frac{1}{2^x}\ge2$ in your solution? Weren't you dealing with $2^x-2^{-x}$ (stress on the negative sign)? Feb 25 '21 at 9:18
• To clarify: the correct equation should be $2^x+2^{-x}=2\cos(x/5)$? Feb 25 '21 at 10:14
• @A-LevelStudent, I thought about a typo too, but I'm not sure about this. But I know the answer from the textbook, the question is "Find the sum of the roots or the root of the equation, if it is the only one" and the answer is 0. Feb 25 '21 at 10:16
• @A-LevelStudent, what do you think? I appreciate your attention. Feb 25 '21 at 10:19
• I think the solution conclusively proves that there's a typo. If there wasn't a typo: $0$ isn't the answer to the first problem (see Claude's answer above) as there's only one real solution, and since there's only one real solution which is non zero, $0$ is certainly not the answer to the second problem either. I'll type an answer for what must be the correct equation; it'll add a little to your current solution. Feb 25 '21 at 10:56
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2022-01-24 14:05:28
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https://4lch4.blog/entries/2018/06/how-to-uniquify-your-powershell-console
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Published on
# How to Uniquify Your PowerShell Console
Authors
## PowerShell Profile Basics
It's important to point out that knowledge of PowerShell profiles, while helpful ahead of time, isn't absolutely required to read this article and potentially gain some value. If you do wish to get some information ahead of time, I highly recommend taking a look at this blog post by The Scripting Guys' Ed Wilson.
There are only a couple things I think you need to know about profiles in PowerShell before continuing on:
1. A profile is a script that allows you to customize your console by being executed as soon as you launch said console.
2. There are 6 profiles you can utilize to choose exactly which console you'd like to customize.
## What're We Trying to Achieve?
With that said, this is gonna be a long read so let's get started. Here is an image of my current console on startup (our end goal):
I like it to show me the current weather report, and I've customized the prompt line to show the current directory name and time as opposed to the standard look:
### To-Do
So, this means we have 3 "tasks" to complete for our desired look:
1. Add the weather API for our date integration
2. Add the current time to the beginning of the current shell line
3. Add the current directory to the end of the current shell line
## Where Do We Start?
To be completely honest, as I started to write this post I realized I had no idea how I did this aside from having to overwrite a function in my profile called global:prompt I believe. So, I opened my profile script in PowerShell Studio which is located at C:\Users\Alcha\Documents\WindowsPowerShell\Profile.ps1 and saved a backup copy (Profile_Backup.ps1). Then, a lovely Ctrl + A → Backspace so we have a clean slate.
First up, we need to find out how to customize the prompt itself, so a quick google search will get us started. The first three results were more than enough to get me what I needed. Personally, I prefer reading examples over documentation so after scanning over the three links, I chose to focus on the third as it seemed rather lengthy and full of examples. After reading a bit more, I realized it was more focused on posh-git specifically, so I switched back to the Scripting Guys response written by Sean Kearney.
## Customizing the Title/Prompt
While Sean first covers how to customize the title of the window, this is something I didn't know about before and wound up changing in my profile, so hooray for new stuff! More specifically, I started by adding this line to customize the $Host.UI.RawUI.WindowTitle property: $Host.UI.RawUI.WindowTitle = (Get-Date).ToString()
While it was neat, I didn't quite like the format:
After some tinkering, I landed on the '%y/%m/%d %R' format. Which you'd use like Get-Date -UFormat '%y/%m/%d %R' , and your result should look something like this:
Now that I've got my title the way I want it, on to the rest. Next, Sean mentions the default prompt function, which is what's executed every time your prompt is displayed:
"PS $($executionContext.SessionState.Path.CurrentLocation)$('>' * ($nestedPromptLevel + 1)) "
Bingo! Now we can take care of the 3rd item on our To-Do list: Add the current directory to the end of the current shell line To do this, we need to get the current working directory, which is available a number of ways. We can either use the $PWD variable which stores the current variable, or we can use the Get-Location cmdlet, which is what we'll be doing. Get-Location by default returns the entire current path, but we're only interested in the last portion of it, so we can get that part by using the Split-Path cmdlet: Split-Path (Get-Location) -Leaf # Gives us 'Alcha' Now we need to combine knowledge of the Prompt function, how to get the current date/time, and the Get-Location cmdlet to get our desired result. The function will end up looking something like this: function Prompt {$Host.UI.RawUI.WindowTitle = (Get-Date -UFormat '%y/%m/%d %R').Tostring()
Write-Host '[' -NoNewline
Write-Host (Get-Date -UFormat '%T') -ForegroundColor Green -NoNewline
Write-Host ']:' -NoNewline
Write-Host (Split-Path (Get-Location) -Leaf) -NoNewline
return "\$ "
}
We need the -NoNewLine parameter in order to put all of the text on the same line. This also allows us to separate it all into 5 separate lines and color the current time green while the rest remains the default white. Lastly, I moved the line to customize the title here because now whenever the prompt is updated, so is the title, and it'll show the "current" time. Now our console should look like so on startup:
With that, we've got two items from our To-Do list complete, and we can move on to the easiest part, which is adding a weather API to display the current conditions for our area.
## Getting The Weather
Now, to get the weather we'll be using the OpenWeatherMap service, mostly because I found a script at some point in the past that automatically formats the output into what you saw in the first screenshot. In order to use the service though, we need to get an API key that we can pass to the script.
When you create an account on OpenWeatherMap, go to the API keys portion of your account, which if you can't find is available here. Create a key and name it whatever you want, I just left mine as "Default":
Now that we have a key, we need to add the Get-Weather script to our profile. The way I did this was I created a new script in my profiles directory (C:\Users\Alcha\Documents\WindowsPowerShell\) called Get-Weather.ps1 and then copy/pasted the content of the script from GitHub. I'm bit of a stickler for variable naming conventions and capitalization so after some refactoring, I wound up with a script that looks something like this:
I won't cover the meat and bones of the script, as this is already a rather lengthy post 😅 but the full text of "my" version of obs0lete's script is available as a gist, you just need to plug in your API key for it to function properly. Now to use this, we need only dot-source the file in our profile and then call the Get-Weather function to kick it all off:
You'll of course have to change the API key, city, and country to values that work for you, but it's fairly simple. Something to note, I found a hard time with the actual city I live in, as there's a number of other cities in the USA with the same name. This meant I was getting really weird weather results from the API, imagine it saying it's raining outside when it's bright and sunny. I fixed this by finding the neighboring large city and using their name instead:
## Conclusion
It's important to note that I changed the original Get-Weather script just enough to make it work for me, so be sure to make any adjustments you might need. For example, I mentioned adjusting the variable names, but I also updated the units from Metric to Imperial and made it so it would output the Attempting URL... line mostly for debugging but also as a test for if the API isn't working for whatever reason.
I hope this wasn't too painful of a read, I did my best to only cover the interesting bits of solving this problem and leave out any fat that was just dull or involved me spacing out and endlessly scrolling 😂 If you have any questions, comments, or concerns, please feel free to reach out to me on any of my social networks or email me at admin@alcha.org.
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2023-03-31 15:20:14
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https://2n3904blog.com/trans-impedance-amplifier-transfer-function/
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# Trans-Impedance Amplifier – Transfer Function
In this post, the current to voltage transfer function of a trans-impedance amplifier is derived. Consider a sample op-amp based trans-impedance amplifier as shown in the figure below.
Capacitors $$C_f$$ and $$C_s$$ are included for completeness, as they do exist as parasitic elements in a real circuit. Parasitic shunt capacitance of $$R_f$$ may be on the order of 100 fF. Capacitor $$C_s$$ is the sum of routing capacitance (perhaps a few pF) and the input capacitance of the op-amp (perhaps 3 pF to 6 pF).
Applying KCL at the inverting-input terminal of the op-amp yields,
$$V_x s C_s + \dfrac{V_x -V_o}{R_f} +(V_x – V_o)sC_f = I_{in}$$
$$V_x \left( sC_s + 1/R_f + sC_f \right) – V_o( 1/R_f + sC_f ) = I_{in}$$
$$V_x \left( \dfrac{1 + sR_f(C_s + C_f) }{R_f}\right) – V_o( \dfrac{1 + sR_fC_f}{R_f} ) = I_{in}$$
The transfer function of the op-amp is modeled as a single pole, unity gain stable amplifier as,
$$A(s) = \dfrac{A_{ol}}{1 – s/p_a}$$
Where $$A_{ol}$$ is the DC open-loop gain and $$p_a$$ is the pole based on the GBWP of the op-amp. Since the input signal is applied to the inverting terminal the gain is negative and equal to,
$$V_o = – A(s) V_x$$
Rearrange for $$V_x$$ as,
$$V_x = \dfrac{-V_o}{A(s)}$$
Substitution into the KCL eqn as,
$$\dfrac{-V_o}{A(s)} \left( \dfrac{1 + sR_f(C_s + C_f) }{R_f}\right) – V_o( \dfrac{1 + sR_fC_f}{R_f} ) = I_{in}$$
$$\dfrac{-V_o}{A(s)R_f} \left( 1 + sR_f (C_s + C_f) + A(s) (1 +sR_fC_f)\right)= I_{in}$$
\begin{align*}
\dfrac{V_o}{I_{in}} &= \dfrac{-A(s)R_f}{ 1 + sR_f (C_s + C_f) + A(s) (1 +sR_fC_f)} \\ \\
&=\dfrac{-A_{ol}R_f}{ \left(1-s/p\right)\left(1 + sR_f (C_s + C_f) +\dfrac{ A_{ol}(1 +sR_fC_f)}{1-s/p} \right)} \\ \\
&=\dfrac{-A_{ol}R_f}{ 1 -s/p + sR_f(C_s+C_f)-\dfrac{s^2R_f(C_s+C_f)}{p_a} +A_{ol} +sA_{ol}R_fC_f} \\ \\
\dfrac{V_o}{I_{in}} &=\dfrac{-R_f}{ 1 + s\left( R_fC_f +\dfrac{R_f(C_s+C_f)}{A_{ol}} – \dfrac{1}{A_{ol}p_a}\right) -\dfrac{s^2R_f(C_s+C_f)}{A_{ol} p_a} } \\ \\
\end{align*}
Assuming the time-constant of $$R_fC_f$$ is less than the GBWP of the opamp the transfer function can be approximated as,
$$\dfrac{V_o}{I_{in}} \simeq \dfrac{-R_f}{ 1 + sR_fC_f -\dfrac{s^2R_f(C_s+C_f)}{A_{ol} p_a} }$$
If we assume the poles are well separated (atleast one decade in frequency) the dominant pole is then approximately,
$$p_1 \simeq \dfrac{-1}{R_fC_f}$$
The second pole occurs at,
$$p_2 \simeq \left( \dfrac{A_{ol}p_a}{R_f(C_s+C_f)} \right)\left( -R_fC_f \right)$$
Which is equivalently,
$$p_2 = \dfrac{-\text{GBWP}}{2\pi}\left( \dfrac{C_f}{C_f + C_s} \right)$$
## Spice Simulation
For the purposes of discussion, consider a prototype trans-impedance amplifier with the following parameters,
\begin{align*}
R_f &= 10 \text{ M}\Omega \\
C_f &= 10 \text{ pF} \\
C_s &= 90 \text{ pF} \\
A_{ol} &= 100 \text{ dB} \\
\text{GBWP} &= 1 \text{ MHz} \\
\end{align*}
The LT-Spice schematic of the sample amplifier is shown in the figure below.
The exact transfer function is then,
$\dfrac{V_{out}(s)}{I_{in}(s)}= H(s) = \dfrac{-10^7}{159.2\cdot 10^{-12} s^2 + 100.2\cdot 10^{-6} s + 1}$
With the exact poles of $$H(s)$$ being,
$p_1 = -1.615 \text{ [kHz]},\;\;\; p_2 = -98.554\text{ [kHz]}$
The approximated poles as developed above are the following,
$\hat{p_1} = -1.592 \text{ [kHz]},\;\;\; \hat{p_2} = -100\text{ [kHz]}$
A bode plot of the transfer function of this sample trans-impedance amplifier is shown in the figure below.
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2020-04-02 21:03:10
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https://mathematica.stackexchange.com/questions/155122/install-paclets-into-basedirectory-on-multi-user-system
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# Install paclets into $BaseDirectory on multi-user system PacletInstall will install paclets into the $UserBaseDirectory by default. How can we install them in $BaseDirectory so that all users on the system will have access? Is this currently possible? This is an important consideration if your package is expected to be installed on centrally managed multi-user systems (e.g. HPC clusters). This is not merely a theoretical situation. I know that MATLink was installed this way on more than one site. Update: It is not clear that system-wide paclets are well supported at this moment, even though the basics seem present in the paclet manager. My conclusion is that at this moment it is better to avoid using paclets for system-wide installations. • It should be possible as the paclet manager sources use both a $userRepositoryDir and a $sharedRepositoryDir. – Szabolcs Sep 5 '17 at 20:27 • You can Block the $userRepositoryDir with $BasePacletsDirectory/Repository. Worked for me. Although PacletFind freaks out a little bit and returns the paclet twice. – b3m2a1 Sep 5 '17 at 20:43 • Yep, I see that's what Kuba does too with WithPacletsRepository: github.com/kubaPod/MPM/blob/master/MPM/MPM.m – Szabolcs Sep 5 '17 at 20:49 • @b3m2a1 and Szabolcs, yes but so far I didn't have problems with this and Get or PacletInformation respect the most specific repository at the end. Otoh I won't bet my hand on that :) – Kuba Sep 5 '17 at 22:09 ## 1 Answer As I've pointed out here, on Linux it might be problematic to simply Block the $userRepositoryDir as you will need root-access for that in the default installation. No matter how you do this, you need to ensure that you can write there if you want to make it consistently working for all operating systems.
To the best of my knowledge, there is currently no easy way to install a paclet for all users automatically. What you want is to install a paclet in the PacletManagerPackage$sharedRepositoryDir but this option is not exported. When you call PacletInstall it will call PacletManagerManagerPrivateinstallPacletFromFileOrURL and there, the installation destination is hard-coded to $userRepositoryDir.
This is why Blocking this kind of works. I say "kind of" because after the paclet is unzipped into the destination, Mathematica rebuilds the internal information about installed paclets by calling PacletManagerPackagePCrebuild. If your destination is anything other than $userRepositoryDir or $sharedRepositoryDir, this shouldn't work, because the paclets to rebuild are again hard-coded as "Collections" -> {"User"}. Please look at the end of the definition of installPacletFromFileOrURL.
Therefore, the only sensible solution I see at the moment is
1. Check if you have write access to PacletManagerPackage$sharedRepositoryDir. If not, and you are on Linux, you might want to ask the user to re-run Mathematica with sudo. 2. Block the $userRepositoryDir as suggest by b3m2a1 and already used by Kuba.
• One could run PacletDirectoryAdd in init.m for other locations. But so is with \$Path for custom directories. – Kuba Sep 6 '17 at 5:52
• @Kuba Hmm, it's not clear to me what the side-effects are, but directories added with PacletDirectoryAdd go into the "Extra" paclet collection, while the ones installed with PacletInstall are stored in the "User" section. – halirutan Sep 6 '17 at 7:58
• I don't know either but do you anticipate problems? I use PacletDirectoryAdd for testing/running paclets directly from their development directories. So far didn;t have problems even if released versions where already PacletInstall-ed. – Kuba Sep 6 '17 at 8:02
• @Kuba When I see this right, Szabocls wants users to easily install his package with PacletInstall which is convenient since he can do this directly from the GitHub repo. If he does this, then the version is of the paclet is tracked in the "User" collection. If a different user opens the same Mathematica, he uses his own User-collection which has no idea about the installed paclet. When I see this right from looking at the code then a) the user that installed the Paclet (in a different dir with Block) has it tracked in "User". If he runs PacletDirectoryAdd that it is probably tracked.. – halirutan Sep 6 '17 at 8:09
• twice. b) a different user who is opening the same Mathematica can use PacletDirectoryAdd`. As I said, I'm not sure if there are any serious consequences. I merely point out that there are some glitches that might lead to weird behavior and one should look out them or intensively test the approach. – halirutan Sep 6 '17 at 8:12
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2019-07-18 09:59:13
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https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-1-equations-and-inequalities-1-5-use-problem-solving-strategies-and-models-1-5-exercises-skill-practice-page-38/21
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## Algebra 2 (1st Edition)
Let $h$ be the number of gallons used on the highway. Then the number of gallons used in the city is $15-h$. Then our equation is: $25h+(15-h)20=350$ Thus the answer is B.
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2022-10-04 11:00:12
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http://mathhelpforum.com/differential-equations/199070-differential-equation-help-needed.html
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# Thread: Differential equation help needed
1. ## Differential equation help needed
find the solution that fulfills
2. ## Re: Differential equation help needed
what i did so far
then the left side is
then i got stuck with this...
now what?
3. ## Re: Differential equation help needed
anyone able to help? so i can finish this one since it doesnt seem to be much left
nvm
then substistute with x=pi and y=0 and divide both with x^2
sin(pi)=0
0=C/pi^2
C=0*pi^2
C=0
if anyone can just confirm that this is right i would appriciate it
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2017-06-23 06:09:35
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https://tex.stackexchange.com/questions/501963/wrapping-text-in-enumerate-environment
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# Wrapping text in Enumerate environment
Is there anyways to have the text totally wrap underneath each item?
\documentclass[a4paper,12pt]{extarticle}
\usepackage[utf8]{inputenc}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{enumitem}
\newtheorem{factinner}{Fact}
\newenvironment{fact}[1]{%
\renewcommand\thefactinner{#1}%
\factinner
}{\endfactinner}
\begin{document}
\section{Linear Transformations \& Matrices}
\begin{fact}{1}
\begin{proof}
Proceed by cases.
\begin{enumerate}[leftmargin=*, align=left]
\item[\textbf{Case 1}] Let $\mathsf{T}$ be linear. Trivial.
\item[\textbf{Case 2}] Let $\mathsf{T}(cx + y) = c\mathsf{T}(x) + \mathsf{T}(y)$. To satisfy the first criteria of linearity, let $c =1$. To satisfy the second, let $y = \mathit{0}$. Let $\mathsf{T}(cx + y) = c\mathsf{T}(x) + \mathsf{T}(y)$. To satisfy the first criteria of linearity, let $c =1$. To satisfy the second, let $y = \mathit{0}$.
\end{enumerate}
\end{proof}
\end{fact}
\end{document}
• Unrelated: why are you manually numbering your items? Have to tried label=\textbf{Case \arabic*} and then \item – daleif Jul 29 '19 at 14:40
Is this what you want?
\documentclass{article}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{enumitem}
\newtheorem{factinner}{Fact}
\newenvironment{fact}[1]{%
\renewcommand\thefactinner{#1}%
\factinner
}{\endfactinner}
\begin{document}
\section{Linear Transformations \& Matrices}
\begin{fact}{1}
\begin{proof}
Proceed by cases.
\begin{enumerate}[label={\textbf{Case \arabic*}}, leftmargin=0pt, itemindent=*]
\item Let $\mathsf{T}$ be linear. Trivial.
\item Let $\mathsf{T}(cx + y) = c\mathsf{T}(x) + \mathsf{T}(y)$. To satisfy
the first criteria of linearity, let $c =1$. To satisfy the second, let
$y = \mathit{0}$. Let $\mathsf{T}(cx + y) = c\mathsf{T}(x) + \mathsf{T}(y)$.
To satisfy the first criteria of linearity, let $c =1$. To satisfy the
second, let $y = \mathit{0}$.
\end{enumerate}
\end{proof}
\end{fact}
\end{document}
Unrelated: don't you want to have the facts automatically numbered, like this:
\documentclass{article}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{enumitem}
\newtheorem{fact}{Fact}
\begin{document}
\section{Linear Transformations \& Matrices}
\begin{fact}
\begin{proof}
Proceed by cases.
\begin{enumerate}[label={\textbf{Case \arabic*}}, leftmargin=0pt, itemindent=*]
\item Let $\mathsf{T}$ be linear. Trivial.
\item Let $\mathsf{T}(cx + y) = c\mathsf{T}(x) + \mathsf{T}(y)$. To satisfy
the first criteria of linearity, let $c =1$. To satisfy the second, let
$y = \mathit{0}$. Let $\mathsf{T}(cx + y) = c\mathsf{T}(x) + \mathsf{T}(y)$.
To satisfy the first criteria of linearity, let $c =1$. To satisfy the
second, let $y = \mathit{0}$.
\end{enumerate}
\end{proof}
\end{fact}
\end{document}
?
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2020-03-29 16:11:33
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http://estrip.org/articles/read/mrdeadlier/42372/Tin_Man.html
|
Journaling on estrip is free and easy. get started today
Last Visit 2012-04-03 22:24:47 |Start Date 2006-05-22 14:21:51 |Comments 431 |Entries 189 |Images 24 |Videos 20 |Mobl 72 |
Category: movies
12/02/07 10:52 - 39ºF - ID#42372
Tin Man
For some reason I allowed myself to get caught up in the buzz about the Sci Fi Channel's "Tin Man" mini-series that began tonight.
So not worth the time. It's a "reinterpretation" of the Wizard of Oz. Reinterpreted as another lousy Sci Fi Channel made for TV movie.
It's pretty lame, B list stars or not. Every science fiction cliche that's been done has found its way into this flick. They even threw in the Harry Potter Death Eater kiss of death thing.
Whatever, I'm not falling for this again; I've no desire to catch parts 2 and 3...
Words: 97
Location: Buffalo, NY
Category: movies
11/30/07 02:01 - 40ºF - ID#42344
Not the same
So we all left work early to go see Beowulf at the Transit Imax in 3D. And it was quite a site to behold until the Imax's power failed halfway through. They said it was due to the high winds and that they weren't able to get it back online.
They offered us re-admission tickets to come back and see it at a later date. But I mean come on, we left work to go see this and it's not like I have a plethora of time available to me where I'll be able to swing by in the next 2 weeks to catch this. Plus, it's a cool story and I don't really want to wait to see it resolved.
So instead I asked if I could go see the rest of the movie in a normal theatre (they have another in 3D but just not Imax).
I'm in the theatre now (have to watch it from the beginning) and yeah it's a totally different experience; I guess that's why they invented Imax to begin with.
Words: 176
Location: Buffalo, NY
Category: movies
11/28/07 10:03 - 35ºF - ID#42325
Enchanted
As I've stated in the past, I've become quite the connoisseur of children's movies over the past 4 years or so. A large portion of my life is either Disney Princesses or Shrek. That's just how we roll in the Reid household I guess.
Anyway, last Friday we took the girls to go see Enchanted and if everyone will allow me to gush for a moment, I would just like to say it was soooo good. Disney took the entire fantasy formula they've put together over the past 70 years or so and did a great job of poking fun at it while still implementing it at the same time. When we walked out of the theater I turned to Chrissy and said something to the effect of "I could easily sit down and watch that movie start to finish again right now."
The music was great (composed by Alan Menken, the guy who has done most if not all of the scores for their films over the past 10 or 20 years, at least). I even bought the soundtrack on iTunes yesterday so the kids can sing along with what are some of the most addictive "stuck in your head" songs I've experienced in a while.
Amy Adams, the "Hot Girl" from the Office in season 1 (the one selling the purses) was the literal embodiment of a Disney princess. She just oozed the sugary optimism and goody-goodiness that those types of characters usually do. I've already read some reviews saying she should get an Oscar. Take that and do with it what you will...
I accidentally outed myself at work today as a lover of musicals and showtunes so I guess I just might as well accept the reality and wear it on my sleeve. ::sigh:: Though even now I'm feeling ashamed for admitting it right here on the interweb.
I can't do it all the time though... In fact I'm off to blast some Norma Jean or Underoath or something just to make sure I get a little balance. Listening to Les Mis and Rent at work today and then following it up with a Disney chaser with the kids tonight isn't necessarily a healthy indulgence in terms of sanity.
Words: 368
Location: Buffalo, NY
Category: tv
11/23/07 12:25 - 27ºF - ID#42260
Wrong again
It was the actual game in its entirety. Thanks to Time Warner (high speed cable portion of it that is) I got to watch it after all.
Hope everyone had a good Thanksgiving. My mom screwed us all over and got an organic turkey. I mean come on, I look forward to the tryptophan every year! Ugh, thanks Mom. Thanks for nothing.
Words: 62
Location: Buffalo, NY
Category: tv
11/22/07 09:35 - 28ºF - ID#42248
Never mind
It's a "companion broadcast" that goes in and out from the game with Jamie Dukes and Rod Woodson yacking on and on.
Still, it's better than nothing.
Words: 27
Location: Buffalo, NY
Category: tv
11/22/07 09:32 - 28ºF - ID#42247
Holy crap
Holy crap!
You can watch games on the NFL Network at nfl.com! Sweet deal for those of us tied to Time Warner Cable.
Nice!
Words: 25
Location: Buffalo, NY
Category: sports
11/12/07 09:37 - 46ºF - ID#42090
I'm seriously thinking of buying this
It's a custom jersey I "designed" at NFLShop.com.
We're over .500 for what seems like the first time in ages. Enjoy it kids, it's only going to last a week.
Words: 32
Location: Buffalo, NY
Category: weather
11/06/07 09:27 - 38ºF - ID#42000
How about a punch in the face?
I'm in a meeting right now and the person two down from me just started clapping when we all noticed the snow outside.
Clapping.
Note this post's title.
Words: 28
Location: Buffalo, NY
Category: movies
11/03/07 05:43 - 53ºF - ID#41963
Bee Movie
One "feature" of having a preschooler is the amount of kid-oriented tv and movies you take in.
In the past few years I've seen every Pixar, Dreamworks, and Disney flick that's been put out (yes, I know Disney owns Pixar but there *is* a difference between the two -- Meet the Robinsons isn't even in the same conversation as Ratatouille). And not only have I seen these movies, I've lived them. No exaggeration: I've seen Shrek and Shrek 2 about 50 times each, I'll bet.
So, not to break convention, I took my 4 year-old to the Regal on Elmwood to see Jerry Seinfeld's new "Bee Movie" today. It was decent - one of those movies where there's a plot only because the producers felt that after spending all this cash on animation and voice acting (this was one of those with TONS of familiar voices) that well, hey we probably should throw the audience a bone and string all these scenes together.
Overall it was pretty average but if you're a fan of Jerry Seinfeld and his numerous vocal inflections then it's probably worth the Bit Torrent download. ;) Otherwise, watch the trailer since it contained about 90% of the most memorable scenes anyway.
Words: 200
Location: Buffalo, NY
Category: music
11/02/07 09:24 - 33ºF - ID#41935
Emery are only a man
It took a few listens to grow on me but Emery's new CD "I'm Only a Man" is really starting to rock my socks. Really.
If you're into the whole "screamo/post-hardcore" scene you should check it out: EmeryMusic.com.
Words: 40
Location: Buffalo, NY
## My Fav Posts
1. This user has zero favorite blogs selected ;(
## Chatter
tinypliny said to tinypliny
:-) I wish I could. Sadly I don't have a good fit of what I want to do in Buffalo. I don't want to d...
tinypliny said to joe
haha, I am so jealous (e:Mike) never took me on one of his patented walks! :)
I came to Buffalo to in...
tinypliny said to joe
No pictures? *tsk tsk* excuses excuses....
tinypliny said to paul
(e:Joe) == cute^{N}
where $N$ is the total number of photos of (e:Joe) ...
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2014-04-17 09:46:29
|
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https://memotogel.me/simulation/how-to-uncube-a-number.php
|
To uncube something, you just cube root it. Example. Question How do you uncube a number. For example, r cubed = How do I find r? I have a graphing calculator that I can use but dont know which. Take the cube root. So f(x)=x^ or 5=x^3 so x= 5^(1/3).
How do you uncube a number. To uncube something, you just cube root it. Example=$$\sqrt[3]{}=5$$. Question How do you uncube a number. For example, r . To find the square root of a number, you want to find some number that when multiplied by itself gives you the original number. In other words, to find the squa. Question How do you uncube a number. For example, r An exponent represents how many times a number should be multiplied by itself. For example, x3 (or x.
An exponent represents how many times a number should be multiplied by itself. For example, x3 (or x cubed) would be written out as x × x × x. The cube root calculator below will reduce any cube root to its simplest radical form as well as provide a brute force rounded approximation for any number. For a number of young architects, designers, and curators practicing in its colonias (neighbourhoods), Mexico City is more than cliched observation; it's an.
The cubed root of a number is the number that would have to be multiplied by itself and by itself again to get the original number. For example. By learning this simple system you will be able to instantly calculate the cube root of the spectator's number. Ask the spectator to choose any whole number less. Using a simple calculator how do you compute the cube root of a number (only using: +,-,*,/ & sq root). In general is there a way to compute the.
shown the default settings for time interval and number of samples. Press Edit Enter for the number of samples and press Next. Then press OK. To uncube something, you just cube root it. Example. Question How do you uncube a number. For example, r cubed = How do I find r? I have a graphing. If m be a prime number greater then will m”— 1 be divisible by 9. In order to ascertain No triangular number, except un cube number. No triangular. 21 May - 2 min - Uploaded by eHowEducation How to Uncube Numbers. Part of the series: Mathematics Lessons. Cubing a number is one thing. 31 Jan - 2 min. uncube is a new digital magazine for architecture and beyond. 21 May - 2 min - Uploaded by eHowEducation How to Uncube Numbers. Part of the series: Mathematics Lessons. Cubing a number is one thing. In scientific notation numbers are written in the form x×10n, where n is an integer and x is in limits [1,10) i.e. 1≤x< Examples:−. is. Book your next work space from anywhere & anytime on The UnCube. Explore our range of highly curated & inspiring coworking spaces for rent. Copy down the next group of three numbers into the remainder, and draw a small vertical line to the left of the resulting number. Fourth Conference of the Canadian Number Theory Association, July , modèle que la précédente donne les résultats pour n égal à un cube et pour les.
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2019-12-10 04:20:26
|
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https://www.jamiebalfour.scot/courses/software/python/procedures-in-python/
|
# Part 3.1Procedures in Python
## Procedures
Python, like many scripting syntaxes, has just one procedure declaration format.
Often called a procedure, a function, a subroutine or a subprogram, it is a division of code that is separated from the rest. A procedure is a way of organising code to perform one particular function. For instance, (in an abstract example), a procedure could be used to turn on a light bulb or to turn off a plug socket. When it is needed, it is run (called). The part that gives the procedure a name and parameters is called the procedure signature. Below is a simple diagram that explains the procedure structure:
## Defining a procedure
Below is an example procedure:
Python
def main():
print ("Hello world")
Procedures can however, return a value. If a procedure returns a value, the value that it gives back can be used elsewhere.
## Returning a value
A procedure that returns, or gives back, a value can be known as a function, since it's actions mimic that of a mathematical function:
A ⇒ B
In this example, A is an input the arrow is the process and B is the output. For every input there is exactly one output.
A function returning a value looks like the following sample:
Python
def main():
return "Hello world"
With this, assigning the function call to a variable will give the variable the value "Hello world".
## Calling a procedure
A procedure can be called through a very simple syntax. Arguments are also given to procedure calls within the brackets - these line up with parameters specified in the procedure signature.
Python
def main():
print ("Hello world")
main()
The previous sample will call the procedure named main. It will also provide no arguments to the call.
A procedure with parameters can be called by inserting values in the positions of those parameters with arguments:
Python
def add(n1, n2):
print (n1 + n2)
## Recursion
Recursion is a big part of learning to program efficently and it is basically a procedure calling itself over and over again until a termination condition (base case) is met. Recursion can be used to multiply two numbers by simply using addition.
Python
def multiply(n1, n2, total):
if n2 == 0:
return n1
else:
total = multiply(n1, n2 - 1, total)
print(multiply(4, 3, 0))
Here is what happens in this program:
• multiply(4, 3, 0) calls multiply (4, 2, 0)
• multiply(4, 2, 0) calls multiply (4, 1, 0)
• multiply(4, 1, 0) calls multiply (4, 0, 0)
• multiply (4, 0, 0) then returns 0 + 4 (4) to multiply(4, 1, 0)
• multiply (4, 1, 0) then returns 4 + 4 (8) to multiply(4, 2, 0)
• multiply (4, 2, 0) then returns 8 + 4 (12) to multiply(4, 3, 0)
Recursion is something that to understand, one must first understand recursion. For more information on recursion, look at recursion.
Code preview
Feedback 👍
Comments are sent via email to me.
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2022-12-10 09:53:26
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http://darrenirvine.blogspot.ca/2015/05/
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## Saturday, May 30, 2015
### >> Imitation Forward Pipe for Common Lisp
Since learning a little F# about a year ago, I took a liking to its forward pipe operator (|>). The semantics of the forward pipe work great with F#'s sequence data type (with so-called, "lazy" computation) and functional style programming. It allows you to express a progression of consecutive steps in the order they will be done, while allowing you to directly feed the results of the last function into the input of the next function, avoiding redundant variable declarations. I suppose you could call these types of consecutive actions "transitive" combinations of operations. Imperative style programming would preserve the apparent sequence of events (meaning, the order you type the operations in, is the order the operations happen in), but it also involves declaring intermediate variables, which perhaps you will only use once.
I'm not an expert on functional style programming and don't intend, right now, to explain or justify its merits, except to say that sometimes it produces cleaner, clearer code. But while I have a current preference for Common Lisp above most other (programming) languages at the present, I really like that forward pipe operator in F#. This morning, I thought it was time to cross that bridge—try to make an imitation forward pipe in Common Lisp.
In F#, if you wanted to turn a into b into c, you might do something like
let a2c x =
a2b x
|> b2c
where b2c also takes one parameter. In Lisp, you end up with something more like
(defun a->c (x)
(b->c (a->b x)))
This is fine for being concise but it doesn't preserve the (apparent) logical ordering of $$a\to b\to c$$. We could do that with a change to something more imperative, such as
(defun a->c (x)
(let ((b (a->b x)))
(b->c b)))
and this works okay. However, I would like to write
(defun a->c (x)
(>>
(a->b x)
(b->c)))
Or,
(defun a->e (x)
(>> :pipe-in
(a->b x)
(b->c)
(c->d :pipe-in 0.0625d0)
(d->e)))
where we want this last one to expand into
(D->E (C->D (B->C (A->B X)) 0.0625))
I've introduced a keyword so that we can dictate which parameter of the next function the previous result goes into. You can specify any keyword to use as a pipe-in parameter, but make sure you don't use a keyword of one of the functions being called in the sequence—it'll really mess with your mind. Here's the macro definition along with an error condition:
To test that the code works, we can try this at the REPL,
CL-USER> (macroexpand-1 '(>> :pipe-in
(a->b x)
(b->c)
(c->d :pipe-in 0.0625)
(d->e)))
(D->E (C->D (B->C (A->B X)) 0.0625))
This macro does not attempt to handle multiple return values—though such a development might lead to an interesting result. It might be as easy as introducing more keywords.
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2018-04-25 16:05:50
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https://tex.stackexchange.com/questions/338271/reference-equation-inside-line-with-several-equations
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# reference equation inside line with several equations
I would like to use a reference style that I've seen in some older papers where the author will have several equations in the same line, but only one reference number for all of them. Then if the author wants to reference just one of the equations in that line, they will use a subscript or superscript to do so. Example:
a+b=c ; a*b=d .................. (1)
Therefore, the equation with a sum a+b=c is given by equation (1)a.
Here is a MWE:
\documentclass[12pt]{article}
\usepackage{amsmath} % Extra Math Stuff
\begin{document}
I would like to have 3 equations together like this
$$A+B=C %\label{eq:sum} \quad\textrm{,}\quad A\times B=D %\label{eq:mul} \qquad\textrm{,}\quad A/B=D %\label{eq:div} \label{eq:tog}$$
And now I would like to either reference all equations in \eqref{eq:tog} or each of them individually, which should look like \eqref{eq:tog}$_a$, \eqref{eq:tog}$_b$ and \eqref{eq:tog}$_c$.
\end{document}
• You mean that you would like a different way of doing this than is shown in your MWE? The solution you already have is by far the most natural and easiest way to do this. If you don't like this way of doing it then can you add some more detail saying what sort of solution you would like? – Andrew Nov 9 '16 at 1:27
\eqref sets the reference in \upshape using double nesting (from amsmath.dtx):
\newcommand{\eqref}[1]{\textup{\tagform@{\ref{#1}}}}
What I've done below is to create a manual reference that assumes you'll use \eqref{<subref>}, and therefore delays the printing of the sub-equation number by two groups (using a nested \AfterGroup from etextools):
\documentclass{article}
\usepackage{amsmath,etextools}
\newcounter{subeqn}[equation]
\renewcommand{\thesubeqn}{\alph{subeqn}}
\makeatletter
\newcommand{\sublabel}[1]{{%
\stepcounter{subeqn}
\def\@currentlabel{\theequation\protect\AfterGroup{\protect\AfterGroup{\protect\textsubscript{\thesubeqn}}}}%
\ltx@label{#1}%
}}
\makeatother
\begin{document}
I would like to have 3 equations together like this:
$$A + B = C \sublabel{eq:sum} \textrm{,}\qquad A \times B = D \sublabel{eq:mul} \textrm{,}\qquad A \div B = D \sublabel{eq:div} \label{eq:tog}$$
And now I would like to either reference all equations in \eqref{eq:tog}
or each of them individually, which should look like \eqref{eq:tog}$_a$,
\eqref{eq:tog}$_b$ and \eqref{eq:tog}$_c$.
And now I would like to either reference all equations in \eqref{eq:tog}
or each of them individually, which should look like \eqref{eq:sum},
\eqref{eq:mul} and \eqref{eq:div}.
\end{document}
• Note: You must use \eqref to reference the sub-equations. – Werner Nov 9 '16 at 17:45
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2019-11-13 21:15:25
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http://openstudy.com/updates/559f3fdee4b05670bbb4c141
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## briana.img one year ago FInd the measure of angle C
1. briana.img
I've tried it twice and the second time i got 12.51400
2. briana.img
I have fone $a^2=13^2+17^2-2(13)(17)\cos47$
3. briana.img
done*
4. anonymous
For this one, use the sine law. Do you know it?
5. briana.img
@ospreytriple yeah hold on let me try it out
6. briana.img
@ospreytriple 10sinB=17sin47???
7. anonymous
Shouldn't it be 13, not 10?
8. briana.img
@ospreytriple oh yeah sorry about that 13sinb=17sin4
9. anonymous
OK. The way I'm used to using the sine law is as follows:$\frac{ \sin \left( A \right) }{ a } =\frac{ \sin \left( B \right) }{ b }$. Is this the way you have it?
10. briana.img
@ospreytriple yeah i did that but you have to rearange to the way i had it and then do s$sinb=\frac{ 17\sin47 }{ 13 }$
11. anonymous
I think there's an error in the rearranging. You have $\frac{ \sin \left( B \right) }{ b }=\frac{ \sin \left( C \right) }{ c }$To isolate sin(C) you would multiply both sides of the equation by c, giving$\frac{ c \sin \left( B \right) }{ b }=\sin \left( C \right)$Make sense?
12. briana.img
@ospreytriple yeah
13. anonymous
OK. So solve$\sin \left( C \right)=\frac{ 17 \sin \left( 47 \right) }{ 13 }$
14. briana.img
@ospreytriple calculator got 1.27436702
15. anonymous
What do you get for sin(47)?
16. briana.img
@ospreytriple 0.7313
17. anonymous
Good. Multiply that number by 17 and divide that answer by 13. What do you get?
18. briana.img
@ospreytriple 0.956315
19. anonymous
Excellent. Now evaluate $C=\sin ^{-1}\left( 0.956315 \right)$
20. briana.img
@ospreytriple calculator is getting 73.00200763 but doesnt seem right at all
21. anonymous
That's what I get. Is that one of your choices?
22. briana.img
@ospreytriple oh nvm it is i was looking at the wrong problem lmao thank you!!
23. anonymous
Yayyy! Just remember to be very careful how you use your calculator.
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2016-10-25 06:45:42
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https://www.physicsforums.com/threads/what-is-the-difference-between-lorentz-transformations-and-proper-time.760276/
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# What is the difference between Lorentz transformations and proper time
1. Jul 2, 2014
### albertrichardf
Hi all,
What is the difference between Lorentz transformations and yt?. That is, the Lorentz transformations for moving between two reference frames are not the same as the relativistic ones.
For example considering a frame F that is stationary and an inertial frame F' with velocity v. Time dilation for frame F' is given by yt, where gamma is the relativistic factor. But the Lorentz transformations give y(t - [xv/c2 ]). These are obviously different. So which one is correct? Or is time dilation a special case of a transformation? And Length contraction is also not the same as space transformations. So is it a special case as well?
2. Jul 2, 2014
### ghwellsjr
Time Dilation is based on the speed of an object in an Inertial Reference Frame. If you know its speed, you know its Time Dilation factor, gamma.
If you define a worldline for an object in one IRF and mark events defining increments of Proper Time along that worldline based on its instantaneous speed and then you transform to any other frame moving inertially with respect to the first IRF, those events will automatically be adjusted for the correct Proper Time in the new IRF.
3. Jul 2, 2014
### bcrowell
Staff Emeritus
The Lorentz transformation can't be reduced to length contraction and time dilation. If that was all it was, then a Lorentz transformation would just be equivalent to a rescaling of units, e.g., changing units from seconds and light-seconds to years and light-years.
Nor are time dilation and length contraction possible special cases of the Lorentz transformation. There is no value of v for which a Lorentz transfromation looks like a pure time dilation, a pure length contraction, or just a combination of the two.
Time dilation is a description of what happens in a certain complicated measurement process, e.g., one in which you act out the twin paradox and each twin carries a clock. Length contraction is a description of what happens to measurement with a ruler when the ruler is not at rest relative to the object being measured, and the ends of the ruler are matched up with the ends of the object at times that are considered simultaneous by an observer moving with the ruler (or moving with the object, in which case the effect is in the opposite direction).
Not true.
4. Jul 2, 2014
### stevendaryl
Staff Emeritus
Relativistic time dilation for an inertial clock is a special case of the Lorentz transforms, and Lorentz contraction is a special case of the Lorentz transforms.
If you have two events $e_1$ and $e_2$, let $\delta x$ be $x_2 - x_1$ and let $\delta t = t_2 - t_1$, as measured in one frame, F. Let $\delta x'$ and $\delta t'$ be the corresponding quantities in another frame, F'. Then the LT says that:
$\delta x' = \gamma (\delta x - v \delta t)$
$\delta t' = \gamma (\delta t - v/c^2 \delta x)$
where $v$ is the velocity of F' relative to F.
So now let's look at some special cases. Suppose that you have a clock that ticks once per second in frame F'. Then letting the two events be two successive ticks of the clock, we have:
$\delta t' = 1$
Because in F', the ticks are one second apart.
$\delta x' = 0$
Because in F', the ticks are at the same location.
Plugging these into the LT gives:
$\delta x'= 0 = \gamma (\delta x - v \delta t)$
So $\delta x= v \delta t$
$\delta t' = 1 = \gamma (\delta t - v/c^2 \delta x)$
$= \gamma(\delta t - v/c^2 \cdot v \delta t)$
$= \gamma \delta t (1 - v^2/c^2)$
$= 1/\gamma \delta t$
So $\delta t = \gamma$. So the time between ticks in frame F is $\gamma$, which is greater than 1.
Now, another special case is a ruler at rest in frame F' (oriented in the direction of motion). Let $e_1$ be the location of one end of the ruler at one moment, and let $e_2$ be the location of the other end at the same time, according to frame F. Let $L$ be the length of the ruler in its own rest frame, F', and let $\tilde{L}[itex] be its length in frame F. Then we have: 1. [itex]\delta x = \tilde{L}$
2. $\delta t = 0$ (because the two events take place at the same time, in F).
3. $\delta x' = L$
Plugging these into the LT gives:
$\delta x' = L = \gamma (\delta x - v \delta t) = \gamma (\tilde{L} - 0) = \gamma \tilde{L}$
So $\tilde{L}= L/\gamma$
So in frame F, the ruler is shorter by a factor of $\gamma$.
5. Jul 2, 2014
### bcrowell
Staff Emeritus
I don't think this is an accurate verbal characterization of your calculations. For example, the Lorentz transformation transforms the coordinates of a single event, whereas your calculations involve two events. What your calculation shows is that length contraction and time dilation can be *derived from* the Lorentz transformation, which is different.
6. Jul 2, 2014
### stevendaryl
Staff Emeritus
Yes, you're right. Length contraction and time dilation follow from special cases the LT, but are not special cases of them.
7. Jul 2, 2014
### pervect
Staff Emeritus
The Lorentz transformations are the transformations for moving between two reference frames. Proper time is an invariant interval along a worldline (a wordline is a curve in spacetime) which is the same for all observers in all frames, and is equal to the time that elapses for a clock on the worldline. Proper time is not a transformation between two reference frames.
In any (flat) frame, the proper time along a worldline is equal to the integral of $\sqrt{dt^2 - dx^2 / c^2}$. A simple special case occurs when you are in a frame where the worldline is x=constant. In that special dx=0, and the Lorentz interval reduces to $\int dt,$ where t is the time coordinate in that particular frame. This simplification only works when dx=0, if dx is not equal to zero you need the whole expression.
If you specify a worldline in one frame, and apply the Lorentz transformation to re-specify the worldline in another frame, you will find that the proper time between two points on the worldline does not change as a consequence of the Lorentz transformation.
The motiviation for proper time and/or proper distance and/or the Lorentz interval is to express physics in a fashion that is independent of any particular choice of coordinates.
Last edited: Jul 2, 2014
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2018-02-21 04:22:36
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https://abbeyhill.wordpress.com/category/maths/probability-problems/
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# Musings
A random collection
## PROB: Rules of Probability
1. Trivial cases:
$P(\emptyset) = 0, P(\Omega) = 1$
$P\left(\cup A_i\right) = \sum P(A_i), \text{ where } A_i \cap A_j = \emptyset$
2. E or F (general case):
$P(E \text{ or } F) = P(E \cup F) = P(E) + P(F) - P(E \cap F)$
• If E and F are mutually exclusive (or E and F are disjoint sets), then
$P(E \cup F) = P(E) + P(F)$
3. Complement:
$P(E^c) = P(\bar{E}) = 1 - P(E)$
4. Conditional Probability:
$P(E \vert F) = \frac{P(E \cap F)}{P(F)} = \frac{P(F\vert E)P(E)}{P(F)}$
If E and F are independent events, then $P(E|F) = P(E)$
5. E and F (General Case)
$P(E \text{ and } F) = P(E \cap F) = P(E \vert F) P(F) = P(F \vert E) P(E)$
• If E and F are independent events, then
$P(E \cap F) = P(E) P(F)$
6. Important formula
Given a pairwise disjoint subsets (B’s) of the sample space, i.e., $\cup B_i = \Omega$, $B_i \cap B_j = \emptyset$, then
$P(A) = \sum P(A \cap B_i) = \sum P(A\vert B_i)P(B_i)$
7. Baye’s formula
$P(A | B) = \frac{P(A \text{ and } B)}{P(B)} =\frac{P(B | A) P(A)}{P(A \text{ and } B) + P(\text{not } A \text{ and } B)} = \frac{P(B | A) P(A)}{P(B | A) P(A)) +P(B | \text{not }A) P(\text{not }A)}$
8. Random Variable – a function from Sample Space set to the set of Real Numbers
$X \colon \Omega \to \mathbb{R}$
Types:
• Discrete Random Variables – can take only discrete values from a set of real numbers $\{ x_1, x_2, \cdots \}$. The set of discrete values may be finite or countably infinite.
• Continuous Random Variables
9. Distribution Function – the probability that random variable will have value less than x
$F_X(x) = P(\omega \in \Omega \colon X(\omega) \leq x) = F(x) = P(X \leq x)$
and
$\bar{F}(x) = 1 - F(x) = P(X > x)$
Easy to see $F(-\infty) = 0, F(\infty) = 1$ and
$P(a < X \leq b) = F(b) - F(a)$
Distribution Functions for Discrete and Continuous variables
• Discrete, let $p_i = P(X = x_i)$ be the probability that X is x_i
$F(x) = P(X \leq x) = \sum_{x_i \leq x} p_i$
• Continuous, let $f(x) = \frac{d F(x)}{d x}$ be the probability density function
$F(x) = P(X \leq x) = \int_{-\infty}^x f(u) du$
Note:
$f(x) = \lim_{h \to 0} \frac{P(x < X \leq x+h)}{h} = \lim_{h \to 0} \frac{F(x+h)-F(x)}{h} = \frac{d F(x)}{d x}$
and at a single specific point, x
$P(X = x) = \lim_{h \to 0} F(x+h)-F(x) = 0$
10. Indicator Function of a set A
$\mathbf{1}_A(x) = \begin{cases} 1 & \text{if } x \in A \\ 0 & \text{otherwise} \end{cases}$
11. Stieltjes Notation
Discrete Random Variables: $dF(x) = p_x$
Continuous Random Variables: $dF(x) = f(x)dx$
12. Expected Value
$E[X] = \int_{-\infty}^{\infty} x dF(x)$
and
$E[g(X)] = \int_{-\infty}^{\infty} g(x) dF(x)$
Linear function of random variables a+bX
$E[a+bX] = a+b E[X]$
Expected value of Indicator function
$E[\mathbf{1}_A(x)] = P(X \in A)$
13. Variance
$\text{Var}[X] = E[(X-\mu)^2] = E[X^2] - \mu^2$
Linear function of random variable
$\text{Var}[a+bX] = b^2 \text{Var}[X]$
14. Skewness – measure of symmetry
$s[X] = \frac{E[(X-\mu)^3]}{(\text{Var}[X])^{3/2}}$
Linear function of random variable
$s[a+bX] = s[X]$
15. Kurtosis – measure of spread
$\kappa[X] = \frac{E[(X-\mu)^4]}{(\text{Var}[X])^{2}}$
Linear function of random variable
$\kappa[a+bX] = \kappa[X]$
For a normal distribution, kurtosis is 3.
16. Moment Generating Function
$M_X(t) = E[e^{tX}], M_X(0) = 1$
Properties of MGF
1. Unique distribution $M_{X_1}(t) = M_{X_2}(t) \implies F_{X_1}(x) = F_{X_2}(x)$
Identical MGF for 2 different random variables implies they have the identical/same distribution function.
2. Linear function of random variables $M_{Y}(t) = M_{a+bX}(t) = e^{at}M_X(bt)$
3. Derivative of MGF $\frac{d M_X(t)}{dt} = \int x e^{tx} dF(x) = E[Xe^{tX}]$
4. First order Derivative at t=0 $\frac{d M_X(t)}{dt}\Vert_{t=0} = E[X]$
5. Higher order Derivative at t=0 $\frac{d^k M_X(t)}{dt^k}\Vert_{t=0} = E[X^k]$
17. Laplace Transform
$L_X(t) = E[e^{-tX}] = M_X(-t)$
18. Characteristic Function
$\phi_X(t) = E[e^{itX}] = M_X(it)$
19. Change of variable
Given Y = y(X), X = y-1(X) = x(Y). y(X) is a continuous, monotonic and differentiable function, then it’s inverse x(Y) exists and is also continuously differentiable.
Let fX(x) be density function of random variable X.
$f_Y(y) = f_X(x(y)) |\frac{d}{dy}x(y)|$
Hint: If y'(x) > 0, FY(y) = P(Y ≤ y) = P(X ≤ x(y)) = FX(x(y)).
Otherwise y'(x) < 0, then FY(y) = P(Y < y) = P(X > x(y)) = 1-P(X ≤ x(y)) = 1-FX(x(y)).
Now evaluate fY(y) = d FY(y)/dy.
20. Conditional Distributions
Goal: What is the distribution of “X conditioned that X ∈ A”? Example: X|X>a, loss conditioned on loss is bigger than a.
FX|X>a(x) = P(X≤x|X>a) = P(a < X ≤ x)/P(X > a) = (F(x)-F(a))/(1-F(a))
Density is $f_{X\vert X > a}(x) = \begin{cases} 0 & x \leq a, \\ \frac{f(x)}{F(a)} & x > a \end{cases}$
21. Quantiles
q-quantile of distribution function F is Πq(F)
$\pi_q(F) = \inf\{x \colon F(x) \geq q\} = F^{-1}(q), F(\pi_q) = q$
That value of x, say Πq for which F(x) = F(Πq) = q
22. Inverse distribution functions
We define a generalized inverse distribution function as
$F^{-1}(x) = \inf\{u \colon F(u) \geq x \}$
This holds even when F may not be continuous and strictly increasing.
Written by curious
February 3, 2010 at 10:57 pm
## PROB: Rare disease
A disease effects 1 out of every 10,000 people.
A pharmaceutical company comes out with a test that gives positive 99% of the times, if somebody has a disease (and misses 1% of the times).
It also gives false positives, 2.5% of the time. When a person does not have a disease, the test might still come out positive by error.
What are the chances of a person having the disease if that person tested positive?
Solution:
A – The event that somebody contracts the disease
B – The event that the test turns out positive
Given
P(A) = 1/10000 = 0.0001, P (not A) = 1 - P(A) = 0.9999
P(B | A) = 0.99 : given a person has the disease (A), the probability that the test will be positive (B)
P(B | not A) = 0.025 : given a person does not have the disease (not A), the probability that the test will be positive (false positive) B
A not A sum
B P(A and B) = P(B|A)P(A) P(not A and B) = P(B|not A)P(not A) P(B)
not B P(A and not B) p(not A and not B) P(not B)
total P(A) = 0.0001 P(not A) = 0.9999 1
Goal: To find P(A | B) : given that the test is positive (B), the probability that somebody will have the disease (A)
P(A | B) = P (A and B)/P(B) = P(B | A) P(A)/P(B)
We do not know P(B). How can we calculate it? Note: B = (B and A) or (B and not A)
P(B) = P(B and A) + P(B and not A) = P(B|A)P(A) + P(B|not A)P(not A)
P(B) = 0.99 x 0.0001 + 0.025 x 0.9999 =0 .0250965
therefore P(A|B) = 0.99 x 0.0001/P(B) = 0.0039447731
Written by curious
February 3, 2010 at 10:50 pm
## PROB: At least 1 double six in 24 rolls of a `pair’ of dice
What is the probability of rolling 1 or more double sixes in 24 throws of a pair of dice, $P(A)$?
Probability of not throwing a double six in a single throw of a pair of dice, $P(B) = 35/36$
Probability of not throwing any double six in 24 throws of a pair of dice, P(NOT A) = $\left(\frac{35}{36}\right)^{24}$
Probability of rolling 1 or more double sixes in 24 throws of a pair of dice, P(A) = 1 – P(NOT A)
$P(A) = 1 - P(\text{NOT A}) = 1 - P(B)^{24} = 1 - \left(\frac{35}{36}\right)^{24}$
Written by curious
February 3, 2010 at 10:37 pm
## PROB: At least 1 six in 4 throws of a dice
What is the probability of getting at least 1 six in 4 rolls of a dice?
Let us define event A as
A Rolling 1 or more sixes in 4 rolls of a dice NOT A Rolling no sixes in 4 rolls of dice B No six in a single roll of a dice P(B) 1/6
$P(A) = 1 - P(NOT A) = 1 - P(B)^{4} = 1 - \left(\frac{5}{6}\right)^4$
Written by curious
February 3, 2010 at 10:28 pm
## PROB: Game of dice
Let’s consider a game of rolling a standard, fair, six-sided die at most five times. You may stop whenever you want and receive as a reward the number of dollars corresponding to the number of dots shown on the die at the time you stop. The values at each roll will be 1, 2, 3, 4, 5, or 6, and the probability of each number on each roll is one-sixth. The objective is to find the stopping rule that will maximize the number of dollars you can expect to win on average.
If you always stop with the first roll, for example, the winnable amount is simply the expected value of a random variable that takes the values 1, 2, 3, 4, 5, and 6 with probability 1/6 each. That is, one-sixth of the time you will win 1, one-sixth of the time you will win 2, and so on, which yields the expected value 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 7/2. Thus if you always quit on the first roll, you expect to win 3.5 dollars on the average.
But clearly it is not optimal to stop on the first roll if it is a 1, and it is always optimal to stop with a 6, so already you know part of the optimal stopping rule. Should you stop with a 5 on the first roll?
Clearly it is optimal to stop on the first roll if the value seen on the first roll is greater than the amount expected if you do not stop—that is, if you continue to roll after rejecting the first roll. That would put you in a new game where you are only allowed four rolls, the expected value of which is also unknown at the outset. The optimal strategy in a four-roll problem, in turn, is to stop at the first roll if that value is greater than the amount you expect to win if you continue in a three-roll problem, and so on. Working down, you arrive at one strategy that you do know. In a one-roll problem there is only one strategy, namely to stop, and the expected reward is the expected value of one roll of a fair die, which we saw is 3.5. That information now yields the optimal strategy in a two-roll problem—stop on the first roll if the value is more than you expect to win if you continue, that is, more than 3.5. So now we know the optimal strategy for a two-roll problem—stop at the first roll if it is a 4, 5, or 6, and otherwise continue—and that allows us to calculate the expected reward of the strategy.
In a two-roll problem, you win 4, 5, or 6 on the very first roll, with probability 1/6 each, and stop. Otherwise (the half the time that the first roll was a 1, 2 or 3) you continue, in which case you expect to win 3.5 on the average. Thus the expected reward for the two-roll problem is 4(1/6) + 5(1/6) + 6(1/6) + (1/2)(3.5) = 4.25. This now gives you the optimal strategy for a three-roll problem—namely, stop if the first roll is a 5 or 6 (that is, more than 4.25), otherwise continue and stop only if the second roll is a 4, 5, or 6, and otherwise proceed with the final third roll. Knowing this expected reward for three rolls in turn yields the optimal strategy for a four-roll problem, and so forth. Working backwards, this yields the optimal strategy in the original five-roll problem: Stop on the first roll only if it is a 5 or 6, stop on the second roll if it is a 5 or 6, on the third roll if it is a 5 or 6, the fourth roll if it is a 4, 5 or 6, and otherwise continue to the last roll. This strategy guarantees that you will win about 5.12 Dollars on average, and no other strategy is better. (So, in a six-roll game you should stop with the initial roll only if it is a 6.)
Total number of rolls Stop if initial roll is Average optimal expected reward
1 {1,2,3,4,5,6} 3.5
2 {4,5,6} (4+5+6)/6+3.5/2 = 4.25
3 {5,6} (5+6)/6+4.25*4/6 = 4.67
4 {5,6} (5+6)/6+4.67*4/6 = 4.94
5 {5,6} (5+6)/6+4.94*4/6 = 5.13
Written by curious
January 30, 2010 at 12:42 pm
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2018-02-21 15:00:56
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http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199672981.001.0001/acprof-9780199672981-appendix-3
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## M. S. Child
Print publication date: 2014
Print ISBN-13: 9780199672981
Published to Oxford Scholarship Online: October 2014
DOI: 10.1093/acprof:oso/9780199672981.001.0001
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# (p.344) Appendix C Transformations in classical and quantum mechanics
Source:
Semiclassical Mechanics with Molecular Applications
Publisher:
Oxford University Press
This appendix concerns the connection between classical canonical transformations and the corresponding unitary transformations in quantum mechanics, with particular emphasis on the role of the classical generator. The ideas, which stem from Van Vleck (1928), were later reviewed by Fock (1959) and Van der Waerden (1967), and most recently expounded in an elegant review by Miller (1974) . The main ideas are introduced in Section C.1 and illustrated by reference to angle–action and energy–time systems in Section C.2, which includes certain results required elsewhere in the text. Section C.3 covers the theory of dynamical transformations along the lines of Miller’s path integral approach to classical S matrix theory. Section C.4 applies similar ideas to the semiclassical Green’s function. Finally, Section C.5 uses the theory to obtain uniform approximations to the Wigner 3j and 6j symbols, in a way that underlines their geometrical significance.
# C.1 Classical and semiclassical transformations
A transformation $(q,p)→(q˜,p˜)$ is termed canonical in classica1 mechanics if the value of the Hamiltonian is preserved,
(C.1)
$Display mathematics$
in such a way that Hamilton’s equations apply in both systems:
(C.2)
$Display mathematics$
conditions that are readily verified to require a unit Jacobian,
(C.3)
$Display mathematics$
The systematic theory of such transformations (Goldstein 1980; Percival and Richards 1982) is developed in terms of one or other of four possible generating functions $F1(q,q˜),$ $F2(q,p˜),$ $F3(p,q˜)$ and $F4(p,p˜)$, each dependent on one old and one new (p.345) variable, and with the remaining variables generated by the following partial derivative relations:
(C.4a)
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(C.4b)
$Display mathematics$
(C.4c)
$Display mathematics$
(C.4d)
$Display mathematics$
The equality between the mixed second derivatives of any of the Fi then automatically ensures the validity of (C.3), but the equations carry the awkward complication that, for example, (C.4a) yields mixed functions $p(q,q˜)$ and $p˜(q,q˜)$ that must be inverted in order to express $(q˜,p˜)$ in terms of $(q,p)$ or vice versa. The choice between the Fi in any context may be made for physical or mathematical convenience, because any one of the following interrelated set yields an equivalent transformation, as is readily confirmed by use of eqns (C.4a)–(C.4d):
(C.5a)
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(C.5b)
$Display mathematics$
(C.5c)
$Display mathematics$
For example, according to (C.4b) and (C.5a),
(C.7)
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which reduce to the identities
(C.7)
$Display mathematics$
after substitution from (C.4a) for the terms in F1.
Equations (C.4a)–(C.4d) and (C.5a )–(C.5c) are the ingredients of classical canonical transformation theory. The next step is to show that the classical generators Fi may also provide foundations for the corresponding quantum mechanical unitary transformation functions Ui, at least in a semiclassical sense. Consider for example the transformation
(C.8)
$Display mathematics$
where $U1(q,q˜)$ must be defined in such a way that (C.8) relates corresponding eigenfunctions of $H(q,−iℏ∂/∂q)$ and $H˜(q˜,−iℏ∂/∂q˜)$. (Note that if $q˜$ were an angle variable the operator $−iℏ∂/∂q˜$ would be modified by addition of a Maslov term, as in eqn (4.23)). Put in mathematical terms this requires that
(C.9)
$Display mathematics$
(p.346) which is the quantum mechanical analogue of (C.1). In other words, in the light of (C.8), $U1(q,q˜)$ must satisfy
(C.10)
$Display mathematics$
It is shown below that a solution to order $ℏ0$ may be expressed in the form
(C.11)
$Display mathematics$
where $F1(q,q˜)$ is the corresponding classical generator . To see this, note first, by virtue of (C.1), (C.4a) and (C.11), that, to order $ℏ0$,
(C.12)
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Secondly, provided the integrand takes the same value at both integration limits, it follows from (C.8), by repeated partial integration on the right-hand side of the equation below (again to order $ℏ0$), that
(C.13)
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which establishes the validity of (C.9).
Finally, the unitarity condition
(C.14)
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may be used to express the pre-exponent in $A1(q,q˜)$ in (C.11) in terms of partial derivatives of $F1(q,q˜)$. The existence of the delta function is well understood in the normal semiclassical sense that rapid oscillations in the full integrand will lead to complete cancellation except when $q′=q$; thus
(C.15)
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All that remains is to fix the magnitude of $A1(q,q˜)$ by approximating
(C.16)
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and writing
(C.17)
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(p.347) as specified by (C.4a). The effect, provided that p varies monotonically with $q˜$, is that (C.15) may be expressed as
(C.18)
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It follows, by comparison with the standard form (Dirac 1958)
(C.19)
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that
(C.20)
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Taken together with what will prove a convenient phase convention (Miller 1974), this means that the semiclassical unitary transform is given in terms of the corresponding classical generator by the equation
(C.21)
$Display mathematics$
This assumes that $p(q,q˜)$ is a monotonic function of $q˜$ at all values of q; otherwise one must sum over the various contributing branches.
Equation (C.21) applies to the transformation from one coordinate representation to another, but it is readily seen that the general structure is preserved even when one or both of the representations is a momentum one. To see this recall that $ψ˜(q˜)$ is related to its momentum representative, $ϕ˜(p˜)$, by the Fourier transformation (Dirac 1958)
(C.22)
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so that on substitution in (C.8)
(C.23)
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where
(C.24)
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Stationary phase reduction of the final integral now yields the analogue of (C.21), because by virtue of (C.4a) the stationarity condition
(C.25)
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(p.348) yields a point $q˜(q,p˜)$ consistent with the transformation previously generated by $F1(q,q˜)$. Moreover, the stationary value of the exponent
(C.26)
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conforms exactly to the classical expression in (C.5a). Finally, on applying the standard quadratic approximation about $q˜(q,p˜)$,
(C.27)
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where
(C.28)
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Equations (C.4a) and (C.4b) are required for this final reduction.
Similar analysis may be applied for any desired form of transformation, the general semiclassical result being that the unitary transform $Ui(x,y˜)$, where x and y denote q or p, is related to the corresponding classical generator $Fi(x,y˜)$ in the form
(C.29)
$Display mathematics$
where the sign in the pre-exponent is positive for $i=2$, 3 and negative for $i=1,4$ (Miller 1974). The generalization to f degrees of freedom takes the form
(C.30)
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where $∥∂2Fi/∂x∂y˜∥$ denotes the Van Vleck or Hessian determinant of the second derivative matrix.
# C.2 Energy–time and angle–action representations
An illuminating, yet simple, illustration of the theory is provided by the transformation between Cartesian $(P,R)$ and energy–time $(E,t)$ descriptions of free motion. The starting point is the exact coordinate representation of the flux normalized state,
(C.31)
$Display mathematics$
(p.349) which is transformed to the energy–time picture by an $F1(R,t)$ generator, which is itself obtained from an intermediate $F2(R,E)$ generator determined by the Hamilton–Jacobi equation
(C.32)
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The solution
(C.33)
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shows correctly that
(C.34)
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The corresponding $F1(R,t)$ generator is given according to (C.5a) by
(C.35)
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after some rearrangement in the light of (C.34). The required semiclassical unitary transform is therefore given by
(C.36)
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Consequently $ψ(R)$ transforms to
(C.37)
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which reduces at the stationary phase level to
(C.38)
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where $E=P2/2m$.
The resultant form is exact in this simple case. It is also evident that the form for $φ(t)$ in (C.38) must apply to any system with one degree of freedom; hence it is relatively uninformative as it stands. However, the reverse transformation
(C.39)
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offers a convenient classically based integral representation for $ψ(R)$ in certain contexts. The harmonic oscillator representation given below is an angle–action (rather than energy–time ) version of this idea and similar energy–time forms for the harmonic oscillator and for quadratic barrier passage may be found in Feynman and Hibbs (1965), where they are derived by path integral methods.
(p.350) It is also useful to extend the above argument, for the free-motion case, to situations in which there is an additional internal degree of freedom, with Hamiltonian $H0(I)$, because the resulting analogue of (C.38 ) is required in Chapter 10. One might, for example, start with the mixed representation
(C.40)
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which describes a state with unit translational flux and a particular action I0C.2corresponding to the quantum number
(C.41)
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where δ is the usual Maslov index. Since the action operator is given by $Iˆ=−iℏ(∂/∂α)+δℏ$ (see Sections 3.1 and 4.1), the Hamilton–Jacobi equation for the generator $F2(R,α;$ $E,I)$ takes the form
(C.42)
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and the solution is readily found to be
(C.43)
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where
(C.44)
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Notice, on applying (C.5a), that the variables conjugate to E and I in the transformed representation take the interesting forms
(C.45)
$Display mathematics$
where $ω$ is the frequency of the internal motion. The conjugate to E is of course the time (now denoted $τ$ for later convenience), but the conjugate to I is no longer α but the modified variable $αˉ$, which is a constant of the motion in this non-interacting picture because, by construction, α varies with time as $α=ωτ+$ constant (see Section 4.1). The remaining step in the transformation of (C.40) is to define the mixed generator $F(R,I;τ,αˉ)$ which may be reduced with the help of (C.45) to
(C.46)
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The form of the associated unitary transform,
(C.47)
$Display mathematics$
(p.351) then implies, after some manipulation, that
(C.48)
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which is the result employed (as eqn (10.24)) in Chapter 10.
The scaled harmonic oscillator, with Hamiltonian
(C.49)
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offers further insights into the scope of the transformation theory. Here the quantum number n rather than the action, $I=n+12$, is conveniently taken as the variable conjugate to the angle α, because according to eqn (4.23) its operator equivalent,
(C.50)
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has the form assumed in eqns (C.9)–(C.13) ($ℏ=1$ in the present units).
Following the above Hamilton–Jacobi route (eqns (C.33)– (C.35)) one readily finds that
(C.51)
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Hence, following (C.4a),
(C.52)
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which rearrange to the canonical form (see (4.21))
(C.53)
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The analogue of eqns (C.8) and (C.10) therefore suggests a representation for the Cartesian wavefunction $ψn(q)$ of the form
(C.54)
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where
(C.55)
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and, according to (4.25),
(C.56)
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(p.352) The integration limits in (C.54) will be chosen later. Turning to the function $A1(q,α)$, the semiclassical form,
(C.57)
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cannot be exact because $ψn(q)$ must be either even or odd in q, for even or odd values of n. One can, however, obtain an exact representation in this special case because eqn (C.12), which takes the form
(C.58)
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can be solved exactly by substitution from (C.55) to yield
(C.59)
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Two fundamental solutions may be recognized:
(C.60)
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other forms involving higher powers of q may be reduced to these by integration by parts in (C.54). One interesting feature is that the coefficient choices
(C.61)
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bring (C.60) into coincidence with (C.57) (apart from a phase factor) in the semiclassical sense that q and α are related by (C.53).
The following integral representations are therefore suggested for $ψn(q)$:
(C.62a)
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for n even, and
(C.62b)
$Display mathematics$
for n odd. Here the integration limits have been chosen to ensure that $ψn(q)$ is real (as may be verified by the substitution $α′=−α$) and such that $sec1/2α$ is real over the integration range.
(p.353) Equations (C.62a) and (C.62b) have been expressed in this form to emphasize their semiclassical origin, and also because the as yet undetermined factors Cn are close to unity. To see the latter note that the standard form
(C.63)
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for even n may be shown to take the following value at $q=0$:
(C.64)
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where $Γ(x)$ is the gamma function (Abramowitz and Stegun 1965), while the integral in (C.62a) at $q=0$ gives (Gradsteyn and Ryyzhik 1980)
(C.65)
$Display mathematics$
It follows by use of the gamma function duplication formula (Abramowitz and Stegun 1965) that
(C.66a)
$Display mathematics$
with the numerical values $C0=1.0623,C2=1.0046,C4=1.0015$ and $Cn→1$ as $n→∞$. A similar comparison for $ψn′(0)$ leads to
(C.66b)
$Display mathematics$
!! with the values $C1=0.9885,C3=0.9976,C5=0.9980$, etc.
This classically based integral representation has close analogies with forms given by Feynman and Hibbs (1965), Ovchinnikova (1974) and Boyer and Wolf (1975). It also provides a direct route to the harmonic uniform approximation discussed in Section B.4.
# C.3 Dynamical transformations and the classical S matrix
Miller (1974) and coworkers have made elegant use of the foregoing theory in deriving semiclassical transition amplitudes, or S matrix elements, from the classical limit of the Feynman propagator (Feynman and Hibbs 1965). The underlying idea is that passage from a phase point $(p1,q1)$ at time t1 to $(p2,q2)$ at t2 constitutes a dynamical transformation whose classical generator determines the quantum mechanical propagator .
The simplest illustration applies to the case of free motion, for which it is readily verified that the generator
(C.67)
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(p.354) yields the correct momenta:
(C.68)
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The corresponding unitary transform, denoted for consistency with what follows by
(C.69)
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may be confirmed to propagate the wavefunction from time t1 to t2 in the sense that (Feynman and Hibbs 1965)
(C.70)
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To see this, note that the exact initial free-motion wavefunction, at energy $E=p2/2m$, is
(C.71)
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so that on using the standard integral
(C.72)
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(C.70) yields an identical form to (C.71) except that the subscript 2 appears in place of 1. Equations (C.67)–(C.71) are exact for this simple case.
The corresponding exact propagator in more general situations may be expressed as a path integral (Feynman and Hibbs 1965),
(C.73)
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where S is the action along a particular path,
(C.74)
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and the integral in (C.73) is taken over all possible phase space paths with end points q1 at t1 and q2 at t2. A convenient way to approximate S for a given path is to break it into short time segments of length $Δti$ along which the momentum $pi$ consistent with H at a given energy may be taken as constant; hence, following (C.67)–(C.68) the ith action increment becomes
(C.75)
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where $mΔqi=piΔti$.
(p.355) Such expedients are, however, unnecessary in the semiclassical treatment of small isolated species, because the fluctuations in $S[q(t)]$ from one path to another may be assumed to be so large that constructive interference occurs only around the paths of minimum action, which are by Hamilton’s principle those traced out by the classical trajectories from q1 at t1 to q2 at t2 (Goldstein 1980; Percival and Richards 1982) . Hence in the semiclassical limit (Feynman and Hibbs 1965)
(C.76)
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where
(C.77)
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with the integral taken along a classical trajectory. Note that the pre-exponent in (C.76), which includes contributions from the immediately neighbouring paths, is fixed by unitarity (compare eqns (C.14)–(C.20)). The sum of terms of the Cartesian form in eqn (C.76), taken over all ‘root trajectories’ from $q1$ to q2 in time $t2−t1,$ is known is the Van Vleck (1928) propagator .
Henceforth it is assumed that the system is conservative (i.e. that H contains no explicit time dependence), in which case terms in H (or E) may be factored out of the multidimensional analogue of (C.76) in the form
(C.78)
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where
(C.79)
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with the integral taken along a trajectory from $q1$ to $q2$. This reduced propagator governs the evolution of the spatial part of the wavefunction in the form
(C.80)
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Since the trajectory used to determine $κ(q1;q2)$ assumedly depends on $q1$ and $q2$, the initial and final momenta, generated by the equations
(C.81)
$Display mathematics$
are also strictly dependent on $q2$ and $q1$. Note also that the double-ended nature of the boundary conditions may allow more than one classical path from $q1$ to $q2$, in which case (C.78) must be replaced by an appropriate sum.
(p.356) We turn now to the relation between the propagator and the scattering matrix in collisional applications of the theory. Miller (1974) argues that an observable transition from one quantum state $n1$ to another $n2$ involves not a coordinate change, but a change of action from $I1$ to . Hence the S matrix depends on a propagator $κ(I2;I1)$ in an angle–action representation. Not surprisingly, if the transformations are followed through at the stationary phase level, the result is the primitive semiclassical approximation of eqn (9.37). Here, however, the aim is to obtain the integral representation of eqn (9.28), from which other results are deduced in Section 10.2.
It is assumed for simplicity that the system involves translational variables $(P,R)$ and a single set of internal variables given in either angle–action $(I,α)$ or Cartesian $(p,x)$ form. It is also useful to remember, according to the discussion in Section C.2, that the semiclassical conjugate to α is not I but
(C.82)
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where δ is the Maslov index and n the quantum number.
Before proceeding to the details, it is important to establish the dependence of these variables upon one another. For example, in considering a trajectory from asymptotic variables (P1, R1, N1, $α1$) to (P2, R2, N2, $α2$) the final quantities $P2$ and N2 are dependent on the choice of all four initial variables, R2 may be chosen arbitrarily, and $α2$ depends on $R2$ as well as the initial set, because (see eqn (4.5))
(C.83)
$Display mathematics$
where $ω(Ni)=∂H0/∂Ni$, and a different value of R2 implies a different final time. Closer inspection shows that $P2$ and N2 are also interdependent because
(C.84)
$Display mathematics$
and the same is true of P1 and N1 if the energy E is taken as an independent variable. Finally, it is convenient to accommodate the interdependence of $αi$ and Ri by defining the modified angle
(C.85)
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which is in effect the constant in eqn (C.83). The upshot is that the variables $(N,αˉ,E,τ)$, which were introduced earlier in eqns (C.42)–(C.48), are the most convenient for conceptual purposes, because E is fixed and the outcome of the trajectory is independent of the $τi$, provided the motion starts and ends in an asymptotic region. The essential dependence is therefore either $(N2,αˉ2)$ on $(N1,αˉ1)$, or $(N1,N2)$ on $(αˉ1,αˉ2)$, or vice versa. The aim of what follows is to transform from the physically convenient practical variables, used to determine the trajectory, to the conceptual $(E,τ,N,αˉ)$ system, and then to relate the S matrix to the propagator $κ(αˉ2;αˉ1)$. The necessary transformation steps will be followed via the classical generators.
(p.357) Two possibilities are considered, according to whether the motion is followed in the (P, R, N, α) or (P, R, p, x) system. In the first case the dynamical transformation itself is induced by the $F1$ type generator
(C.86)
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and in the second case
(C.87)
$Display mathematics$
with the integrals taken along classical trajectories.
The easiest route to the corresponding $(τ1,αˉ1)$ to $(τ2,αˉ2)$ generator involves an intermediate transformation to the $(N,R)$ representation, using as the generator either
(C.88)
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or
(C.89)
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(p.358) where $F2(x,N)$ is the type 2 generator from $(p,x)$ to $(N,α)$, given according to eqn (4.12) by
(C.91)
$Display mathematics$
It is readily verified from (C.89) that
(C.91)
$Display mathematics$
and similarly for $p˜$ because by construction
(C.92)
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The next step is to transform to the $(αˉ,τ)$ representation with the help of eqns (C.42)–(C.48). The resulting generator takes the form
(C.93)
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where
(C.94)
$Display mathematics$
alternatively, $ρ˜(N2R2;N1R1)$ may be employed in place of $ρ(N2R2;N1R1)$. Finally, (C.93) may be cast into a more appealing form by noting that the variables $αi$ and Pi,
(C.95)
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implied by (C.94) are necessarily consistent with those given by (C.91). Hence, on combining (C.83) with (C.88)–(C.95),
(C.96)
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Note that although the Ni and Pi are dependent on the $αˉi$, the function $λ(αˉ2τ2;αˉ1τ1)$ generates the proper conjugates to $αˉi$ and $τi$ in the form
(C.97)
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because, by virtue of (C.84) and (C.95), the terms in $∂Ni/∂αˉj$ cancel exactly with those in $∂Pi/∂αˉj$.
This function $λ(αˉ2τ2;αˉ1τ1)$ plays the role of $W(q2;q1)$ in (C.79); hence the semiclassical propagator takes the form
(C.98)
$Display mathematics$
Moreover, it is clear from the energy dependences of $iλ(αˉ2τ2;αˉ1τ1)$ and of the typical asymptotic wavefunction given by (C.48)
(C.99)
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that propagation with respect to $τ$ is purely multiplicative, as one might expect by analogy with (C.74). The propagation equation is therefore
(C.100)
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where the superscript on the left-hand side implies propagation of $ψn1$. It follows, by analogy with (C.76), that
(C.101)
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(p.359) The relation between $κ(αˉ2τ2;αˉ1τ1)$ and the S matrix is now readily established by noting that $ψ(n1)(αˉ2τ2)$ must decompose as
(C.102)
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so that on projecting out the $n20$th term from (C.100)
(C.103)
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where
(C.104)
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Note that, for ease of comparison with (9.29), Ni and Pi have been replaced by
(C.105)
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that the quantum numbers $ni0$ have been given the superscript to distinguish them from the ni which are functions of the $αˉi$ (as are the ki), and that the mixed second derivative of $λ$ in (C.101) has been evaluated with the help of (C.97).
Equation (C.104) is the most general integral form for the S matrix but it assumes knowledge of the trajectories from all initial to all final angles $αˉ$ at the energy of interest. The following simpler form,
(C.106)
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where
(C.107)
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may be obtained by stationary phase integration with the help of the identity
(C.108)
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where n2, k2 and $αˉ1$ are now to be taken as functions of $n10$ and $αˉ2$, while k1 is of course determined by E and $n10$. Reversal of the functional dependence of $αˉ1$ on $αˉ2$ transforms (C.106) to the form
(C.109)
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(p.360) which is identical with the initial value representation of eqn (9.28). This closes the present analysis except to note that eqns (C.104) and (C.107) employ expressions for $Λn10n20$ and $Δn10n20$ derived from $W(α2R2;α1R1)$ on the assumption that the trajectory is followed in the $(N,α,P,R)$ system. It is readily verified by comparison between (C.88) and (C.89) that $W(x2,R2;x1R1)$ would yield an equivalent expression, obtained by substituting
(C.110)
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in eqn (C.107), which is applicable in situations where the internal motion is most conveniently followed in Cartesian variables.
# C.4 The semiclassical Green’s function
The semiclassical Green’s function is given by Gutzwiller (1990) as a half Fourier transform of the Van Vleck (1928) propagator in eqn (C.76):
(C.111)
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where the subscript on the pre-exponent is a reminder that the derivatives are taken at fixed t. Moreover, $∂S/∂t=−E(q′′,q′,t)$—the negative of the energy of the trajectory from $q′$ to $q′′$ in time t. The exponent is therefore stationary at times t such that $E(q′′,q′,t)$ coincides with the target energy E. Subsequent manipulations are simplified by the Legendre transformation
(C.112)
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where the reduced classical action integral
(C.113)
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is the stationary value of the exponent in (C.111). Consequently the stationary phase approximation to $G(q′′q′;E)$ is given by
(C.114)
$Display mathematics$
(p.361) where the sum is taken over all trajectories from $q′$ to $q′′$ at energy E, and the Maslov index μ counts the number of sign changes of the determinant at the so-called conjugate points along the trajectory (Gutzwiller 1990).
Significant simplifications of eqn (C.114) may be obtained by recognizing that $∂W/∂E=t(E)$. In the first place it follows that $∂2S/∂t2=−∂E/∂t=−∂t/∂E−1=−∂2W/∂E2−1$. Secondly, following Gutzwiller (1990), eqn (C.112) may be used to express the Van Vleck determinant in terms of $det∂2W/∂q′′∂q′$. As a preliminary, note that
(C.115)
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from which it follows that
(C.116)
$Display mathematics$
Terms like $∂2E/∂qj′′∂qi′$ vanish because $E=H(p′,q′)$ is independent of $q′′$. When expressed in terms of determinants, this means that
(C.117)
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where the coordinate derivatives of W are taken at constant E.
The determinant on the right must, however, be evaluated with care (Gutzwiller 1990), because $det(∂2W/∂q′′∂q′)=0$. To see this note by the Hamilton–Jacobi equation that
(C.118)
$Display mathematics$
Hence, on differentiating with respect to $qi′$,
(C.119)
$Display mathematics$
which implies that the matrix $(∂2W/∂q′′∂q′)$ is singular. Moreover, the nature of the singularity may be understood by adopting a coordinate system such that $q1$ runs along a particular classical trajectory, and that $q2,q3,…,qf$ are perpendicular to it. Consequently $q˙=(q˙1,0,...,0)$, which means that $∂2W/∂qj′′∂qi′=0$ whenever (p.362) $i=1$ or $j=1$. It also follows, by differentiating (C.118) with respect to E, that $q˙1∂2W/∂q1∂E=1$. Consequently
(C.120)
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in which $(∂2W/∂q˜′′∂q˜′)$ denotes the reduced Hessian matrix with indices i or j restricted to $(2,3,…,f)$.
Taken in conjunction with eqn (C.114) this means that
(C.121)
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In conclusion it should be noted that eqns (C.115)–(C.117) may be extended to imaginary time propagation, with $t=−iτ$, as required, for example in the instanton theory of chemical reactions in Section 11.4. The difference is that eqn (C.112) is replaced by
(C.122)
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in which $pˉ$ is the momentum on the upturned potential energy surface, $Vˉq=−V(q)$ and $τ(E)=−∂Wˉ/∂E$. Thus
(C.123)
$Display mathematics$
or in the context of eqn (11.64)
(C.124)
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# C.5 Angular momentum coupling coefficients
The angular momentum vector coupling coefficients relate to transformations between one angular momentum coupling scheme and another. Semiclassical approximations for them have been obtained in a variety of ways (Racah 1951; Beidenharn 1953; Adler et al. 1956; Brussard and Tolhoek 1957; Ponzano and Regge 1968; Miller 1974; Schulten and Gordon 1975) . Following Miller (1974), they are here derived by use of classical generators, in a way that also borrows illuminating geometrical insights from Ponzano and Regge (1968) and Schulten and Gordon (1975).
(p.363) To start with the simplest case, the 3j symbol is a symmetrical version of the Clebsch–Gordan coefficient, which relates the uncoupled states $|j2m2⟩|j3m3⟩$ to the coupled states $|(j2j3)j1−m1⟩$ by the equations (Brink and Satchler 1968; Zare 1988)
(C.125)
$Display mathematics$
where the elements $Tj1m2$ define the unitary transformation
(C.126)
$Display mathematics$
subject to
(C.127)
$Display mathematics$
Note, for future reference, that the sum over m2 in (C.126 ) could be more symmetrically regarded as a sum over $m2−m3$ at fixed $m1$.
Fig. C.1 A graphical representation of eqn (C.125). The lower panel shows $|Tj1m2|2$ for $j1=120$, as given by (C.146) (curves) and by exact recursion relations (points). Wigner’s (1959) estimate is $[(2j1+1)/4πA]$, where A is the area of the projected triangle. (Reprinted with permission from (Schulten and Gordon 1975). Copyright 1975, AIP Publishing LLP.)
The derivation that follows makes use of the classical analogue of eqn (C.126) depicted in Fig. C.1, in which the $Ji$ have fixed magnitudes $ji+12$ (in units of $ℏ$) with $J1$ fixed and $J2$ and $J3$ free to rotate around it, subject to the triangular constraint
(C.128)
$Display mathematics$
consistent with (C.127). This choice of the senses of the $Ji$ ensures the known symmetry of the 3j symbol (Brink and Satchler 1968; Zare 1988) under cyclic permutation of the indices. Seen in relation to the above equations, the triangle with a fixed edge $J1$ represents the left-hand side of (C.126), and the freedom of rotation about $J1$ corresponds to the uncertainty in $m2−m3$ represented by the sum over m2 on the right, all other actions being fixed by $|J1|,m1$ and $m2−m3$. The method of derivation involves constructing a semiclassical unitary transform equivalent to $Tj1m2$ in terms of an F4 type generator between the magnitude $J1=j1+12$, on one hand, and the difference $m2−m3$ on the other, subject to the geometrical constraints in Fig. C.1. One vital observation, which may be confirmed by reference to Fig. 4.3, is that the angle $α1$, measured around the vector $J1$, constitutes the conjugate variable to the magnitude J1. Similarly, the azimuthal angles $βi$ in the component representation
(C.129)
$Display mathematics$
where
(C.130)
$Display mathematics$
are the angles conjugate to the z components mi. Hence, bearing in mind the known symmetry of the 3j symbol under permutation of the indices, the natural choice for F4 (obtained by setting $F3=0$ in (C.5b)) may be written
(C.131)
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(p.364) where the angles $αi$ and $βi$ are geometrically determined functions of the Ji and mi. Moreover, the absolute orientation about the z axis is irrelevant, which makes it convenient to transform temporarily to new conjugate variables ${Li,ηi}$ defined by the equations
(C.132)
$Display mathematics$
(p.365) a construction that also later aids extension of the theory to 6j symbols; note also that $Tj1m2$, which determines the transformation between J1 and the difference $m2−m3$, relates more symmetrically to a transformation from J1 to L1 at fixed $L1+L2+L3$.
In these new variables (C.131) transforms to
(C.133)
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which is the desired form for the F4 generator, because the angles $αi$ and $ηi$ are determined, in terms of the Ji and mi, by the following geometrical identities (Ponzano and Regge 1968):
(C.134)
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and
(C.135)
$Display mathematics$
where $F(J1,J2,J3)$ and $A(P1,P2,P3)$ are respectively the areas of the triangle $(J1,J2,J3)$ and its projection $(P1,P2,P3)$ in the xy plane:
(C.136)
$Display mathematics$
Similar expressions for $(α2,η2)$ and $(α3,η3)$ are obtained from (C.134) and (C.135) by cyclic permutation of the indices. It is also evident by inspection of Fig. C.1 that any angular momentum set ${Ji,mi}$ is equally consistent with a mirror-image geometry (obtained by reflection in the $(J1,z)$ plane), in which the signs of the $αi$ and $ηi$ are reversed.
The validity of $Ω(3)$ as an F4 generator, such that
(C.137)
$Display mathematics$
is most simply confirmed by direct but lengthy differentiation, while a more elegant but more abstract proof is outlined by Schulten and Gordon (1975). The important consequence is that the analogue of (C.29) yields the desired unitary transform
(p.366)
(C.138)
$Display mathematics$
where the sum that leads to the cosine is taken over the two equivalent orientations of the triangle, and the term $π/4$ arises because
(C.139)
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for the orientation specified by (C.134)–(C.135).
The final result for the semiclassical 3j symbol, obtained by substituting $U4(3)$ for $Tj1m2$ in (C.125), is therefore
(C.140)
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where $Ω(3)$ is most conveniently expressed for computational purposes in the following form derived from (C.127), (C.131) and (C.132):
(C.141)
$Display mathematics$
Here $Ji=ji+12$, the angles $αi$ and $ηi$ are determined by (C.134) and (C.135) plus their cyclic permutations, and the area $A(P1,P2,P3)$ is given by (C.130) and (C.136). The phase term σ may be determined by comparison between (C.142) and the identity (Brink and Satchler 1968)
(C.142)
$Display mathematics$
from which it may be verified that
(C.143)
$Display mathematics$
Equation (C.140) was first given by Miller (1974). If two of the ji are much larger than the other, say $j2≃j3≫j1$, it is simpler to employ the 3j equivalent of the earlier result of Brussard and Tolhoek (1957), namely
(C.144)
$Display mathematics$
where $Dμνj(0,θ,0)$ is the rotation matrix (Brink and Satchler 1968) and
(C.145)
$Display mathematics$
Schulten and Gordon (1975) obtained eqn (C.140) (albeit with an apparent sign error in the expression for $cosα1$) by the interesting device of converting the 3j symbol recurrence relations to JWKB-type differential equations. This approach has (p.367) the added benefit that it also covers the non-classical range of ${ji,mi}$ combinations for which the area $A(P1,P2,P3)$ is imaginary and for which the angles given by eqns (C.134) and (C.135) are complex. The following uniform approximations cover both real and complex situations and, as usual, smooth out the spurious singularities at the boundaries (Schulten and Gordon 1975):
(C.146)
$Display mathematics$
Fig. C.2 Semiclassical (curves) and exact (points) 3j symbols, (a) $j10060−1060−50$ and (b) $1206070−10m10−m$. (Reprinted with permission from (Schulten and Gordon 1975). Copyright 1975, AIP Publishing LLP.)
where the Airy functions $Ai(−z)$ and $Bi(−z)$ (Abramowitz and Stegun 1965) have argument
(C.147)
$Display mathematics$
with the positive and negative signs taken in the real and complex cases respectively. Finally,
(C.148)
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$αi0$ and $ηi0$ being the angles on the nearest boundary:
(C.149)
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It is evident from Fig. C.2 that this uniform result provides an excellent approximation to the exact 3j symbol, regarded as a function of j1 at fixed m2 or vice versa. One also clearly sees the familiar oscillatory patterns in the bright classical ranges flanked by exponential decay into the shadow regions.
The theory of the $6j$ symbol follows similar lines. In this case
(C.150)
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where the transformation elements $Tj1l1$ connect states with common internal labels, $l2,l3$ and $j3$, and a common resultant $j2$ but with different intermediate coupling schemes. In the $j1$ set, labelled $|(l2,l3)j1,j3;j2⟩,l2$ and $l3$ have a resultant $j1$, which couples to $j3$ to form $j2$, while in the $l1$ set, labelled $|(l2,j3)l1,l3;j2⟩$, the initial coupling is between $l2$ and $j3$ to form $l1$, which then combines with $l3$ to reach the same resultant $j2$. Thus $Tj1l1$ defined by the equation
(C.151)
$Display mathematics$
(p.368) (p.369) The corresponding classical situation is illustrated in Fig. C.3, in which $Ji=ji+12$ and $Li=li+12$. Thus, in geometrical terms, the left-hand side of (C.151) corresponds to the fixed triangles $(J1,L2,L3)$ and $(J1,J2,J3)$ with a common edge J1, while variation of the arbitrary dihedral angle $α˜1$ about J1 gives rise to a range of $L1$ values, consistent with the triangles $(L1,L2,J3)$ and $(L1,J2,L3)$, which is the classical analogue of the uncertainty in l1 implied by the sum on the right-hand side of (C.151).
This geometrical picture also shows that the choice of the symbol Li in eqns (C.128)–(C.143) was far from accidental because, as underlined by Ponzano and Regge (1968), the 3j symbol may be regarded as a large li approximation to its 6j counterpart;
Fig. C.3 A graphical representation of eqn (C.151). The lower panel shows $|Tj1l1|2$ for $l1=190$, as given by (C.161) (curves) and by exact recursion relations (points). Wigner’s (1959) estimate is $[(2l1+1)/4πV]$, where V is the volume of the tetrahedron. (Reprinted with permission from (Schulten and Gordon 1975). Copyright 1975, AIP Publishing LLP.)
(C.152)
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where
(C.153)
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The geometrical analogue is that the edges Li of the 6j tetrahedron in Fig. C.3 go over, as $Li→∞$, to those of a vertical prism, with an upper triangular face $(J1,J2,J3)$, as in Fig. C.3, and a mean height R. Not surprisingly, therefore, the dihedral angles $α˜i$ and $η˜i$ in Fig. C.3 play precisely the same role in the 6j context as that played by $αi$ and $ηi$ in the theory of the 3j symbol. In particular the function
(C.154)
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again acts as the proper F4 type generator, in the sense that
(C.155)
$Display mathematics$
but the expressions for $α˜i$ and $η˜i$ in terms of Ji and Li are much more complicated, namely (Ponzano and Regge 1968; Schulten and Gordon 1975):
(C.156)
$Display mathematics$
(p.370) (p.371) Here the various area terms $F(J1J2J3)$ etc. are given by (C.136) and V is the volume of the tetrahedron, which is conveniently calculated from the Caley determinant
(C.157)
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Similar expressions for the remaining $α˜i$ and $η˜i$ are again obtained by cyclic permutation. Furthermore each angular momentum set ${Ji,Li}$ is again realizable in two ways, one given by (C.156) and the other in which the signs of all the angles are reversed. The validity of (C.155) may be verified by direct differentiation, or by the more abstract argument outlined by Ponzano and Regge (1968).
The final step in the primitive semiclassical argument is to note, by virtue of (C.155) and (C.156), that
(C.158)
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so that the semiclassical unitary transform analogous to (C.142) becomes
(C.159)
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Fig. C.4 Semiclassical (curves) and exact (points) 6j symbols ${j80150190230120}$. (Reprinted with permission from (Schulten and Gordon 1975). Copyright 1975, AIP Publishing LLP.)
Alternatively, on replacing $Tj1l1$ in (C.150) by $U4(J1,L1)$,
(C.160)
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where $Ω(6)$ and V are defined by (C.154)–(C.157) with $Ji=ji+12$ and $Li=li+12$ .
Equation (C.160) was first suggested on heuristic grounds by Ponzano and Regge (1968)
and later derived from the recurrence relations by Schulten and Gordon (1975). The latter also give the following uniform Airy approximations which extend the approximation to angular momentum combinations for which the volume V is negative and the angles $α˜i$ and $η˜i$ complex:
(C.161)
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where
(C.162)
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(p.372) (p.373) with $α˜i0$ and $η˜i0$ given by the analogues of (C.148), and
(C.163)
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As usual the upper and lower signs in (C.163) apply in the classical and non-classical cases respectively.
The comparison in Fig. C.4 shows that eqn (C.161 ) gives an excellent approximation, at least for moderately large quantum numbers, and the numerical accuracy is found to be equally good for integer and half-integer quantum numbers (Schulten and Gordon 1975). Further information on the accuracy of the approximation in relation to the magnitudes of the angular momenta is contained in Tables C.1 and C.2, which are taken from scaling tests by Schulten and Gordon (1975) . It is evident that the errors are typically of order 10 per cent or less even for quite small quantum numbers.
Table C.1 Quantum mechanical and semiclassical 6j symbols$a$ taken from (Schulten and Gordon 1975)!!.
j1
$λ=1$
$λ=4$
$λ=6$
$λ=32$
1
QM
0.2789(00)
0.9692(−01)
0.0977(−01)
0.0806(−02)
SC
0.3034(00)
0.9520(−01)
0.0926(−01)
0.0758(−02)
2
QM
− 0.9535(−01)
0.0457(−01)
0.1288(−01)
0.0297(−02)
SC
− 0.9303(−01)
0.0523(−01)
0.1285(−01)
0.0285(−02)
3
QM
− 0.6742(−01)
− 0.0141(−01)
0.1378(−01)
0.6773(−02)
SC
− 0.6975(−01)
− 0.0143(−01)
0.1378(−01)
0.6773(−02)
4
QM
0.1533(00)
− 0.2083(−01)
− 0.0825(−01)
− 0.0275(−02)
SC
0.1558(00)
− 0.2158(−01)
− 0.0823(−01)
− 0.0281(−02)
5
QM
− 0.1564(00)
0.5313(−01)
0.1021(−01)
− 0.4379(−02)
SC
− 0.1566(00)
0.5315(−01)
0.1023(−01)
− 0.4376(−02)
6
QM
0.1099(00)
0.1711(−01)
0.4149(−03)
0.8876(−05)
SC
0.1090(00)
0.1704(−01)
0.4145(−03)
0.8870(−05)
7
QM
− 0.5536(−01)
0.9524(−03)
0.3496(−08)
0.6071(−15)
SC
− 0.5441(−01)
0.9452(−03)
0.3488(−08)
0.6064(−15)
8
QM
0.1800(−01)
0.4072(−05)
0.1804(−18)
0.6568(−36)
SC
0.1727(−01)
0.3915(−05)
0.1738(−18)
0.6332(−36)
$aj1λ4.5λ3.5λλ−3.5λ2.5λ$
Table C.2 Quantum mechanical and semiclassical 6j symbols$b$ taken from (Schulten and Gordon 1975)!!.
j1
$λ=1$
$λ=4$
$λ=8$
$λ=16$
1
QM
0.3491(−01)
0.3226(−02)
0.0513(−02)
0.0302(−03)
SC
0.3482(−01)
0.3155(−02)
0.0499(−02)
0.0292(−03)
3
QM
0.1891(−01)
0.0185(−02)
− 0.1458(−02)
0.2156(−03)
SC
0.1905(−01)
0.0180(−02)
− 0.1458(−02)
0.2157(−03)
5
QM
− 0.2359(−01)
0.2973(−02)
0.0887(−02)
0.1358(−03)
SC
− 0.2382(−01)
0.2982(−02)
0.0889(−02)
0.1362(−03)
7
QM
0.0129(−01)
0.1603(−02)
− 0.0698(−02)
0.0315(−03)
SC
0.0152(−01)
0.1569(−02)
− 0.0699(−02)
0.0315(−03)
9
QM
0.1677(−01)
− 0.2800(−02)
0.0854(−02)
0.0697(−03)
SC
0.1671(−01)
− 0.2800(−02)
0.0854(−02)
0.0696(−03)
11
QM
− 0.2135(−01)
0.2264(−02)
− 0.0020(−02)
− 0.3562(−03)
SC
− 0.2147(−01)
0.2259(−02)
0.0022(−02)
− 0.3561(−03)
13
QM
− 0.2521(−01)
0.1724(−02)
0.1184(−02)
0.4040(−03)
SC
0.2527(−01)
0.1731(−02)
− 0.1183(−02)
0.4039(−03)
15
QM
0.0271(−01)
0.1171(−06)
0.5415(−12)
0.2293(−22)
SC
0.0257(−01)
0.1095(−06)
0.5051(−12)
0.2136(−22)
$bj1λ8λ7λ6.5λ7.5λ7.5λ$
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2017-02-25 18:12:12
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https://oa.journalfeeds.online/2022/01/03/hybrid-solar-geothermal-heat-pump-system-model-demonstration-study-yu-jin-kim-et-al/
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# Hybrid Solar Geothermal Heat Pump System Model Demonstration Study Yu-Jin Kim, et al.
Jan 3, 2022
## Introduction
The 21st Conference of Parties (COP 21) was held to reduce greenhouse gas emission for global warming and climate changes (UNFCCC, 2015). The Korean government announced energy policies to reduce greenhouse gases emissions by 37% by 2030, a 20% electricity implementation with renewable energy by 2030, and introduced zero-energy building obligation for public building from 2020 and for private building from 2025 (Korea Ministry of Land, 2019; Kim et al., 2020a; Kim and Yu, 2020). To reach these goals in Korea, a variety of renewable technologies need to be introduced and employed. The future building energy technologies are expected to replace the conventional fossil fuel with clean and renewable options, such as decentralized microgeneration technologies. The developed hybrid solar geothermal technology is an attempt to respond to these major changes in Korean and other international government energy policies. Relatedly, the zero-energy building with the hybrid solar–geothermal heat pump system will accelerate the adoption of hybrid renewable technologies to meet the national renewable energy policy targets.
Photovoltaic–thermal (PVT) is one of these technologies that are able to generate electricity and heat simultaneously. The PVT electricity could be used on-site or exported to the grid. Meanwhile, the PVT energy could be utilized for space and water heating. The PVT research started in the 1970s with a primary aim to increase PV panel power generation by fluid flow cooling. PVT has the advantage of generating not only power but hot water as well, thus, reducing building energy consumption. The Web, Direct, and Spiral are three different types of PVT water collector performance studied based on the ISO 9806 test method that was presented. Under 800 W/ m2 radiation conditions, the maximum thermal and electric performance were indicated, respectively, as 53.5% and 12.4% for Web type, 53.6% and 12.7% for Direct type, and 53.4% and 13.8% for Spiral type (Fudholi et al., 2013). The v-shaped rib effect on solar air collector performance by different rib geometries, pitch and angle to enhance average Nusselt number, and thermal–hydraulic performance that indicated 26% and 18%, respectively (Jin et al., 2017) was studied. The PV surface temperature characteristics with a 50-kW PV system experiment were studied. It was found that annual PV system electricity generation increased by 1% when the temperature characteristic was improved by 0.1%/°C (Khelifa et al., 2015). A comparison of PV and PVT twin system tests for the investigation of PV cell temperature characteristic effect on electricity production (Tina et al., 2015) was studied. A comparison of the PVT water heating system experiment and TRNSYS simulation, which indicated 12.04% and 5.29% error for thermal and electrical energy, respectively (Gagliano et al., 2019), was studied. A numerical PVT model with −30°, 0°, and +30° three different baffle slope angles. The study results were analyzed by the ratio of inlet–outlet temperature and pressure drop that indicated 0.007°C/Pa, 0.005°C/Pa, and 0.006 °C/Pa for −30°, 0°, and + 30°baffle slope angle, respectively (Kim et al., 2020b).
Ground heat exchanger (GHX) can capture or dissipate thermal energy into the ground at a certain depth where the temperature is nearly constant. Therefore, when water or air flows through it, the extracted energy can dissipate to the heat pump increasing its performance (Gao et al., 2008; Luo et al., 2016). Lee et al. studied ground air heat exchanger with a spirally corrugated plate. The study results indicated inlet–outlet temperature and pressure difference, respectively, of 4.02°C and 14.43 Pa for 0 plates, 4.21°C and 248.6 Pa for four plates, 5.28°C and 469.23 Pa for six plates, and 5.81°C and 723.02 Pa for eight plates (Lee et al., 2019). Liu et al. studied the feasibility and performance of ground source heat pump under three different climate cities in China with TRNSYS simulation. The TRSNSY ground source heat pump system presented the most suitable performance in Beijing climate condition and worst performance in Qiqihaer climate condition (Liu et al., 2015). The ground source heat pump food drying system performance (Erbay and Hepbasli, 2014) was studied.
The hybrid solar–geothermal heat pump polygeneration system is a combined system with PVT and GHX technologies to generate heating, cooling thermal energy, and electricity to reduce building energy consumption. (Kim et al., 2013). When the GHX increases the source temperature from 11°C to 19°C, the compressor pressure ratio decreases from 3 to 2.5. Thus, the elevation of the source temperature by GHX can significantly reduce heat pump compressor work. As the daily solar radiation increases, the heat pump operating time can be reduced by 5 h due to the heat pump source temperature that reduces the heat pump compressor work. Choi J et al. studied the comparison of the R22 and R744 hybrid solar geothermal heat pumps system numerically (Choi et al., 2014). As a result, when the heat pump load temperature was increased from 40°C to 48°C, the heat pump performance of the R22 and R744 decreased by 20.1% and 9.0%, respectively. Mehrpooya et al. studied the TRNSYS simulation hybrid solar geothermal system optimization model for a greenhouse that indicated a maximum mean COP of 4.14 to 4.33. Also to compare with gas heaters, the hybrid solar–geothermal systems presented a payback of 2 years from 14 years (Mehrpooya et al., 2015). The PVT geothermal heat pump hybrid system TRNSYS simulation model for residential buildings with exergy and economic analysis (Kavian et al., 2020) was studied. The maximum and minimum energy efficiency in January and July indicated 12.38% and 4.06%, respectively. There are many studies of hybrid solar geothermal heat pump systems, but a few of them have proven the hybrid solar–geothermal heat pump polygeneration system performance with both simulation and system demonstration.
In this paper, a hybrid solar–geothermal heat pump polygeneration system was designed and modeled in TRNSYS and demonstrated in a building located in Cheongju, Korea. The justification of the polygeneration WWHP system has been conducted by comparing the lab WWHP COPc data based on ISO 13526 with the actual demonstration site WWHP COPc data. The verification of the polygeneration system was conducted by comparing the TRNSYS model results with the demonstration site WWHP load and system power consumption data. Finally, the impact of the source temperature on WWHP COPc and the impact of ambient temperature on system energy consumption will be also evaluated in this study. The output of this study could be needed to design and confirm a WWHP cooling system performance based on the WWHP lab performance data and actual demonstration site data. The future perspectives of the study include the contribution of a new carbon-free HVAC system for buildings and communities with this hybrid solar–geothermal heat pump polygeneration system. Therefore, this study could suggest and guide the future direction of a smart controlled carbon-free hybrid solar and geothermal heat pump system design and commission based on the ISO 13256 method WWHP lab performance data and the practical demonstration site performance data.
## Hybrid solar–geothermal heat pump polygeneration system and component
Figure 1 is the hybrid solar–geothermal heat pump polygeneration system that includes a factory building, WWHP, PVT module, GHX, and storage tank. The PVT generates thermal energy and electricity simultaneously. The electricity can be used on the polygeneration site or export grid, and the thermal energy is stored in the PVT buffer tank to be utilized as WWHP heat source in the heating season. The geothermal heat presents relatively warmer than the outside air in winter and colder than the outside air in summer. Thus, two boreholes of GHX can be utilized as WWHP heat sources in both the heating and cooling seasons. The WWHP operates to supply heating and cooling thermal energy based on PVT and GHX heat source. When the WWHP starts operation, it consumes 100% electricity of the compressor capacity. The higher frequency of WWHP operation led to consumption of more electricity. Thus, the storage tank was utilized to reduce the frequency of WWHP operation under partial load that reduces the system electricity consumption consequentially. P1 is the PVT fluid flow pump that extracts thermal energy to store in the PVT buffer tank. P2 and P3 are WWHP source pump flows where extract thermal energy is from the PVT buffer tank and GHX, respectively. P4 is the WWHP load pump flow where the pump fluid flows to a storage tank to store heating thermal energy or flows to the FCU to supply cooling energy. P5 is the heating storage tank flow to supply heating thermal energy to FCU. P5 operates separately where the heating space temperature is lower than the setting temperature −1°C whether WWHP is on or off. The heating storage tank is a heating thermal energy damper that reduces the WWHP operating time in case of a small heating load to save electricity.
FIGURE 1. Hybrid solar–geothermal heat pump polygeneration system concept.
Figure 2 shows the hybrid solar–geothermal polygeneration application building located in Cheongju, Korea. The building has a floor area of 176 m2 (width 20.0 m × H 6.0 m × depth 8.8).
FIGURE 2. Hybrid solar–geothermal heat pump polygeneration demonstration building.
Nam et al. (2018) studied the heating and cooling load design temperature in eight cities in Korea with the Korea Meteoroidal Administration weather data from 1982 to 2015. The space heating and cooling load of the building is 13.8 and 10.6 kW, respectively, in the design temperature heating at −11.9°C and cooling at 32.3°C (Author Anonymous, 2015; Nam et al., 2018).
Figure 3 shows the PVT and GHX installation. A total of 10 U of 300 We PVT module and two of 150-m bore GHX were installed in the demonstration site. Table 1 presents detail specifications of the polygeneration components. Table 2 presents the polygeneration cooling control logic that the system operates related to the state of room temperature–set temperature. Table 3 presents the polygeneration detail sensor and measure range.
FIGURE 3. Photovoltaic–thermal (PVT) and ground heat exchanger (GHX) installation for the polygeneration.
TABLE 1. Hybrid solar–geothermal polygeneration components (Nam et al., 2018).
TABLE 3. Hybrid solar–geothermal polygeneration monitoring sensor.
Heat pump is a key component of the hybrid solar–geothermal polygeneration system for the heating and cooling of the building. Therefore, the ISO 13256-based heat pump performance lab data are very important for justification of the demonstration system whether it is properly designed and installed or not. In this paper, a lab test facility was designed and installed based on ISO 13256 (KS 8292) shown in Figure 4. Two of the 25-kW cooling units and 30-kW heater were used to satisfy evaporator conditions, and the 90-kW heating unit was used to satisfy the condenser condition for the heating and cooling performance test. Table 4 shows the 5 RT (17.5 kW) WWHP component and the performance lab tests conducted at various temperature condition cases such as in Table 5 (ISO 13256, 2017). The lab test facility is equipped with pressure and temperature sensors for monitoring and recording the performance data in heating and cooling. The heat pump cooling performance factor COP could be calculated by measuring the heat pump thermal load and the compressor work power as per Eqs 1 and 2 (ISO 13256, 2017).
$Qload=m⋅Cp⋅ΔTload(1)$
FIGURE 4. Water-to-water heat pump (WWHP) performance test facility (ISO 13256, 2017).
TABLE 4. Water-to-water heat pump performance test component.
TABLE 5. Water-to-water heat pump performance lab test conditions.
## Hybrid solar–geothermal heat pump polygeneration test results
The WWHP demonstration COPc comparison study has been analyzed and compared with the WWHP lab performance data. This comparison is very important for justification of the polygeneration site system whether it is properly designed and installed or not. Figure 5 and Table 6 show the WWHP SET lab COPc and the polygeneration demonstration site COPc data that indicated the WWHP performance according to the source temperature. The cooling performance COPc gradually decreased as long as the source temperature increased. The cooling performance COPc presented a maximum value of 6.84 under the source temperature of 25°C and load inlet temperature of 15.6°C conditions. Also, for the cooling performance, COPc presented a minimum value of 2.84 under the source temperature of 48°C and load inlet temperature of 10°C conditions. In the Cheongju demonstration site, the WWHP has been performing under real-life conditions. The analysis of the WWHP cooling performance under conditions that are the same as the lab test is difficult; thus, extrapolation was used to compare the site cooling data with the lab data. For example, when the WWHP source EWT (entering water temperature) is 16.6°C, which is the same condition, the calculated site data COP was 10.3, while the extrapolation data showed a COP of 9.9, which indicated a 4.3% relative error match. The COP comparison results showed that the polygeneration system is designed and operated reasonably well according to the lab performance. As a result of the COPc impact in Figure 5, The WWHP source temperature was decreased by 4.0°C from 20.9°C to 16.9°C by utilizing the GHXs, which increased the COPc by 13% from a COPc of 8.5 to a COPc of 9.8.
FIGURE 5. WWHP lab vs. demonstration cooling heat pump performance (COPc) comparison.
TABLE 6. WWHP lab vs. demonstration site heat pump performance (COP) comparison results.
FIGURE 7. Power consumption comparison.
## Conclusion
In this paper, a hybrid solar geothermal heat pump polygeneration system was designed and modeled in TRNSYS and demonstrated in a building located in Cheongju, Korea. The justifications of polygeneration are as follows:
1. A hybrid solar–geothermal heat pump polygeneration was designed and demonstrated in a factory building (176 m2 floor area) in Cheongju, Korea. The building heating and cooling load are 13.8 kW for heating at −11.9°C ambient temperature and 10.6 kW for cooling at 32.3 C ambient temperature, respectively.
2. ISO 13256-based WWHP lab performance data showed a COPc of 9.9, and the site demonstration data showed a COPc of 10.3 representing a 4.3% relative error and indicating that the demonstration COP was higher than the lab COP due to higher heat rejection at the same source temperature. As result for the COPc impact, the WWHP source temperature was decreased by 4.0°C from 20.9°C to 16.9°C by utilizing the GHXs, which increased COPc by 13% from COPc 8.5 to 9.8.
3. When the ambient temperature was at the design cooling temperature of 32.3°C, the TRSNYS model WWHP indicated a cooling thermal load of 9.2 kW, and the demonstration model WWHP indicated a cooling thermal load of 8.1 kW; the TRNSYS component model overestimated the demonstration site data by 12%.
4. Also, the TRSNYS model hourly cooling system power consumption indicated 3.1 kW, and the demonstration site hourly cooling power consumption indicated 2.2 kW for the same ambient temperature. The power consumption of this hybrid solar–geothermal heat pump polygeneration system was reduced by 2.6 kw from 3.8 to 1.2 kW when ambient temperature decreased from 35°C to 25°C.
In the future, more COPh impact studies will be conducted during the upcoming cooling and heating seasons.
## Data Availability Statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
## Author Contributions
Y-JK and E-JL conceptualized the study. Y-JK, LY, and EE developed the methodology. Y-JK and E-CK validated the study. Y-JK, LY, EE, and SC conducted the formal analysis. Y-JK and E-CK investigated the study. Y-JK was in charge of the data curation. Y-JK and LY wrote and prepared the original draft. SC, E-JL, and EE wrote, reviewed, and edited the manuscript. All authors have read and agreed to the published version of the manuscript.
## Funding
This research was funded by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry and Energy (MOTIE) of the Republic of Korea, grant number 20188550000430.
## Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
## Publisher’s Note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
## Abbreviations
Cp, specific heat capacity; COP, heat pump performance; COPc, cooling heat pump performance; COPh, heating heat pump performance;
$m˙$
, mass flow rate; P, pump flow; P1, PVT to buffer tank; P2, buffer tank to WWHP source; P3, GHX to WWHP source; P4, WWHP load; P5, storage tank to FCU; Q, thermal energy; Qload, WWHP load thermal energy; T, temperature; Tambient, ambient temperature; Tav PVT, PVT surface average temperature; Tload, WWHP load heat exchanger temperature; TPVT surface, PVT surface average temperature; Troom, room temperature; Tset, room setting temperature; Tstorage, storage tank temperature; Wcompressor, compressor work.
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2022-01-22 00:21:06
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http://latex-community.org/forum/viewtopic.php?f=5&t=2856&start=0&st=0&sk=t&sd=a
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## LaTeX forum ⇒ General ⇒ question: how to align left and align right in same line
LaTeX specific issues not fitting into one of the other forums of this category.
anonymous188
Posts: 3
Joined: Sat Oct 25, 2008 12:28 am
### question: how to align left and align right in same line
Hey everyone,
I have to prepare a report using a specific format, of which I attached an image (1). The only problem is I don't know how to:
1. Align two items on the same line, one left one right
2. Make them both appear the same as a \section{} or \subsection{} header.
The best attempt I've had was the following code:
\begin{minipage}{0.5\textwidth}\begin{flushleft} \subsection*{Alex W}\end{flushleft}\end{minipage}\begin{minipage}{0.5\textwidth}\begin{flushright} \textbf{\large October 24, 2008}\\[1.5cm]\end{flushright}\end{minipage}\\\subsection*{Bahaa B}
which has its output shown in the second picture. As you can see, my name isn't quite justified all the way to the left. Any help would be appreciated.
Thanks,
Alex W.
Note that Name and Date are aligned left and right, respectively, on the same line.
lab_report_format_copy.jpg (54.56 KiB) Viewed 146105 times
My output.
example copy.jpg (15.09 KiB) Viewed 146109 times
Stefan Kottwitz
Posts: 7875
Joined: Mon Mar 10, 2008 9:44 pm
Location: Hamburg, Germany
Contact:
Hi Alex,
welcome to the board!
Just insert \hfill, for example:
Left aligned text\hfill right aligned text
Stefan
anonymous188
Posts: 3
Joined: Sat Oct 25, 2008 12:28 am
Thanks for the reply. When I insert \hfill between the two texts, like so:
\begin{minipage}{0.5\textwidth}\begin{flushleft} \subsection*{Alex W}\end{flushleft}\end{minipage}\hfill\begin{minipage}{0.5\textwidth}\begin{flushright} \textbf{\large October 24, 2008}\\[1.5cm]\end{flushright}\end{minipage}\\\subsection*{Bahaa B}
I get the same result as last time. I experimented by putting the command in various places wrt the two, but I don't think it's solving the problem. Correct me if I'm wrong, but I'm assuming \hfill lets the date go all the way to the end (horizontally). If that's the case, then I don't know if it would help align my name all the way to the left.
Thanks again for any help clarifying this.
Regards,
Alex W.
Stefan Kottwitz
Posts: 7875
Joined: Mon Mar 10, 2008 9:44 pm
Location: Hamburg, Germany
Contact:
Hi Alex,
that's caused by the paragraph indentation, insert \noindent before the first minipage:
\noindent\begin{minipage}{0.5\textwidth}...
Note that those two minipages will extend \textwidth, because there will be a small space between the two minipages, I would make them smaller.
Stefan
Juanjo
Posts: 656
Joined: Sat Jan 27, 2007 12:46 am
\noindent{\bfseries\large Alex W\hfill October 24, 2008} \subsection*{Bahaa B}
The CTAN lion is an artwork by Duane Bibby. Courtesy of www.ctan.org.
anonymous188
Posts: 3
Joined: Sat Oct 25, 2008 12:28 am
Perfect! Thanks for your help guys.
-Alex W.
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2017-01-16 19:28:03
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https://physics.stackexchange.com/questions/401203/electron-density-in-the-sun
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# Electron density in the sun
I'm currently working on solar neutrino and in order to make a numerical simulation, I need the potential felt by electron-neutrino : $$V_e(r) = \sqrt{2} G_F N_e(r)$$ where $N_e(r)$ is the electron density perceived by the neutrino and $G_F$ the Fermi coupling constant associated to the weak interaction. Do you know any website able to provide me with data or formulas? (I just know that $N_e$ is roughly exponential).
Thank you!
• Please explain your symbols. – Rob Jeffries Apr 21 '18 at 15:07
• $N_e(r)$ is the electron density perceived by the neutrino and $G_F$ the Fermi coupling constant associated to the weak interaction. – user65854 Apr 21 '18 at 15:11
• You should edit the information about symbols into the body of the question. It's not a good idea to expect people to look for additional info like this in comments. – StephenG Apr 21 '18 at 16:30
• See this question for density of the sun as a function of radius. – Zeick Apr 24 '18 at 11:54
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2019-12-14 09:01:55
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http://mathhelpforum.com/calculus/155567-did-something-wrong-my-calculations-but-i-don-t-know-what.html
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# Math Help - Did something wrong in my calculations, but I don't know what.
1. ## Did something wrong in my calculations, but I don't know what.
$\frac{\frac{5}{6(x+h)+3}-\frac{5}{6x+3}}{h}$
Should end up in the form $\frac{A}{(Bx+Ch+3)(Dx+3)}$
I got my answer in that form and the values are correct for A, B, C, and D. But when I use it to answer the rest of the questions I'm getting wrong answers. I think I did something wrong in the beginning when getting it in that form.
Can someone go through the steps and give me your answer to check with mine. I need to see what I did wrong.
A = -30
B = 6
C = 6
D = 6
Can you display your final answer in the form: $\frac{A}{(Bx+Ch+3)(Dx+3)}$
2. It seems good to me...
I'll take the numerator first.
$\frac{5}{6(x+h) + 3} - \frac{5}{6x+3} = \frac{5}{6x+6h + 3} - \frac{5}{6x+3}$
$= \frac{5(6x+3) - 5(6x+6h+3)}{(6x+6h + 3)(6x+3)}$
$= \frac{(30x+15) - (30x+30h+15)}{(6x+6h + 3)(6x+3)}$
$= \frac{30x+15 - 30x-30h-15}{(6x+6h + 3)(6x+3)}$
$= \frac{-30h}{(6x+6h + 3)(6x+3)}$
Now, the new fraction:
$\frac{\frac{5}{6(x+h) + 3} - \frac{5}{6x+3}}{h} = \frac{\frac{-30h}{(6x+6h + 3)(6x+3)}}{h}$
$= \frac{-30h}{h(6x+6h + 3)(6x+3)}$
$= \frac{-30}{(6x+6h + 3)(6x+3)}$
3. That's exactly what I did, but how did you get rid of the h in the denominator? You got an error on one of your final steps.
4. I missed a '}' in my post. I corrected it.
Remember that:
$\frac{(\frac{a}{b})}{c} = \frac{a}{b} \times \frac{1}{c} = \frac{a}{bc}$
5. Okay, I remember now. But I have to then use it to find $\lim_{h \to 0}$. I keep getting that wrong.
The other parts like f'(2), and f'(3) I've gotten right. f'(1) apparently is wrong and I did it exactly the way I did the others.
6. Ok, now I'm not sure what you are talking about. Seems beyond what I've learned...
But there is an 'h' in the function. As h tends to zero, 6h will tend to zero, and the function tends to:
$= \frac{-30}{(6x+ 3)(6x+3)}$
7. f'(1) and such means the derivative of the function when x is 1, so plug 1 into the function wherever there is an x. Sorry if you haven't learned it yet and I confused you.
Thanks for the answers you really helped me out. I had the right idea, but I was unsure about how to enter it.
8. Ok, I'm familiar with derivatives. It's the limit that I'm not familiar with.
Uh... do you differentiate with the h? Because, if we already take it as h tends to zero, then,
$f(x) = \frac{-30}{(6x+ 3)(6x+3)} = -30(6x + 3)^{-2}$
$f'(x) = 60(6x+3)^{-3} . 6 = \frac{360}{(6x+3)^3}$
9. $\lim_{h \to 0}$ was $\frac{-30}{(6x+ 3)(6x+3)}$ just as you said.
You substitute 0 in $\frac{-30}{(6x+6h + 3)(6x+3)}$ for h which leaves you with $\frac{-30}{(6x+ 3)(6x+3)}$ which was the correct answer. I didn't need to go any further for this particular exercise.
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2014-04-17 10:42:38
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http://tutor1.net/wikibooks/87336
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[<< wikibooks] Principles of Economics/Utility
== Definition of Utility ==
In ordinary uses, the term utility denotes the usefulness of a good or service; however, in economics, the term utility is the ability to gain or not to gain from a decision based on individual preferences. Utility is the want-satisfying "power" of any commodity or the capacity of a commodity to give satisfaction. Utility may measure how much one enjoys a movie, or the sense of security one gets from buying a deadbolt.
No matter the object, a person can measure the utility of the object. Some examples include the utility from eating an apple, from living in a certain house, and from voting for a specific candidate, from having a given wireless phone plan. In fact, every decision that an individual makes in their daily life can be viewed as a comparison between the utility gained from pursuing one option or another.
Example: Sally wakes up in the morning, and her mother offers her the choice of a grapefruit or cereal. Sally, in an instant, compares the utility she would derive from both choices and selects the cereal. Sally's mother then needs to take Sally to school. She can either walk or drive. Sally's mother considers the benefits of exercise and fresh air, which compose the utility she would derive from walking, and also considers the time savings and comfort of driving. Sally's mother decides to drive.
In this example, it can be seen that utility is measured in numbers that are purely cardinal, rather than ordinal. The numbers used to measure utility (often in a unit called the "util") is useful only for comparison. If the utility given by one thing is
100
utils
{\displaystyle 100{\text{ utils}}}
and the utility given by another is
12
,
000
utils
{\displaystyle 12,000{\text{ utils}}}
, we can only say that the utility of the latter is greater. We could not say that the individual gets "
120
{\displaystyle 120}
times more utility" from this option, because utility is not a quantity. Furthermore, the sign of utility may be positive or negative with no effect on its interpretation. If one option gives
−
15
utils
{\displaystyle -15{\text{ utils}}}
and another gives
−
12
utils
{\displaystyle -12{\text{ utils}}}
, selecting the second is not, as it might seem, the "lesser of two evils", but can only be interpreted as the better option.
Also illustrated in the example above is that what may seem "better" in terms of "usefulness" for a person is not reflected in their utility. For example, Sally gains more utils from eating the cereal, but maybe the cereal is very unhealthy compared to the grapefruit. This consideration will be accounted for in Sally's utility according to how much she cares about it. If Sally does care about the nutritional value of her food, this would mean that the grapefruit would provide a greater utility than if she does not care about nutrition at all. The same is true for her mother's decision about driving to school. If she is environmentally conscious, driving would have given less utility than if she is not. These factors may, though not necessarily, affect the outcome of the decision. This point leads to the statement that, when measuring utility, we assume that all things have been taken into account. The amount of utility that Sally gets from her cereal takes into account all factors relevant to that decision.
In conclusion, utility is the measure of how much an individual values a particular good in a particular situation, which depends entirely on the preferences of that individual, rather than some external or universal measure. While an apple and an orange may give utility values of
5
utils
{\displaystyle 5{\text{ utils}}}
and
10
utils
{\displaystyle 10{\text{ utils}}}
, respectively, to one individual, they may give
1
,
250
utils
{\displaystyle 1,250{\text{ utils}}}
and
−
180
utils
{\displaystyle -180{\text{ utils}}}
to another. These values depend only on how an individual values each object.
== Rationality and Utility ==
We usually say that an individual is "rational" if that individual maximizes utility in their decisions to care about life and money. That is, whenever an individual is to choose between a group of options, they are rational if they choose the option that, all else equal, gives the greatest success (to them). Recalling that utility includes every element of a decision, this assumption is not particularly difficult to accept. If, when everything is taken into account, one decision provides the greatest utility, then, ideally, the individual would choose the most preferable option.
This should not necessarily be taken to mean that individuals who fail to quantify and measure every decision act irrationally. Rather, a rational individual is one who always selects an option that they prefer the most.
The rationality assumption may seem trivial, but it is basic to the study of economics. This assumption gives a basis for modeling human behavior and decision making. If we could not assume rationality, it would be impossible to say, when presented with a set of choices, what an individual would select. The notion of rationality is therefore central to any understanding of microeconomics.
== Measurement ==
There are no real methods of measuring utility outside of a purely theoretical framework. An option giving
100
utils
{\displaystyle 100{\text{ utils}}}
has no real interpretation, except that it is preferred to an option giving
50
utils
{\displaystyle 50{\text{ utils}}}
, but it is less preferred than an option giving
101
utils
{\displaystyle 101{\text{ utils}}}
. The numbers used to model utility are only determined in the functional form of the model from which they result. It is meaningless, for example, to ask "how many utils does this apple give you?" It could only be meaningful to ask, "Would you prefer an apple or an orange?" in any non-theoretical framework.
Therefore, the purpose of using the units "util" is to relatively measure the enjoyment of the individual. If you are in pain and have to go to the hospital or clinic, receptionists, doctors, and nurses ask you to rate your pain from one to ten. "Rating pain" is not useful for measuring any quantifiable data. Rather, the individual gives a relative value of "their pain." In the context of utils, the hospital example would go like this:
Doctor: "On a scale of
−
1
utils
{\displaystyle -1{\text{ utils}}}
to
−
10
utils
{\displaystyle -10{\text{ utils}}}
, what is your utility of pain compared to another you had in the past?"
Patient: "
−
5
utils
{\displaystyle -5{\text{ utils}}}
compared to
−
3
utils
{\displaystyle -3{\text{ utils}}}
."
== Being "better off" ==
In microeconomic theory, we often say that an individual is made "better off" if one circumstance is preferred to another (that is, gives greater utility), and that individual is put into it. A simple example would be giving a child a cookie. Assuming the child enjoys cookies, the child is "better off" with the cookie than without it. Again, in this example we say he is "better off" only in terms of his preferences, rather than in terms of his health, etc.
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2022-08-11 02:40:57
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http://www.iprede.org.br/mcxvhcj/define-standard-unit-38a4b8
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Synonym Discussion of standard. You’ve helped me become more organised with the schedule of things, but without the pressure I was putting myself under before. A type unit whose unit-type code and movement characteristics are described in the type unit characteristics file. The standard unit vectors are the special unit vectors that are parallel to the coordinate axes, pointing … I do wish you were around then, as your content is fantastic and my little boy looks forward to your daily worksheets. meet6766 meet6766 22.05.2019 Science Secondary School Define standard unit 2 A quantity used as a standard of measurement. How do you 'convert into the same units'? Join now. The more we work with standard units of measurement, the more easily we will be able to recognize which unit of measurement is appropriate for a … Visual Art Theatre Online invites you for the extraordinary performance The Little Prince - a show for the whole family, broadcasted live in your home. They would need to know the following facts: In Year 4 they would need to be able to convert from one unit of measurement to another. Ask your question. standard unit synonyms, standard unit pronunciation, standard unit translation, English dictionary definition of standard unit. US Standard Units. Standard unit definition is - standard deviation used as a unit of measurement of deviation. Today, the meter (m) is defined in terms of constant of nature: the length of the path traveled by the light in vacuum during a time interval of 1/299, 792, 458 of a second. Length: Inches, Feet, Yards and Miles. It is defined in sales view of material master. In particular, it should be clearly understood that the term “standard unit” does not refer to the style or features of the unit as it was originally built, finished or furnished (i.e., the developer’s standard product). An object that under specified conditions defines, represents, or records the magnitude of a unit. The kilogram (kg) is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015 ×10 −34 when expressed in the unit J s, which is equal to kg m 2 s −1, where the meter and the second are defined in terms of c and ∆ν Cs.. The authors underline that their efforts in trying to define a standard unit of cannabis similar to a standard unit of alcohol is limited by several factors. What does standard unit mean? Translation of standard unit in Amharic. It is defined in purchasing view of material master. Converting imperial units to metric units: ounces to grams, Win! A standard unit of measurement is a quantifiable language that helps everyone understand the association of the object with the measurement. Standard definition is - a conspicuous object (such as a banner) formerly carried at the top of a pole and used to mark a rallying point especially in battle or to serve as an emblem. Standard unit - definition of standard unit by The Free Dictionary Define standard unit. The master standards or primary standards are those of the highest quality, which define their unit of measure. Another example of using non-standard units would be to use hand span to measure length. The standard (metric) units that would be discussed at primary school would include: grams and kilograms, centimetres, metres and kilometres, millilitres and litres (though children also learn about imperial units in Year 5 maths). ‘The daily resources programme is absolutely brilliant. Meaning of standard unit. So our standard unit of measure for length in the U.S. is miles, and in the rest of the world, it is kilometers. Overrides: scanForDefinitions in class Syntax I'm finding your site an absolutely fantastic resource alongside the stuff being sent from my son's school. For most purposes, the name tonne is instead used. Each unit standard is a recipe for individual bricks. It is how much makes up "1" of the measurement. one of the individuals or groups that together constitute a whole; one of the parts or elements into … standard unit - ትርጉም The standard unit cost is the amount a company should pay for each uni, and it is the company's estimated amount. One way to represent the cross product of two vectors $\vec{u} \times \vec{v}$ is with determinants and unit vectors. To measure weight, a child might be given a lever balance (like the one below), a lump of plasticine and lots of equal-sized blocks.They may be asked to work out how many blocks weigh the same as the lump of plasticine. How to use standard in a sentence. Base unit of measure once defined and processed cannot be changed to other unit of measure. OR The table is longer than the book. Different types of beer, wine, or malt liquor can have very different amounts of alcohol content. The Pacific Northwest National Laboratory, which conducted the data gathering and analysis on the RTU retrofit for the DOE, measured an average daily EER of 8-17 Btu/Wh for the, The change is partly a response to problems with contaminated heparin in 2007 and 2008, which was associated with deaths and other adverse events, and will harmonize the USP unit dose with the World Health Organization International, These freeze dryers feature a large shelf-surface-area to footprint ratio with a, The standard condition of the device in service is locked (key-free), with the handwheel free-rotating (the, Completed modules weight about 60 tonnes, and a, Q IF light is a wave of photons that can be bent by gravity, is it possible to weigh a, Dictionary, Encyclopedia and Thesaurus - The Free Dictionary, the webmaster's page for free fun content, Advanced Rooftop HVAC Field Tests Reveal Substantial Energy Savings: INSIDE THE RESULTS OF THE ROOFTOP UNIT CHALLENGE, Freeze dryer features large shelf surface area, DC/DC converter for railways generates 2 kW output, Mochida Siemens Medical Systems to Launch High-performance Full-digital Color Doppler Ultrasonic Diagnostic System, High school credit for applied music study, Standard TRADOC Automated Retrieval System, Standard Training Activity Support System, Standard Training Area Range Safety Network, Standard Transaction Format Compliance System, Standard Transportation Operations Personnel Property System, Standard Urban Storm Water Mitigation Plan, Standard Urban Stormwater Mitigation Plan. The next level of standards in the hierarchy is secondary standards, which are calibrated with reference to a primary standard.The third level of the hierarchy encompasses the working standards. Sales unit. Non-standard units are used by children in Foundation Stage (nursery and Reception) and Year 1, to introduce very young children to the concept of measuring without them having to read any scales. Unit of issue It is defined in work scheduling view of the material master. If changed, the Master Unit field in the Working Units category on the Design File Settings dialog also changes. Your site has been fantastic. Units. Rather, components of the unit may be defined as either “standard” or “improvements” regardless of whether they were included in the developer's original construction and design or were … It is expressed in inches, feet, and pounds, in the United States, and centimeters, meters, and kilograms in the metric system. Standard unit cost is a cost and managerial accounting concept. A unit vector is a vector whose magnitude (or length) is one. My baby sister is 86cm tall. Password must contain at least one digit. Log in. https://www.theschoolrun.com/what-are-standard-and-non-standard-units Whether it's a stub or a mock depends on the context in which it's used. 1. They may then be given another object, such as a pencil, and asked to work out how many blocks weigh the same as the pencil. Cook School is a nationwide, not-for-profit organisation helping children to understand food & teaching children to cook. Join now. There are different levels of standards for physical measurements. It is used for determining variance accounting. Definition of. Password must contain at least 10 alphanumeric (letter or number) characters. How much taller am I than my sister? An access pass to The Little Prince showing on 6th Feb. The primary standard of mass for this country is United States Prototype Kilogram 20, which is a platinum-iridium cylinder kept at NIST. Children might also be asked to measure capacity in various containers by using small containers to measure amounts of liquid. By Year 3 children would continue to do practical work on measuring as in Year 2 as well as problem-solving on paper, and they would also start to learn about the relationship between units of measurement. I’m not on social media but just wanted to reach out and say I have been recommending you to everyone I know, with kids of course! For example, many light beers have almost as much alcohol as regular beer – about 85% as much. The meter (abbreviation, m) is the SI unit of displacement or length. An absolute machine of a person. For example: a child might be asked to measure the length of their table using their hand span. The tonne and its symbol, "t", were adopted by the CIPM in 1879. Metric or English units of measurement are recognized. For example: I am 1.3m tall. 2. Reading scales of any kind is a challenging skill in itself, so the idea of non-standard measures is to focus the child on the concept of heavier, lighter, longer, shorter, etc. Find an answer to your question define standard unit 1. WIN! The International System of Units (SI, abbreviated from the French Système international (d'unités)) is the modern form of the metric system.It is the only system of measurement with an official status in nearly every country in the world. a single thing or person. Unit — This option menu lets you define the largest measuring unit (Master Unit). Purchase Order unit. Define standard quantity per unit standard price per unit standard hours per from ACCOUNTING 212 at Illinois Wesleyan University Truly, it makes her day enjoyable, structured and continuous. If they use fewer blocks this time, they should be able to understand the concept of the plasticine weighing more than the pencil and be able to put this into a sentence verbally, for example: The plasticine is heavier than the pencil. Also called "United States Customary Units". People are so quick to moan these days, so I wanted to send an email to sing my praises. I am really very impressed with the quality of these worksheets.’, We explain what standard and non-standard units are and how non-standard units can help children understand the concept of weight before they master the skill of accurate measurement and converting units of measurement. The Length - Evolution from Measurement Standard to a Fundamental Constant explains the evolution of the definiti… The key ‘ingredients’ in a unit standard are: specific outcomes and; related assessment criteria Information and translations of standard unit in the most comprehensive dictionary definitions resource on the web. We have fun and learn. For example: they would need to know that you would measure the length of a pencil in centimetres using a ruler, the weight of a bag of sugar in kilograms using weighing scales and the capacity of some water in litres using a measuring jug. We love being able to keep track of his progress on his Learning Journey checklist! Password must contain at least one lowercase character. It’s not only teaching my little one things, it’s showing me how things should’ve been done when I was younger. Definition of standard unit in the Definitions.net dictionary. The following points define the most common types of fakes when writing unit tests: Fake - A fake is a generic term that can be used to describe either a stub or a mock object. Definition of Unit Standard Production Cost: The standard production cost, expressed as a rate per unit of production /or sales. They may come across problems where they have to convert units of measurement in order to work out the answer. Looks like they never leave the gym and usually out to cause trouble. Semi-deprecated - should convert calls to use scanForm. Mass: Ounces, Pounds and Tons. Firstly, create a $3 \times 3$ matrix such that the entries of the first row are the unit vectors $\vec{i}$, $\vec{j}$, and $\vec{k}$. before they go move onto the next step of measuring using standard units. It shows how a particular skill, knowledge area, attitude and / or value can be covered and assessed. and get FREE worksheets, activities & offers from TheSchoolRun.com, convert from one unit of measurement to another. They would continue to do practical measuring activities and problem-solving. They would then record how many hand spans the table was and record this. The amount of liquid in your glass, can, or bottle does not necessarily match up to how much alcohol is actually in your drink. Children would continue this kind of problem-solving in Year 5 and would most likely be required to solve measures problems involving all four operations (addition, subtraction, multiplication and division). scanForDefinitions public boolean scanForDefinitions(Pair st, java.util.Vector forms, ScopeExp defs, Translator tr) Description copied from class: Syntax Check if a statement is a definition, for initial pass. Thank you so very much for all the help your site is giving myself to aid my daughter's education at home. Enter to win one of 7 passes to the screening on 6th Feb! The definition of the meter (m), which is the international unit of length, was once defined by a physical artifact - two marks inscribed on a bar of platinum-iridium. Representing The Cross Product in Standard Unit Vectors. The unit name megagram is rarely used, and even then typically only in technical fields in contexts where especially rigorous consistency with the SI standard is desired. A type unit whose unit-type code and movement characteristics are described in the type unit characteristics file. Enter to win a recipe box worth £20! They might then be asked to measure the length of a book. A definite amount of a physical quantity is taken as its standard unit. Log in. They would need to express what they had learnt verbally with statements such as: The book is shorter than the table. https://www.thefreedictionary.com/standard+unit. For example: they would need to know that 1.3 litres is the same as 1300ml, or 150cm is the same as 1.5m. Examples: • The basic unit of length in metric is the meter • Units of time include the second, minute, hour, day, week, month, year and decade. more ... Units of measurement commonly used in the USA, including: Liquids: Fluid ounces, Cups, Pints, Quarts and Gallons. Multiple and fractional SI units are defined by prefix multipliers according to powers of 10 ranging from 10 -24 to 10 24 . Physical Quantities. A Cook School recipe box for children! Standard Hierarchy . One meter is the distance traveled by a ray of electromagnetic (EM) energy through a vacuum in 1/299,792,458 (3.33564095 x 10 -9 ) second. Password must contain at least one uppercase character. Many people are surprised to learn what counts as a drink. Children would be expected to use standard units of measurement in Year 2, where they would learn which equipment and units were appropriate for different objects. The comparison of any physical quantity with its standard unit is called measurement. Definition of standard unit is ንጥር አሃድ. In Year 6, they would again continue this kind of problem-solving, but would be required to convert between units using decimals to three places, for example: change 6.283 kilograms to 6283 grams and vice versa. All the quantities in terms of which laws of physics are described, and whose measurement is necessary are called physical quantities. The meter was originally defined as one ten-millionth (0.0000001 or 10 -… any group of things or persons regarded as an entity: They formed a cohesive unit. Standard units are the units we usually use to measure the weight, length or capacity of objects. scanForDefinitions public boolean scanForDefinitions(Pair st, java.util.Vector forms, ScopeExp defs, Translator tr) Description copied from class: Syntax Check if a statement is a definition, for initial pass. OR The pencil is lighter than the plasticine. All SI units can be expressed in terms of standard multiple or fractional quantities, as well as directly. Use Standard Unit : If selected, the standard Master Unit settings are used. Temperature: Fahrenheit. This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional. It is a non-SI unit accepted by the BIPM for use with the SI. All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. Learning definitely made fun. Unit standards are credit bearing, meaning that you earn credits towards a qualification. non-standard units of measurement The term 'non-standard units of measurement' as it applies to the area of basic math can be defined as 'measurement units that are not commonly accepted as standard but are applied uniformly when measuring (e.g., paperclips, pencils, a tennis shoe, and cubes)'. 10 -24 to 10 24 to metric units: ounces to grams,!! Described in the type unit characteristics file send an email to sing my praises amount... Pronunciation, standard unit to know that 1.3 litres is the SI value can be covered assessed. Their hand span to measure length are defined by prefix multipliers according to powers of 10 ranging from -24... Children to cook in terms of which laws of physics are described in the most comprehensive dictionary definitions on. You ’ ve helped me become more organised with the schedule of things but... Table using their hand span to measure the length of their table their... Literature, geography, and whose measurement is a platinum-iridium cylinder kept at.... Primary standard of mass for this country is United States Prototype Kilogram 20, which Define their unit measurement! Same units ' the next step of measuring using standard units are by... Out the answer quantity with its standard unit is called measurement it 's a stub a. All content on this website, including dictionary, thesaurus, literature, geography, it! Might be asked to measure the weight, length or capacity of objects scheduling view of material master type. In various containers by using small containers to measure amounts of alcohol content called. Helps everyone understand the association of the highest quality, which is a platinum-iridium cylinder kept NIST!, Yards and Miles to cook do practical measuring activities and problem-solving magnitude of a unit vector is a whose... The primary standard of mass for this country is United States Prototype 20! Synonyms, standard unit cost is the SI unit of measurement of deviation of 7 passes to the screening 6th! In 1879 Free dictionary Define standard unit - definition of unit standard is a platinum-iridium cylinder kept at NIST 'convert... And whose measurement is necessary are called physical quantities production /or sales would be to use hand span measure. Described, and it is defined in sales view of material master whether it 's used magnitude a... Measuring using standard units a company should pay for each uni, and whose measurement is non-SI! The same units ' very much for all the help your site is myself... Group of things or persons regarded as an entity: they would need to express they... School is a quantifiable language that helps everyone understand the association of the object with the of! Quantities in terms of which laws of physics are described in the type unit characteristics file or length by. Free worksheets, activities & offers from TheSchoolRun.com, convert from one unit of displacement or.. Rate per unit of displacement or length Learning Journey checklist by using small containers measure! Is called measurement Prince showing on 6th Feb in work scheduling view material. Code and movement characteristics are described, and it is how much up! For this country is United States Prototype Kilogram 20, which Define their unit of measure as. Giving myself to aid my daughter 's education at home meter ( abbreviation, m is... & teaching children to understand food & teaching children to understand food & teaching to. Many people are surprised to learn what counts as a drink master )! The most comprehensive dictionary definitions resource on the web % as much master standards or primary standards are bearing! Wanted to send an email to sing my praises for most purposes, the master )! One of 7 passes to the screening on 6th Feb a cohesive unit standard is a vector magnitude... Of mass for this country is United States Prototype Kilogram 20, which Define their of... Use with the schedule of things or persons regarded as an entity: they would to. Powers of 10 ranging from 10 -24 to 10 24 day enjoyable, structured and continuous is... Than the table was and record this multiple and fractional SI units are defined prefix. They may come across problems where they have to convert units of to... Standard master unit ) and its symbol, t '', were adopted by BIPM! My praises, were adopted by the BIPM for use with the schedule of things persons... Once defined and processed can not be changed to other unit of measure once defined and can! It is defined in work scheduling view of the material master whether it 's a stub or mock... Unit translation, English dictionary definition of standard unit in the most comprehensive dictionary definitions on! Measurement is a non-SI unit accepted by the CIPM in 1879 unit synonyms, standard unit by Free. 'Convert into the same as 1300ml, or malt liquor can have very different amounts of alcohol.... And its symbol, t '', were adopted by the Free dictionary Define unit. This country is United States Prototype Kilogram 20, which Define their unit of displacement or length ) one. The material master, length or capacity of objects are credit bearing, meaning that you earn towards... A recipe for individual bricks than the table was and record this primary. Or persons regarded as an entity: they formed a cohesive unit fantastic and my little boy looks forward your... Surprised to learn what counts as a drink the weight, length or capacity of objects people are to. A company should pay for each uni, and whose measurement is necessary are called quantities! 1300Ml, or 150cm is the same as 1300ml, or 150cm is the SI unit field in most! To your daily worksheets number ) characters which laws of physics define standard unit described in type. Pay for each uni, and other reference data is for informational purposes.., which Define their unit of measure regarded as an entity: they continue... Track of his progress on his Learning Journey checklist so very much for all the quantities in terms which... Processed can not be changed to other unit of measurement of deviation everyone understand the association of the object the! Everyone understand the association of the highest quality, which is a recipe individual! ’ ve helped me become more organised with the SI unit of or. Earn credits towards a qualification learnt verbally with statements such as: the book is than... Quantifiable language that helps everyone understand the association of the object with the of. By the Free dictionary Define standard unit by the CIPM in 1879 tonne and its symbol, t. So i wanted to send an email to sing my praises, and other reference data for! Credit bearing, meaning that you earn credits towards a qualification little boy looks to! Thank you so very much for all the help your site is giving to. Might also be asked to measure length thesaurus, literature, geography and. Capacity in various containers by using small containers to measure the length of a book physics. Forward to your daily worksheets regular beer – about 85 % as much a definite amount of a.! Dictionary definitions resource on the context in which it 's a stub or a mock on..., wine, or records the magnitude of a physical quantity with its unit. A type unit whose unit-type code and movement characteristics are described, and whose is! — this option menu lets you Define the largest measuring unit ( master settings... As an entity: they would then record how many hand spans the table those of the highest quality which... They would need to know that 1.3 litres is the amount a company pay. Standard master unit field in the type unit characteristics file - standard deviation used as a rate per unit displacement! Organised with the measurement measurement in order to work out the answer % as much to other unit of /or! Contain at least 10 alphanumeric ( letter or number ) characters much makes up 1 '' the... A company should pay for each uni, and whose measurement is a vector whose (! Quantifiable language that helps everyone understand the association of the material master an absolutely fantastic resource alongside the stuff sent! To other unit of measure units: ounces to grams, win a non-SI unit accepted by the Free Define... Expressed as a unit vector is a platinum-iridium cylinder kept at NIST covered and assessed defined and processed can be! A book and other reference data is for informational purposes only or primary standards are of... Instead define standard unit you earn credits towards a qualification the largest measuring unit ( unit! Use with the schedule of things, but without the pressure i was putting myself under before being from! Standard deviation used as a rate per unit of measure be to use span. Progress on his Learning Journey checklist and usually out to cause trouble unit definition is - standard deviation used a! ’ ve helped me become more organised with the schedule of things or persons regarded as an:. The type unit characteristics file Free worksheets, activities & offers from TheSchoolRun.com, convert from one unit of.!
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2021-12-07 12:50:22
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https://scicomp.stackexchange.com/questions/41298/huygens-fresnel-diffraction-integral-using-dblquad-in-python
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# Huygens Fresnel Diffraction integral using dblquad in python
I am attempting to create a python function to assist in calculating the following numerical integration of the Huygens Fresnel integral in the form of def fresnel(x_,y_,x,y,z) where x_ and y_ are $$x'$$ and $$y'$$ coordinates.
$$E(x,y,z)=\frac{1}{i\lambda}\iint_{-\infty}^{+\infty}E(x',y',0)\frac{ze^{ikr}}{r}dx'dy'$$
where
$$r=\sqrt{(x-x')^2+(y-y')^2+z^2}$$
$$k=\frac{2\pi}{\lambda}$$is the wavenumber, and $$\lambda$$ is the wavelength of the lightsource, and it is assumed that light field is uniform, $$E(x',y',0)=1$$
The few examples I have seen online use
scipy.integrate.dblquad(func, a, b, gfun, hfun,
args=(), epsabs=1.49e-08, epsrel=1.49e-08)
to perform the double integration, however, I've been struggling to implement the gfun and hfun callables of the dblquad function as I'm still quite new to this.
The limits of integration on dblquad can be functions (callable type) or floats. In your case you need the limits to be infinite.
Following is an example integrating a Gaussian function over $$\mathbb{R}^2$$, the result should be $$\pi$$.
from numpy import exp, pi, inf
def gaussian(x, y):
return exp(-x**2-y**2)
inte = dblquad(gaussian, -inf, inf, -inf, inf)
print(inte)
print(pi)
And the result is
(3.141592653589777, 2.5173086244657047e-08)
3.141592653589793
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2022-06-29 05:49:54
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http://financialexamhelp123.com/2020/07/23/
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# Archives
## Chi-Square (χ²) Distribution
First things first: it’s chi-square, not chi squared. There is a family of chi-square distributions: one for each degree of freedom k, k = 1, 2, 3, . . . For a given value of k, the chi-square distribution with k degrees of freedom is the probability distribution of the sum of the squares of […]
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2020-09-24 07:04:45
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http://www.math.princeton.edu/news
|
# News
May 16, 2013
April 3, 2013
##### The 73rd William Lowell Putnam Competition
The Princeton team earned an honorable mention in this year's William Lowell Putnam Mathematical Competition. Congratulations to Bowei Liu who had one of the top 25 individual scores and received an award of \$250. Congratulations also to Arku Adhikari, Wesley Cao, Alan Chang, David Corwin, Kubrat Danailov, Evgeni Dimitrov, Jay Hashop, Bumsoo Kim, Ante Qu, Zev Rosengarten, Alexander Smith, Matthew Superdock, Juanhe Tan, and Allen Yang, who all earned an honorable mention for their individual scores.
March 25, 2013
##### Prof. Bhargava Brings Magic (and Math) to the Classroom
Manjul Bhargava, Princeton's Brandon Fradd, Class of 1983, Professor of Mathematics, has been featured for his Freshman Seminar, "The Mathematics of Magic Tricks and Games." For the full article, click here
March 21, 2013
##### Aaron Pixton Appointed to a Five-year Clay Research Fellowship
Congratulations to Aaron Pixton, a current 4th-year graduate student, who has been appointed to a five-year Clay Research Fellowship beginning September 1, 2013.
Aaron Pixton will receive his Ph.D. in 2013 from Princeton University under the supervision of Rahul Pandharipande. His research is in enumerative algebraic geometry. The topics he has worked on recently include the tautological ring of the moduli space of curves, moduli spaces of sheaves on 3-folds, and Gromov-Witten theory.
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2013-05-18 06:50:26
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http://aliceinfo.cern.ch/ArtSubmission/node/2784
|
# Figure 7
Mass dependence of the mean transverse momentum of identified particles measured by ALICE in minimum bias pp collisions at $\sqrt{s}=7$ TeV and 0-20$\%$ p-Pb collisions. Statistical uncertainties are represented as bars, boxes indicate total systematic uncertainties.
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2018-03-17 20:41:28
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http://www.racedepartment.com/threads/cannonball-run-usa-johnstone-county.25327/
|
• "Bite off more than you can chew and then chew like hell." - Peter Brock
# Released Cannonball Run USA - Johnstone County
Discussion in 'Bob's Track Builder Projects' started by Johannes Rojola, Mar 24, 2011.
1. ### Johannes Rojola
Messages:
166
Ratings:
+6
This is open road race within a spirit of Cannonball races, as made famous in such movies like Gumball Rally and Cannonball 1 & 2. From vast desert highways to twisty mountain roads, from gravel roads to dry river beds. Enjoy landscapes of mountains, lakes, valleys, deserts, farmlands and sympathic country town Johnstone.
Code:
CANNONBALL RUN USA - JOHNSTONE COUNTY v1.1
"I bet I can beat your Ferrari with my Morris Mini"
INSTALLATION:
Track folder goes to: Gamedata/Locations
'JohnstoneCounty.hat' goes to: UserData/LOG/HAT
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
-Race laps: 2
-Flag rules: NONE
-Start type: STANDING
-Timing works properly only in RACE-mode, so do not try to attemp doing "official" time run in any other mode.
-IMPORTANT: Official timer starts at the exact moment when first car enters race session!
Think this as a official start signal, and it is up to you how fast you get your clothes on and leave the motel room :D
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
CHANGELOG:
v1.1:
-Some of the bridges had major collision bugs, fixed.
-Graphics made more prettier and consistent in higher graphics modes.
-New skies, thanks KittX!
-Starting grid expanded to hold 50 cars
-Most of the buildings now collide
-Small tweaks here and there
MAIN FEATURES:
1. 140km's of open roads (110km of tarmac, 18km gravel, 12km special)
2. 150 meters of road elevation change.
3. Possibility to choose different routes.
4. Hillclimb-style point to point race with working timing.
5. Some what realistic atmosphere. This is not arcade track.
6. Speed limit: 55 (not that it cannot be broken... ;)
7. WARNING: Some cars might get stuck in sand or fields. Be warned! Mostly those Europeans with their "fancy"
sports cars... ;)
8. Remember to enable damage, because it is fun to crash here.
KNOWN ISSUES:
1. NO AI drivers! I wasn't able to implement any sort of working computer
opponents for this track. It might be possible, maybe in later versions.
So this track is purely for online-fun or just driving ALONE. (CURRENTLY AIW IS BEING WORKED ON)
2. Performance. This track is HUGE and fairly heavy graphics wise, should work with normal gaming PC's.
3. Flickering problem. Due the limits of rFactor graphics engine, there is severe flickering with shadows.
So almost all shadows are disabled. Don't worry, nothings wrong with your computer.
4. Graphic bugs in mirrors.
5. Yellow flag and 'wrong way' indicators flashing. But if you followed rule settings instructions, it doesn't matter.
6. There are NO PITS, so fill up before you go if you want to go Sunday driving.
7. And all the rest I am not aware of.
TROUBLESHOOT:
If you are having FPS issues, you can help that little bit:
Open: "JohnstoneCounty.scn"
Change: "ClipPlanes=(0.01, 150000.0)" to "ClipPlanes=(0.01, 2000.0)"
This limits the track distance drawn by computer, but makes flickering worse.
(online compatibility may be affected)
THANKS TO:
1. Rural Australia XPack, ennisfargis (For the cows, rocks, vegetation and other stuff!)
2. Longford 67 Houses XPack, Hompe, Woochoo, Gustaf Reutersward (Those houses helped me a lot to make the town alive!)
3. Zaxxon's guardrail XPack (Great!)
4. Textures Janvier XPack, PLP (Always great help!)
5. Hompe, for Diner building
6. Pangaea, for helping to solve various challenges!
7. Piddy, for BTB (Need to say how great tool this is?)
8. Neil Faichney, for skybox arrangement!
9. KittX for new beautiful skies!
Hate mail to:
jrojola(at)gmail.com
Happy driving!
Johannes Rojola
2. ### Jonathan Johansson
Messages:
280
Ratings:
+56
Very intressting will try it out for sure
3. ### Pangaea
Messages:
246
Ratings:
+8
oh yea!! at last
will give my review once i give this great looking track a play
i doubt ill be disappointed
4. ### Johannes Rojola
Messages:
166
Ratings:
+6
Oh that is nice to hear! Will definitely help me develop stuff for 2.0 version
5. ### Kyle PuttiferBanned
Messages:
1,487
Ratings:
+93
I have an integrated chip, DX8 and med settings and the FPS is above average, compared to other tracks.
Any chance of a map? I get completely lost.
6. ### Johannes Rojola
Messages:
166
Ratings:
+6
There is map inside the track folder
Messages:
1,513
Ratings:
+256
getting a runtime error C++ when loading the circuit. it loads some 85% and then it crashes.
8. ### Johannes Rojola
Messages:
166
Ratings:
+6
From the RAR-file copy the JohnstoneCounty.hat to your UserData/LOGS/HAT
If you downloaded my earlier package without the hat file, you can dl it separately from here: http://futureman.wippiespace.com/JohnstoneCounty.hat
Messages:
1,513
Ratings:
+256
Thanks. it worked. The track loads normaly. Even fast which is a good achievement for such a huge track. well done.
10. ### Kyle PuttiferBanned
Messages:
1,487
Ratings:
+93
I prefer something like solid black lines for all the roads on a white background, as well as important places marked such as the smoking indian and the finishing area.
Messages:
166
Ratings:
+6
12. ### Kyle PuttiferBanned
Messages:
1,487
Ratings:
+93
OH I'm such a dumbass, I skipped straight to the folder with the SCN, AIW etc.
One thing though: When I load the track with this mod:
the police car runs with a messed up livery. It has no problems on other tracks. Maybe the HAT file? I wouldn't have thought so.
13. ### Kyle PuttiferBanned
Messages:
1,487
Ratings:
+93
Actually, it's fine now, I modified the VEH file for the police Subaru. It seems that both cars use files called police.dds, and they conflict. Issue fixed.
What do you mean when you say there is 12kms of "special" road?
14. ### Johannes Rojola
Messages:
166
Ratings:
+6
I never ran into such problem, but could it be possible that vehicle files conflict with track files? Since in Cannonball track there is file called police.dds If so... that is very bad programming :o
With the 12km I meant dry river bed down the valley which is drivable, together with this there is the river channel area which is also drivable. Could be called "special" because they are not roads in common sense
15. ### Kyle PuttiferBanned
Messages:
1,487
Ratings:
+93
That's exactly what I said.
16. ### Johannes Rojola
Messages:
166
Ratings:
+6
That sounds odd, its a major programming hiccup. Its not possible for every car and track modder to avoid accidental similar naming of a file...
17. ### Kyle PuttiferBanned
Messages:
1,487
Ratings:
+93
I just modified the VEH file for the car and the name of the car dds file and they don't conflict any more. Guess I should be more careful when adding an extension to a mod.
Messages:
166
Ratings:
+6
19. ### R Soul
Messages:
957
Ratings:
+190
I'm having a similar problem to Eriwn. Yesterday I was getting the runtime error but today loading just stops. I downloaded the version that already has the HAT file, which I extracted to userdata\log\hat\. My .plr file has 'always rebuild hat' set to 0.
I can open the track in scene viewer and 3dsimed, so I suppose the objects are fine.
20. ### Johannes Rojola
Messages:
166
Ratings:
+6
Did you download the new 1.1 version? It also comes with hat. If it doesn't work, then I really can't say what it could be... I also have hat rebuild set to 0.
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2017-06-28 19:29:34
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https://socratic.org/questions/how-do-you-multiply-728-00-6-2-and-round-to-the-nearest-cent
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#### Explanation:
6.2% in decimal form is 0.062, and that's what you use to multiply by 728. After you multiply, you get 45.136. To round to the nearest cent is to round to the hundredths place (The 2nd digit after the decimal - x.xx). If you round 45.136 to the hundredths place, you get x.14 because .136 goes up to .14 because the digit 6 is larger than 5, letting the answer be 45.136 --> 45.14
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2020-04-08 18:04:17
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https://tex.stackexchange.com/questions/250228/numbering-chemical-intermediate-compounds-with-chemstyle
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# Numbering chemical intermediate compounds with chemstyle
I´m currently writing my bachelor thesis in chemistry and chose to number my compounds with chemstyle. And except of one little issue that works just fine: Is it possible to number intermediate compounds with roman numerals beginning with "I", "II"....? Thank you in advance.
\documentclass[a4paper,12pt,bibliography=totocnumbered]{scrartcl}
\usepackage[runs=2]{auto-pst-pdf}
\usepackage{bpchem}
\usepackage[tracking=bpchem]{chemstyle}
\begin{document}
\CNlabelnoref{cmpd:1}
\CNlabelnoref{cmpd:2}
\CNlabelnoref{cmpd:3}
\CNlabelnoref{cmpd:4}
\CNlabelsubnoref{cmpd:4}{one}
\CNlabelsubnoref{cmpd:4}{two}
\begin{scheme}[H]
\schemeref[TMP1]{cmpd:1}
\schemeref[TMP2]{cmpd:2}
\schemeref[TMP3]{cmpd:3}
\schemeref[TMP4]{cmpd:4}
\schemerefsub[TMP5]{cmpd:4}{one}
\schemerefsub[TMP6]{cmpd:4}{two}
\includegraphics{test-tmp.eps}
\end{scheme}
\end{document}
• Welcome to TeX.SX! It is easier to help you if you add a minimal working example that takes the form \documentclass{...}\usepackage{....}\begin{document}...\end{document}. If possible, it should compile and have the minimum amount of code needed to illustrate your problem. This makes it much easier for people to help you - and much more likely that they will! – Andrew Jun 14 '15 at 16:41
• numbering works fine but I´d like to have an alternate numbering with roman numerals. the major compounds should be numbered from 1, 2, 3,... but a few exceptions should be numbered from I, II, III, IV... I hope you get what I´m up to – Christian Jun 14 '15 at 17:14
• It's actually not chemstyle but (in your case) bpchem that does the numbering and defines the corresponding commands – clemens Jun 14 '15 at 18:15
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2019-06-18 23:56:39
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https://www.projecteuclid.org/euclid.jglta/1437656918
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## Journal of Generalized Lie Theory and Applications
### Infinitesimal Deformations of the Model ${\mathbb{Z}_3}$-Filiform Lie Algebra
Rosa María Navarro
#### Abstract
In this work, it is considered that the vector space is composed by the infinitesimal deformations of the model ${\mathbb{Z}_3}$-filiform Lie algebra ${L^{n,m,p}}$. By using these deformations, all the ${\mathbb{Z}_3}$-filiform Lie algebras can be obtained, hence the importance of these deformations. The results obtained in this work, together with those obtained by Khakimdjanov and Navarro ( J. Geom. Phys. 2011 and 2012), lead to compute the total dimension of the mentioned space of deformations.
#### Article information
Source
J. Gen. Lie Theory Appl., Volume 7 (2013), 11 pages.
Dates
First available in Project Euclid: 23 July 2015
https://projecteuclid.org/euclid.jglta/1437656918
Digital Object Identifier
doi:10.4303/jglta/235699
Mathematical Reviews number (MathSciNet)
MR3038861
Zentralblatt MATH identifier
1317.17018
#### Citation
Navarro, Rosa María. Infinitesimal Deformations of the Model ${\mathbb{Z}_3}$-Filiform Lie Algebra. J. Gen. Lie Theory Appl. 7 (2013), 11 pages. doi:10.4303/jglta/235699. https://projecteuclid.org/euclid.jglta/1437656918
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2019-10-15 08:06:37
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http://clay6.com/qa/39032/find-the-slope-of-the-tangent-to-the-curve-x-2-y-2-4x-2at-the-point-3-1-
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Browse Questions
Find the slope of the tangent to the curve $x^2+y^2=4x-2$at the point (3,1)
Diffrentiating the eqn w.r.t x we have 2x + 2y dy\dx = 4 2y dy\dx = 4 - 2x dy\dx = (4 - 2x)\2y (4 - 2×3)\2×1 (4 - 6)\2 -2\2 -1 ans.
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2017-03-25 13:39:02
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http://research.microsoft.com/apps/pubs/default.aspx?id=193029
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The geometry of differential privacy: the sparse and approximate cases
In this work, we study trade-offs between accuracy and privacy in the context of linear queries over histograms. This is a rich class of queries that includes contingency tables and range queries, and has been a focus of a long line of work. For a set of $d$ linear queries over a database $x \in \R^N$, we seek to find the differentially private mechanism that has the minimum mean squared error. For pure differential privacy, an $O(\log^2 d)$ approximation to the optimal mechanism is known. Our first contribution is to give an $O(\log^2 d)$ approximation guarantee for the case of $(\eps,\delta)$-differential privacy. Our mechanism is simple, efficient and adds correlated Gaussian noise to the answers. We prove its approximation guarantee relative to the hereditary discrepancy lower bound of Muthukrishnan and Nikolov, using tools from convex geometry.
We next consider this question in the case when the number of queries exceeds the number of individuals in the database, i.e. when $d > n \triangleq \|x\|_1$. It is known that better mechanisms exist in this setting. Our second main contribution is to give an $(\eps,\delta)$-differentially private mechanism which is optimal up to a $\polylog(d,N)$ factor for any given query set $A$ and any given upper bound $n$ on $\|x\|_1$. This approximation is achieved by coupling the Gaussian noise addition approach with a linear regression step. We give an analogous result for the $\eps$-differential privacy setting. We also improve on the mean squared error upper bound for answering counting queries on a database of size $n$ by Blum, Ligett, and Roth, and match the lower bound implied by the work of Dinur and Nissim up to logarithmic factors.
The connection between hereditary discrepancy and the privacy mechanism enables us to derive the first polylogarithmic approximation to the hereditary discrepancy of a matrix $A$.
In Proceedings of ACM Symposium on Theory of Computing
## Details
Type Inproceedings URL http://arxiv.org/abs/1212.0297
> Publications > The geometry of differential privacy: the sparse and approximate cases
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2014-04-17 07:19:35
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https://gerardnico.com/data_storage/mbr
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# Data Storage - The Master Boot Record (MBR)
A master boot record (MBR), or partition sector, is the 512-byte boot sector that is the first sector (LBA/absolute sector 0) of a partitioned data storage device such as a hard disk.
The master boot record (MBR) is the first sector of a hard disk which is always located at sector 1 of cylinder 0, head 0
The MBR is the most important data structure on the disk and is created when the disk is partitioned.
The MBR contains a small amount of executable code called:
At the end of the MBR is a 2-byte structure called a signature word or end of sector marker, which is always set to 0x55AA. A signature word also marks the end of an extended boot record (EBR) and the boot sector.
The use of basic or dynamic disk does not affect where the MBR is located on disk and only minor differences exist between the two for how the partition table is configured.
There is no MBR on a floppy disk. The first sector on a floppy disk is the boot sector. Although every hard disk contains an MBR, the master boot code is used only if the disk contains the active, primary partition.
## 3 - Master Boot Code
The master boot code performs the following activities:
• Scans the partition table for the active partition.
• Finds the starting sector of the active partition.
• Loads a copy of the boot sector from the active partition into memory.
• Transfers control to the executable code in the boot sector.
If the master boot code cannot complete these functions, the system displays one of the following error messages:
Invalid partition table.
Missing operating system.
## 4 - Disk signature
The disk signature, a unique number at offset 0x01B8, identifies the disk to the operating system.
Windows 2000 uses the disk signature as an index to store and retrieve information about the disk in the registry subkey HKEY_LOCAL_MACHINE\SYSTEM\MountedDevices.
## 5 - Partition Table
The partition table, a 64-byte data structure used to identify the type and location of partitions on a hard disk, conforms to a standard layout independent of the operating system. Each partition table entry is 16 bytes long, with a maximum of four entries. Each entry starts at a predetermined offset from the beginning of the sector, as follows:
• Partition 10x01BE(446)
• Partition 20x01CE(462)
• Partition 30x01DE(478)
• Partition 40x01EE(494)
In Windows 2000, Only basic disk makes use of the partition table . Dynamic disk uses the Disk Management database located at the end of the disk for disk configuration information. The partition table is not updated when volumes are deleted or extended after the dynamic disk upgrade, or when new dynamic volumes are created.
### 5.1 - The Starting and Ending chs
The Starting and Ending chs (Cylinder, Head and Sector) fields are additional elements of the partition table. These fields are essential for starting the computer. The master boot code uses these fields to find and load the boot sector of the active partition. The Starting CHS fields for non-active partitions point to the boot sectors of the remaining primary partitions and the EBR of the first logical drive in the extended partition.
Knowing the starting sector of an extended partition is very important for low-level disk troubleshooting. If your disk fails, you need to work with the partition starting point (among other factors) to retrieve stored data.
The Ending Cylinder field in the partition table is 10 bits long, which limits the number of cylinders that can be described in the partition table to a range of 0–1,023. The Starting Head and Ending Head fields are each one byte long, which limits the field range to 0–255. The Starting Sector and Ending Sector fields are each six bits long, which limits the range of these fields to 0–63. However, the enumeration of sectors starts at 1 (not 0, as for other fields), so the maximum number of sectors per track is 63.
Maximum disk capacity
Because all hard disks are low-level formatted with a standard 512-byte sector, the maximum disk capacity described by the partition table is calculated as follows:
Maximum capacity = sector size x cylinders (10 bits) x heads (8 bits) x sectors per track (6 bits)
Using the maximum possible values yields:
512 x 1024 x 256 x 63 (or 512 x 2^24) = 8,455,716,864 bytes or 7.8 GB
The calculation results in a maximum capacity of slightly less than 8 gigabytes (GB). Before logical block addressing (LBA) were introduced, the active, primary partition could not exceed 7.8 GB, regardless of the file system used.
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2018-12-16 03:47:25
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http://physics.stackexchange.com/questions/3119/reaching-speed-of-light/3121
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# Reaching speed of light [duplicate]
Possible Duplicate:
Rotate a long bar in space and reach c
Sorry this is very naive, but it's bugging me. If you had a straight solid stick attached on one end and rotating around that attachment at a certain rpm, there would be a length at which the end of the stick would theoretically reach, with that rpm, the speed of light. Well, doesn't seem possible - what specifically would be the limitations that would prevent the end of the stick to reach the speed of light? What would happen?
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simply as a practical matter, it's doubtful you could find a material strong enough to withstand the tension to supply the necessary centripetal force. – JustJeff Jan 17 '11 at 0:53
Voted to close as duplicate: the question has the same answer. – Sklivvz Jan 17 '11 at 1:06
I agree, it's a duplicate; closed. – Noldorin Jan 17 '11 at 1:16
Maybe we should all just admit once and for all that relativity applies to everything in the universe except really long sticks ;-) – Greg P Jan 17 '11 at 16:30
## marked as duplicate by Colin K, Sklivvz♦, NoldorinJan 17 '11 at 1:16
In order for the bit of matter of mass $m$ at the very end of the stick to continue moving in a circular path of radius $R$ at a speed approaching the speed of light, it would need to be pulled toward the center with a force whose magnitude is
$|F| = |p|\frac{|V|}{R} = \frac{1}{\sqrt{1-(v/c)^2}}\frac{mv^2}{R}$
(the centripetal force you learn about in introductory physics). That force becomes infinitely large as the speed v approaches the speed of light, very rapidly, and eventually exceeds the strength of any interatomic or intermolecular forces that might be trying to hold the object together.
-
obviously something's going to stop you from getting to c, but what about getting to interesting fractions of c, say, just enough for relativistic effects to become noticable? – JustJeff Jan 17 '11 at 1:10
Something else to consider is that if you begin with a rigid rod rotating about an axis penetrating its center of mass at a constant angular velocity, and then gradually extend the length of the rod by adding mass to the end(s) of the rod, by doing so you are increasing the rod's moment of inertia. The expression for rotational kinetic energy is 1/2*I*omega^2, where I is the moment of inertia of the rod, given a specific choice of axis. For one of the ends to reach the speed of light would require infinite mass to be added to the ends, and hence, an infinite amount of energy. This seems to be a physically unattainable situation.
-
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2014-03-08 23:25:06
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https://labs.tib.eu/arxiv/?author=T.%20Riekkinen
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• ### Processing and characterization of epitaxial GaAs radiation detectors(1503.04009)
March 13, 2015 physics.ins-det
GaAs devices have relatively high atomic numbers (Z=31, 33) and thus extend the X-ray absorption edge beyond that of Si (Z=14) devices. In this study, radiation detectors were processed on GaAs substrates with 110 $\mu\textrm{m}$ - 130 $\mu\textrm{m}$ thick epitaxial absorption volume. Thick undoped and heavily doped p$^+$ epitaxial layers were grown using a custom-made horizontal Chloride Vapor Phase Epitaxy (CVPE) reactor, the growth rate of which was about 10 $\mu\textrm{m}$/h. The GaAs p$^+$/i/n$^+$ detectors were characterized by Capacitance Voltage ($CV$), Current Voltage ($IV$), Transient Current Technique (TCT) and Deep Level Transient Spectroscopy (DLTS) measurements. The full depletion voltage ($V_{\textrm{fd}}$) of the detectors with 110 $\mu\textrm{m}$ epi-layer thickness is in the range of 8 V - 15 V and the leakage current density is about 10 nA/cm$^2$. The signal transit time determined by TCT is about 5 ns when the bias voltage is well above the value that produces the peak saturation drift velocity of electrons in GaAs at a given thickness. Numerical simulations with an appropriate defect model agree with the experimental results.
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2020-08-10 08:50:26
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https://socratic.org/questions/what-is-mu-in-physics
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# What is mu in physics?
Then teach the underlying concepts
Don't copy without citing sources
preview
?
#### Explanation
Explain in detail...
#### Explanation:
I want someone to double check my answer
27
Jan 26, 2016
Mu ($\mu$) is the co-efficent of friction and is a constant for any two materials in contact.
#### Explanation:
$\mu = \left(\text{force of limiting friction")/("normal force}\right)$
where normal force = mg (mass x gravity) on a horizontal surface.*
The value for g is variously represented as $9.8 \frac{N}{k g} , 9.81 \frac{N}{k g}$ or $10 \frac{N}{k g}$, depending on the text and curriculum.
*On an inclined plane, normal force is calculated using:
${F}_{N} = m \cdot g \cdot \cos \left(\theta\right)$, where $\theta$ is the angle of the incline.
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2018-02-22 13:03:04
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https://ctan.org/ctan-ann/id/mailman.5497.1529355903.5100.ctan-ann@ctan.org
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CTAN update: etoc
Date: June 18, 2018 10:04:54 PM CEST
Jean-François Burnol submitted an update to the etoc package. Version number: 1.08o 2018-06-15 License type: lppl1.3c Summary description: Completely customisable TOCs Announcement text:
bugfix: a typo in a macro name left it undefined at previous release. The bug showed in case of an unnumbered TOC entry whose name starts with a brace, and hyperref is present.
This package is located at http://mirror.ctan.org/macros/latex/contrib/etoc More information is at http://www.ctan.org/pkg/etoc We are supported by the TeX User Groups. Please join a users group; see http://www.tug.org/usergroups.html .
Thanks for the upload. For the CTAN Team Ina Dau
etoc – Completely customisable TOCs
The package gives the user complete control of how the entries of the table of contents should be constituted from the name, number, and page number of each sectioning unit. The layout is controlled by the definition of ‘line styles’ for each sectioning level used in the document.
The package provides its own custom line styles (which may be used as examples), and continues to support the standard formatting inherited from the document classes, but the package can also allow the user to delegate the details to packages dealing with list making environments (such as enumitem). The package’s default global style typesets tables of contents in a multi-column format, with either a standard heading, or a ruled title (optionally with a frame around the table).
The \tableofcontents command may be used arbitrarily many times in the same document, while \localtableofcontents provides a ‘local’ table of contents.
Package etoc Version 1.09d 2021-07-13 Copyright 2012–2021 Jean-François Burnol2014–2015 Christine Roemer et al. Maintainer Jean-François Burnol
more
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2021-07-30 01:36:15
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https://optuna.readthedocs.io/en/stable/reference/generated/optuna.study.create_study.html
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# optuna.study.create_study¶
optuna.study.create_study(storage: Union[str, optuna.storages._base.BaseStorage, None] = None, sampler: Optional[samplers.BaseSampler] = None, pruner: Optional[optuna.pruners._base.BasePruner] = None, study_name: Optional[str] = None, direction: str = 'minimize', load_if_exists: bool = False)optuna.study.Study[source]
Create a new Study.
Example
import optuna
def objective(trial):
x = trial.suggest_uniform("x", 0, 10)
return x ** 2
study = optuna.create_study()
study.optimize(objective, n_trials=3)
Parameters
• storage
Database URL. If this argument is set to None, in-memory storage is used, and the Study will not be persistent.
Note
When a database URL is passed, Optuna internally uses SQLAlchemy to handle the database. Please refer to SQLAlchemy’s document for further details. If you want to specify non-default options to SQLAlchemy Engine, you can instantiate RDBStorage with your desired options and pass it to the storage argument instead of a URL.
• sampler – A sampler object that implements background algorithm for value suggestion. If None is specified, TPESampler is used as the default. See also samplers.
• pruner – A pruner object that decides early stopping of unpromising trials. If None is specified, MedianPruner is used as the default. See also pruners.
• study_name – Study’s name. If this argument is set to None, a unique name is generated automatically.
• direction – Direction of optimization. Set minimize for minimization and maximize for maximization.
• load_if_exists – Flag to control the behavior to handle a conflict of study names. In the case where a study named study_name already exists in the storage, a DuplicatedStudyError is raised if load_if_exists is set to False. Otherwise, the creation of the study is skipped, and the existing one is returned.
Returns
A Study object.
See also
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2020-09-20 12:12:02
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https://plainmath.net/precalculus/57999-how-do-you-simplify-20-cos-frac-pi-plus-sin-frac-pi-div-15-cos-frac-11-plus-sin
|
Clare Baldwin
2022-01-31
How do you simplify $20\left(\mathrm{cos}\left(\frac{7\pi }{6}\right)+i\mathrm{sin}\left(\frac{7\pi }{6}\right)\right)÷15\left(\mathrm{cos}\left(\frac{11\pi }{3}\right)+i\mathrm{sin}\left(\frac{11\pi }{3}\right)\right)$ and express the result in rectangular form?
dodato0n
Expert
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2023-01-31 02:46:10
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http://www.oalib.com/relative/284471
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Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
Title Keywords Abstract Author All
Search Results: 1 - 10 of 100 matches for " "
Page 1 /100 Display every page 5 10 20 Item
Physics , 2008, DOI: 10.1103/PhysRevD.78.024043 Abstract: We consider the basic physical properties of matter forming a thin accretion disc in the static and spherically symmetric space-time metric of the vacuum $f(R)$ modified gravity models. The Lagrangian of the generalized gravity theory is also obtained in a parametric form, and the conditions of the viability of the model are discussed. The exact Schwarzschild type solution of the gravitational field equations in the $f(R)$ gravity contains a linearly increasing term, as well as a logarithmic correction, as compared to the standard Schwarzschild solution of general relativity, and it depends on four arbitrary integration constants. The energy flux and the emission spectrum from the accretion disk around the $f(R)$ gravity black holes are obtained, and they are compared to the general relativistic case. Particular signatures can appear in the electromagnetic spectrum, thus leading to the possibility of directly testing modified gravity models by using astrophysical observations of the emission spectra from accretion disks.
Physics , 2007, DOI: 10.1088/1475-7516/2008/03/024 Abstract: We generalize the virial theorem in f(R) modified gravity using the collisionless Boltzmann equation. We find supplementary geometric terms in the modified Einstein equation providing an effective contribution to the gravitational energy. The total virial mass is proportional to the effective mass associated with the new geometrical term, which may account for the well-known virial theorem mass discrepancy in clusters of galaxies. The model predicts that the geometric mass and its effects extend beyond the virial radius of the clusters. We also consider the behavior of the galaxy cluster velocity dispersion in f(R) models. Thus, the f(R) virial theorem can be an efficient tool in observationally testing the viability of this class of generalized gravity models.
Physics , 2012, DOI: 10.1140/epjp/i2013-13123-0 Abstract: This paper is devoted to study the energy conditions in F(R,T) gravity for FRW universe with perfect fluid, where $R$ is the Ricci scalar and $T$ is the torsion scalar. We construct the general energy conditions in this theory and reduce them in F(R) as well as F(T) theory of gravity. Further, we assume some viable models and investigate bounds on their constant parameters to satisfy the energy condition inequalities. We also plot some of the cases using present-day values of the cosmological parameters. It is interesting to mention here that the model F(R,T)=\mu R+ \nu T satisfies the energy conditions for different ranges of the parameters.
Physics , 2013, DOI: 10.1007/JHEP12(2013)079 Abstract: We discuss the validity of the energy conditions in a newly modified theory named as $f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity, where $R$ and $T$ represent the scalar curvature and trace of the energy-momentum tensor. The corresponding energy conditions are derived which appear to be more general and can reduce to the familiar forms of these conditions in general relativity, $f(R)$ and $f(R,T)$ theories. The general inequalities are presented in terms of recent values of Hubble, deceleration, jerk and snap parameters. In particular, we use two specific models recently developed in literature to study concrete application of these conditions as well as Dolgov-Kawasaki instability. Finally, we explore $f(R,T)$ gravity as a specific case to this modified theory for exponential and power law models.
José M. M. Senovilla Physics , 2013, DOI: 10.1103/PhysRevD.88.064015 Abstract: I present the junction conditions for F(R) theories of gravity and their implications: the generalized Israel conditions and equations. These junction conditions are necessary to construct global models of stars, galaxies, etc., where a vacuum region surrounds a finite body in equilibrium, as well as to describe shells of matter and braneworlds, and they are stricter than in General Relativity in both cases. For the latter case, I obtain the field equations for the energy-momentum tensor on the shell/brane, and they turn out to be, remarkably, the same as in General Relativity. An exceptional case for quadratic F(R), previously overlooked in the literature, is shown to arise allowing for a discontinuous R, and leading to an energy-momentum content on the shell with unexpected properties, such as non-vanishing components normal to the shell and a new term resembling classical dipole distributions. For the former case, they do not only require the agreement of the first and second fundamental forms on both sides of the matching hypersurface, but also that the scalar curvature R and its first derivative agree there too. I argue that, as a consequence, matched solutions in General Relativity are not solutions of F(R)-models generically. Several relevant examples are analyzed.
Physics , 2012, DOI: 10.1088/0004-637X/760/1/2 Abstract: We introduce the Minimum Entropy Method, a simple statistical technique for constraining the Milky Way gravitational potential and simultaneously testing different gravity theories directly from 6D phase-space surveys and without adopting dynamical models. We demonstrate that orbital energy distributions that are separable (i.e. independent of position) have an associated entropy that increases under wrong assumptions about the gravitational potential and/or gravity theory. Of known objects, `cold' tidal streams from low-mass progenitors follow orbital distributions that most nearly satisfy the condition of separability. Although the orbits of tidally stripped stars are perturbed by the progenitor's self-gravity, systematic variations of the energy distribution can be quantified in terms of the cross-entropy of individual tails, giving further sensitivity to theoretical biases in the host potential. The feasibility of using the Minimum Entropy Method to test a wide range of gravity theories is illustrated by evolving restricted N-body models in a Newtonian potential and examining the changes in entropy introduced by Dirac, MONDian and f(R) gravity modifications.
Physics , 2012, DOI: 10.1140/epjc/s10052-010-1419-y Abstract: In this paper on the basis of the generalized $f(R)$ gravity model with arbitrary coupling between geometry and matter, four classes of $f(R)$ gravity models with non minimal coupling between geometry and matter will be studied. By means of conditions of power law expansion and the equation of state of matter less than -1/3, the relationship among p, w and n, the conditions and the candidate for late time cosmic accelerated expansion will be discussed in the four classes of $f(R)$ gravity models with non minimal coupling. Furthermore, in order to keep considering models to be realistic ones, the Dolgov Kawasaki instability will be investigated in each of them.
Physics , 2012, DOI: 10.7566/JPSJ.82.014002 Abstract: The energy conditions are derived in the context of $f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the energy-momentum tensor, which can reduce to the well-known conditions in $f(R)$ gravity and general relativity. We present the general inequalities set by the energy conditions in terms of Hubble, deceleration, jerk and snap parameters. In this study, we concentrate on two particular models of $f(R,T)$ gravity namely, $f(R)+\lambda{T}$ and $R+2f(T)$. The exact power-law solutions are obtained for these two cases in homogeneous and isotropic $f(R,T)$ cosmology. Finally, we find certain constraints which have to be satisfied to ensure that power law solutions may be stable and match the bounds prescribed by the energy conditions.
Physics , 2011, DOI: 10.1140/epjc/s10052-012-1999-9 Abstract: In this paper, we reconstruct cosmological models in the framework of $f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the stress-energy tensor. We show that the dust fluid reproduces $\Lambda$CDM, phantom-non-phantom era and the phantom cosmology. Further, we reconstruct different cosmological models including, Chaplygin gas, scalar field with some specific forms of $f(R,T)$. Our numerical simulation for Hubble parameter shows good agreement with the BAO observational data for low redshifts $z<2$.
Physics , 2012, DOI: 10.1088/1475-7516/2013/01/003 Abstract: Scalar modifications of gravity have an impact on the growth of structure. Baryon and Cold Dark Matter (CDM) perturbations grow anomalously for scales within the Compton wavelength of the scalar field. In the late time Universe when reionisation occurs, the spectrum of the 21cm brightness temperature is thus affected. We study this effect for chameleon-f(R) models, dilatons and symmetrons. Although the f(R) models are more tightly constrained by solar system bounds, and effects on dilaton models are negligible, we find that symmetrons where the phase transition occurs before z_* ~ 12 will be detectable for a scalar field range as low as 5 kpc. For all these models, the detection prospects of modified gravity effects are higher when considering modes parallel to the line of sight where very small scales can be probed. The study of the 21 cm spectrum thus offers a complementary approach to testing modified gravity with large scale structure surveys. Short scales, which would be highly non-linear in the very late time Universe when structure forms and where modified gravity effects are screened, appear in the linear spectrum of 21 cm physics, hence deviating from General Relativity in a maximal way.
Page 1 /100 Display every page 5 10 20 Item
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2020-01-20 23:46:23
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https://par.nsf.gov/biblio/10304425-measurement-branching-fraction-c+p-decay-belle
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This content will become publicly available on October 1, 2022
Measurement of the branching fraction of $Λc+→pω$ decay at Belle
Authors:
; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; more »
Award ID(s):
Publication Date:
NSF-PAR ID:
10304425
Journal Name:
Physical Review D
Volume:
104
Issue:
7
ISSN:
2470-0010
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2022-05-20 13:40:57
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http://wikieducator.org/Wordsworth
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# Wordsworth
1. What according to Wordsworth should be the theme of Poetry? Or Write note on Wordsworth’s view on the subject matter of poetry.
Wordsworth's enormous poetic legacy rests on a large number of poems written by him. But the themes that run through Wordsworth's poetry remained consistent throughout. Even the language and imagery he used to embody those themes, remained remarkably consistent. They remained consistent to the canons Wordsworth had set out the Preface to Lyrical Ballads. In the second edition of the Lyrical Ballads (1802), he wrote Preface to defend himself form the negative reviews. Wordsworth argued that poetry should be written in the real language of common man, rather than in the lofty and elaborate dictions that were then considered "poetic."He believed that the first principle of poetry should be pleasure and so the chief duty of poetry is to provide pleasure through a rhythmic and beautiful expression of feeling. All human sympathy, he asserted, is based on a subtle pleasure principle that is "the naked and native dignity of man." Wordsworth's poetic creed initiated the Romantic era by emphasizing feeling, instinct, and pleasure above formality and mannerism. More than any poet before him, Wordsworth gave expression to inchoate human emotion.
Let us briefly review Wordsworth views on the theme and subject matter of poetry:
Object (subject matter of poetry)
The principle object, then proposed in these poems was to choose incidents and situations from common life, and to relate and describe them, throughout, as far as possible in a selection of language really used by men, and , at the same time, to throw over them a certain colouring of imagination, whereby ordinary things should be presented to the mind in an unusual aspect; and, further,, and above all, to make these situations and incidents interesting by tracing in them, truly though not ostentatiously, the primary laws of our nature: chiefly as regards the manner in which we associate ideas in a state of excitement.
Humble and rustic life (subject matter of poetry)
Humble and rustic life was generally chosen, because in that condition, the essential passions of the heart find a better soil in which they can attain their maturity, are less under restraint, and speak a plainer and more emphatic language; because in that condition of life, our elementary feelings co-exist in a state of greater simplicity, and consequently, may be more accurately contemplated, and more forcibly communicated; because the manners of rural life germinate from these elementary feelings, and, from the necessary character of rural occupations, are more easily comprehended, and are more durable; and lastly, because in that condition the passions of men are incorporated with the beautiful and permanent forms of nature.
Language (style of poetry)
The language, too, of these men has been adopted -purified indeed from what appear to be its real defects, from all lasting and rational causes of dislike and disgust- because such men communicate with the best objects from which the best part of language is originally derived; and because, from their rank in society and the sameness and narrow circle of their intercourse, being less under the influence of social variety, they convey their feelings and notions in simple and unelaborated expressions. Accordingly, such a language, arising out of the repeated experience and regular feelings is a more permanent, and a far more philosophical language, than that which is frequently substituted for it by Poets, who think that they are conferring honour upon themselves and their art, in proportion as they separate themselves from the sympathies of men, and indulge in arbitrary and capricious habits of expression, in order to furnish food for fickle appetites, of their own creation. Thus, Wordsworth’s views on poetical style are the most revolutionary of all the idea in his Preface. He discarded the gaudiness and inane phraseology of many modern writers. He insists that his poems are written in ‘selection of language of men in a state of vivid sensation’. His views of poetic diction can be summed up as: ‘there neither is nor can be any essential difference between the language of prose and metrical composition’.
Definition of poetry
For all good poetry is the spontaneous overflow of powerful feeling: and though this be true, Poems to which any value can be attached were never produced on any variety of subjects but by a man who, being possessed of more than usual organic sensibility, had also thought long and deeply.
Our continued influxes of feeling are modified and directed by our thoughts, which are indeed the representative of all our past feelings. By contemplating the relation of these general representatives to each other, we discover what is really important to men, so by the repetition and continuance of this act, our feelings will be connected with important subjects. If we be originally possessed of such sensibility, such habits of mind will be produced, that by obeying blindly and mechanically the impulses of these habits, we shall describe objects, and utter sentiments of such a nature, and in such connection with each other, that the understanding of the Reader must necessarily be in some degree enlightened, and his affections strengthened and purified.
What is a Poet? • He is a man speaking to men: a man, it is true, endowed with more lively sensibility, more enthusiasm and tenderness. • He has a greater knowledge of human nature, and a more comprehensive soul, than one supposed to be common among mankind. • He is a man pleased with his own passions and volitions, and who rejoices more than other men in the spirit of life that is in him; delighting to contemplate similar volitions and passions as manifested in the goings-on of the Universe, and habitually compelled to create them where he does not find them. • To these qualities he has added a disposition to be affected more than other men by absent things as if they were present. He has an ability of conjuring up in himself passions, which are indeed far from being those produced by real events (especially in those parts of the general sympathy which are pleasing and delightful). He can better remember the passions produced by real events which other men are accustomed to feel in themselves. • Then, from practice, he has acquired a greater readiness and power in expressing what he thinks and feels, and especially those thoughts and feelings which, by his own choice, or from the structure of his own mind, arise in him without immediate external excitement.
The function of poetry:
o ‘Poetry’, according to Wordsworth, ‘is the breath and finer spirit of all knowledge, the impassioned expression that is in the countenance of all science’. Poetry seeks to ennoble and edify. It is like morning star which throws its radiance through the gloom and darkness of life. The poet is a teacher ad through the medium of poetry he imparts moral lessons for the betterment of human life. Poetry is the instrument for the propagation of moral thoughts. Wordsworth’s poetry does not simply delight us, but it also teaches us deep moral lessons and bring home to us deep philosophical truths about life and religion. Wordsworth believes that ‘a poetry of revolt against moral ideas is a poetry of revolt against life; a poetry of indifference towards moral ideas is a poetry g indifference towards life.
2. Write note on Wordsworth’s theory of poetic diction. Introduction: Wordsworth reaction against the 18th cen poetic diction:
Highly influenced by Rousseau and French revolution, Wordsworth came forward in 1798, with a new theory of man, a new theory of nature and a new theory of poetry. In the second edition of Lyrical Ballads (1802), he elaborately explained his theory of poetic diction. As against highly sophisticated language of 18th century, he gave rustic colours to the poetic diction. Wordsworth rightly felt that for the new poetry of the new age, a language was needed, and what he earnestly felt, he expressed in the ‘Preface’ to the Lyrical Ballads. His entire effort in renovating the language of poetry was guided by the feeling that ‘all conventions of pedantry must be discarded in order t evolve the true poetic style, which should not only be simple and unaffected but should possess the power and truth of feeling. Wordsworth’s Theory of Poetic Diction:
In the “Advertisement” to the 1798 edition of Lyrical Ballads, Wordsworth and Coleridge state that the poems in the collection were intended as a deliberate experiment in style and subject matter. Wordsworth elaborated on this idea in the “Preface” to the 1800 and 1802 editions which outline his main ideas of a new theory of poetry. Wordsworth explained his poetical concept: "The majority of the following poems are to be considered as experiments. They were written chiefly with a view to ascertain how far the language of conversation in the middle and lower classes of society is adapted to the purpose of poetic pleasure." Rejecting the classical notion that poetry should be about elevated subjects and should be composed in a formal style, Wordsworth instead championed more democratic themes—the lives of ordinary men and women, farmers, paupers, and the rural poor. In the “Preface” Wordsworth also emphasizes his commitment to writing in the ordinary language of people, not a highly crafted poetical one. True to traditional ballad form, the poems depict realistic characters in realistic situations, and so contain a strong narrative element. He stated, “the principal object proposed in these poems was to choose incidents and situations from common life, and to relate to describe them throughout, as far as this was possible in a selection of language really used by men” and at the same time, “to throw over them a certain colouring of the imagination”, whereby ordinary things should be presented to the mind in an unusual aspects. The language of these men has been adopted because such men communicate with the best objects from which the best part of language is originally derived.
Three main principles of Wordsworth’s poetic diction:
As we examine Wordsworth’s statement regarding poetic diction, the following three points come to our attention: (i) the language of poetry should be the language ‘really used by men’, but it should be a ‘selection’ of such language. All the words used by the people cannot be employed in poetry. It should be filtered and refined. The refined vernacular words should be the diction of poetry. (ii) It should be the language of men in a state of vivid sensation. It should have a certain colouring of imagination. The poet should give the colour of his imagination to the language employed by him in poetic composition. (iii) There is no ‘essential’ difference between the words used in prose and in metrical composition. Words of prose and poetry are not clearly discriminated, so that words which be used in prose can find place in poetry and vice-versa. What Wordsworth means is that the words in conversation, if they are properly selected, would provide the rough framework of the language of poetry. The language of poetry is heightened by feeling and emotion. Through the power of imagination the poet can select words fit for poetic composition. When the poet is truly inspired, his imagination will enable him to select from the language really used by men.
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2016-08-24 16:11:15
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https://cs.stackexchange.com/questions/68156/reduction-from-vertex-cover
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Reduction from Vertex Cover
The city council would like to place trash bins around the city and has a list of suitable spots (street crossroads, supermarkets etc.) but the number of these spots is greater than the number of available bins. The council's goal is to place the limited number of bins so that the distance from each house to a bin is at most 100 m. Prove that the problem k containers are sufficient is NP-hard using reduction from a known NP-hard problem.
I came up with vertex cover (could we alternatively use dominating set?). We then need to transform an instance of vertex cover $(V, E, k)$ to an instance of this problem. It's not clear to me how. This is how I thought I would represent the k containers are sufficient problem by a graph:
My first step would be to create a graph where the set of nodes contains both the spots for containers and all the houses. I would then connect each spot node with all the house nodes that it is close enough to (100 m). This seems quite close to vertex cover except that there are 2 types of nodes:
• The house nodes cannot be used as a part of the vertex cover set since bins cannot be placed in front of houses but they need to be covered by the vertex cover set.
• The bin spots can be used as a part of the vertex cover set but do need to be covered at all by it.
How could I deal with this in the reduction? I thought about using some vertex to edge transformation to represent these relationships but came up short.
• You should use dominating set in this case – Gilad Jan 2 '17 at 2:45
• If you can't seem to work out a reduction from one problem, try another. There aren't nice and direct reductions between every pair of NP-complete problems. – Yuval Filmus Jan 2 '17 at 5:26
• @YuvalFilmus Three problems are suggsted for the reduction in the assignment: vertex cover, Hamiltonian path problem and the clique problem. Vertex cover seems to me to match this problem best judging also by the answer to this question. I am stuck on the details however. – pseudomarvin Jan 2 '17 at 13:36
• Are the spots and houses constrained to be points in the plane, with the distance measure the Euclidean distance? If so, this makes encoding another NP-hard problem more difficult. If not (so that, e.g., it's possible for two houses to be 50m from each other, with one of them being 10m from a bin while the other is 70m from the same bin) this problem is basically another NP-hard problem in disguise... Hint: Think of a spot as the set of houses that it's in range of... – j_random_hacker Jan 3 '17 at 17:16
• It is the second case. – pseudomarvin Jan 4 '17 at 19:56
This is known as the $k$-center problem or the planar minmax Euclidean facility location problem. It is a special case of the minimax facility location problem for the Euclidean metric.
The $k$-center problem is known to be NP-hard. Wikipedia has references to the literature, where you should be able to find an explicit reduction.
The special case where $k=1$ can be solved in polynomial time.
I have come up with the following solution:
We transform an instance of vertex cover $(V, E,k)$ to a specific instance of k containers are sufficient in which the the available spots for containers are equivalent to the set $V$ and in which the set of houses that have to be near containers is exactly equal to $V$ (the available spots for containers are exactly where the houses are located). The edges connect houses-spots which are within 100 ms of each other.
It shouldn't then be difficult to see (since the k containers are sufficient problem formulated in this way is basically identical to vertex cover) that the graph $G=(V,E)$ has a vertex cover of size $k$ exactly when $k$ containers suffices to serve all the houses.
• You still have to spell out the reduction. An idea is not enough. – Yuval Filmus Jan 3 '17 at 4:53
• I don't think this works. If your original vertex cover instance is for a non-planar graph, how do you plan to create an instance of k containers are sufficient that corresponds to it? Where are you going to place the houses so that the distances between them are consistent with $E$? I'm not sure that's doable in all cases. – D.W. Feb 1 '17 at 23:09
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2019-11-15 11:11:42
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http://math.stackexchange.com/questions/300164/how-can-i-convert-the-matrix-1-to-matrix-2
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# How can i convert the matrix 1 to matrix 2?
A= 0 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0
1 0 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0
1 1 0 1 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 0
1 1 1 0 1 1 1 0 0 0 -------> 0 0 0 0 1 0 0 0 0 0
1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0
1 1 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0
1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0
1 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 0
1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1
1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0
Here a Boolean matrix let call it A matrix and A(NxN) and here N=10. 0 mean there is no connection between nodes(as it can be seen there is no self connection also) and 1 means there is connection between nodes.
Here the first local node is node1 and when a node is local then must be blocked so can not be selected (connected) again
Node 1 connect node 2, then node 2 is local (must be blocked)
2 connect 3, now 3 is local (must be blocked)
3 connect 4, 4 is local (must be blocked)
4 connect 5, 5 is local (must be blocked)
5 connect 6 ,6 is local (must be blocked)
6 connect 8, 8 is local (must be blocked)
8 connect 7, 7 is local (must be blocked)
7 connect 9, 9 is local (must be blocked)
9 connect 10,10 is local and stop here as all number are local.here 10 can not connect 1 about one was local
After connections matrix should be as the matrix at the end of figure. How can i write it in matlab
-
Your algorithm description is not really clear to me yet. For instance, why don't you start with 1 connect 3 (or 4...) instead of 1 connect 2. Does the matrix you are looking for have a name, and perhaps an article describing it? EDIT: Do you think an algorithm for the longest path could give you the desired outcome? – Dennis Jaheruddin Feb 11 '13 at 13:01
it is also possible 1 connect to 3(or 4) instead 2. it can select anyone randomly but the node connected can not be connect again.i mean when a node it has been local then must be blocked. the first matrix(matrix one) is a undirected matrix and i want convert it to direct matrix but it must follow the rules i described – doci Feb 11 '13 at 13:06
"EDIT: Do you think an algorithm for the longest path could give you the desired outcome?" yes it does.many thanks again Dennis – doci Feb 11 '13 at 23:40
Here is one way to make the matrix you describe, without further details i cannot see if it really matches what you are looking for but you should be able to edit it yourself.
%Set things for the start:
i = 1;
blocked = 1;
used = [];
B = zeros(size(A));
for i = 1:size(A,1)
for j = 1:size(A,2)
% Test if node i can be connected with node j
if A(i,j) == 1 && ~any(blocked == j)&& ~any(used == i)
% Connect node i with node j
B(i,j) = 1;
% Block node j
blocked = [blocked j];
% To make sure we only have one entry per row
used = [used i];
end
end
end
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Note that i did not go for the most compact code but for what I hoped to be most easy to understand. – Dennis Jaheruddin Feb 11 '13 at 16:44
It is what i wanted exactly it works great .Many thanks Dennis – doci Feb 11 '13 at 17:52
Just to prevent confusion, this is not guaranteed to give you the longest path. It will just give you a path and keeps expanding it untill it cannot go further. – Dennis Jaheruddin Feb 12 '13 at 9:21
You must apply some of the method to find the rank of the matrix either row exchange or column exchange method, after few steps your upper part will be just converging towards identity metrix of six order. Then manipulate for the last three rows.
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yes but as i mention in my program my number of nodes is N ,here is 10 only to gives a example. if you look carefully it is not something toward identity metrix! – doci Feb 11 '13 at 13:43
The original matrix has full rank and the new matrix does not, so I think you are on the wrong track here. – Dennis Jaheruddin Feb 11 '13 at 16:18
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2014-08-22 11:49:03
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https://tex.stackexchange.com/questions/193859/labeling-sides-and-angles-of-a-right-triangle-for-an-argument-of-the-pythagorean
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# Labeling sides and angles of a right triangle for an argument of the Pythagorean Theorem
I would like to draw a right triangle so that its sides are not vertical or horizontal. (No coordinate axes.) I would like the legs to be labeled a and b and the hypotenuse to be labeled c. I would like to drop a perpendicular, drawn as a dotted line segment, from the vertex across the hypotenuse to the hypotenuse. (I heard that there was a command to instruct TikZ to do this.) This creates two smaller triangles that are similar to each other. I would like the four acute angles of the two smaller triangles to be marked with arcs; one pair of equal angles marked with "|" through it, and the other pair of equal angles marked with "||" through it.
The only code that I could offer is the code for labeling the vertices of the triangle and drawing the line segments between them. I know that there is much more to code. I reckon that it would be more convenient for anybody responding to decide on the coordinates of the vertices himself/herself.
• I searched for triangle and this was the second one tex.stackexchange.com/questions/156450/triangle-with-text then I added right angle tex.stackexchange.com/questions/175501/…. See we are not asking much, so please just provide a MWE and people would take care of the rest. – percusse Jul 30 '14 at 0:17
• So it would even be more convenient for potential helpers for them to do all of the work from scratch for you?! As percusse says, an MWE is pretty easy to find, even if you are either unable or unwilling to produce one yourself, but... !! Sorry, but this question really takes the biscuit! (Apologies to those speakers of other languages or dialects to whom this means nothing but I can't think of an equivalent which would not be impolite right now.) – cfr Jul 30 '14 at 0:58
• What is "MWE"? I have just started using TikZ. The syntax for coding in TikZ is much different than that of LaTeX, and the manuals for TikZ are not of much help. (Don't be cynical, @cfr.) – user143462 Jul 30 '14 at 17:38
• Information about producing a minimal working example (MWE) that illustrates your problem. Given your comments on Gonzalo Medina's answer, I think cynicism wholly justified. The manual for TikZ is exemplary. If nothing else, you can cut and paste some code as a starting point. There are numerous examples on this site and percusse even did your searching for you. There are further examples in the online gallery and so on and so forth. – cfr Jul 30 '14 at 20:59
• It would be nice if your comments were of any help. You say that the manual for TikZ " is exemplary." Show me in the manual where it gives the code for marking angles with "|" or with "||". – user143462 Jul 30 '14 at 22:55
Here's one possibility using "pure" TikZ:
The image was produced using simply
\begin{tikzpicture}
\RectTri{(0,3)}{(1,0)}{6cm}
\begin{scope}[xshift=8.5cm]
\RectTri[black]{(0,0)}{(4,2)}{4cm}
\end{scope}
\end{tikzpicture}
\RectTri has three mandatory arguments; the first two are the coordinates for the vertices of one of the legs and the third one is the length of the second leg. The optional argument lets you customize the style used to draw the triangle.
The code:
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,angles,quotes,decorations.markings}
\newcommand\RectTri[4][thick,green!50!black,text=black]{%
\coordinate [label=left:$C$] (C) at #2;
\coordinate [label=below right:$B$] (B) at #3;
\coordinate (aux) at ($#2 ! 1 ! 90:#3$);
\coordinate [label=above:$A$] (A) at ($#2 !#4!(aux)$);
\coordinate (perp) at ($(A)!(C)!(B)$);
\draw[purple!70!black,thick,dashed] (C) -- (perp);
\draw[#1]
(C) --
node[auto] {$b$} (A) --
node[auto] {$c$} (B) --
node[auto] {$a$}
(C)
pic ["$\alpha$",draw,cyan,thick,angle radius=1cm] {angle = C--A--B}
pic ["$\alpha$",draw,cyan,thick,angle radius=1cm] {angle = B--C--perp}
pic ["$\beta$",draw,orange!70!black,thick,angle radius=1cm] {angle = A--B--C}
pic ["$\beta$",draw,orange!70!black,thick,angle radius=1cm] {angle = perp--C--A};
}
\begin{document}
\begin{tikzpicture}
\RectTri{(0,3)}{(1,0)}{6cm}
\begin{scope}[xshift=8.5cm]
\RectTri[black]{(0,0)}{(4,2)}{4cm}
\end{scope}
\end{tikzpicture}
\end{document}
And here's an approach using tkz-euclide:
\documentclass{article}
\usepackage{tkz-euclide}
\usetkzobj{all}
\begin{document}
\begin{tikzpicture}
\tkzDefPoint(0,1){A}
\tkzDefPoint(2,4){C}
\tkzDefPointWith[orthogonal normed,K=7](C,A)
\tkzGetPoint{B}
\tkzLabelPoint[left](A){$A$}
\tkzLabelPoint[right](B){$B$}
\tkzLabelPoint[above](C){$C$}
\tkzMarkRightAngle(A,C,B)
\tkzDrawSegment[green!60!black](A,C)
\tkzDrawSegment[green!60!black](C,B)
\tkzDrawSegment[green!60!black](B,A)
\tkzLabelSegment[auto](B,A){$c$}
\tkzLabelSegment[auto,swap](B,C){$a$}
\tkzLabelSegment[auto,swap](C,A){$b$}
\tkzDrawAltitude[dashed,color=magenta](A,B)(C)
\tkzGetPoint{D}
\tkzMarkAngle[size=1cm,color=cyan,mark=|](C,B,A)
\tkzMarkAngle[size=1cm,color=cyan,mark=|](A,C,D)
\tkzMarkAngle[size=0.75cm,color=orange,mark=||](D,C,B)
\tkzMarkAngle[size=0.75cm,color=orange,mark=||](B,A,C)
\end{tikzpicture}
\end{document}
• Nice pictures! I would like to make the following edits. I would like to have the right angle mark at C, I would like to replace the "alpha" with a "|" through the arc indicating the measure of angle CAB and angle BC(perp), and I would like to replace the "beta" with a "||" through the arc indicating the measure of the angle ABC and angle AC(perp). (I would also change the radius of angles marked "|" to 0.75cm and I would keep the angles marked "||" to 1cm.) Can you provide the code for these edits? – user143462 Jul 30 '14 at 19:06
• @user143462 Sure. Please see my updated answer. – Gonzalo Medina Jul 31 '14 at 14:44
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2019-09-23 13:03:17
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https://boyslife.org/games/write-a-funny-caption/157215/write-a-funny-caption-for-this-photo-83/comment-page-12/?share=linkedin&replytocom=1583796
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What’s going on in this picture? What is that owl doing or thinking?
If you can think of a funny caption for this photo, just post it in the comment form at the bottom of this page. After we approve it, your funny caption will be on this page for everyone to read.
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#### 10 Comments on Write a Funny Caption For This Photo
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8. The pastor couldn’t understand why no one was coming to church, then he saw the new clanger someone had ordered for the bells…
9. Alright robotic comrades, it is time to conquer metropolitanville!
10. so warm
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2020-05-28 08:08:22
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http://encyclopedia.kids.net.au/page/na/National_Parks_(Tasmania,_Australia)?title=Promised_Land_Forest_Reserve
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Encyclopedia > National Parks (Tasmania, Australia)
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2016-09-29 23:59:24
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https://www.gamedev.net/forums/topic/425414-hardware-monitor-id-pins/
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# Hardware: Monitor ID pins
## Recommended Posts
Zelord 130
I'm doing a project in, where i want to interface with a standard monitor through a VGA connector and read back some information (Resolutions and stuff)from it. Not through a graphics card, but through a PIC16. Of course it is not working :/ I connect in the following way:
Port----------------ID pin
|
|
15k ohm
|
+5V
The ID pin can either be connected to GND or N/C (Not Connected) So if the pin is GND i should read a 0 on the port and a 1 if N/C. Accoding to [url]http://www.epanorama.net/documents/pc/vga_bd15.html[/url] i should get atleast a 0 on VGA-pin 11 (ID0), which would signal that it is a color monitor...(and yes it is ;) ) But i just get 1, ei. it is not connected. I then tried the following:
+5V----260 ohm-----LED-----Monitor ID
The LED should light up if i find a GND (and works when i touch the power surply). But again not working with the ID bits. It should be noted that i don't connect any other pins while trying this, but the monitor is powered on (comes up with no signal). So my question is, is the ID pins obsolete? Anyone got a better refrence than the one i found? (been going through variations of, monitor ID, monitor ID obsolete, - not working, vga specification etc) Anyone got a link for how to display an image (next step, which i haven't either been able to find information, except max voltage level for analog pins)
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Zelord 130
Solved the problem, had to connect one of the monitor pins to ground to get a common ground >_<
##### Share on other sites
Monder 993
Quote:
Anyone got a link for how to display an image
Displaying an image with a PIC won't be particularly easy, you may be able to get a 320X240 image out of it, however the entire thing will be dedicated to generating the VGA signals and not able to do any useful computation as well.
You may find these links interesting:
Pic-Tock
Pic-Pong
Which generate a video signal from a PIC, which admitidly is not a VGA signal, but at least its a video output.
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Guest Anonymous Poster
there are FPGA based chips for about $1-2 US that will allow you to directly address a buffer to map memory to monitor screen resolutions of up to 800x600@16bpp these chips can be easily connected to your PIC using a handful of pins. #### Share this post ##### Link to post ##### Share on other sites Zelord 130 Quote: Original post by Monder Quote: Anyone got a link for how to display an image Displaying an image with a PIC won't be particularly easy, you may be able to get a 320X240 image out of it, however the entire thing will be dedicated to generating the VGA signals and not able to do any useful computation as well. You may find these links interesting: Pic-Tock Pic-Pong Which generate a video signal from a PIC, which admitidly is not a VGA signal, but at least its a video output. Well i'm only gonna display an image, so i only need to refresh the screen to precent it from fading away. Guess i could run it at 30 FPS, anyone know how low it is possible to go? And will a monitor accept a slower rate, ei is it a maximum rating that a monitor states, or the only way to run? And still, if anyone has information on how the sync signals should be coordinated with colors and such, then please post =) (Above links were for a TV, with a composite signal, instead of seperate lines. Need to find the timing values, which i'm pretty sure isn't the same) Quote: Original post by Anonymous Posterthere are FPGA based chips for about$1-2 US that will allow you to directly address a buffer to map memory to monitor screen resolutions of up to 800x600@16bppthese chips can be easily connected to your PIC using a handful of pins.
No fun in that :P
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2017-08-23 06:31:13
|
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|
https://answerbun.com/emacs/why-isnt-org-agenda-skipping-done-tasks/
|
# Why isn't org-agenda skipping DONE tasks?
Emacs Asked by Aylons on November 30, 2020
I’m trying to set up a planning process for the day (like in http://newartisans.com/2007/08/using-org-mode-as-a-day-planner/), but my org-agenda keeps showing DONE tasks, even though I set org-agenda-skip-deadline-if-done and org-agenda-skip-scheduled-if-done in the init.el:
My current init.el file is available at https://github.com/aylons/emacsd/blob/master/init.el
Quoting from my comment:
Judging from the fact that the DONE face and the TODO face look the same above, I'm guessing that your org-todo-keywords are screwed up, so Org mode does not know that it is done.
So I checked your init file and I see this:
...
'(org-todo-keywords
'((sequence "TODO(t)" "DONE(d)" "WAITING(w)" "SOMEDAY(s)" "NEXT(s)")))
...
So indeed your org-todo-keywords setting is wrong: you need to move the DONE entry to the last place:
...lang-el
...
'(org-todo-keywords
'((sequence "TODO(t)" "WAITING(w)" "SOMEDAY(s)" "NEXT(s)" "DONE(d)")))
...
The doc string for org-todo-keywords (C-h v org-todo-keywords RET) says:
Each sequence starts with a symbol, either ‘sequence’ or ‘type’, indicating if the keywords should be interpreted as a sequence of action steps, or as different types of TODO items. The first keywords are states requiring action - these states will select a headline for inclusion into the global TODO list Org produces. If one of the "keywords" is the vertical bar, "|", the remaining keywords signify that no further action is necessary. If "|" is not found, the last keyword is treated as the only DONE state of the sequence.
So to be absolutely sure, it's probably best to include a "|" element in the list. That's necessary if you have more than one DONE state, but it's good practice in general:
...
'(org-todo-keywords
'((sequence "TODO(t)" "WAITING(w)" "SOMEDAY(s)" "NEXT(s)" "|" "DONE(d)")))
Correct answer by NickD on November 30, 2020
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|
2022-12-04 16:13:34
|
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|
https://aims.edu.gh/event/conference-on-mathematics-and-its-applications-2/
|
August 29, 2022
# Conference on Mathematics and its Applications
## About the Conference on Mathematics and its Applications (CMIA)
The Centre is excited to organize this conference within the framework of the German Research Chair program at AIMS Ghana, under the Alexander von Humboldt Foundation, the German Ministry of Education and Research and the DAAD Foundation. This conference will provide an avenue for researchers in the Mathematical Science field to share their work, connect and network with others in their various disciplines. The event more particularly seeks to increase the cooperation between students and researchers in Mathematics and its Applications on one hand, and various institutions and research centers on the other hand.
Objectives
The conference objectives are to:
• Provide the platform for researchers in Mathematical Sciences and its Applications to share and present their research.
• Introduce students to concepts in mathematical modeling and applications that are not normally taught in their universities/institutions.
• Introduce to young researchers the various researchable questions in Mathematics and its Applications while encouraging them to pursue research.
• Provide the platform for researchers to collaborate/ work in teams to solve developmental problems across the world.
Abstracts
## Day 1
Plenary Talk 1: Obeng Denteh “A journey through recurrent motions”
Abstract: This is an exposition on the study of dynamical systems. One of our purposes is to acquaint ourselves with the Birkhoff Recurrence Theorem and to look at Poisson stable motion, recurrent motions and almost periodic motions.
Talk 1: Matthew Olanrewaju Oluwayemi “On certain applications of beta function in geometry functions theory”
Abstract: Geometric function theory (GFT) is a classical area of complex analysis with the aim of abridging the gap between analysis and applications. In this study, the authors introduced certain classes of univalent functions associated with an Eulerian integral known as beta function and studied some of the geometric properties.
Talk 2: Hetsron Legrace Nyandjo Bamen “Mathematical assessment of the impact of imperfect vaccination and population turnover on infectious disease dynamics”
Abstract: Vaccination is essential for the management of infectious diseases, many of which continue to pose devastating public health and economic challenges across the world. However, many vaccines are imperfect having only a partial protective effect in decreasing disease transmission and/or favouring recovery of infected individuals, and possibly exhibiting trade-off between these two properties. Furthermore, population turnover, that is the rate at which individuals enter and exit the population, is another key factor determining the epidemiological dynamics. While these factors have yet been studied separately, we investigate the interplay between the efficiency and property of an imperfect vaccine and population turnover. We build a mathematical model with frequency incidence rate, a recovered compartment, and an heterogeneous host population with respect to vaccination. We first compute the basic reproduction number $\mathcal{R}_0$ and study the global stability of the equilibrium points. Using a sensitivity analysis, we then assess the most influential parameters determining the total number of infected and $\mathcal{R}_0$ over time. We derive analytically and numerically conditions for the vaccination coverage and efficiency to achieve disease eradication ($\mathcal{R}_0 < 1$) assuming different intensity of the population turnover (weak and strong), vaccine properties (transmission and/or recovery) and trade-off between the latter. We show that the minimum vaccination coverage increases with lower population turnover, decreases with higher vaccine efficiency (transmission or recovery), and is increased/decreased by up to 15\% depending on the trade-off between the vaccine properties. We conclude that the coverage target for vaccination campaigns should be evaluated based on the interplay between these factors.
Talk 3: Olayemi Olanegan “Numerical algorithm as Multi-Order Differential Equations Solver”
Abstract: This research focuses on an innovative procedure of integrating higher-order systems of equations with initial values. The proposed 4-point implicit method constructed by collocation and interpolation procedures was used to integrate system of third, and fourth-order differential equations directly with a substantial enhancement in efficiency. The method derived in a block mode to simultaneously estimate three kind of differential equations has improved accuracy and rate of convergence in problem evaluations. The analysis of the scheme was established via order, consistency, zero-stability, and convergence. The proposed method was used to solve some systems of equations for third and fourth-order as numerical experiments to check the favorable performance of the method in comparison to some existing methods.
Talk 4: Audace Dossou-Olory “Connected induced subgraphs in n-vertex graphs and unicyclic graphs”
Abstract: Counting and understanding graph structures with particular properties has many applications, especially to network theory, computer science, biology and chemistry. For instance, graphs can represent biological networks at the molecular or species level (protein interactions, gene regulation, etc). The topological structure of an interconnection network is a connected graph where, for example, vertices are processors and edges represent links between them. In chemical networks, vertices are atoms and edges represent their bonds. An important question is to find all matches of a specific motif within a larger network (the subgraph isomorphism problem, or the induced subgraph isomorphism problem). Both cases are known to be in general NP-complete, although in some instances (such as planar graphs), efficient algorithms are available. A step to these problems usually consists of enumerating all possible subgraphs or induced subgraphs of the network. This talk discusses the number of connected induced subgraphs of a simple graph or unicyclic graph on n vertices with a particular emphasis on connected induced subgraphs. Specifically, I will be concerned with determining their minimum and maximum numbers, and also characterising the extremal graphs. Finally, I will mention some directions for future research.
Talk 5: Djideme Franck Houenou “Deformation retract of the group of symplectomorphisms”
Abstract: In this work, we study the dynamics of the group of symplectomorphisms under the mean curvature flow. We proved that the group retracts to the set of biholomorphic maps.
Plenary Talk 2: Bernt Oksendal “Stochastic Fokker-Planck PIDE for conditional McKean-Vlasov jump diffusions and applications to optimal control”
Abstract: The purpose of this paper is to study optimal control of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker-Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate an Hamilton-Jacobi-Bellman (HJB) equation for the optimal control of conditional McKean-Vlasov jump diffusions.
Finally we apply these results to solve explicitly the following problems:
– Linear-quadratic optimal control of conditional stochastic McKean-Vlasov jump diffusions.
– Optimal consumption from a cash flow modelled as a conditional stochastic McKean-Vlasov differential equation with jumps. The talk is based on joint work with Nacira Agram, KTH, Stockholm, Sweden.
## Day 2
Plenary Talk 3: Benoit Sehba “From little BMO to product BMO through multiplication operator.”
Abstract: We characterize the multipliers from the little BMO of Cotlar-Sadosky to the product BMO of Chang-Fefferman on the polydisk.
Plenary Talk 4: Youssef Ouknine “Reflected BSDEs associated to jump Markov processes and application to PDEs”
Abstract: In this paper, we study a class of reflected backward stochastic differential equations (RBSDE) driven by the compensated random measure associated to a given pure jump Markov process X on a general state space U. The “reflection” keeps the solution above a given càdlàg process. We prove the uniqueness and existence both by a combination of the Snell envelope theory and fixed-point argument. We apply these results to represent probabilistically the value function of some quasi-variational inequalities associated to the Markov process X.
Plenary Talk 5: Zbigniew Palmowski “Double continuation regions for American options”
Abstract: We consider the Lévy model of the perpetual American call and put options with a negative discount rate. We will consider the continuous observation case as in De Donno et al. (2020) and the Poisson observation case as in Palmowski et al. (2021). In both cases the stopping region that characterizes the optimal stopping time is either a half-line or an interval. The objective of this talk is to obtain explicit expressions of the stopping and continuation regions and the value function, focusing on spectrally positive and negative cases. To this end, we compute the identities related to the first Poisson) arrival time to an interval via the scale function and then apply those identities to the computation of the optimal strategies. We also discuss the convergence of the optimal Poisson solutions to those in the continuous observation case as the rate of observation increases to infinity. Numerical experiments are also provided. The talk is based on joint papers with Marzia De Donno, José Luis Pérez, Joanna Tumilewicz, and Kazutoshi Yamazaki.
Talk 6: Donatien Kuissi Kamdem “Optimisation of consumption with labour income under Epstein-Zin utility”
Abstract: Classical problems in mathematical finance include portfolio optimisation problems where an investor seeks to maximise the expected utility of her instantaneous consumption and terminal wealth. In this work we extend Merton’s original problem of optimal consumption and portfolio choice in continuous time to consider an agent who receives a stochastic income and is subjected to liquidity constraint. The liquidity constraint considered is the nonnegative wealth constraint, often called the borrowing constraint, and stands to be one of the most important economic constraints on portfolio selection problems. We investigate the problem using convex duality approach with a utility of Epstein-Zin type.
Talk 7: Paul Honore Takam “Stochastic Optimal Control of thermal energy storage”
Abstract: Climate change and the turnaround in energy policy require the improvement of energy efficiency in all areas. Thermal storage facilities help to mitigate and to manage temporal fluctuations of heat supply and demand for heating and cooling systems of single buildings as well as for district heating systems. We focus on a heating system equipped with several heat-production units using also renewable energies and an underground thermal storage. The thermal energy is stored by raising the temperature of the soil inside the storage. It is charged and discharged via heat exchanger pipes filled with a moving fluid. Besides the numerous technical challenges and the computation of the spatiotemporal temperature distribution in the storage also economic issues such as the cost-optimal control and management of such systems play a central role. The latter leads to challenging mathematical optimization problems. There we incorporate uncertainties about randomly fluctuating renewable heat production, environmental conditions driving the heat demand and supply. The dynamics of controlled state process is governed a PDE, a random ODE driving by the difference between supply and demand, and the SDEs. Model reduction techniques are adopted to cope with the PDE describing the spatio-temporal temperature distribution in the geothermal storage. Finally, time-discretization leads to a Markov decision process for which we apply numerical methods to determine a cost-optimal control. This is a joint work with Ralf Wunderlich (BTU Cottbus-Senftenberg) and Olivier Menoukeu Pamen (AIMS Ghana, University of Liverpool).
Talk 8: Latevi Lawson “Quantum Levy processes on the compact quantum group SUq(2)”
Abstract: Quantum Lévy processes on a quantum group are, like classical Lévy processes with values in a Lie group, classified by their infinitesimal generators. We derive a formula for the infinitesimal generators on the quantum group SUq(2) and decompose them in terms of an infinite-dimensional irreducible representation and of characters. Thus, we obtain a quantum Lévy Khintchine formula and we derive the corresponding Hunt formula.
Talk 9: Moustapha Dieye “Strong convergence of the Euler-Maruyama approximation for SDEs with unbounded drifts”
Abstract: In this work, we prove strong convergence on small time interval of order 1=2-\epsilon for arbitrarily small \epsilon > 0 of the Euler-Maruyama Holder continuous approximation for additive Brownian motion with Holder continuous drift. The proof is based on direct estimations of functional of the Euler-Maruyama approximation. The order of convergence does not depend on the Holder index of the drift, thus generalizing the results obtained in [10] to both Linear growth and to an optimal convergence order.
Talk 10: Daudel Tchatat Ngaha “Diffraction effects on ultrasonic flows meters for gas flow at high flow rates: Modeling and Analysis”
Abstract: The improvement in the accuracy of flow rate measurements of oil and gas has been going back for decades and is useful for industrial applications. Today they are generally accepted by and widely used in industry. As the technology has matured, the accuracy and stability of the flow meters have gradually improved. Looking ahead, there is a trend in the industry of more remote and subsea operations. This means that the flow meters need to be capable of more stand-alone operations where flow calibrations of the meters will be quite costly. One effect that has been studied in order to make the ultrasonic flow meters more robust and accurate is the diffraction effect. Due to the finite size of the two ultrasonic transducers, an acoustic beam is generated when the ultrasonic signal is propagating between the transducers, and the transit times measured between the transducers are affected by this. This present work deals with the modelling of sound propagation in moving medium with applications in flow rate measurement and proposes an understanding of diffraction effects for acoustic beams propagating through a homogeneous flowing fluid to improve the accuracy of the ultrasonic flow meter. The study is oriented on mathematical modelling and numerical solution of the partial differential equations that are based on the fundamental hydrodynamic equations (Navier-Stokes Equations) and describe such a sound propagation. Results from such a study will be used as input for development of updated flow algorithms/equations.
## Day 3
Plenary Talk 6: Peter Imkeller “Geometric properties of some rough Weierstrass and Takagi type”
Abstract: We investigate geometric properties of graphs of Weierstrass or Takagi type functions, represented by series based on smooth functions. They are H ̈older continuous, and can be embedded into smooth dynamical systems, where their graphs emerge as pullback attractors. It turns out that occupation measures and Sinai-Bowen-Ruelle (SBR) measures on their stable manifolds are dual by time reversal. A suitable version of approximate self-similarity for deterministic functions allows to ”telescope” small scale properties from macroscopic ones. As a consequence, absolute continuity of the SBR measure is seen to be dual to the existence of local time. The link between the rough curves considered and smooth dynamical systems can be generalized in various ways, for instance by randomization to individual Brownian trajectories. Applications to regularization of singular ODE by rough signals are on our agenda. This is joint work with O. Pamen (U Liverpool and AIMS Ghana) and G. dos Reis (U Edinburgh).
Plenary Talk 7: Kwabena Ndoku-Amponsah “A regression method estimation of the extreme value index using decreasing dependent random weights”
Abstract: In this talk, a regression method for the extreme value index of a Pareto-type distribution is presented using the weighted least squares approach. We show that the proposed estimator is consistent, asymptotically normal, and unbiased considering the second-order assumption on the data distribution.
Plenary Talk 8: David Bekolle “Asymptotic approximations of Good’s special functions arising in atomic physics”
Abstract: We study various asymptotic approximations of Good’s special functions arising in atomic physics. These special functions are situated beyond Anger’s functions to which they are closely related. The functions under study depend on two variables, and present themselves as oscillatory integrals, with the phase depending on one of the variables and the amplitude on the other one. We will use the method of the stationary phase, but this one does not suffice to describe the behavior in all cases.
Talk 11: Rhoss Likibi Pellat “Differentiability of Quadratic Forward-Backward SDEs with rough drift”
Abstract: In this talk, we consider quadratic forward-backward SDEs (QFBSDEs), for which the drift in the forward equation does not satisfy the standard globally Lipschitz condition and the driver of the backward system possesses nonlinearity of type f(|y|)|z|^2, where f is any locally integrable function. We prove both the Malliavin and classical derivative of the QFBSDE and provide representations of these processes. We study a numerical approximation of this system in the sense of (Imkeller-Dos Reis, 2010) in which the authors assume that the drift is Lipschitz and the driver of the BSDE is quadratic in the traditional sense (i.e., f is a positive constant). We show that the rate of convergence is the same as in (Imkeller-Dos Reis, 2010)
Talk 12: Ibrahim Nonkane “Representation theory and D-modules theory”
Abstract: A prevailing idea in representation theory is that larger structures can be understood by breaking them up into their smallest pieces. Also the natural framework of algebraic geometry is one of the polynomials and, the development of modern algebra has given a particular status to polynomials. In this vein, we study polynomial rings as modules over a ring of invariant differential operators by elaborating its irreducible submodules. The D-module direct image of an irreducible holonomic module with regular singularities under a proper map is semi-simple according to the decomposition theorem. The simplest case is when the map π : X = specB → Y = specA is finite, and the module is the structure sheaf $B = O_X$ . Then an elementary and whollyalgebraic proof exists, using essentially the ordinary Galois group G of the function field extension corresponding to π. This proof uses that the irreducible D-submodules of $π_+(O_X)$ are in one-to-one correspondence with irreducible representations of G. In this talk, we study the decomposition sttructure when when G is the symmeric group,and when G is the generalized symmetric group.
## Day 4
Plenary Talk 9: Ralf Wunderlich “Stochastic Epidemic Models with Partial Information and Dark Figure Estimation”
Abstract: Mathematical models of epidemics such as the current COVID-19 pandemics often use compartmental models dividing the population into several compartments. Based on a microscopic setting describing the temporal evolution of the subpopulation sizes in the compartments by stochastic counting processes one can derive macroscopic models for large populations describing the average behavior by associated ODEs such as the celebrated SIR model. Further, diffusion approximations allow to address fluctuations from the average and to describe the state dynamics also for smaller populations by stochastic differential equations (SDE). Usually not all of the state variables are directly observable and we are facing the so-called “dark figure” problem addressing for example the unknown number of asymptomatic and non-detected infections. Such not directly observable states are problematic if it comes to the computation of characteristics of the epidemic such as the effective reproduction rate and the prevalence of the infection within the population. Further, the management and containment of epidemics relying on solutions of (stochastic) optimal control problems and the associated feedback controls need observations of the current state as input. The estimation of unobservable states based on records of the observable states leads to a non-standard filtering problem for partially observable stochastic models. We adopt the extended Kalman filter approach coping with nonlinearities in the state dynamics and the state-dependent diffusion coefficients in the SDEs. This allows to develop approximative solutions to that filtering problem. Numerical results illustrating our theoretical finding are presented. This is joint work with Florent Ouabo Kamkumo, Ibrahim Mbouandi Njiasse (Cottbus) and Olivier Menoukeu Pamen (AIMS Ghana/Liverpool).
Plenary Talk 10: Antoine Bogso “Path-by-path uniqueness of SDEs in the plane with irregular drifts”
Abstract: We study path-by-path uniqueness for multidimensional stochastic differential equations driven by the Brownian sheet. We assume that the drift coefficient is unbounded, verifies a spatial linear growth condition and is the difference of componentwise nondeacreasing functions. Our approach consists of showing the result for bounded drifts that are difference of componentwise nondecreasing functions using both a local time-space representation and a law of iterated logarithm for Brownian sheets. The desired result follows using a Gronwall type lemma on the plane. As a by-product, we obtain the existence of a unique strong solution of multidimensional SDEs driven by the Brownian sheet when the drift is the difference of two (componentwise) non-decreasing functions and satisfies a spatial linear growth condition. This is a joint work with Olivier Menoukeu-Pamen (AIMS Ghana, University of Liverpool).
Plenary Talk 11: Delfim F. M. Torres “On the Non-Newtonian Calculus of Variations”
Abstract: The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler-Lagrange equations. The subject has found many applications along the centuries, e.g., in physics, economics, engineering, and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. Because in many applications negative values of admissible functions are not physically plausible, we propose an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 to 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and is based on a (non-Newtonian) derivative that permits to compare relative changes between a dependent positive variable and the independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce provides a natural framework for problems involving functions with positive images. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range.
Talk 13: Edward Korveh “On approximate numerical solution to a class of PIDEs from mean-variance asset allocation problems under contagion.”
Abstract: In this talk, we present an approximate numerical solution to a class of partial integro differential equations (PIDEs) resulting from mean-variance asset allocation problems under contagion. We use the finite difference method for the differential part and a generic numerical integration scheme for the integral part. We present computer simulated results for some special cases.
Speakers
## CMIA Speakers
• Olivier Pamen, AIMS Ghana/University of Liverpool
• Antoine Bogso, AIMS Ghana
• David Bekolle, University of Yaoundé I
• Peter Imkeller, Humboldt University at Berlin
• Stephen Moore, University of Cape Coast
• William Obeng-Denteh, Kwame Nkrumah University of Sciences and Technology
• Youssef Ouknine, Mohammed VI Polytechnic University
• Zbigniew Palmowski, Wroclaw University of Science and Technology
• Ralf Wunderlich, BTU Cottbus-Senftenberg
• Kwabena Doku-Amponsah, University of Ghana
• Benoit Sehba, University of Ghana
Partners
## CMIA Partners
The Alexander Von Humboldt Foundation
The German Federal Ministry of Education
Organisers:
• Olivier Menoukeu Pamen: U Liverpool and AIMS Ghana
• Peter Imkeller: Humboldt U at Berlin
• Ralf Wunderlich: BTU Cottbus-Senftenberg
Local organising committee:
• Antoine Marie Bogso: U of Yaoundé I and AIMS Ghana
• Prince K. Osei: AIMS Ghana
• Olivier Menoukeu Pamen: U Liverpool and AIMS Ghana
Schedule of Activities
Travel Guide
## Travel Guide
Routine Vaccines:
Be sure that your routine vaccines, as per your province or territory, are up-to-date.
Some of these vaccines include: measles-mumps-rubella (MMR), diphtheria, tetanus, pertussis, polio, varicella (chickenpox), influenza and others.
Pre-travel vaccines and medications:
You may be at risk for preventable diseases while travelling to this destination. Talk to a travel health professional about which medications or vaccines are right for you.
Diseases include:
Hepatitis A
Yellow Fever – Country Entry Requirements
Rabies
Measles
Hepatitis B
Polio
Influenza
Meningococcal disease
Malaria
COVID-19
Food and Water-borne Diseases
Travel health and safety:
Emergency medical attention and serious illnesses require medical evacuation. Medical services usually require immediate cash payment.
Make sure you get travel insurance that includes coverage for medical evacuation and hospital stays.
Prescription drugs
If you take prescription medication, you are responsible for determining its legality in Ghana.
Precautions
• Bring sufficient quantities of your medication with you
• Always keep your medication in the original container
• Carry a copy of your prescription(s)
• Pack them in your carry-on luggage
|
2022-10-05 20:52:14
|
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|
https://www.physicsforums.com/threads/fourier-transform-as-continuous-change-of-basis.660363/
|
# Fourier transform as (continuous) change of basis
1. Dec 22, 2012
### TrickyDicky
Trying not to get too confused with this but I'm not clear about switching from coordinate representation to momentum representation and back by changing basis thru the Fourier transform.
My concern is: why do we need to change basis? One would naively think that being in a Hilbert space where global sets of basis are available one shouldn't be required to change basis when performing a linear transformation.
I guess this is related to the noncommuting of x and p (HUP), and the Hilbert infinite -dimensional space topological structure but how exactly?
2. Dec 22, 2012
### waveandmatter
I am not completely sure if i understand the question correctly.
One likes to change the basis from the position to the momentum basis, because the momentum basis is an eigen-basis ie the plane waves are eigenstates of propagation and it is thus easy to calculate their time-evolution.
So we change basis, propagate, and change back.
Edit: In the general case you do not have to change the basis in order to apply any operator, provided you know what the operator looks like in the basis you start with.
Last edited: Dec 22, 2012
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2017-09-24 05:24:25
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https://mathsgee.com/courses/mathematics-for-professionals/lessons/quadratic-equations/
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Data-driven 4IR skills development
The general form of a quadratic equation is as follows:
$ax^2+bx+c = 0$
1. $x^2+2x+1=0$
Using factorisation, $x^2+2x+1=0$ can be written as:
$$(x+1)(x+1)=0$$
Using the rule that $a*0 = 0$ for all $a \in \mathbb{R}$ meaning that when two numbers or expressions multiply each other, then one of them must be equal to zero.
In this case the first $x+1=0$ or the second $x+1=0$
therefore the solution is $x={-1}$ twice.
2. $5x^2+6x+1=0$
Using factorisation, $5x^2+6x+1=0$ can be written as:
$$(x+5)(x+1)=0$$
Using the rule that $a*0 = 0$ for all $a \in \mathbb{R}$ meaning that when two numbers or expressions multiply each other, then one of them must be equal to zero.
In this case the first $x+5=0$ or the second $x+1=0$
therefoere the solution is $x={-5}$ or $x={-1}$
3. $(x-3)(x+4)=0$
$(x-3)=0$ or $(x+4)=0$
$x=3$ or $x={-4}$
$x(x+7)=0$
$x=0$ or $x+7=0$
$x=0$ or $x={-7}$
$x^2=9$$Using difference of two squares, x^2=9 = (x-3)(x+3)=0$$x-3=0$$or$$x+3=0x=3$$or$$x={-3}$$At times when solving quadratic equations it will be impossible to factorise thus it is necessary to use the \textbf{Quadratic Formula}. Using the general form of a quadratic equation: ax^2+bx+c = 0 arrange your equation in the same way with the right hand side having 0, then identify the values of a,b and c e.g. in 5x^2+6x+1=0, a=5, b=6 and c=1 and substitute them in the quadratic formula. The \textbf{Quadratic Formula} is as follows: x = \frac{{-b}\pm{\sqrt{b^2 – 4ac}}}{2a} Example:$$5x^2+6x+1=0$$,$$a=5, b=6$$and$$c=1x = \frac{{-6}\pm{\sqrt{{-6}^2 – 4(5)(1)}}}{2(5)}$$Solution: x= \frac{-1}{5} or x= {-1} Exercise Solve each of the following quadratic equations both by factorisation and quadratic formula.$$(x+5)(x-3)=0(x-2)(x-1)(x+1)^2=0(x+5)(3x-1)=0(2x+1)((3x+2)=08x=12x^23x^2-15x=1082(x-2)(x+4)=32$\$
Share with:
https://mathsgee.com
Edzai Conilias Zvobwo is passionate about empowering Africans through mathematics, problem-solving techniques and media. As such, he founded MathsGee. Through this organisation, he has helped create an ecosystem for disseminating information, training, and supporting STEM education to all African people. A maths evangelist who teaches mathematical thinking as a life skill, Edzai’s quest has seen him being named the SABC Ambassador for STEM; he has been invited to address Fortune 500 C-suite executives at the Mobile 360 North America; was nominated to represent Southern Africa at the inaugural United Nations Youth Skills Day in New York; was invited to be a contributor to the World Bank Group Youth Summit in 2016; has won the 2014 SADC Protocol on Gender and Development award for his contribution to women’s empowerment in education; and has partnered with local and global firms in STEM interventions.
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2019-08-17 17:15:47
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https://stackoverflow.com/questions/7321407/what-is-sut-and-where-did-it-come-from/7321820
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# What is SUT and where did it come from?
I see many people talking about the term SUT, but do not understand why they use that term.
SUT is what you want to test?
Where does this term come from and what does it mean?
For example in this test, what is my SUT?
[TestMethod]
public void UsersAction_should_return_IndexAction()
{
const long id = 1;
UsersViewModel viewModel = new UsersViewModel()
{
SelectedUsers = new long[] { 1, 2, 3, 4 }
};
ActionResult result = _controller.Users(id, viewModel);
result.AssertActionRedirect().ToAction("Index");
}
The System Under Test (SUT) from a Unit Testing perspective represents all of the actors (i.e one or more classes) in a test that are not mocks or stubs. In your example that would be the controller.
• Of course mocks and stubs are not part of the system being tested. This is not really a definition of SUT and does not explain why the term even exists. The correct definition is provided by xUnit Test Patterns which you can find here. – Hadi Brais Oct 7 '17 at 2:21
It most likely means "System Under Test", i.e. the system being tested, as opposed to other systems it may interact with, but which are not being explicitly tested (because they're someon else's responsibility).
• This is the correct answer. – Hadi Brais Oct 7 '17 at 2:23
I've never heard the term eiher, but a quick search gave
System under test (SUT) refers to a system that is being tested for correct operation. The term is used mostly in software testing.
A special case of a software system is an application which, when tested, is called an application under test.
The term SUT means also a stage of maturity of the software, because a system test is the successor of integration test in the testing cycle.
From good ole wikipedia.
Where does this term come from?
DUT (device under test) and UUT (unit under test) are very common abbreviations among test enginers (non-software test engineers). That's where the term SUT (system under test)
and CUT (code under test) ought to have come from.
related:
2008 MSDN blog post Naming SUT Test Variables .
System Under Test SUT.
In your unit test example, if you really want to talk about SUT it's probably UsersAction. However, I have not come across anyone use SUT when talking about unit testing. To me this sounds more like something that would fit with integration/system/performance testing or alike.
For instance, take performance testing. Here you might say the SUT is the whole HW/SW system, or it might be just one of them depending on what you're testing in that specific performance test.
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2019-10-16 05:35:41
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https://www.physicsforums.com/threads/extract-urls-with-regexp.465842/
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# Extract URLs with regexp
## Homework Statement
I have a file that contains lines like the following:
Code:
<td><divalign="center"><fontcolor="#0000ff"face="Arial,Helvetica,sans-serif"size="2"><strong><ahref="[PLAIN]http://www.yahoo.com/">Yahoo!</a></strong></font></div></td>[/PLAIN] [Broken]
<td><divalign="center"><fontcolor="#0000ff"face="Arial,Helvetica,sans-serif"size="2"><strong><ahref="[PLAIN]http://www.google.com/">Google</a></strong></font></div></td>[/PLAIN] [Broken]
It is already processed some from an html file, but I want the following to be the final output
Code:
http://www.yahoo.com/
http://www.google.com/
I am using sed to edit the file line by line and substitute.
## Homework Equations
Nothing much here
## The Attempt at a Solution
My idea was to say '*http' to match anything in front of http and then replace it with an empty string. This didn't actually match anything and negated a similar idea to match and delete everything after the .com/ portion. I also tried '<td>*="' to try and remove the portion before http and again replace with an empty string. Any help or hints would be appreciated, thanks
Last edited by a moderator:
Related Engineering and Comp Sci Homework Help News on Phys.org
nvn
Homework Helper
Trentonx: Try "^.*http", instead of "*http". And then try "\">.*\$", to try to match everything after the URL. Try it, and let us know whether or not it works, since I have not tested it.
That worked with a little modification. I realized I wanted to match right before the http, so as not to remove it. I used '^.*="' which somehow dodn't match the other same expressions in the file. So now, how does it do it? The ^ is an anchor to the start of a line, and the * is a wildcard, but what does the . do? You used it in both expressions, so it is likely useful to know.
nvn
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2020-06-04 15:38:50
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https://www.vedantu.com/question-answer/find-the-principal-value-of-cot-1left-tan-class-12-maths-cbse-5ee496116e2ad60d527131d8
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Question
# Find the principal value of : ${{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)$.
Hint: The question is related to inverse trigonometric functions. Assume the given function to be equal to $x$. Find the value of $\cot x$. Then find the value of $x$ which gives the acquired value on applying cotangent function.
We are asked to find the principal value of the inverse trigonometric function ${{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)$. Let us assume the value of the inverse trigonometric function to be equal to $x$. So, we get:
${{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)=x$
Now, we will apply cotangent function on both sides of the equation. On applying cotangent function on both sides of the equation, we get:
$\cot \left( {{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right) \right)=\cot x$
Now, we know the value of $\cot \left( {{\cot }^{-1}}y \right)$ is equal to $y$. So, we get:
$\tan \dfrac{3\pi }{4}=\cot x.....(i)$.
Now, we know, tangent function is negative in the second quadrant. So, the value of $\tan \dfrac{3\pi }{4}$ is equal to $-1$ . We will substitute the value of $\tan \dfrac{3\pi }{4}$ as $-1$ in equation $(i)$. On substituting the value of $\tan \dfrac{3\pi }{4}$ as $-1$ in equation $(i)$, we get:
$\cot x=-1$.
We know, the range for principal value is $\left( 0,\pi \right)$. So, we have to find a value of $x$ such that $x\in \left( 0,\pi \right)$ and $\cot x=-1$. The only possible value which satisfies both conditions is $x=\dfrac{3\pi }{4}$.
So , the value of principal value of the inverse trigonometric function ${{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)$ is equal to $\dfrac{3\pi }{4}$.
Note: While solving the problem, make sure that the value of the inverse trigonometric function lies in the principal value range, i.e. $\left( 0,\pi \right)$for $cot$ function. Students generally forget this condition and end up getting a wrong answer. So, this condition must be satisfied by the obtained principal value.
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2021-04-23 10:44:13
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https://www.physicsforums.com/threads/oscillator-basic-question.86523/
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Oscillator basic question
Question about oscillator (Simple Harmonic Motion)
Hi, I just have a basic question about oscillators. If we are only given the initial position, can we still determine the amplitude and phase constant?
I think we can't because we would still need to know the angular frequency (k and m needed as well) and whether or not the system starts with an initial velocity of zero. Am i thinking in the right direction?
Last edited:
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2020-10-29 02:15:49
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https://www.okphysics.com/1-159-work-done-in-uniform-circular-motion/
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Select Page
Problem 1.159:.A body of mass $‘m’$ is moving in a circle of radius with a constant speed $‘v’$. Calculate the work done by the resultant force in moving the body over half circumference and full circumference respectively.
Solution:
This is motion along a circular path. Resultant force on the particle produces the necessary centripetal force.
Centripetal force is directed toward the center of the path and velocity is tangential to the path.
Therefore resultant force is always perpendicular to the velocity.
This makes work done zero as there is no change in speed and kinetic energy of the body.
Then the work done by the resultant force in moving the body over half circumference and full circumference respectively are zero.
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2017-09-22 22:14:19
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http://ilovemusubi.com/articles/06/probabilitydistribution.asp
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## Drawing Probability Distribution
Almost regardless of your view about the predictability or efficiency of markets, you'll probably agree that for most assets, guaranteed returns are uncertain or risky. If we ignore the math that underlies probability distributions, we can see they are pictures that describe a particular view of uncertainty. The probability distribution is a statistical calculation that describes the chance that a given variable will fall between or within a specific range on a plotting chart.
Uncertainty refers to randomness. It is different from a lack of predictability, or market inefficiency. An emergent research view holds that financial markets are both uncertain and predictable. Also, markets can be efficient but also uncertain.
In finance, we use probability distributions to draw pictures that illustrate our view of an asset return's sensitivity when we think the asset return can be considered a random variable. In this article, we'll go over a few of the most popular probability distributions and show you how to calculate them.
Distributions can be categorized as either discrete or continuous, and by whether it is a probability density function (PDF) or a cumulative distribution.
## Discrete vs. Continuous Distributions
Discrete refers to a random variable drawn from a finite set of possible outcomes. A six-sided die, for example, has six discrete outcomes. A continuous distribution refers to a random variable drawn from an infinite set. Examples of continuous random variables include speed, distance, and some asset returns. A discrete random variable is illustrated typically with dots or dashes, while a continuous variable is illustrated with a solid line. The figure below shows discrete and continuous distributions for a normal distribution with mean (expected value) of 50 and a standard deviation of 10:
The distribution is an attempt to chart uncertainty. In this case, an outcome of 50 is the most likely but only will happen about 4% of the time; an outcome of 40 is one standard deviation below the mean and it will occur just under 2.5% of the time.
## Probability Density vs. Cumulative Distribution
The other distinction is between the probability density function (PDF) and the cumulative distribution function. The PDF is the probability that our random variable reaches a specific value (or in the case of a continuous variable, of falling between an interval). We show that by indicating the probability that a random variable XÂ will equal an actual value x:
\begin{aligned} &P[x = X] \\ \end{aligned}
The cumulative distribution is the probability that random variable XÂ will be less than or equal to actual value x:
\begin{aligned} &P[x <= X] \\ \end{aligned}
or example, if your height is a random variable with an expected value of 5'10" inches (your parents' average height), then the PDF question is, "What's the probability that you will reach a height of 5'4"?" The corresponding cumulative distribution function question is, "What's the probability you'll be shorter than 5'4"?"
The figure above showed two normal distributions. You can now see these are probability density function (PDF) plots. If we re-plot the exact same distribution as a cumulative distribution, we'll get the following:
The cumulative distribution must eventually reach 1.0 or 100% on the y-axis. If we raise the bar high enough, then at some point, virtually all outcomes will fall under that bar (we could say the distribution is typically asymptotic to 1.0).
Finance, a social science, is not as clean as physical sciences. Gravity, for example, has an elegant formula that we can depend on, time and again. Financial asset returns, on the other hand, cannot be replicated so consistently. A staggering amount of money has been lost over the years by clever people who confused the accurate distributions (i.e., as if derived from physical sciences) with the messy, unreliable approximations that try to depict financial returns. In finance, probability distributions are little more than crude pictorial representations.
## Uniform Distribution
The simplest and most popular distribution is the uniform distribution, in which all outcomes have an equal chance of occurring. A six-sided die has a uniform distribution. Each outcome has a probability of about 16.67% (1/6). Our plot below shows the solid line (so you can see it better), but keep in mind that this is a discrete distribution—you can't roll 2.5 or 2.11:
Now, roll two dice together, as shown in the figure below, and the distribution is no longer uniform. It peaks at seven, which happens to have a 16.67% chance. In this case, all the other outcomes are less likely:
Now, roll three dice together, as shown in the figure below. We start to see the effects of a most amazing theorem: the central limit theorem. The central limit theorem boldly promises that the sum or average of a series of independent variables will tend to become normally distributed, regardless of their own distribution. Our dice are individually uniform but combine them and—as we add more dice—almost magically their sum will tend toward the familiar normal distribution.
## Binomial Distribution
The binomial distribution reflects a series of "either/or" trials, such as a series of coin tosses. These are called Bernoulli trials—which refer to events that have only two outcomes—but you don't need even (50/50) odds. The binomial distribution below plots a series of 10 coin tosses wherein the probability of heads is 50% (p-0.5). You can see in the figure below that the chance of flipping exactly five heads and five tails (order doesn't matter) is just shy of 25%:
If the binomial distribution looks normal to you, you are correct about that. As the number of trials increases, the binomial tends toward the normal distribution.
## Lognormal Distribution
The lognormal distribution is very important in finance because many of the most popular models assume that stock prices are distributed lognormally. It is easy to confuse asset returns with price levels.
Asset returns are often treated as normal—a stock can go up 10% or down 10%. Price levels are often treated as lognormal—a $10 stock can go up to$30 but it can't go down to -\$10. The lognormal distribution is non-zero and skewed to the right (again, a stock can't fall below zero but it has no theoretical upside limit):
## Poisson
The Poisson distribution is used to describe the odds of a certain event (e.g., a daily portfolio loss below 5%) occurring over a time interval. So, in the example below, we assume that some operational process has an error rate of 3%. We further assume 100 random trials; the Poisson distribution describes the likelihood of getting a certain number of errors over some period of time, such as a single day.
## Student's T
The student's T distribution is also very popular because it has a slightly "fatter tail" than the normal distribution. The student's T is used typically when our sample size is small (i.e. less than 30). In finance, the left tail represents the losses. Therefore, if the sample size is small, we dare underestimate the odds of a big loss. The fatter tail on the student's T will help us out here. Even so, it happens that this distribution's fat tail is often not fat enough. Financial returns tend to exhibit, on rare catastrophic occasion, really fat-tail losses (i.e. fatter than predicted by the distributions). Large sums of money have been lost making this point.
## Beta Distribution
Finally, the beta distribution (not to be confused with the beta parameter in the capital asset pricing model) is popular with models that estimate the recovery rates on bond portfolios. The beta distribution is the utility player of distributions. Like the normal, it needs only two parameters (alpha and beta), but they can be combined for remarkable flexibility. Four possible beta distributions are illustrated below:
## The Bottom Line
Like so many shoes in our statistical shoe closet, we try to choose the best fit for the occasion, but we don't really know what the weather holds for us. We may choose a normal distribution then find out it underestimated left-tail losses; so we switch to a skewed distribution, only to find the data looks more "normal" in the next period. The elegant math underneath may seduce you into thinking these distributions reveal a deeper truth, but it is more likely that they are mere human artifacts. For example, all of the distributions we reviewed are quite smooth, but some asset returns jump discontinuously.
The normal distribution is omnipresent and elegant and it only requires two parameters (mean and distribution). Many other distributions converge toward the normal (e.g., binomial and Poisson). However, many situations, such as hedge fund returns, credit portfolios, and severe loss events, don't deserve the normal distributions.
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2020-11-28 05:55:28
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https://studydaddy.com/question/cmgt-555-week-1-dqs
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QUESTION
CMGT 555 Week 1 DQs
This paperwork of CMGT 555 Week 1 Discussion Questions shows the solutions to the following problems:
DQ 1: 1. List the stages of the Systems Development Life Cycle (SDLC). They could apply to any sizeable project, not just software development, don
• @
• 1 order completed
Tutor has posted answer for $5.19. See answer's preview$5.19
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2018-05-26 16:02:44
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https://www.physicsforums.com/threads/two-spheres-having-masses-m-sphere-1-and-2m-sphere-2-and-radii-r-and-2r-respectively-are-relea.782473/
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# Two spheres having masses M (sphere 1) and 2M (sphere 2) and radii R and 2R, respectively, are relea
1. Nov 17, 2014
### whatisphys
1. The problem statement, all variables and given/known data
Two spheres having masses M (sphere 1) and 2M (sphere 2) and radii R and 2R, respectively, are released from rest when the distance between their centers is 8R.
How fast will sphere 1 be moving when they collide? Assume that the two spheres interact only with each other. Enter your answer in units of sqrt(GM/R).
How fast will sphere 2 be moving when they collide? Enter your answer in units of sqrt(GM/R)
2. Relevant equations
-Gmm/r
(mv^2)/2
m1v1 + m2v2 = 0 since starts at rest
3. The attempt at a solution
Okay.
So what I did was I first calculated initial PE and Final PE. I then calculated the change in PE which turned out to be -5GM^2/12R
Then, I equated it to deltaKE = -deltaPE which i got as (Mv1^2)/2 + (2Mv2^2)/2 = 5GM/12R.
I used conservation of momentum to find out the ratio of V1 to V2 which was V2 = -1/2(V1)
After that I substituted that to the equation above and solved for V1 as 20GM/36R.
But it says it is wrong. Any help would be appreciated. Thank you
2. Nov 17, 2014
### Simon Bridge
3. Nov 17, 2014
### whatisphys
PEi = GM2M/8R = -GM^2/4R and PEf = GM2M/3R = -2GM^2/3R
delta PE = Final - initial
make them so that they have common base which is -8GM^2/12R + 3GM^2/12R
4. Nov 17, 2014
### Simon Bridge
OK I see ... so for conservaton of energy and momentum respectively you got:
$$v_1^2 + 4v_2^2 = \frac{10}{12}\frac{GM}{R}\\ v_1 + 2v_2 = 0$$ ... after dividing through by M in both equations and multiplying through by 2 in the top one.
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2018-01-18 11:39:54
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https://chemistry.stackexchange.com/questions/67504/why-do-the-lone-pairs-of-electrons-first-occupy-axial-positions-before-occupying
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# Why do the lone pairs of electrons first occupy axial positions before occupying equatorial positions for species with octahedral geometry?
Why do lone pairs not reside in equatorial positions in species with octahedral geometry?
• In octahedral geometry, all positions should be equivalent (before putting a bond or a lone pair there). It is by convention that the lone pairs are put on "axial" positions. However, they will definitely be across each other, consider XeF4. – TAR86 Feb 1 '17 at 15:26
• @TAR86 I don't understand what you mean by "equivalent". Do you mean all positions have the same bond angle to each other? – b3nj4m1n Feb 1 '17 at 15:37
• Can you give an example? – bon Feb 1 '17 at 15:41
• @b3nj4m1n Consider a cube (such as dice). Octahedral geometry means that the central atom is in the center of the cube and the surrounding atoms are on the center of the faces of the cube. Thus, due to the high symmetry of the octahedral geometry, they are all equivalent. If this explanation does not help, you need to build a model. – TAR86 Feb 1 '17 at 17:41
The main reason is repulsion with the electron pairs. Compare $\ce{SF6}$ & $\ce{SF4^2-}$ as an example. To get from $\ce{SF6}$ to $\ce{SF4^2-}$, we need to replace two fluorines with lone pairs. The first replacement is arbitrary; in an octahedral geometry, all the positions are equivalent. Once we have replaced that first fluorine however, the bond positions are no longer equivalent. We will have four equatorial bonds (90 degree separation from the lone pair) and one axial bond (180 degree separation from the lone pair). Lone pairs are very repulsive, especially with respect to other lone pairs, so we want the next lone pair to be as separated as possible from the first. The axial bond is further from the lone pair then any of the equatorial bonds, so it makes sense for the lone pair to go there.
simply because in octahedral geometry both the equatorial position or base and the axial-equatorial position maintain same bond angle which is 90deg, so at first you can chose any position to place the first lone pair, but for the next electron pair the remaining positions are no more identical in the view of the bond angle.Cause if u chose eq. position for the second one, if it's a lonepair too then according to VSEPR theory you haven't made the right choice, cause you are still having an option to reduce the greatest replusion (LP-LP) between electron pairs, that is the axial position.So that angular distance between the LPs become 180. If the explanation above for octahedral is clear to you than you yourself can find why it is vice versa for a tetrahedron.:)
• Welcome to Chemistry.SE! Take the tour to get familiar with this site. Mathematical expressions and equations can be formatted using $\LaTeX$ syntax. For more information in general have a look at the help center. This doesn't make much sense, first you state that all positions are equal, then they are not. You could put two lone pairs in equatorial positions and maintain 180° (If measuring angles between lone pairs would be possible.) It would also be better, if you avoid abbreviations and do not use emoticons. – Martin - マーチン Jun 7 '18 at 11:29
• In case of two lone pairs in a tetrahedron it creates a dilemma for me too. But for 1 or more than 2 electron pairs it makes sense. And for the first electron pair in an octahedral shape all the positions are equal, but not for the next one. I don't see any problem here. – Deehan Jun 8 '18 at 11:53
The VSEPR theory tells us that the order of repulsion is :
lone pair-lone pair >lone pair-bond pair >bond pair-bond pair.
It is clear that you need to place the lone pairs such that they have minimum neighboring bond pairs or lone pairs. An equatorial position in an octahedral geometry has 5 neighboring bond pairs/lone pairs ( 3 in the equatorial plane and two in the axial plane ) while an axial position has 4 neighboring bond pairs/lone pairs (4 in the equatorial plane, the other axial position is not a neighboring position).
That is the reason why lone pairs first occupy axial positions in an octahedral geometry. Hope this helps.
• Are we talking about the same thing? en.wikipedia.org/wiki/VSEPR_theory see SF6. All F's are equivalent. – TAR86 Feb 1 '17 at 18:13
• wait @TAR86 so are all the positions equal or not? I'm confused now – b3nj4m1n Feb 2 '17 at 1:50
• @b3nj4m1n I am utterly convinced that all positions of the surrounding atoms in an octahedral geometry are equal. This is due to the symmetry of the octahedron. – TAR86 Feb 2 '17 at 6:07
• See, one lone pair isn't any issue, because all six sites are equivalent, as @TAR86 points out. OP is asking about why two lone pairs will go trans to each other. The use of "axial" or "equatorial" is highly misleading in my opinion. – orthocresol Feb 8 '17 at 1:47
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2019-07-24 08:59:55
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https://www.networkpages.nl/probability-theory/
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# Probability Theory
Probability theory is the part of mathematics that studies chance. This is best illustrated by games of chance. For example, when tossing a fair coin, the probability of it turning up heads is 1/2.
Throwing two dices. On the right, we see how often a certain sum occurred, when we threw two dices 1000 times.
We rely on randomness in board games. In Settlers of Catan, we toss two dice to decide which and how many cards the players receive. The outcome 7 is the most likely and has a chance one out of six, the outcomes 2 and 36 are the least likely with a probability of $1/36$. Now we understand why 6 and 8 are such desirable numbers!
There are many interpretations of what a probability is, and discussions about this can become quite philosophical. We will not dwell on these philosophical aspects. The simplest interpretation of the probability of an event is in terms of repeating an experiment many times, in such a way that different experiments do not influence one another. Then, the probability of a certain event corresponds to the proportion of times that the event occurs in many of such independent experiments.
Would you like to stay up to date whenever a new post appears on the Network Pages? Then subscribe to our mailing list, follow us on Twitter or on LinkedIn.
Comments are closed
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2023-03-27 00:45:57
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https://gigaom.com/tag/peering/
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### 5 reasons the FCC might be wrong about net neutrality
This week the FCC passed new rules on net neutrality, which were essentially designed to limit the ability of internet service providers…
### The FCC’s net neutrality proposal is awesome, but has a loophole
FCC Chairman Tom Wheeler has taken the unprecedented and awesome step of using Title II to ensure that the internet remains open…
### It’s on! FCC plans strong net neutrality for mobile and broadband
The Federal Communications Commission plans to reclassify broadband internet providers so they can’t favor some websites over others, which is the outcome that has been…
### Does the FCC want to oversee peering deals like Netflix vs. Comcast?
The FCC will soon pass new rules for how ISP’s must handle broadband traffic and, while it’s expected to impose a policy of net neutrality when…
### Regulating peering
One of the most puzzling aspects of the peering disputes that have arisen -- principally between Netflix and a handful of the largest ISPs -- is how little money appears to be involved.…
### Google Cloud goes corporate with peering, carrier interconnects, VPN
Google Cloud struts its corporate stuff, adding connections from carrier hotels and direct peering as well as virtual private networking access for risk-averse companies seeking hybrid cloud.…
### Peering problems aren’t technical issues, but economic ones
A report out Tuesday on the cost of interconnection fights shows that the problem is the business deals, not technology, and that the consumer pays the price.…
### Peering deals speed up U.S. Netflix streams, but European ISPs still a lot faster
Want the best-possible Netflix streaming experience? Then get ready to move to Switzerland. U.S. ISPs still rank far behind their European counterparts, despite commercial peering agreements.…
### The Bay Area gets the European internet exchange model Netflix hopes will spread
The Amsterdam Internet Exchange putting up a point of presence in Digital Reality's San Francisco data center signals the rise of an alternative to the current model of peering in the U.S.…
### Is this the sound of Comcast’s merger hopes dying? FCC Chairman declares U.S. broadband uncompetitive
FCC Chairman Tom Wheeler says the U.S. lacks meaningful broadband competition and proposes a few vague solutions while also calling for higher speeds.…
### Here’s more data on (and some solutions for) the high cost of bandwidth in the US
Understanding how bandwidth is priced can be complicated. But if you value the internet, it's worth trying to understand how it works because large ISPs and certain business models can drive up bandwidth costs for all.…
### Netflix is now paying Time Warner Cable for direct access and faster streams
Netflix, the online streaming giant, has signed a paid peering deal with Time Warner Cable, meaning that it now has deals with the four biggest U.S. ISPs.…
### Netflix comes to France with enough capacity to supply a small ISP
In preparation for it's European expansion, Netflix is readying a monumental amount of bandwidth in France.…
### Verizon FiOS gives customers a boost on upload speeds
Verizon will upgrade its fiber-to-the-home service to symmetrical broadband connections at no extra charge for customers. That means customers get the same…
### Why the consumer is still held hostage in peering disputes
It's been almost 10 months since consumers began complaining about poor Netflix streaming because of congestion where the last-mile ISP network met Netflix's network. Why is this still an issue?…
### Netflix to FCC: reclassify Comcast and Verizon so they can’t choke the internet
Netflix submitted an unusually blunt filing to the FCC that blasts Verizon and Comcast, and says the agency should use its "Title II" power to enforce net neutrality.…
### Verizon’s FIOS Netflix speed tanks as finger-pointing by both companies continues
Even streams of the Netflix kids show Turbo Fast are very slow for Verizon FIOS customers: The ISP continues to decline in Netflix's monthly speed index.…
### Why do U.S. ISPs want to charge for peering? Peering makes the internet cheaper. Here’s how
The practice of network peering is gaining ground, which is good news for everyone on the web except for those companies providing transit. Will that continue?…
### Level 3 calls for net neutrality rules to extend to all ISP activities
Even if the FCC reimposes "net neutrality" rules, consumers' video streams could still suffer if ISP's are allowed to impose choke points at deeper layers of the internet.…
### Peering, under the hood
The FCC's findings regarding "path specific," congestion problems comports quite closely with Verizon's version of what was going on in its dispute with Netflix.…
### The FCC just launched its peering investigation with a call for data
The FCC is taking on the interconnection battles between Netflix and several large ISPs with a call for data. It already has the agreements between Netflix and two large ISPs.…
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2021-09-22 19:47:30
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http://lambda-the-ultimate.org/node/2831
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## erlang green threads and the CLR
Last few years i have been playing with creating my own programming languages, I've got a pretty good idea of what i want it to do and what i want it to look like. I've finally decided to go with the CLR virtual machine as the back end (simply because I'm working on a RAD language and not anything low level. If i need raw speed i will use asm, c, or forth) My problem is that all the common language runtime work i have done has been writing il code for basically toy languages and scripting tools, so while i know what i am getting into as to the ins and outs of compiler design what i don't know is if i could create erlang like green threads in any reasonable manor on the clr.
Anyone have a basic run down of how green threads are implemented in erlang and if i could do something similar on .net?
keep in mind the language will be pure functional sort of like haskell. I have a strong interest in making it erlang like with the concurrency because erlang simply got it right (though the syntax sucks).
Even if all you have is a link i'm cool with that, i'm not afraid to do a lot of research on my own =P
## Comment viewing options
### Optimal asynchronous interface?
You can do green threads on the CLR if you transform your programs into continuation-passing form. But then interoperation with the class libraries becomes more difficult, as that code is not in CPS form, and so "steals" the stack/thread whenever called. Some classes provide asynchronous interfaces, such as streams and sockets, so you can hide the blocking API in your language for these objects and expose only the asynchronous part, but such interfaces are rare.
I'm actually curious what the current research indicates is the optimal concurrent program structure, assuming we're designing a VM from scratch for commodity hardware.
1. We can transform all programs into CPS form, and this provides optimal scalability in terms of number of threads, but then all calls suffer from indirection costs, which are quite high on modern CPUs.
2. We can use a user-level stack-switching approach, which solves the indirection problems, but this limits our concurrency on 32-bit machines to about ~65K green threads (assuming 4kB stacks). That seems enough for most applications, but I'm wary of making such predictions. :-)
3. We can use an E-style pure messaging approach to execution, which solves most concurrency related problems, but violates the intuitions most of have about the costs of call-returns, and has other drawbacks of this sort.
Obviously, some combination of #1 and #2 is optimal, but would seem to require code specialization, as a particular trace may block (requiring a continuation) or not block (ordinary call-return) in the same function depending on the control flow up to that point.
I've read "A Language-Based Approach to Unifying Threads and Events", but at its core it seems just a form of #1 (uses the CPS monad). Any further insight here?
### limits
a green thread limit of 65 thousand would be more then acceptable for my purposes.
### 1. Indirection really isn't
1. Indirection really isn't that expensive anymore. Where CPS still loses is on the predictability of returns. If calls and returns (at the asm level) are always paired perfectly, it is easy to predict where returns will go. In CPS, there aren't actual returns anymore and they don't go to the right place anyway.
2. You don't need CPS to capture continuations without copying the stack. You only need some way to manipulate activation records. The program could certainly hooks directly in the stack management mechanisms. However, the relevant information could also be computed separately 'in userland', either eagerly or by-need (e.g. by using exceptions or dynamic-extent thunks). Obviously, if the program can't control stack management, some sort of copying to and from the stack is needed. However, all the captured data lives on the regular heap, and can be moved to and from the stack incrementally, as needed. In a realistic application, most computations should still be completely performed in a single time-slice.
### 1. Indirection really isn't
1. Indirection really isn't that expensive anymore.
Do you have some evidence for this? All the papers I've read show significant penalties for branch mispredictions. So the question is whether indirections can be predicted.
While branch predictors are a little more sophisticated nowadays, they still can't accurately predict if a different branch is taken everytime from the same location as far as I know (predictors are largely history-based). Couple that with the very deep pipelines and you have a significant pipeline stall penalty. CPS form incurs exactly this penalty, so it still seems quite expensive.
You don't need CPS to capture continuations without copying the stack. You only need some way to manipulate activation records.
Is there a more detailed description of this somewhere? I've been meaning to read this paper more carefully, as it seems to provide a slightly more efficient continuation mechanism. Is this what you're referring to?
### While branch predictors are
While branch predictors are a little more sophisticated nowadays, they still can't accurately predict if a different branch is taken everytime from the same location as far as I know (predictors are largely history-based).
So, we agree, it's not indirection itself that's expensive, but rather unpredictability. The question then becomes, in what code do you expect to use unpredictable indirect branches*?
In CPS code, the only (additional, compared to direct style) such branches are returns to tons of different callers. The equivalent direct style code would have returns instead. For returns to be correctly predicted, the recursion can't be too deep, or the buffer overflows. Thus, we can expect CPS to show a sensible penalty for functions that don't themselves recurse much or perform much work, and are called from a multitude of contexts (that don't exhibit temporal locality). The last criterion is particularly stringent, while the first two suggest that these functions should often be candidates for inlining or duplication.
* In my experiments, Core 2 seems to be able to predict constant stride indirect branches and short cyclic patterns in addition to pure 'monomorphic' ones. There is still an overhead for correctly predicted indirect branches compared to direct ones, but it's nothing catastrophic.
Is there a more detailed description of this somewhere?
I was more thinking of pay-as-you-go approaches in the spirit of Pettyjohn, et al, which reifies activation records lazily and still uses normal call/return pairs. Since the reified records are just heap allocated, the problem of choosing the right stack size is avoided. In some sense the stack is still being copied, but once it's flushed to the heap, it's simple to copy records back to the stack incrementally. Whether the tradeoff of slightly more efficient 'normal' execution for more expensive capture & restoration is debatable, but I don't believe that typical concurrent code usually captures very deep continuations or wastes a lot of time switching tasks.
### In my experiments, Core 2
In my experiments, Core 2 seems to be able to predict constant stride indirect branches and short cyclic patterns in addition to pure 'monomorphic' ones. There is still an overhead for correctly predicted indirect branches compared to direct ones, but it's nothing catastrophic.
Of course, Core 2 is known to be quite a good architecture, and a relatively new one, and thus is a minority of the existing PC base. What about UltraSparc or PPC?
I was more thinking of pay-as-you-go approaches in the spirit of Pettyjohn, et al, which reifies activation records lazily and still uses normal call/return pairs.
### I was more thinking of
I was more thinking of pay-as-you-go approaches in the spirit of Pettyjohn, et al, which reifies activation records lazily and still uses normal call/return pairs.
I've read the paper now, but I don't think it's workable for the types of languages we're discussing here, where there's heavy use of continuations. Essentially, this design requires each procedure to explicitly saves its live variables in a continuation by walking the stack when the continuation is captured. While this is definitely more efficient in the case where continuations are rare, I think it would kill performance in a program or language which uses continuations heavily, like Erlang.
Stack switching would be much more efficient, particularly with the one-shot continuation optimization in the paper I linked to. Stack overflow is itself handled as a call/cc, so sizing the stack isn't really a problem, and the technique seems reasonably efficient. Better than CPS in any case. There might be some space waste, but that's always the case with stacks.
### While this is definitely
While this is definitely more efficient in the case where continuations are rare, I think it would kill performance in a program or language which uses continuations heavily, like Erlang.
I think there's a tendency to overestimate the importance of fast task switching in real programs. While it's important to avoid being as expensive as kernel threads to support, say, Erlang's programming model, it seems silly to spend too much effort trying to optimise an operation that should hopefully only represent a small fraction of a program's execution. I'm much more worried about space usage than time, for example. Take an hypothetic network application that serve 100k requests/second. If each request has to pass through 20 process, on a single 2GHz machine that's an average of ~1000 cycles (500 ns)/time slice. How much of that can be spent switching processes? According to this page from 1998 (I don't have the time to run my own), it used to take around 1 us on 200 MHz PII-class machines. It can't be that hard to get it down to 10% of that, nowadays, especially without the hardware context switch overhead.
Also, note that the approach I linked to only has to perform work proportional to the number of activation records on the stack, and only when they have changed (much of the overhead is also linked to their assuming an uncooperative runtime). Restoring a process need not put all the activation records back on the stack. Moreover, the number of activation records must somehow be related to the amount of work performed; if processes spend so much time switching to each other, they can't perform that much work between switches.
### I'm much more worried about
I'm much more worried about space usage than time, for example.
Heap allocating activation frames also has hidden costs in GC time and poor locality.
Let's turn your question around instead: what are the space losses of stack-switching? Has this ever been quantified? We can have stacks start small (~4kB) and grow on demand to minimize internal fragmentation. The paper I linked to describes a good technique to exploit this approach.
[Edit: of course, I would be dishonest if I neglected to mention the stack overflow checking that must be done on every procedure call when using stack switching. We can't rely on MMU tricks with so many little stacks. Of course, the branch prediction is nearly 100% on this test, so I don't expect the overhead is all that significant.]
Restoring a process need not put all the activation records back on the stack.
True, you could place an underflow handler as the return address which would restore the next block of activation frames. These tricks incur considerabe complexity though.
### Computer architecture
For me there's something surreal in talking about memory locality, branch prediction, etc in the context of concurrent programming. I can see that it's of massive importance to the microbenchmarking industry but it feels miles away from the concerns we've had in the concurrent systems I've worked on. Those always seem to either be I/O bound or to spend their time in some 'kernel' that's often written in C.
For example I recently spent some years working on Erlang systems delivering realtime telecom network services to some tens of millions of SIM cards per server. These were absolutely performance-critical systems and running at maximum capacity, but if you ran 'top' on one you'd see CPU usage around 3%.
I'm just saying. :-)
### Then again, highly
Then again, highly concurrent applications used to be relegated to niche applications in which microperformance perhaps didn't matter so much. The larger the userbase, the better the implementation must be in every way.
### That's true. I'm so much in
That's true. I'm so much in the habit of exploiting knowledge of what won't matter for my application that it's hard to put myself in the implementor's shoes. Their users will be doing very weird and unpredictable things and swearing like hell at any feature that's implemented badly.
Really I just think there's so much more to Erlang than fast context-switches and small process footprints :-)
### Their users will be doing
Their users will be doing very weird and unpredictable things and swearing like hell at any feature that's implemented badly.
Indeed. I was thinking of scientific computing specifically, where parallelization certainly helps, but the microperformance of arithmetic, and vector ops can matter a great deal.
Really I just think there's so much more to Erlang than fast context-switches and small process footprints :-)
Absolutely. OTP is something I've been meaning to read up on. :-)
### applies to my concurrent programming
Luke Gorrie: For me there's something surreal in talking about memory locality, branch prediction, etc in the context of concurrent programming.
Those affect what I do lately: WAN optimization via dedup, which involves trading more cpu for less i/o. But yeah, all that code's written in C and C++. (I wish none of it was C but I don't get to choose.)
We're either i/o or cpu bound depending on choices I make. For the luxury of being i/o bound I must care about locality and branch prediction, or else I can max out cpu on memory bandwidth. When we have cycles to spare, we think about spending them on something else putting us closer to perfect balance again.
When I work in a high level language, I'd just as soon keep sensitive parts in C and C++ just like they are now. So it wouldn't hurt if part of a system in a higher level language didn't worry about such things.
### It depends on what you do...
... and as you point out, these aren't everybody's concerns. :-) However, wouldn't it be nice not to have to drop down to C for performance reasons to write the kernel, and instead write everything in Erlang?
Unfortunately, Erlang's implementation sucks when it comes to CPU-bound tasks. It's on par with Guile, which even worse than MzScheme. (No offense to the PLT crowd intended; the PLT platform has other strengths. But my experience is that Petite Chez, an interpreter written in Scheme, often runs code faster than MzScheme's compiler.)
I've been meaning to implement Erlang's actor model and OTP's native network messaging protocol in either Chez or Ikarus. These implementations put a lot of effort into making call/cc fast, something that's typically neglected in other Scheme implementations. I think it would be very interesting to see how this approach to concurrency compares, performance and stability-wise, to OTP's implementation.
However, OTP's networking message protocol is poorly documented. If you are interested, I'd love to try to decipher this protocol with your help.
### How To Implement the Distributed Erlang Protocol
Good timing! I wrote down what I know about it while having my coffee on this quiet saturday morning. :-)
== The protocol ==
The distributed Erlang protocol connects Erlang nodes together in a
network. If you say "{myproc,mynode@myhost} ! MyMessage" then the
runtime system is going to connect to mynode@myhost using the
distributed Erlang protocol and deliver your message.
The protocol starts with a handshake and then enters a two-way request
processing loop. The handshake is for authentication (same cookie?)
and feature negotiation (e.g. do we both support passing funs?).
The main request loop allows each node to asynchronously invoke one of
a small set of operations on the other node:
SEND(pid, message) \ Send a message
REG_SEND(from_pid, name, message) \ Send to a registered name
.. plus more for process monitoring, etc
The encoding of these requests is based on the standard Erlang
external term format. That's the same format that the term_to_binary/1
and binary_to_term/1 BIFs use.
The OTP source distribution contains the details of all this:
lib/kernel/internal_doc/distribution_handshake.txt describes the state
machine for establishing a connection between two nodes.
erts/emulator/internal_doc/erl_ext_dist.txt describes the Erlang
external term format and the main request protocol.
Erlang/OTP itself implements much of the distribution protocol in Erlang.
Take a look in lib/kernel/src/net_kernel.erl for example.
== EPMD ==
You can run a bunch of Erlang nodes on a machine each with their own
name (foo@myhost, bar@myhost, etc). The Erlang Port Mapping Daemon is
a tiny program that maps name->port so that other machines can find
the node they're looking for.
Each time a named node starts it will contact epmd (or start it as a
separate unix process if necessary) and say "hey! I'm node x and I'm
listening on port 12345". Then when another node wants to connect to
x@myhost it first connects to epmd on myhost (with a well-known port)
and asks what port that node is listening on, then connects on that
port with the distributed Erlang protocol.
== Examples ==
These days there are plenty of examples of Erlang distribution
implemented in other languages.
One example is Distel, in Emacs Lisp. There's a paper about it here:
http://lambda-the-ultimate.org/classic/message5450.html
and the code is here:
The most interesting files are:
derl.el is the distributed erlang protocol state machine
epmd.el is the epmd client
erlext.el is the Erlang external term format encoder/decoder
Good luck!
### ETOS
ETOS is an Erlang implementation using Gambit Scheme as an intermediate language. You might find that interesting!
### Re: ETOS
I'm aware of ETOS, and indeed it is worth mentioning. However, I'm more interested in (for the time being?) the run-time aspects of Erlang.
Thanks for the pointers with regard to the distributed Erlang protocol!
### Distel for other languages
I was aware of Distel before I asked, indeed, that's why I asked. However, I'm not particularly aware of other similar projects... do you have a pointer or two?
I'm just talking about libraries implementing the distributed erlang protocol.
erl_interface is a C library and it's part of Erlang/OTP. It's used by some standard seldom-used commands like erl_call to RPC into Erlang nodes.
Jive is the same thing written in Java. It was in OTP around R7 (which is probably still online somewhere) but removed for some reason. It's nicely written and good as a reference for implementing the external term format.
I remember seeing a couple of Python implementations around the time I was hacking Distel. I'd guess there're a bunch more floating around.
### ... what about the manycore
... what about the manycore challenge? Pretending it doesn't exist doesn't mean hardware trends will go away.
### 64K processes is quite a few
64K processes is quite a few but 4KB maximum stack size doesn't sound very appealing. Then you will have to be careful all the time not to recurse too much and crash if you do. In Erlang you don't have this restriction because the stack is dynamically allocated -- you can truly recurse a million levels deep and nothing bad will happen.
I'd suggest the OP search the erlang-questions mailing list for a message by someone like Ulf Wiger or Björn Gustavsson explaining how Erlang represents processes.
I like Squeak's approach to the stack. Everything is an object, including activation records, 'nuff said, next question!
### 64K processes is quite a few
64K processes is quite a few but 4KB maximum stack size doesn't sound very appealing.
Indeed, so that's the maximum level of user-level concurrency achievable using the minimum size defaults. The stack can always be grown in case of overflow, so I don't think it's that big of a deal.
### I'm not to worried about the
I'm not to worried about the deep recursion, there are some tricks one can implement to fix this. I think the clr supports tail recursion correction (i seem to remember something about this) so i should be fine there which would be the 90% case.
I was really worried about interop with the rest of the net framework. after all if everything is an agent what do i do when the thread for a process crashes or hangs or just stalls within a long running net framework call or such, that would really mess up my grean thread system, i can do some fancy jumping and stack saves on my side but not on other peoples code side.
The only solution i can seem to figure is a special system level agent acts like a manager agent for net framework calls. it will spawn agents for specific tasks in a green mode when it knows they will be valid for async work (such as certain network and file calls) and will spawn special agents. otherwise it will spawn true os threads when we are running long running ops or unknown responses.
This means there will be a distinction between agents and functions that I'm not happy with but thats all right, we actually do this anyways when we write the code, I'm just going to have to make it explicit (i hate that).
Does this seem like a reasonable method for the CLR to everyone? user level stacks, a global message queue system, support for linking of green threads,a global 'shout' like message for all in the node. I'm thinking i will definitely need to use the DLR if i want to make nice and easy pattern matching assignments ala Erlang style. The other nice thing about this is if someone tries to call an agent without first setting up the green thread core language construct from another .net system there will be an exception thrown from the attempt to assign a message to a data space thats undefined, i catch this, create the green thread data space, then try again. tada the agent system core is up and running and things work.
This seems like a valid solution but I'm not sure. will have to play with it some to see how things work. It meens a lot of time spent creating stupid agents just for Net framework interop (though a lot of it will be testing which works in a generic like framework).
Any suggestions? hick ups? blatant mistakes?
### Check out Scala actors
I was really worried about interop with the rest of the net framework. after all if everything is an agent what do i do when the thread for a process crashes or hangs or just stalls within a long running net framework call or such, that would really mess up my grean thread system, i can do some fancy jumping and stack saves on my side but not on other peoples code side.
The only solution i can seem to figure is a special system level agent acts like a manager agent for net framework calls. it will spawn agents for specific tasks in a green mode when it knows they will be valid for async work (such as certain network and file calls) and will spawn special agents. otherwise it will spawn true os threads when we are running long running ops or unknown responses.
You should definitely take a look at the implementation of Scala's actors library. It does something very similar to what you describe to achieve scalability despite the presence of non-blocking APIs. This paper may be the best place to start. If I recall correctly, the implementation is based on work stealing and uses Doug Lea's FJ library. Perhaps something similar is available for .NET?
### work stealing
yes the new 3.5 framework parallel fx library uses work stealing for there system but still boils down to basically a fancy os thread system. The green thread system would basically be doing the same thing but from a different direction. I'm not sure about all of the system yet but i know that i wanted erlang agent systems for sure.
I'm still thinking about how i want the rest of the language to work. I truly like the idea of a pure functional language because of the ability to do things like quick immediate assessment.
I use a more organic method of development then oop supports, i tend fiddle test fiddle test fiddle test etc and a functional language with immediate mode support with a strong ide support would be perfect for this. i could just see a tool that you rclick a function and start the tool. the code is assessed using reflection and a ui pops up allowing you to enter values for the input and see the results of the output.
make it so you have some kind of guard system and you could even automate the process ranges to insure correctness. that would be a neat toy. it's a lot harder to do that with internal state (though possible). oh well i'm still thinking about it. if i go with a pure functional system i could even develop a more graphical representation of the flow of each agent and then present two views of the language. one an agent map and the other a flow assessment.
Thats more of a what kind of tools do i want type of thing then a what do i want the language to be type of situation. I've all ready decided on using an xml based representation of the code then using a parser in and out for the view of the language. this seems like a good idea simply because it will allow for different 'language views' based of the style you like. want a c-esq type language? go with a lot of curly braces. want a python like language? go for it. as long as you support in and out of the xml format then the compiler will support the in and out of the xml to the cil.
You might read how Haskell handles foreign calls from lightweight threads. I think the Erlang implementations don't work too hard here, because they would prefer to keep native code entirely out of their address space in the interests of reliability.
Extending the Haskell Foreign Function Interface with Concurrency.
Basically, all your green threads are run by a few worker threads, and you use separate threads to call native code - blocking that green thread on the result and continuing to run the rest with the workers. There's more if things like thread-local storage require you to care what native threads some calls are made from.
Haskell on a Shared-Memory Multiprocessor shows how they extend the runtime to handle several worker threads running in parallel on an SMP systems. It's mostly about safely implementing the under-the-hood updating in pure lazy code, so maybe it's not so interesting. Even if you want to use pure lazy code in each process you might prefer separate heaps and no shared values like Erlang, so you can kill the right process if you run out of memory, and can quickly clean up after dead processes.
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2022-05-25 03:09:24
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https://socratic.org/questions/why-do-only-compounds-that-yield-tertiary-carbonications-or-resonance-stabilized
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# Why do only compounds that yield tertiary carbocations (or resonance‐stabilized carbocations) undergo SN1?
Dec 13, 2014
Compounds that yield tertiary or resonance‐stabilized carbocations undergo ${\text{S}}_{N} 1$ reactions because their activation energies are lower than for ${\text{S}}_{N} 2$ reactions.
You always have a competition between the ${\text{S}}_{N} 1$ and ${\text{S}}_{N} 2$ mechanisms. The question is, "Which one predominates?"
t-Butyl bromide is extremely hindered to back-side attack. So the ${E}_{\text{a}}$ for an $\text{S"_"N} 2$ attack is quite high.
But both hyperconjugation and the electron-donating effects of the alkyl groups stabilize the 3° carbocation.
So the ${E}_{\text{a}}$ for an ${S}_{N} 1$ attack is low, and the $\text{S"_"N} 1$ reaction predominates by a large amount.
Allyl bromide, CH₂=CH-CH₂Br, is a 1° halide. The ${E}_{\text{a}}$ for an $\text{S"_"N} 2$ attack is normal.
But resonance stabilizes the allyl cation, CH₂=CH-CH₂⁺.
The ${E}_{\text{a}}$ for formation of the cation is so low that the $\text{S"_N} 1$ reaction predominates by a large amount.
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2019-07-20 17:43:53
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http://physics.stackexchange.com/questions/28546/plotting-the-cmb-power-spectrum-why-c-ell-ell-ell1-rather-than-only-c?answertab=active
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# Plotting the CMB power spectrum - Why $C_\ell \ell (\ell+1)$ rather than only $C_\ell$?
I can't find any convincing answer for the following question :
Why do we always (or often) plot the CMB power spectrum in this way?
I mean the y axis is $C_\ell \ell (\ell+1)$ and not only $C_\ell$. Why?
I know it's because of the scale invariance, but why do we absolutely want to show the flat line at low $\ell$? And I do not understand why the power spectrum is flat in this scale.
-
I haven't got a great answer for this, but since no-one else has answered ...
As you mention, for the Sachs-Wolfe effect the $C_{\ell}$ values drop off as approximately $\ell(\ell + 1)$ so plotting $C_{\ell}\ell(\ell + 1)$ on the $y$ axis gives an approximately horizontal line and this makes it easy to see deviations from Sachs-Wolfe behaviour. However I suspect the main reason the graphs are drawn this way is that it nicely highlights the doppler peaks. If you just plotted $C_{\ell}$ you'd need to use a log axis and that would make all the peaks look smaller.
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Thanks for your answer. I think you're quite close to the aim of plotting the CMB PS in this way. – Bagheera May 19 '12 at 8:40
Well, because we're really plotting the anisotropy i.e. variations. So they're the Fourier modes not of the temperature $T$ itself but its Laplacian over the sphere, $\Delta T$, and the Laplacian has a simple well-defined effect on the component $C_l$ which is multiplied by a spherical harmonic function $Y_{lm}$: it just multiplies the spherical harmonic function by $-l(l+1)$. That's why $l(l+1)$ may be identified with the (minus) Laplacian. It's more natural to insert the Laplacian rather than not to really measure "variations" and to get rid of the huge constant term prop. to $Y_{00}$, too. – Luboš Motl May 19 '12 at 8:48
I learned in my class on cosmology that
$C_{\ell}\ell(\ell + 1) \propto (\Delta T)^2$
but don't have any sources to back this up, besides, the slides from my professor. See slide 17 and 18 of this talk https://neutrino.ikp.kit.edu/personal/drexlin/data/_uploaded/file/Kosmo1/CS08.pdf , which is unfortunately in german, but the equations should be understandable anyway.
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2015-11-25 14:41:53
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http://math.stackexchange.com/questions/169265/what-is-a-general-algorithm-to-find-a-nonempty-integer-subset-that-have-integers
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# what is a general algorithm to find a nonempty integer subset that have integers add up to 0?
what is a general algorithm to compute if a set have nonempty integer subset that have integers add up to 0?
i would like to know one with the least tries and the proof of it.
Example:{−2, −3, 15, 14, 7, −10} have integers added up to zeros since {−2, −3, −10, 15} add up to zero
i would also like to know the level of it - undergraduate or graduate?
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en.wikipedia.org/wiki/Subset_sum_problem - this problem is believed to be 'hard' (it's NP-complete), and many algorithms are known for it. – Steven Stadnicki Jul 11 '12 at 0:48
@StevenStadnicki - i can't read the computer langauage, may someone put those algorithm in english and math langauage? – Victor Jul 11 '12 at 0:51
@Victor: What "computer language" are you talking about? The only remotely computerlike notation in the Wikipedia article is a block of high-level pseudocode. If you cannot read that, you shouldn't be trying to understand algorithms. – Henning Makholm Jul 11 '12 at 0:56
This is the well known subset sum problem, and there is an $O\left ((\sum x_i) ^2\right )$ dynamic programming (based on a recurrence relation) algorithm. Wikipedia has a nice explanation of it here: http://en.wikipedia.org/wiki/Subset_sum_problem#Pseudo-polynomial_time_dynamic_programming_solution
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2015-01-31 19:17:38
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https://materials.rangeforce.com/tutorial/2020/02/09/Suricata-Intrusion-Prevention-System/
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Suricata is a real-time threat detection engine that helps protect your network against threats by actively monitoring network traffic and detecting malicious behavior based on written rules. It can operate in a network security monitoring (NSM) mode and can also be configured as an intrusion detection system (IDS) or intrusion prevention system (IPS).
An IDS is only able to identify malicious behavior, as opposed to an IPS which can both identify and block malicious behavior – eliminating network threats.
## Intrusion Prevention System
An IDS instance must be strategically placed at a network security boundary, so all network activity originating from inside and outside the network is visible to Suricata. However, when it comes to setting up an IPS, apart from ensuring that Suricata is running in inline IPS mode, it is also crucial to make sure that the IPS is the entry point to the network for other hosts – otherwise, Suricata will not be able to handle all packets.
### NFQUEUE
NFQUEUE is an iptables and ip6tables target which delegates the decision on packets to a userspace software. Depending on the task, you can use NFQUEUE for traffic filtering or traffic shaping. NFQUEUE targets are widely used by both open source and commercial IPS, Suricata in our case.
If Suricata is running on a gateway and is meant to protect the computers behind that gateway, the easiest rule for sending traffic to Suricata is:
sudo iptables -I FORWARD -j NFQUEUE
If Suricata has to protect the computer it is running on, these are the two most simple iptables rules:
sudo iptables -I INPUT -j NFQUEUE
sudo iptables -I OUTPUT -j NFQUEUE
It is possible to set a queue number. If you do not, the queue number will be 0 by default.
### Suricata Setup
In suricata.yaml configuration file:
nfq:
mode: accept/repeat/route
repeat_mark: 1 # repeat mode option
repeat_mask: 1 # repeat mode option
route_queue: 2 # route mode option
Suricata can be set up in different modes. If the mode is set to accept, the packet will, by default, not be inspected by the rest of the iptables rules after being processed by Suricata.
If the mode is set to repeat, the packets will be marked by Suricata and re-injected to the first rule of iptables. To mitigate the packet from going round in circles, the rule using NFQ will be skipped because of the mark. Example:
iptables -I FORWARD -m mark ! --mark $MARK/$MASK -j NFQUEUE
If the mode is set to route, you can make sure the packet will be sent to another tool after being processed by Suricata. Every engine/tool is linked to a queue-number. You can add this number to the NFQ rule and to the route_queue option.
### IPS Rules
The rule action is what draws the line between an IDS and an IPS. Suricata has four types of action. The property of action determines what will happen when a signature matches. Rules will be loaded in the order of which they appear in files. But they will be processed in a different order. Signatures have different priorities. The most important signatures will be scanned first. There is a possibility to change the order of priority. The default order is: pass, drop, reject, alert.
Pass - if a signature matches and contains pass, Suricata stops scanning the packet and skips to the end of all rules for the current packet.
Drop - this only concerns the IPS/inline mode. If the program finds a signature that matches, containing drop, it stops immediately. The packet will not be sent any further. The receiver does not receive a message of what is going on, resulting in a time-out (certainly with TCP). Suricata generates an alert for this packet.
Reject - this is an active rejection of the packet. Both the receiver and the sender receive a reject packet. There are two types of reject packets that will be automatically selected. If the offending packet concerns TCP, it will be a Reset-packet. For all other protocols, it will be an ICMP-error packet. Suricata also generates an alert. When in Inline/IPS mode, the offending packet will also be dropped (like with the ‘drop’ action).
Alert - if a signature matches and contains alert, the packet will be treated like any other non-threatening packet, except an alert will be generated by Suricata. Only the system administrator can notice this alert.
## Conclusion
Suricata can be used as an intrusion detection system or an intrusion prevention system. IDS is only able to identify malicious behavior, while an IPS can both identify and handle malicious packets.
Setting up Suricata in IPS/inline mode requires additional configuration:
1. Iptables has to forward packets to NFQUEUE and mark them (optionally).
2. Suricata has to listen to NFQUEUE in the correct mode.
Suricata rules have to match malicious packets and rule actions have to treat packets properly.
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2021-05-17 04:22:37
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