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# Mirror equation derivation

It begins with the linearization of the ﬁeld equations, demonstration of gauge transformations in the linearized regime, and the writing of a wave equation for small deviations from the background spacetime. This \law" was obtained by William Lawrence Bragg in 1912 (published in 19131) while trying to decipher The Doppler Effect associated with the reflection on a moving mirror is reduced to two Doppler Effect experiments involving the incoming incident ray and the outgoing reflected ray or vice-versa. The sign conventions for the given quantities in the mirror equation and magnification equations are as follows: f is + if the mirror is a concave mirror; f is - if the mirror is a convex mirror; d i is + if the image is a real image and located on the object's side of the mirror. 5 cm tall is placed 30 cm from a Deriving the mirror equation help? Can someone please help me with how you would derive the mirror equation Si*So= f^2 with a Ray diagram for a convex mirror? Follow This law of reflection can be derived from Fermat's principle. Optical Systems. The derivation of mirror formula is one of the most common questions asked in various board examination as well as competitive examinations. Step 1: Find the focal length of the mirror (remembering that convex mirrors have negative focal lengths, by convention). _____ Lensmaker’s equation. Derivation of modular anomaly equation in compact elliptic Calabi-Yau spaces Minxin Huang University of Science and Technology of China atEast Asia Joint Workshop on Fields and Strings 2016 Based on MH, S. Everything can be taken from a man but one thing; the last of the human freedoms — to choose one’s attitude in any given set of circumstances, to choose one’s own way. In general, the light will enter the interface between the two medii at an angle. Precautions in “Michelson interferometer diagram and derivation”. Since the OP is eager to know the answer and I already prepared a detailed explanation, I decided to create a new question, asking the following: a 256 The Mirror Equation and the Magnification Equation 779 The Mirror from PHYS 152 at College of Southern Nevada Chapter 10: Mirror Reflection. Consider an object AB placed in front of a concave mirror M beyond the centre of curvature C (see figure below). NOW THE DERIVATION MIRROR EQUATION + = Exterior Angle of Triangle = Sum of Opposite Internal Angles . Due to the geometrical symmetry, the thin lens equation can be used for a convex and concave mirrors. 16-1 Two steps in the derivation of the general gas law; . u is the distance between the object and the mirror and v is the distance between the image and the mirror. F u f v object image Active Loads and IC MOS Amplifiers ECE 102, Winter 2011, F. 6, using the geometry of similar concave mirror. Appendix: Solving the diﬀerential equation, eq. Najmabadi Sedra & Smith Sec. ~~~~~ The first kind we will be looking at is a concave mirror, which would mean that you made the inside of the ball shiny. (7). Back to Course Grade10-CBSE-Physics 0% Complete 0/143 Steps Reflection and Refraction 40 Topics A Review of Reflection Review of the Laws of Reflection of Light… Refraction at Spherical Surface. (f) is given by mirror equation f d d. com. In most monochromators, the input slit and collimating mirror fix the direction of the input beam that strikes the grating. •Geometry can be used to Convex Mirror, Equations . Indeed, y(x) is a parabola, whose symmetry axis coincides with the y-axis. That is, under these conditions a grating made of grooves coated on a transparent substrate, such as optical glass, does not diffract and exhibits the refraction properties of the glass. For I = 100 µA, find the V SD and V SG for R = 0, 10k, 30k, 100k. If the Equation 13 is called the lensmaker’s equation. A Section 2: The Lens Equation 6 2. Combining ray tracing with the mirror equation is a good way to analyze mirror systems. We are assuming that a lens or mirror will form a point image of a point object, and that a parallel beam entering a lens will come to a point focus. Cascode Amplifiers and Cascode Current Mirrors ECE 102, Fall 2012, F. )Here is an example: Graphing. 0. In the following table, we can examine the effect of each component's reliability on the overall system reliability. The Mirror Formula (also referred to as the mirror equation) gives us the relationship between the focal length (f), distance of object form the mirror (u) and the distance of image form the mirror (v). Measuring the index of refraction of air In this part of the experiment you are aiming to carefully control the vacuum within the air cell that you should place in the beam path in between the fixed mirror and the beam splitter. A mirror formula can be defined as the formula which gives the relationship between the distance of object ‘u’, the distance of image ‘v’, and the focal length of the mirror ‘f’. The Lens Equation An image formed by a convex lens is described by the lens equation 1 u + 1 v = 1 f where uis the distance of the object from the lens; vis the distance of the image from the lens and fis the focal length, i. Equations. We can also calculate these things precisely, using something known as the mirror equation. Ray diagrams are useful for determining the general location and size of the image formed by a mirror. . Despite the heroic efforts of Valerie Censabella (for which I am very grateful) to translate my hand-written materials into LaTeX, and extensive editing on my part, I don't doubt that there are many typographical errors. Synopsis and Discussion of Derivation of Line! of!Sight Stabilization Equations for Gimbaled!Mirror. Spherical coordinates with azimuthal symmetry about the polar axis were used. The equation accounting for the index of refraction, , is therefore: In the equation above, L represents the geometrical length of the air cell. In order to show that the Einstein di usion equation (3. com In 1905 Albert Einstein published his derivation of what has become known as Special Relativity. This chapter is concerned with the Special Theory of Relativity, which dates from 1905. A convex mirror is a spherical reflecting surface, where the bulging side of the mirror faces the source of light. how to derive the mirror formula for convex mirror - Physics - Ray Optics And Optical Instruments Synopsis and Discussion of "Derivation of Line-of-Sight Stabilization Equations for Gimbaled-Mirror Optical Systems Keith B. , (r(t)−r(t0))2 ˘ t. For an object which is very far away from the mirror (i. eq. ConnectedObjects. Now from A, walk a little away from the mirror and mark it  Solving each equation for L yields: Therefore. Ray Optics 06 : Mirror's Formula -Derivation for Concave & Convex Mirror +Basic Numericals + Concept video for JEE is made by best teachers who have written some of the best books of JEE. Webster a1rational@gmail. . The technique was applied to a magnetic-mirror system for confining charged par- Derivation of Lens Equation: The magnification M of a lens is the ratio of the image size or height to the object height. (7) Focal length and radius of curvature of a concave mirror are positive where as that of convex mirror negative. 1 Derivation of the Gain of a Cascode Stage I went over a derivation of the gain of a cascode stage during discussion that was incorrect (the end result was correct, but the methodology was ﬂawed). The light then returns from each mirror and is recombined at the output and viewed on a screen. A general principle for light reflecting from a uniformly moving mirror: A relativistic treatment J. 2 Prelab Questions 1. To obtain this type of numerical information, it is necessary to use the Mirror Equation and the Magnification Equation. There are really two Einstein theories of relativity. the negative supply rail. The mirror equation. Katz and A. o Derive the Poiseuille equation for fluid flow in . 7 (MOS portion) (S&S 5. 2 2 2 2 2 2 a h x a a h h y − − − = , 4. To measure the position of the focal point relative to the 10cm predicted by the lens makers equation the stick was moved into position along the axis of the lens. By convention, distances are measured, along the central axis, as positive from the mirror in the direction of the object and A multi-scale model of a Micro-Mirror Array and an automatic model derivation tool heat equation in a Micro-Mirror Array developed for astrophysics is presented together with some key elements Provide the time dilation equation found in Section 15. 1 2x2 matrix The 2x2 Jones matrix is just a simple shorthand way to represent the polarization state of light. (Jones matrix). When source and observer are stationary, observer sees waves of frequency ν or wavelength λ. Let us do a quick activity. 3 x 1014 m2 P abs = W eA c PDIFFERENT DOMAINS AND DETERMINATION OF THE RELATED arXiv . " in Physics if there is no answer or all answers are wrong, use a search bar and try to find the answer among similar questions. Because the two transistors have their base Conceptually, an ideal current mirror is simply an ideal current amplifier with a gain of -1. However, the mirror equation and magnification equation give a more accurate mathematical description of the image light source to the rotating mirror and the rotating mirror the second reflected beam is related to the time that was required by the light to travel the distance between the fixed and rotating mirrors. by Ron Kurtus (20 February 2015) By examining the relationships between the various factors in the Cavendish Experiment, you can derive the equation for the Universal Gravitational Constant, G. Keith B. Derivation of the Boltzmann integral 6. If we position a hyperbola in the plane with its center at the origin and its foci along the x axis we can obtain a nice equation for a hyperbola. L´evya Laboratoire de Physique Nucl´eaire et de Hautes Energies, CNRS - IN2P3 - Universit´es Paris VI et Paris VII, Paris. In the derivation of the time dilation equation, what are the two events? O the emission of light on source B and the reflection of the light on mirror M O the reflection of light on mirror M and the absorption of light on source B. How to derive the kinetic gas equation . Since in case of convex mirror, image is always virtual and according to sign conventions, distance of virtual image and focal length of convex mirror is taken as negative. First time I've experienced a real problem rearding equations in physics, so: As conclusions to the lab on focal points and and center of curvatures (which will be explained at a later date), we were given the following relationships: So = Do - f Si = Di - f And, we were ably to verify the Gaussian Hello everyone, I really run into a problem here. Fizeau). Spherical Mirror Equation #1. It is beyond the scope of this applet to derive the Gaussian lens formula. The remainder of the light is scattered and absorbed inside the monochromator. 8 nm. Consider three components arranged reliability-wise in series, where , and (for a given time). Deriving Kepler’s Laws from the Inverse-Square Law . Powell. ) and Ro . One of the easiest shapes to analyze is the spherical mirror. 10 meter. Smith Oregon Center for Optics and Department of Physics University of Oregon, Eugene, Oregon 97403 ABSTRACT James Clerk Maxwell unknowingly discovered a correct relativistic, quantum theory for the light quantum, forty-three years before Einstein postulated the photon’s existence. This expression relates the object distance, the image distance, and the radius of curvature of the mirror. Equation definition, the act of equating or making equal; equalization: the symbolic equation of darkness with death. Light is an electromagnetic wave: this was realized by Maxwell circa 1864, as soon as the equation c = 1/(e 0 m 0) 1/2 = 2. There are two types of curved mirrors; concave and convex mirror. d i is - if the image is a virtual image and located behind the As opposed to merely calculating the magnification of a mirror system, this equation is often used to compare the sizes of objects and images with their locations. The law of reflection gives the familiar reflected image in a plane mirror where the image distance behind the mirror is the same as the object distance in front of the mirror. We will not go through all the details here but set up the machinery for you to obtain the equation. 50% of this beam passes straight through beam splitter and reaches the screen. Current mirrors are basic building blocks of analog design. There are many ways to derive the Lorentz transformations utilizing a variety of physical principles, ranging from Maxwell's equations to Einstein's postulates of special relativity, and mathematical tools, spanning from elementary algebra and hyperbolic functions, to linear algebra and group theory. Object AB is place beyond C and the real image formed by it is A'B' between F and C. Substituting 13) into 12) gives. Why E = mc2 Let us today discuss and derive an interesting concept in relativity called the length contraction. 2). I know many questions have been asked already regarding this circuit on EE SE, but I didn't find an answer to my question. The Distance of the principal focus from the pole of the mirror is called the Focal length (f). Nov 1, 2016 Mirror formula is the relationship between object distance (u), image distance (v) and What is the derivation of the mirror formula? Equation (2) becomes. W. Nov 9, 2009 An image formed by a convex lens is described by the lens equation. FIGURE 1 (Encyclopedia Britannica) To understand why this apparatus measures the speed of light, the primary equation this experiment is centered around was derived. The Physics Classroom, 2009. derivation of the equation using simple techniques of conservation of momentum and the Reynolds transport theorem, along with a basic knowledge of singularities and circulation. focal length, f, is the distance between the Michelson interferometer. Powell OPTI-521 Project 1 Steward Observatory, University of Arizona Abstract This paper is a synopsis and commentary on the technical paper by DeBruin (1991) on the derivation of stabilization equations for gimbaled The Mirror Equation and Magnification Example. Simpler to compute open-loop gain (Avo. If propagating need to have distance d Formulas for Bayesian A/B Testing. Online physics calculator that calculates the concave mirror equation from the given values of object distance (do), the image distance (di), and the focal length (f). From equation 1 we can develop a relation to determine the index of refraction of air. The reflection of light from a moving mirror has been studied in many different contexts in the last century [1–11]. Derivation of Time Dilation! Imagine a light clock which consists of two mirrors and beam of light reﬂecting back and forth between the mirror! One “tick” is when the light goes from one mirror to the other and back again! Scenario 1! You are in the same inertial frame as the light clock! You are therefore measuring the proper time A formal derivation of the quasi-linear equation of a plasma in a magnetic field is presented. The mirror equation, giving the relationship between the object distance, the tangential image distance, the radius of curvature of a concave mirror, and the angle of incidence, is derived in a way which is different from that appearing in the literature. jpg Download Citation on ResearchGate | Reflection from a moving mirror - a simple derivation using the photon model of light | We present an elementary analysis of the effects on light reflected from Derivation of the equation For the derivation of the equation Hooghoudt used the law of Darcy, the summation of circular potential functions and, for the determination of the influence of the impermeable layer, de method of mirror images and superposition. Derive the mirror formula for concave mirror. How to solve the Mirror and Lens Problems. See more. Image in the Public Domain . Find the location of the image and the magnification. The demonstration includes isolated bod- ies, infinite cascades with application to rotating fluid machines, and pairs of identical or mirror-image bodies A Bragg mirror (also called distributed Bragg reflector) is a mirror structure which consists of an alternating sequence of layers of two different optical materials. Let us now see the refraction of light at the spherical surface. Mirror Equation. Derive the above equation and calculate λ and compare to the specified value of 632. Moment of Inertia and Rotational Dynamics. pdf; Friction Practice with answers. -M. Experiment 8 Michelson Interferometer 1 Introduction The Michelson interferometer uses light interference to measure distances in units of the wavelength of light from a particular source. Half of the light is sent along path B towards the movable mirror and the other half is sent down path A towards a fixed mirror. In this video David derives the mirror equation and magnification equation. known signal path mirror signal arm light source beam splitter unknown path lens photo-detector where Ar, As are the amplitudes and φr, φs the phase delays introduced after one round trip Paraxial Ray Tracing Derivation of refraction and transfer equations y & d in meters U, u’ in radians ϕIn diopters Want to know what happens to rays as they propagate in air and interact with lens and mirrors etc. The following information is provided to the student :. to order 1. The mirror formula for a concave mirror is given below. The relation to GW theory arises from mirror symmetry, which is a duality between type IIA and type IIB string theories. For example, when a problem states that a real image is twice as large as an object this requires that you use the relationship d i = 2d o in the mirror equation. By this convention, a positive height indicates the arrow is erect and a negative height means the arrow is pointing down (valid for both object and image). Towards an automatic tool for multi-scale model derivation illustrated with a micro-mirror array⋆† Walid Belkhir 1,2, Nicolas Ratier3, Duy Duc Nguyen 1, Bin Yang , Michel Lenczner , Fred´ eric Zamkotsian´ 4 and This entry was posted in physics, science, society and tagged Albert Einstein, derivation, Einstein, explanation, Mirror, photon, popularization, Speed of light, Theory of relativity, Thought experiment, time, time dilation on May 2, 2013 by David Yerle. 1 The focal point of a spherical concave mirror is (approximately) halfway between the center of curvature of the mirror (C) and the middle of the mirror at B. 5) reproduces the well-known di usive behaviour of particles we consider the mean square displacement of a particle described by this equation, i. Snell’s Law describes how a light ray behaves when it passes from a medium with index of refraction n 1, to a medium with a different index of refraction, n 2. e. The Newtonian Lens Equation We have been using the “Gaussian Lens Formula” An alternate lens formula is known as the Newtonian Lens Formula which can be easily verified by substituting p = f + x 1 and q = f + x 2 into the Gaussian Lens Formula. Ed: Sec. The formulas on this page are closed-form, so you don’t need to do complicated integral evaluations; they can be computed with simple loops and a decent math library. The standard symbols for object space distance and image space distance are s o and s i, respectively. The equation for image formation by rays near the optic axis (paraxial rays) of a mirror has the same form as the thin lens equation if the cartesian sign convention is used: From the geometry of the spherical mirror, note that the focal length is half the radius of curvature: The sign conventions for the given quantities in the mirror equation and magnification equations are as follows: f is + if the mirror is a concave mirror; f is - if the mirror is a convex mirror; d i is + if the image is a real image and located on the object's side of the mirror. Kahn, "Geometric Optical Derivation of Formula for the Variation of the Spot Size in a Spherical Mirror Resonator," Appl. A Quadratic Equation in Standard Form (a, b, and c can have any value, except that a can't be 0. Q. Concave Mirrors. The Gaussian mirror equation, also known as the mirror and lens equation, relates the object distance d o {\displaystyle . Using the relations of the experimental setup, Equation 1 was used to determine the speed of light. that the image is formed v cm from the mirror, then u, v and f are related by the equation;. 5) If v=speed of object towards the mirror then 2v is the speed of the image towards the object. You must be astonished to know that when an object moves with a speed comparative to the speed of light, its length contracted or more appropriate term is that it is apparent length contraction in relativity. 50% of the returning beam is then reflected by the beam splitter and strikes the screen, E. Once these two functions are known, the optical properties of the LCD can be calculated. Image Formation by Reflection—The Mirror Equation. As the derivation by Schawlow and Townes largely builds on previous results, which are not well available, it is desirable to have a more comprehensive and complete derivation, which can also generalize the results. The image on a convex mirror is always virtual, diminished and upright. 1 The physics of mirror reflection. 1. pdf; Friction WS ANSWERS. If z = f(x) for some function f(), then –z = jf0(x)j–x: We will justify rule 1 later. It was developed by Albert Michelson and used, in 1893, to measure the standard meter in units of the wavelength of the red line in the cadmium spectrum. G. Derivation of formula for convex mirror: Let AB be an object placed on the principal axis of a convex mirror of focal length f. The lens equation relates the focal length, determined by lens shape, to the distances between an object, the lens and the projected image. Fig. 7. of KMT for ideal gasses and a rigorous derivation of the kinetic gas equation from . The derivation given here is for an ideally shaped mirror and involves no approximations. This equation is non-linear in Q, but it reduces to familiar linear equations for growth and decay near the beginning at Q ˇ 0 and near the end at Q ˇ Q1. This derivation is not as simple as the title of Pr. In setting up the circuit, the two posts at one side should be joined by a fairly long wire. Proof:- To prove this formula, let AB is an object lying beyond centre of curvature (C) on the principle axis of concave mirror, then image will be formed between centre of curvature (C) and focus(F) of the mirror. 08 since the focal length of convex mirrors are negative . - Solution λ = 0 (no channel length modulation) Graphing Quadratic Equations. 4, 758-759 (1965) Export Citation BibTex We will assume this “law of reflection” of photons from a mirror surface is valid for the below calculations of a photon reflection from a mirror. The . Because the the derivations are simplified by the use of VCE. 1 Setup A formal derivation of the quasi-linear equation of a plasma in a magnetic field is presented. (For paraxial rays, MP can be considered to be a straight line perpendicular to CP). Gradient flow around a cyclone is sub-geostrophic, and that gradient flow around an anti-cyclone is super-geostrophic. Derivation of the mirror equation Derivation of the mirror equation Burrows, John W. • A concave mirror caves in on the object; whereas • A convex mirror ﬂexes away from the object. The magnification equation for mirrors describes such a relation: M=-distance of image/distance of object = height of image/height of object. Conclusions The derivation of Lorentz transformation is the keystone of the Relativity The-ory. 2 Never touch the lens or mirrors with your fingers, when you working with them. Lloyd’s Mirror is used to produce two-source interference patterns, similar to the pattern produced by laser light passing through two slits. ΔK = W = FΔs = maΔs. There are really two basic magnification equations: the lens equation and the magnification equation. This can lead an inaccurate measurement for your experiments. Note mirror image nature of force balance across the equator. Derivation of Bragg’s Law, also known as the Bragg scattering condition There are (at least) three ways to derive the Bragg scattering condition, applicable to electromagnetic waves, usually X-rays, scattering from a crystalline solid. DIFFERENT DOMAINS AND DETERMINATION OF THE RELATED . The formula for spherical mirrors is. [Hint: Draw a ray that is not one of the three principal rays, as was done in the derivation for a concave mirror. The distance of the Image from the Pole of the Mirror is called the Image distance (v). Photons are often considered to have no inertial mass because their invariant mass m is zero, as calculated from the relativistic energy-momentum equation E 2 = p 2 c 2 + m 2 c 4 . It is referred to as a diverging or a curved mirror. In Chapter 8 we explored the transistor and you should recall that the BJT device is a current amplifier of sorts (current controlled current source) in that the collector current is β times the base current. Here I will present a method that still considers the stages as a cascade of a common-emitter and common-base stage, but does so correctly solutions to the electromagnetic wave equation exist that take the form of narrow beams { called Gaussian beams. It is straightforward to show that there are several functions that are good candidates for . Using geometry the two right-angled triangles A'B'F and MPF are similar. We rst These are transcriptions of the notes from which I teach the single semester course "Introduction to Plasma Physics". October 1, 2015 . [math]\displaystyle\frac{1}{v} + \frac{1}{u} = \frac{1}{f}[/math should be clearly labeled with a letter. c = 5. Mirror Formula And Magnification In a Spherical Mirror, The distance of the object from its pole is called the Object distance (u). 4) The image distance equals to the object distance. Both are needed to compute the magnification of an object by a convex lens. The derivation given here is for an ideally shaped mirror and involves no approximations Bipolar Junction Transistor or BJT Current Mirror An often-used circuit applying the bipolar junction transistor is the so-called current mirror, which serves as a simple current regulator, supplying nearly constant current to a load over a wide range of load resistances. The differences between the various types of Cassegrains are the figures of the primary and secondary mirrors, and they will be covered in the next section,  Sep 5, 2019 We will start with reflection of light and spherical mirrors and gradually proceed to cover advanced topics such as refraction, prisms, total  I present a new derivation of the analytic form for the phase shift near resonance and the optical equation for LM under the assumption of mirror ge- ometries  equation describing each loss is given. 2. The mirror equation expresses the  Jul 28, 1998 The mirror equation, giving the relationship between the object distance, the tangential image distance, the radius of curvature of a concave  Use the mirror equations to determine location, size, orientation, and nature of images . Refraction at Spherical Surfaces. important derivations from optics for 12th . Of course, Kepler’s Laws originated from observations of the solar system, but Newton ’s great achievement was to establish that they follow mathematically from his Law of Universal Gravitation and his Laws of Motion. For a circular lens in air, the diameter of the spot is given by method or to stick to conventional equations such as equation 2. Athermal Bonded Mounts Chris Monti 5 Solving for h, it can be seen that the thickness of the bond is a function of the optical radius and the three CTEs, as follows: b c h ro c o α α α α − − = ( ) (2) Equation (1) can also be used to calculate the radial strain, if it is not assumed to be zero, where radial strain is a function of Perhaps most dramatically, the Dirac equation predicted the existence of antimatter – the mirror image of all known particles. by the Wien displacement law. Najmabadi Placing a current mirror in the source circuit will not Work! equation for Q1 is (note . In this section, we will discuss single refraction for a circular boundary, where a beam of light passes from one medium into another. (1) From similar triangles A'B'F and DNF or EVERYTHING HERE IS FOR CONCAVE MIRROR. The focusing mirror and exit slit fix the output direction. However, the question was initially unclear and the thread was put on hold. 1 shows the Note that this equation predicts that IO < IREF unless β → ∞. compared with that of the Fokker-Planck equation. 1. Like refraction, mirror reflection is a phenomenon that occurs at the boundary between two substances. Step 3: Use the magnification equation to relate the object distances and heights. What is meant by an ultrasonic sound-wave and what is the di erence between a sound-wave and an electromagnetic wave? The Metric and the Wave Equation • There is a long chain of reasoning that leads to the notion of gravitational waves. 4 . Since these are complex ratios, they describe not only the relative amplitude, but phase shifts between the waves. Walter K. IMPORTANT DERIVATIONS (3 marks derivations) 1. As can be seen the simulation gives very close agreement to the lensmakers equation. The equivalence of mass and energy expressed by the relation E = mc 2 is a fairly direct consequence of Maxwell’s conception of electromagnetic waves. Derivation of Gravitational Constant from Cavendish Experiment. 2. In 1915 Einstein published an additional theory, called the General Theory of Relativity. Inside, a light is shone upwards to a mirror on the ceiling, where the light reflects back down. 5. Derivation of the velity profile . Mirror Formula (Concave Mirror) Mirror formula is the relationship between object distance (u), image distance (v) and focal length. Mirror formula is the relationship between object distance (u), image distance (v) and focal length. Hagen Poiseuille Equation Derivation Pdf 12 -- DOWNLOAD (Mirror #1) 3b9d4819c4 20120903 P620 13C Lec 09 (Work) Mod2 PtrPhy 03 Perm Dev . In both cases, a portion of a light ray's energy is sent in a new direction when that ray hits a point on the boundary. We assume the origin (0,0) of the coordinate system is at the parabola's vertex. DERIVATION OF MIRROR FORMULA - Physics - Ray Optics And Optical Instruments. (and a complicated equation). The surface may be either convex . Thus, the parabola is the unique curve such that all incoming on-axis parallel rays pass through the focal point F after reﬂection oﬀ the mirror. The radius of curvature R is positive for a convex surface and negative for a concave surface. Stand in front of a mirror and mark your position with a colored tape as A. Now, in order to derive the magnification formula in terms of the object distance and image distance, we will draw the ray-diagram by using the incident rays of light (i) one which strikes at the pole of concave mirror, and (ii) another which passes through the centre of curvature An object is located at the center of curvature C of a concave spherical mirror with principal focus F. The derivation of Equation 3-4 is straightforward, but a bit tedious. First, the relativity principle can be restated as a mirror principle based on the space–time exchange-inversion operation. 6 MOS portion & ignore frequency mirror reference arm The relative length of the unknown signal path compared to the reference path can be established (mod 2π). The focal length of the mirror is 46 cm in back of the mirror. Now, the change in direction or bending of a light wave passing from one transparent medium to another caused by the change in wave’s speed is the Refraction. Spherical mirror equation Relation for single spherical surface Lens maker’s formula (thin lenses) Equations for focal length and principal planes (thick lenses) Practicing: Sign conventions for radii and focal points of curved reflecting and refracting surfaces. The equation to the incoming light ray is just y = h, and the equation to the mirror surface is x2 + y2 = a2. Mirror Formula: 1/u + 1/v = 1/f where in u = object distance to the mirror v = image distance to the mirror f = focal length of the mirror \snc2D\optics\Mirror Equation Questions. Typically such a mirror is not a complete sphere, but a spherical cap — a piece sliced from a larger imaginary sphere with a single cut. The textbook does a nice job of deriving this equation in section 25. Only wavelengths that satisfy the grating equation pass through the exit slit. The theory accounts for spatial inhomogeneity in one direction and for bounce motion of particles confined in a mirror magnetic field. The constitutive equation depends on load • This test is a mirror image of stress equation. Mirror loss, Fa: The ellipsoidal mirror will absorb some of the flux  If the device is composed of thin lenses and mirrors that obey the thin lens equations, then it is not difficult to describe their behavior numerically. Everywhere I see the derivation of the mirror equation is given by placing an object before the focus and then proving similarity of the triangles to get to the form derivation of mirror formula as well as lens formula - Science - Light - Reflection and Refraction Mirror Equations of Curved Mirrors We use the given picture below to derive the equations of concave and convex mirrors. Wien's Displacement Law Derivation Pdf 14 >> DOWNLOAD (Mirror #1) 3b9d4819c4 Chemistry 795T Black body Radiation - Nc State UniversityBlack body Radiation NC State University . Step 1) The concept of flats mirror,concave mirror/lens and convex mirror/lens should be clear. A little bit of coordinate geometry will enable us to determine that the equation to the reflected ray is [(2 )]. The formula used are Mirror equation/lens maker equation and Magnification equation. Since LCD is based on polarization manipulation, the Jones A curved mirror is a mirror with a curved reflecting surface. Debye Huckel Onsager Equation Derivation Pdf Download >> DOWNLOAD (Mirror #1) Debye Huckel Onsager Equation Derivation Pdf Download >> DOWNLOAD (Mirror #1) Derivation of time dilation Applying the above postulates, consider the inside of any vehicle (usually exemplified by a train) moving with a velocity v with respect to someone standing on the ground as the vehicle passes. 38 Gradient Wind Equation The previous slide shows us the balance of forces required to make Gradient flow occur. What is the diffraction limit of a telescope? The minimum angular separation of two sources that can be distinguished by a telescope depends on the wavelength of the light being observed and the diameter of the telescope. More specifically, the OP wanted to know how to find the equation of the mirror image of the given parabola. Lévy’s article suggests it. This page collects a few formulas I’ve derived for evaluating A/B tests in a Bayesian context. Microscopes  Basic BJT Current Mirror. •The geometric derivation of the equations is very. When light from the laser encounters the beam splitter it is divided in half. Mirror Formula for Convex Mirror when Real Image is formed. Therefore, A'B' B'F MP PF = or A mirror. general form of the gas law from the above equations. 1986-05-01 00:00:00 The mirror equation, giving the relationship between the object distance, the tangential image distance, the radius of curvature of a concave mirror, and the angle of incidence, is derived in a way which is different from that appearing in the literature. Antimatter was later found to exist in the real world. Ronald Galli and Farhang Amiri Weber State University, Ogden, Utah 84408-2508 (Received 14 September 2011; accepted 8 May 2012) We investigate the reﬂection of light from a mirror moving at relativistic speeds and introduce a The current balance is shown in Fig. The complete derivation of each loss is given in [15]. Derivation of Mirror equation Mirror equation is the relation between the object distance (u), image distance (v) and the focal length (f). Explain each step of the derivation. Algebra Based Physics. 6 MOS portion & ignore frequency 6. Teacher's Guide. 1/f=1/p+1/q, Where f is the focal length and p is the distance from the object to the lens and q is the distance from the lens to the image. Once again, the Green's function satisfies the homogeneous Helmholtz equation (HHE). doc SNC 2D Mirror Equation Questions • based on the geometry of the rays theoretically reflecting from a curved mirror, the mirror equation is derived to be: d o = distance to object d i = distance to image f = focal length • the conventions for this equation include: • all distances are measured 2 LIGHT: REFLECTION AND REFRACTION DERIVATION OF MIRROR FORMULA. The mirror equation relates the image and object distances to the focal distance and is valid only in the small-angle approximation (Equation \ref{sma}). The image A 1 B 1 is formed at a distance 'v' from the mirror. Another Derivation of Mass-Energy Equivalence . Brooklyn College. The image is real if the rays actually converge to the point; it is virtual if the rays do not actually meet but appear to diverge from the point when produced backwards . By Evan Miller. Start from the work-energy theorem, then add in Newton's second law of motion. Let, P = pole and F=principal focus. The Maxwell wave function of the photon M. Step 2) The sign convention and ray diagram should also be clear The simplest object is a mirror, and the law for a mirror is that when the light hits the mirror, it does not continue in a straight line, but bounces off the mirror into a new straight line, which changes when we change the inclination of the mirror. th. i. Thus VCE is  Lengthy analysis to find Av. Answer to derivation of mirror formula for concave mirror when the image is virtual calculation (1820's). Deriving the mirror equation help? Can someone please help me with how you would derive the mirror equation Si*So= f^2 with a Ray diagram for a convex mirror? Follow Derive the magnification equation, m = h’ / h =-q / p, for a convex mirror. The derivations of the ray matrices of a convex spherical mirror and a thin lens are very similar. Aspheric Surfaces • Simple optics uses spherical surfaces • Spherical surface is defined by the radius of curvature only • But to correct many aberrations need aspheric surface • Aspheric from Greek: a means not: thus not spherical • Must have curvature different with radius r from optic axis The Schrödinger-like equations for the nonadiabatic escape of charged particles from magnetic mirror traps are derived starting from the integral equation for the evolution of the quantum mechanical wavefunction for the problem. In this paper, Einstein's simple derivation of the LT is revisited and analysed. The bracketed term in the above equation ( ) referred to as the complex amplitude, contains all the information needed to determine the state of polarization of the optical field. A Virtual Image Formed by a Convex Mirror A convex mirror is used to reflect light from an object placed 66 cm in front of the mirror. This is a derivation of the time dilation formula. Similar right triangles should be identified; algebraic manipulations should be shown in an organized, step‐by‐step fashion. Let us study the most basic Current Mirror circuit, consisting of Design of a Current Mirror DC Sink We now divide the two equations and use all the given. The reflected beam (R) travels to mirror M 2, where it is reflected. This formula is called mirror equation. Please note the following so you can understand how the thin len equation can be derived: This derivation is not fundamental, but it is more intuitive. f = ½ R with R being the radius of the mirror. 3 mirror, and the number of teeth in the wheel, the speed of the light pulses can be formulated (H. Given a parabola with focal length f, we can derive the equation of the parabola. Using triangles 2 and 3, (1) Derivation of mirror equation formula video for Class 10 is made by best teachers who have written some of the best books of Class 10. 1) Plane mirror gives a virtual image 2) Laterally inverted image 3) The minimum size of mirror must be half the size of the object to get the full image of the object in the mirror. The derivation of nouns ending in -er from verbs is productive in English, but the derivation of nouns in -th from adjectives is not: it is hard to expand the set of words of this type such as depth, health, length, strength, and wealth. The size of the spot is determined by the diameter of the lens or mirror. It can be shown that equations 12), 13), and 14) are applicable to all thin lenses through the use of the following sign conventions: 1. The simplest way to solve for the output current is to sum the currents at the node where IREFenters the mirror. ” Back to Course Grade10-CBSE-Physics 0% Complete 0/143 Steps Reflection and Refraction 40 Topics A Review of Reflection Review of the Laws of Reflection of Light… If a lens is thicker than that measure, the thin lens equation cannot be used. odt; ConnectedObjects. Some authors use the same convention for mirrors as for lenses but this results in formula 1) being different for mirrors. Michael Fowler, UVa. Let AB be an object lying beyond the focus of a concave mirror. The emitters of the two transistors are shown connected to ground. 3 Always in touch with your instructor for further information. A negative magnification signifies an inverted image. 3 MIRROR FORMULA A' 4 LI = Lr [laws of reflect] R. Draw a ray diagram as part of the solution. physics derivation for optics . AB,- Again, [Sides are proportional] In C & ,B,C LB LB'[both 90] AB = [Sides are proportional] From (i) and (ii) -(ii) B'C AB BC BP. , ), so that light-rays from the object are parallel to the principal axis, we expect the image to form at the focal point of the mirror. The conducting ground plane is a mirror that creates the lower half of the dipole as the mirror image of the upper half. The derivation of the equation is almost identical to the derivation of the equation for the ellipse. , Oct 20, 2006. I understand how this formula can be proved using a ray diagram for concave mirrors simply In this video David solves a few exmaple problems involving concave and convex mirrors using the mirror equation and magnification equation. Concave Mirror Equation Calculator. To follow ray you have to have y (ray height) and u ray slope. This can be seen from this equation itself, or from the coming proof that equation (1) is the solution to equation (3). Thomas Riepe's answer explains some of the more classical reasons why we might be interested in period integrals and Picard-Fuchs equations, so let me say a few words about the relation to Gromov-Witten theory. Take the the appropriate equation from kinematics and rearrange it a Refraction and Snell’s Law Reading - Shen and Kong – Ch. In contrast to the usual problem of reflection from a stationary mirror, the situation with a moving mirror gives rise to several important deviations from the classical case . Mirror definition, a reflecting surface, originally of polished metal but now usually of glass with a silvery, metallic, or amalgam backing. A better aspheric equation There is growing momentum in the optical engineering community to change the way we describe aspheric surfaces. Although it was derived for a concave mirror, it also holds for convex mirrors (proving this is left as an exercise). pdf; ConnectedObjectsANSWERS. Derivation using algebra alone (and assuming acceleration is constant). Printer Friendly Version: Refer to the following information for the next four questions. This approach places more emphasis on the physics and conceptual understanding of the Friis equation than is provided by the traditional derivation based on antenna impedance. Opt. The justiﬁcation is easy as soon as we decide on a mathematical deﬁnition of –x, etc. While deriving equations we use the similarities of triangles given picture above. The Fresnel equations describe the ratios of the reflected and transmitted waves' electric fields to the incident wave's electric field (the waves' magnetic fields can also be related using similar coefficients). Debye Huckel Onsager Equation Derivation Pdf 35 DOWNLOAD (Mirror #1) 99f0b496e7 Using a carefully justified development of. 10. The other two posts are used to connect the balance in the circuit (Fig. Materials Required: Diagram of concave mirror with Curved mirrors come in two basic types: those that converge parallel incident rays of light and those that diverge parallel incident rays of light. "Morphological patterns that can be systematically extended are called productive. good mirror-like reflections, even. g. com Reflection by spherical mirror, sign DERIVATION OF PHYSICS EQUATIONS Tom Goodnews; The Wave Equation: Derivation (Walter Lewin, MIT) Ray Tracing, Mirror Equation, Magnification by Yau-Jong Twu. AB BC B P PB PB-PC P B PC-PB Dividing whole equation by uvR Derivation or Proof-of- the relation between focal length and radius of curvature(X) physics [R=2f] Consider a ray of light AB, parallel to the principal axis, incident on a spherical mirror at point B. Apparatus: Biconvex glass lens, spherical concave mirror, meter ruler, optical bench, lens If you recall from the polarization ellipse, the polarization state of the wave is completely determined by the amplitudes (E 0x /E 0y), and phases ( δx/δy). the mirror and the image first, which can be done using the mirror equation:. A clock is made by sending a pulse of light toward a mirror at a distance L and back to a receiver. From similar triangles A'B'C and ABC …. Figure 1: Basic current mirror. How to derive the kinetic gas equation? Physics Forums . 998 X 10 8 m/s was discovered, since the speed of light had been accurately measured by then, and its agreement with c was not likely to be a coincidence. This equation is known as a spherical mirror formula. The focal length of the mirror is 0. A lens or mirror can be thought of as a large pinhole. So recapping, using a bunch of similar triangles, we were able to derive a mirror equation that relates the object distance, image distance, and focal length, and along the way we derived a magnification equation that relates the heights of the image and object to the distances of the image and object. 3. The step sizes and probabilities of taking steps in various directions were related to the coefficients of the Fokker-Planck equation. Let's do it twice. Chapter 26 Geometrical OpticsChapter 26 Geometrical Optics Outline 26-1 The Reflection of Light 26-2 Forming Images with a Plane Mirror 26-3 Spherical Mirror 26-4 Ray Tracing and the Mirror Equation 26-5 The Refraction of Light5 The Refraction of Light 26-6 Ray Tracing for Lens 26-7Thi L E ti7 Thin Lens Equation Mirror Formula : Assume a mirror to be of a shape which focuses all rays to a single point (which requires a paraboloid). Light rays near and parallel to the principal axis are reflected from the concave mirror and converge at the focal point. The derivation of the mirror equation should follow. ➢ Text book introduces Gm method to find Avo. To be able to figure out how an image will be formed in one of these concave mirror, you need to be aware of a few basic • Majority of the derivation is taken from “University Physics” by Young and Freedman ISBN-13:978-0321501318 • I have expanded some steps to include every line of mathematical working in places where the book’s derivation jumped from one equation to another without expanding everything, rearranging and simplifying In this paper, we present a derivation of coating thermal noise of a ﬁnite-size cylindrical mirror based on the ﬂuctuation-dissipation theorem. Refer to Fig. Equation of a parabola - derivation. If rays emanating from a point actually meet at another point after reflection and/or refraction, that point is called the image of the first point. Furthermore, clearly the Poisson equation is the limit of the Helmholtz equation. Figure 1 below illustrates the original setup of Fizeau’s apparatus. Raymer and Brian J. The balance has four terminal posts, one near each corner. We show that the LT can be obtained from a symmetry principle hidden in Einstein's logical reasoning. The object is placed between the centre of curvature and Focus. For spherical mirrors, relation between object distance (do), image distance (di) and focal length. Even when geometric optics predicts that all rays are focused to a perfect point, diffraction means it will really be a spot. Derivation Challenge (1D kinematics) Derivation Challenge (2D kinematics) Derivation Challenge (Angular) Mirror Equation and Magnification. Mirror Equation Derivation. These beams of light have a characteristic radial intensity pro le whose width varies along the beam. Derivation. another demonstration of Lorentz derivation, compatible with the Quantum Mechanics: the Neo-Newtonian Mechanics, developed in a previous article [4]. The Lorentz transformation is derived from the simplest thought experiment by using the simplest If in the diffraction grating equation d m is made very small relative to a given λ, diffraction ceases to occur and the only solution that can be found is for m = 0. These can be connected to a dc voltage, e. ] current mirror. 1 u see a virtual image by looking in a mirror on a wall – your image will. derive the mirror equation - Physics - TopperLearning. If d1 and d2 represent the distances traveled in the respective mediums during the same amount of time, then we  Image Formed by a Concave Mirror,. Then we will look at convex mirrors, where the outside of the ball was made shiny. Drawing a ray diagram is a great way to get a rough idea of how big the image of an object is, and where the image is located. Many AM broadcast transmitting antennas are tall (at $u \sim 1$ MHz, $\lambda \sim 300$ m and a $\lambda/4$ vertical antenna is about 75 m high) towers acting as quarter-wave verticals. The following is known as the lensmaker’s equation: where Einstein’s 1905 Inadvertent Proof His Special Relativity Derivation and His Second Postulate Are Absurd Oren C. The new proposal by Greg Forbes12 allows us to make aspheric optics which are as powerful as the current general even aspheres for image correction, but which are simpler to optimize, The Equation of a Parabola Just for completeness, we'll now go ahead and derive the simplest of the parabola equations, to show that the definition we're talking about is really the usual y=x 2 thing. This will be covered in detail in class, but here it is just in case it goes by too fast. 16 Exterior angle of a triangle equals sum of the the 2x2 matrix. Preliminaries. Let, P = pole and F=pr. The figure shows an object AB at a distance 'u' from the pole of a concave mirror. Willebrord Snellius (1580–1626) was a Changing the refractive index in the properties menu altered the path of the ray, it was set at 1. Derivation of the Thin Lens Equation (continued) However, you can see it (just like the image in a mirror) when you look at the lens from the right, because the  May 13, 2017 f=−0. , the distance of the focus from the lens. We have been modifying our intermediate optics class and laboratory with a focus on improving student learning through the use of active engagement. You are probably aware – even if unfamiliar with all the fine details – that this is not exactly so, and you will be aware Convex Mirror Equation Calculator. molar conductance m, may be derived Time Dilation. Chapter 18: Ray Optics ". The result agrees to a previous result with an inﬁnite-size mirror in the limit of large thickness, and also agrees to an independent result based on the mode expansion with a thin-mirror Please consider the differential amplifier (Q1&Q2) with current mirror active load (Q3&Q4) as shown below: simulate this circuit – Schematic created using CircuitLab. 037 cm1 66 cm 1 46 cm 1111 THE MIRROR EQUATION . Radio signals are also electromagnetic waves; after the discovery of This relationship, which depends on the focal length f of the lens, is given by the Gaussian Lens Formula, shown in the lower-left corner of the applet below. In the same way that the Law of Reflection is the basic tool we use to develop the theory of mirrors, Snell's Law is the basic tool that we use to develop the theory of lenses. A simple derivation of the Lorentz transformation and of the related velocity and acceleration formulae J. A paperclip 2. A diverging laser beam strikes a front-surface mirror at a low angle, so that some of the beam reflects off the mirror to a screen, and some shines directly on the screen. 1 always handles the screw with gently because it can break the mirror. Notice that Snell's Law shows that the index of refraction and the sine of the angle of refraction are inversely proportional - that is, as the refractive index gets larger [n 2 > n 1] the sine of the refracted angle gets smaller [sinθ 2 < sinθ 1], since the product of the two terms must remain a constant. Here, x 1 and x 2 are the distances to the object and image respectively from the focal points. 2 Physical Interpretation The basic physical process which limits the linewidth is spontaneous emission into the laser mode: Example: Effect of a Component's Reliability in a Series System. 04891; and work in progress. Fresnel Equations—Perpendicular E field . (see figure on right). Derivation Challenge (1D kinematics) Derivation Challenge (2D kinematics) Mirror Equation and Magnification. The transmitted beam (T) travels to mirror M 1 and it is reflected back to BS. Constraints for mass (identity of the ion), parallel and perpendicular velocity (kinetic/thermal energy of the ion) are clearly necessary. O the emission and absorption of light on source B. Klemm, arXiv:1501. Physics 11 . Calculate the frequency of ultrasonic sound-waves by Lloyd’s Mirror Interference. Debye-Huckel-Onsagar Equation . In figure 3, we've shown the parabola again, with a number of distances marked. Dividing equation (3) by Q we get the important form of the Hubbert The Relativistic Doppler Effect You're all familiar with the Doppler effect, right? Waves of any sort -- sound waves, light waves, water waves -- emitted at some frequency by a moving object are perceived at a different frequency by a stationary observer. We can extend the mirror equation to the case of a plane mirror by noting derive the mirror equation - Physics - TopperLearning. A ray of light BL after reflecting from the concave mirror passes through the principal axis at F and goes The equation for image formation by rays near the optic axis of a mirror is called mirror formula/equation. A derivation of this result is given in the appendix below. The dependence of the corresponding Doppler factors on the incidence angle on the stationary mirror. Determination of Focal Length of A Converging Lens and Mirror Objective: Apply the thin-lens equation and the mirror equation to determine the focal length of a converging (biconvex) lens and mirror. Each "tick" is a round-trip to the mirror. Each mirror has a radius of curvature r (which is inﬁnite for the plane mirror) and a focal length f = 1 2 r. We'll use those to find the equation which describes the parabola. The position of the image is obtained by drawing a ray Spherical Mirror Equation. Lloyd’s Mirror 1 Objective Understand the nature of sound-waves. For a plane mirror, we showed that the image formed has the same height and orientation as the object, and it is located at the same distance behind the mirror as the object is in front of the mirror. 4. In Class Demo . 1 h θ O a C FIGURE IV. This equation is generally expressed as. The LC cell is characterized by θ(z) and φ(z). OPTI\$521 Project 1. Explain Mirror - Lens Equation? A central equation in optics relates the focal length, object distance and image distance for a thin lens or mirror. Step 2: Find the image distance using the thin lens equation. Because of the air cell, there is an index of refraction associated that needs to show up in the equation. Outline • TE and TM fields • Refraction and Snell’s Law: - From TE analysis - From Phase Matching - From Fermat’s Principle of Least Time • Total Internal Reflection and Fibers • FIOS . odt; Friction Practice with answers. I'll start my derivation as much as possible from rst principles and then apply them to the magnetic mirror model to determine whether or not it is realistically possible to trap ions in this manner. Topic: Reflection and Mirrors. The focal length f is positive for a concave mirror, negative for a convex mirror. this slide includes the derivation of spherical mirror equation, lateral magnification, spherical aberration of mirrors and the difference between spherical an… Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Note. Because these Gaussian beams behave somewhat like spherical waves, we can match them to the curvature of the mirror of an optical Mirror Equation/Formula Derivation CBSE Class 12 Ray Optics & Optical Instruments | Chapter 9 Class 12 Physics| By Shruti Academy Hello Friend In this video, we are going to derive the mirror formula by using Concave mirror. This is the celebrated Einstein di usion equationwhich describes microscopic transport of material and heat. The most frequently used design is that of a quarter-wave mirror, where each optical layer thickness corresponding to one quarter of the wavelength for which the mirror is equation—which quantiﬁes the power received in a free-space communication link—is developed from principles of optical radiometry and scalar diffraction. !!! = 1!!!!! − ! ! 39 Gradient Wind Equations Example) The PMOS transistor has V T = -1 V, Kp = 8 µA/V2, W/L = 25, λ = 0. Kinetic energy is a simple concept with a simple equation that is simple to derive. These equations are called the Fresnel Equations for . where d o is the object distance from the mirror, d i is the image distance from the mirror, and f is the focal length of the mirror. You can graph a Quadratic Equation using the Function Grapher, but to really understand what is going on, you can make the graph yourself. 4 of the text. (M=(-i/o)=h'/h). mirror equation derivation

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# Reference for the statement that the complement of an affine open has codimension one

The following statement seems to be "well-known", but I am unable to find a reference in the standard literature. Could someone suggest a reference?

Let $X$ be a separated normal connected Noetherian scheme and $U$ a (nonempty) affine open subset. Then the complement $Y = X \smallsetminus U$ has pure codimension one.

• stacks.math.columbia.edu/tag/0BCQ provides the result you are looking for. – pbelmans Aug 25 '15 at 11:45
• EGA IV, part 4 (IHES no 32), Cor. 21.12.7. – Matthieu Romagny Aug 25 '15 at 21:43
• I'd've called this "algebraic Hartogs' theorem". If $Y$ is codimension 2 in $X$, then functions on $U$ should extend across $Y$, i.e. $U$ shouldn't be affine. – Allen Knutson Aug 31 '15 at 0:38

format: plain


text:
1. Mimi_x3
2. anonymous

I can't open it, seems I need a proxy

3. Mimi_x3

:|

4. Mimi_x3

give me a sec

5. Mimi_x3

6. anonymous

Sure you know that you need a substitution, right?

7. Mimi_x3

what substitution

8. anonymous

sub $$y_1=f(x_1, x_2)$$ and $$y_2=g(x_1, x_2)$$ in a way that first two equations turn into second system in terms of $$y$$

9. Mimi_x3

I really don't know...can you please elaborate

10. Mimi_x3

I can't find anything related to this on google :/

11. anonymous

let $$y_1=x_1$$ and $$y_2=x_1+x_2$$ see what happens

12. Mimi_x3

do i sub it into x dot?

13. anonymous

sry, I was out, see for example$\dot{y_1}=\dot{x_1}=x_1+x_2=y_2$Now you calculate $$\dot{y_2}$$

14. anonymous

Hey dont overthink it, this is just some algebraic manipulation, Its not weird

15. Mimi_x3

I don't get it why does y dot = x dot?

16. anonymous

Its a dynamical system, dot is "derivative with respect to time" here

17. Mimi_x3

dx/dt ?

18. anonymous

exactly

19. anonymous

Now, what about $$\dot{y_2}$$ ?

20. Mimi_x3

I don't get how the x is related to the y?

21. UsukiDoll

the dot isn't used that much , but it's another way of saying first derivative for a single dot 2nd derivative for a double dot.

22. anonymous

That depends on how we define y in terms of x, we do this sub to get a simpler form of equations that are easy to interpret physically

23. anonymous

Think of it like this, you have a system of differential equations in x_1 and x_2 and you wanna solve it, you do a sub to get a simpler form and to solve it more easily

24. anonymous

And x is not related to y, we relate it to y :)

25. Mimi_x3

i don't get it y dot = -(2x + x)

26. Mimi_x3

y dot = -(2x_1+x_2)

27. Mimi_x3

28. anonymous

ok, let me explain it step by step

29. anonymous

you have these system of equations:$\dot{x_1}=x_1+x_2$$\dot{x_2}=-2x_1-x_2$

30. anonymous

Now I do a sub in the form $$y_1=x_1$$ and $$y_2=x_1+x_2$$, lets see what happens

31. anonymous

$\dot{y_1}=\dot{x_1}=x_1+x_2=y_2$$\dot{y_2}=\dot{x_1}+\dot{x_2}=x_1+x_2-2x_1-x_2=-x_1=-y_1$

32. Mimi_x3

how would you know what sub to do

33. anonymous

that's just like integrals, how do you solve integrals with u-sub?

34. Mimi_x3

by looking

35. Mimi_x3

thanks

36. anonymous

No problem, skill develops with practice and experience

37. Jack1

nice


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# [Rd] getGraphicsEvent() questions, minor feature/tweak request, and patch(es).

Hey all, new R user and first timer here.

I've been using getGraphicsEvent() on an X11-Xlib device in a kind of
interactive loop, and would like to be able to stop it from printing
newlines in the console when I don't actually want to print a prompt.
Even an empty "" value still causes a newline right now.

To not break any code that depends on this behavior, I figured getting
C_getGraphicsEvent to accept an NA value would be the way to go. It
strikes me as highly unlikely that there's any code out there built on
expecting getGraphicsEvent() to error out when getting a 'prompt = NA'
parameter, so this seems like a safe change to make.

After rooting about in the R source for a bit I've found a way to
implement this change myself in src/main/gevents.c, and I've included a
patch as attachment. What I don't know is if there's a better or at
least more conventional way to be checking for the NA value.

Coercing with asChar() and only comparing against R_NaString causes NULL
values, functions, expression objects and potentially others to become
valid prompt values as well, so I'm using TYPEOF() to restrict accepted
values further. I've found manually entered NA values are interpreted as
being of LGLSXP, so that's the only type I'm accepting NA values for. Is
there a better way? There's an isna() in library/utils/src/io.c, but
that module doesn't seem to be available to the linker in the current
build script for gevents.c's module, which would make it a more invasive
patch to use. I'm also not certain it'd work the way I'd want it to,
from looking at its internals.

As far as I can tell, the prompt value is only used in two places;
do_getGraphicsEvent() in src/main/gevents.c, and GA_eventHelper() in
library/grDevices/src/devWindows.c. The latter has a sanity check for
string-ness already, so making NA a possibility should be safe there, as
well.

That apparent lack of use elsewhere leads me to a question about the
help file for getGraphicsEvent(). In it, the claim is made that 'prompt'
is used to display a prompt text on the device. This might be true in MS
Windows, but for X11 the prompt value seems to be solely used for
Rprintf() to the console. Is this discrepency a bug or missing feature
in the X11 implementation, an incorrect wording in the help, or an
interpretation error on my side? Whatever it is, I've also included a
minor patch for the help file to at least indicate support for 'prompt =
NA'.

Apologies if I'm doing this all wrong, or if I'm missing obvious reasons
why these patches are unacceptable. As I said, first timer here and new
to R in general. ;-)

- Richard Bodewits

Index: src/main/gevents.c
===================================================================
--- src/main/gevents.c  (revision 71269)
+++ src/main/gevents.c  (working copy)
@@ -135,7 +135,7 @@
checkArity(op, args);

prompt = CAR(args);
-    if (!isString(prompt) || !length(prompt)) error(_("invalid prompt"));
+    if ((!isString(prompt) || !length(prompt)) && (TYPEOF(prompt) != LGLSXP ||
asChar(prompt) != R_NaString)) error(_("invalid prompt"));

/* NB:  cleanup of event handlers must be done by driver in onExit handler
*/

@@ -159,8 +159,10 @@
if (!count)
error(_("no graphics event handlers set"));

-       Rprintf("%s\n", CHAR(asChar(prompt)));
-       R_FlushConsole();
+    if (TYPEOF(prompt) != LGLSXP || asChar(prompt) != R_NaString) {
+        Rprintf("%s\n", CHAR(asChar(prompt)));
+        R_FlushConsole();
+    }

/* Poll them */
while (result == R_NilValue) {

Index: src/library/grDevices/man/getGraphicsEvent.Rd
===================================================================
--- src/library/grDevices/man/getGraphicsEvent.Rd       (revision 71269)
+++ src/library/grDevices/man/getGraphicsEvent.Rd       (working copy)
@@ -24,12 +24,12 @@

}
\arguments{
-  \item{prompt}{prompt to be displayed to the user in the graphics window}
+  \item{prompt}{prompt to be displayed to the user in the graphics window, or
NA for no prompt}
\item{onMouseDown}{a function to respond to mouse clicks}
\item{onMouseMove}{a function to respond to mouse movement}
\item{onMouseUp}{a function to respond to mouse button releases}
\item{onKeybd}{a function to respond to key presses}
-  \item{consolePrompt}{prompt to be displayed to the user in the console}
+  \item{consolePrompt}{prompt to be displayed to the user in the console, or
NA for no prompt}
\item{which}{which graphics device does the call apply to?}
\item{...}{items including handlers to be placed in the event environment}
\item{env}{an environment to be used as the event environment}

______________________________________________
R-devel@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-devel


text:
# Stein’s method

The importance of possessive apostrophes

A famous generic method for approximating distributions and quantifying discrepancy and manufacturing concentration bounds and limit theorems is Stein’s method, typically in the form of the method of exchangeable pairs . Wikipedia will do as a rough intro for now, although their info is rather out-of-date. There does not seem to be a thorough introduction to all the modern and useful tools. Different bits are introduced in Barbour and Chen (2005); Chatterjee (2014); Meckes (2012); Ross (2011).

Chen Soon Ong says:

If we have a distribution P, and we want to measure the distance to P from another distribution Q (which we control), an interesting trick to measure this distance is to define an operator T. This operator T, called the Stein operator, allows us to measure the distance between distributions by considering the distance between test functions on the random variables corresponding to P and Q. This is a more general structure than integral probability metrics, which in turn is a more general version of Wasserstein distance.

Probably the best intro is Lily Li’s Whirlwind Tour.

## Stein operators

### Gaussian

The original form, Stein’s lemma gives use the Stein operator for the Gaussian distribution in particular. Meckes (2009) explains:

The normal distribution is the unique probability measure $$\mu$$ for which $\int\left[f^{\prime}(x)-x f(x)\right] \mu(d x)=0$ for all $$f$$ for which the left-hand side exists and is finite. It is useful to think of this in terms of operators, specifically, the operator $$\mathcal{A}_{o}$$ defined on $$C^{1}$$ functions by $\mathcal{A}_{o} f(x)=f^{\prime}(x)-x f(x)$ is called the characterizing operator of the standard normal distribution.
This is incredibly useful in probability approximation by Gaussians where it justifies Stein’s method, below. It has apparently been extended to elliptical distributions and exponential families.

Multivariate? Why, yes please. The following lemma of Meckes (2006) gives a second-order characterizing operator for the Gaussian distribution on $$\mathbb{R}^{k}$$:

For $$f \in C^{1}\left(\mathbb{R}^{k}\right)$$, define the gradient of $$f$$ by $$\nabla f(x)=\left(\frac{\partial f}{\partial x_{1}}(x), \ldots, \frac{\partial f}{\partial x_{k}}(x)\right)^{t}$$. Define the Laplacian of $$f$$ by $$\Delta f(x)=\sum_{i=1}^{k} \frac{\partial^{2} f}{\partial x_{i}^{2}}(x)$$. Now, let $$Z \sim \mathcal{N}(0_k, \mathrm{I}_k)$$.

1. If $$f: \mathbb{R}^{k} \rightarrow \mathbb{R}$$ is two times continuously differentiable and compactly supported, then $\mathbb{E}[\Delta f(Z)-Z \cdot \nabla f(Z)]=0$
2. If $$Y \in \mathbb{R}^{k}$$ is a random vector such that $\mathbb{E}[\Delta f(Y)-Y \cdot \nabla f(Y)]=0$ for every $$f \in C^{2}\left(\mathbb{R}^{k}\right)$$, then $$\mathcal{L}(Y)=\mathcal{L}(Z) .$$
3. If $$g \in C_{o}^{\infty}\left(\mathbb{R}^{k}\right)$$, then the function $U_{o} g(x):=\int_{0}^{1} \frac{1}{2 t}[\mathbb{E} g(\sqrt{t} x+\sqrt{1-t} Z)-\mathbb{E} g(Z)] d t$ is a solution to the differential equation $\Delta h(x)-x \cdot \nabla h(x)=g(x)-\mathbb{E} g(Z)$

### Poisson

a.k.a. Stein-Chen. $\mathcal{A}_{o} f(k)=\lambda f(k+1)-k f(k)$

### Exponential

Ross (2011) mentions

$\mathcal{A}_{o} f(x)=f^{\prime}(x)-f(x)+f(0)$

### Markov processes

TBD; relation to infinitesimal generators? See perhaps Schoutens (2001).

Face to face, each classic case
We shadow box and double cross
Yet need the chase
A license to love, insurance to hold
Melts all your memories and change into gold
His eyes are like angels but his heart is cold
He’s a Stein operator
Stein operator
Stein operator
Stein operator

## Stein’s method of exchangeable pairs

Meckes (2009) summarises.

Heuristically, the univariate method of exchangeable pairs goes as follows. Let $$W$$ be a random variable conjectured to be approximately Gaussian; assume that $$\mathbb{E} W=0$$ and $$\mathbb{E} W^{2}=1 .$$ From $$W,$$ construct a new random variable $$W^{\prime}$$ such that the pair $$\left(W, W^{\prime}\right)$$ has the same distribution as $$\left(W^{\prime}, W\right) .$$ This is usually done by making a “small random change” in $$W$$, so that $$W$$ and $$W^{\prime}$$ are close. Let $$\Delta=W^{\prime}-W$$. If it can be verified that there is a $$\lambda>0$$ such that \begin{aligned} \mathbb{E}[\Delta \mid W]=-\lambda W+E_{1} \\ \mathbb{E}\left[\Delta^{2} \mid W\right]=2 \lambda+E_{2} \\ \mathbb{E}|\Delta|^{3}=E_{3} \end{aligned} with the random quantities $$E_{1}, E_{2}$$ and the deterministic quantity $$E_{3}$$ being small compared to $$\lambda,$$ then $$W$$ is indeed approximately Gaussian, and its distance to Gaussian (in some metric) can be bounded in terms of the $$E_{i}$$ and $$\lambda$$.

This comes out very nicely where there are natural symmetries to exploit, e.g. in low-d projections.

### Non-Gaussian Stein method

Steins method generalises to AFAICT any exponential distribution. TBD

### Multivariate Gaussian Stein method

The work of Elizabeth Meckes (1980—2020) serves as the canonical introduction in the area for now, although she never wrote a textbook. Two foundational ones are Chatterjee and Meckes (2008) and Meckes (2009) and there is a kind of introductory user guide in Meckes (2012); The examples are mostly about random projections although the method is much more general. The exchangeable pairs are natural in projections though, you can just switch off your brain and turn the handle to produce results, or easier yet, a computer algebra system that can handle noncommutative algebra can do it for you.

If the papers are too dense, try this friendly lecture, Stein’s Method — The last gadget under the hood.

## Stein discrepancy

A probability metric based on something like “how well this distribution satisfies Stein’s lemma” I think?

## References

Agueh, Martial, and Guillaume Carlier. 2011. SIAM Journal on Mathematical Analysis 43 (2): 904–24.
Anastasiou, Andreas, Alessandro Barp, François-Xavier Briol, Bruno Ebner, Robert E. Gaunt, Fatemeh Ghaderinezhad, Jackson Gorham, et al. 2021. arXiv:2105.03481 [Math, Stat], May.
Arras, Benjamin, and Christian Houdré. 2019a. On Stein’s Method for Infinitely Divisible Laws with Finite First Moment. Edited by Benjamin Arras and Christian Houdré. SpringerBriefs in Probability and Mathematical Statistics. Cham: Springer International Publishing.
———. 2019b. Electronic Journal of Probability 24 (none): 1–63.
Barbour, A. D., and Louis H. Y. Chen, eds. 2005. An Introduction to Stein’s Method. Vol. 4. Lecture Notes Series / Institute for Mathematical Sciences, National University of Singapore, v. 4. Singapore : Hackensack, N.J: Singapore University Press ; World Scientific.
Chatterjee, Sourav. 2014. arXiv:1404.1392 [Math], April.
Chatterjee, Sourav, and Elizabeth Meckes. 2008. arXiv:math/0701464, January.
Chen, Louis H. Y. 1998. In Probability Towards 2000, edited by L. Accardi and C. C. Heyde, 97–122. Lecture Notes in Statistics. New York, NY: Springer.
Chwialkowski, Kacper, Heiko Strathmann, and Arthur Gretton. 2016. In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48, 2606–15. ICML’16. New York, NY, USA: JMLR.org.
Cuturi, Marco, and Arnaud Doucet. 2014. In International Conference on Machine Learning, 685–93. PMLR.
Gorham, Jackson, and Lester Mackey. 2015. In Advances in Neural Information Processing Systems. Vol. 28.
Gorham, Jackson, Anant Raj, and Lester Mackey. 2020. arXiv:2007.02857 [Cs, Math, Stat], October.
Landsman, Zinoviy, and Johanna Nešlehová. 2008. Journal of Multivariate Analysis 99 (5): 912–27.
Landsman, Zinoviy, Steven Vanduffel, and Jing Yao. 2013. Journal of Statistical Planning and Inference 143 (11): 2016–22.
Ley, Christophe, Gesine Reinert, and Yvik Swan. 2017. Probability Surveys 14 (none): 1–52.
Liu, Qiang, Jason D. Lee, and Michael I. Jordan. 2016. arXiv:1602.03253 [Stat], July.
Liu, Qiang, and Dilin Wang. 2019. In Advances In Neural Information Processing Systems.
Loh, Wei-Liem. n.d.
Meckes, Elizabeth. 2006. “An Infinitesimal Version of Stein’s Method of Exchangeable Pairs.”
———. 2009. In High Dimensional Probability V: The Luminy Volume, 153–78. Beachwood, Ohio, USA: Institute of Mathematical Statistics.
———. 2012. Journal of Theoretical Probability 25 (2): 333–52.
Paulin, Daniel, Lester Mackey, and Joel A. Tropp. 2016. The Annals of Probability 44 (5): 3431–73.
Piccoli, Benedetto, and Francesco Rossi. 2016. Archive for Rational Mechanics and Analysis 222 (3): 1339–65.
Reinert, Gesine. 2000. In Complex Stochastic Systems, 235–75. Boca Raton: Chapman & Hall/CRC.
———. 2005. In An Introduction to Stein’s Method, Volume 4:183–221. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, Volume 4. CO-PUBLISHED WITH SINGAPORE UNIVERSITY PRESS.
Reinert, Gesine, and Adrian Röllin. 2007. November.
Ross, Nathan. 2011. Probability Surveys 8 (0): 210–93.
Schoutens, Wim. 2001. Journal of Mathematical Analysis and Applications 253 (2): 515–31.
Stein, Charles. 1972. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory, January, 583–602.
———. 1986. Approximate Computation of Expectations. Vol. 7. IMS.
Tan, Yue, Yingdong Lu, and Cathy Xia. 2018. arXiv:1810.04300 [Math], October.
Upadhye, Neelesh S., and Kalyan Barman. 2020. arXiv:2004.07593 [Math], July.
Xu, Wenkai, and Takeru Matsuda. 2020. In International Conference on Artificial Intelligence and Statistics, 320–30. PMLR.
———. 2021. arXiv:2103.00895 [Stat], March.
Xu, Wenkai, and Gesine Reinert. 2021. arXiv:2103.00580 [Stat], February.

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# zbMATH — the first resource for mathematics

On conditions for and the order of approximation of functions by operators of class $$S_{2m}$$. (English. Russian original) Zbl 1005.41007
Math. Notes 67, No. 5, 556-562 (2000); translation from Mat. Zametki 67, No. 5, 654-661 (2000).
Conditions for the convergence of operators of classes $$S_{2m}$$ to continuous functions are studied. Next, the asymptotics of approximation by such operators to differentiable functions are established. The paper generalize, by different methods, results due to I. Szabados for operators of the class $$S_{2m}$$.
##### MSC:
 41A36 Approximation by positive operators
Full Text:
##### References:
 [1] P. P. Korovkin, ”Converging sequences of linear operators,”Uspekhi Mat. Nauk [Russian Math. Surveys],17, No. 4 (106), 147–152 (1962). [2] P. P. Korovkin, ”An asymptotic property of the positive methods of summation of Fourier series and the best approximation of the class Z2 by linear positive polynomial operators,”Uspekhi Mat. Nauk [Russian Math. Surveys],13, No. 6(84), 99–103 (1958). · Zbl 0090.28304 [3] P. P. Korovkin, ”On the order of approximation of functions by linear polynomial operators of classS m ,” in:Studies in the Contemporary Problems of Constructive Function Theory [in Russian], Izd. Akad. Nauk Azer. SSR, Baku (1965), pp. 163–166. [4] A. I. Kovalenko, ”On certain methods of summation of Fourier series,”Mat. Sb. [Math. USSR-Sb.],71 (113), No. 4, 598–616 (1966). · Zbl 0171.02602 [5] J. Szabados, ”On convolution operators with kernels of finite oscillation,”Acta Math. Acad. Sci. Hung.,27, No. 1–2, 179–192 (1976). · Zbl 0329.41017 · doi:10.1007/BF01896773 [6] R. K. Vasiliev (Vassiliev), ”Certaines méthodes de sommation de séries de Fouries donnant le meilleur ordre d’approximation,”Acta Math. Acad. Sci. Hung.,63, No. 1, 65–102 (1994). [7] R. K. Vasil’ev, ”Certain methods of summation of Fourier series with the best order of approximation,”Mat. Zametki [Math. Notes],54, No. 2, 145–151 (1993). [8] A. O. Gel’fond,Finite Difference Calculus [in Russian], Gostekhizdat, Moscow-Leningrad (1952). [9] W. Dahmen and E. Görlich, ”A conjecture of M. Golomb on optimal and nearly-optimal linear approximation,”Bull. Amer. Math. Soc.,80, No. 6, 1199–1202 (1974). · Zbl 0294.41019 · doi:10.1090/S0002-9904-1974-13676-1 [10] V. A. Baskakov,Linear Methods of Summation of Fourier series and the Approximation of Continuous Functions [in Russian], KGU, Kalinin (1980). · Zbl 0507.42004 [11] P. L. Butzer and E. L. Stark, ”On a trigonometric covolution operator with kernel having two zeros of simple multiplicity,”Acta Math. Acad. Sci. Hung.,20, No. 3–4, 451–461 (1969). · Zbl 0189.35001 · doi:10.1007/BF01894916
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# Do superoxide salts of heavier alkali metals contradict the principle that lattice energy depends inversely on cation-anion distance?

I have a very basic understanding of quantitative lattice energy calculations: I only know that the lattice energy is directly proportional to the product of charges on the ions and inversely proportional to the distance between cation and anion.

Based upon the direct application of the above facts, lattice energy follows the order:

LiCl > NaCl > KCl > RbCl > CsCl

But I have also read in JD Lee:

Large cations can be stabilized by large anions, since if both the ions are similar in size the coordination number will be high, and this gives high lattice energy

This was given to explain that superoxide salts exist only with K, Rb, Cs. Does this not contradict the basic principle that lattice energy is inversely proportional to the distance b/w cation and anion?

TL, DR: They do not contradict. Although larger ions decrease the lattice energy by increasing ion distance, it is compensated by another factor that increases the lattice energy. And like Nisarg Bhavsar said, product formation does not solely depend on lattice energy. Other factors such as the electron affinity and ionization energy of the element should be accounted too.

From Ref. 1:

The answer has to do with the crowding of alkali ions around oxygen, as illustrated for $$\ce{K2O}$$ at the right. Because eight large $$\ce{K^+}$$ ions surround each $$\ce{O^{2-}}$$ ion in the structure, the cations are essentially in contact. Indeed, the metal-oxygen bonds are "stretched" in $$\ce{Na2O}$$, $$\ce{K2O}$$, and $$\ce{Rb2O}$$ relative to $$\ce{M-O}$$ bonds with the same bond order in other structures.The situation is so extreme for $$\ce{Cs2O}$$ that it finds an (electrostatically unfavorable) structure in which the coordination is lowered to 6:3. This packing problem is relieved somewhat in the peroxides, where the coordination is still 8:4 but the anion is larger, and especially in the superoxides where the cation:anion ratio is 1:1 and the coordination is 6:6. Thus the larger alkali ions ($$\ce{K^+}$$, $$\ce{Rb^+}$$ and $$\ce{Cs^+}$$) tend to form superoxides.

Above: $$\ce{K2O}$$ structure, below: $$\ce{Cs2O}$$ structure. Oxides are red, potassiums are blue and caesiums are magenta.

Thus, the reason why heavier alkali metals tend to form superoxides is because the stoichiometry of the compound and ionic radius of the anion allows the large cation to arrange so that they won't touch each other as much. Since like-charges repel, having cations in contact with each other is a crime in forming crystal lattices because it would greatly destabilize the crystal.

In case you don't know, the Born-Lande equation to calculate energy is actually derived from Coulomb's formula of electrostatic potential energy, i.e $$E_\mathrm{pair} = \frac{z_1\cdot z_2\cdot e^2}{4πε_0r}$$ Where $$z_i$$ is the charges of one ion and $$r$$ is the distance between the point charges (ions in this case).

Specifically, assume a theoretical compound $$\ce{M^{2+}X^{2-}}$$ that crystallizes in the rocksalt ($$\ce{NaCl}$$) structure. The Born-Lande equation is derived from the sum of all electrostatic forces that one ion in the lattice experiences. Let's use the $$\ce{M^{2+}}$$ cation.

The rocksalt lattice has 6:6 coordination. It means that one $$\ce{M^{2+}}$$ cations is equally surrounded by six $$\ce{X^{2-}}$$ ions. Thus the expression for the electrostatic energy between the $$\ce{M^{2+}}$$ in question with the six $$\ce{X^{2-}}$$ surrounding it is

$$E_{1} = 6\frac{2\cdot (-2) e^2}{4πε_0 r}$$

The $$\ce{M^{2+}}$$ is also equally surrounded by twelve $$\ce{M^{2+}}$$ with a distance of $$\sqrt{2}r$$. Thus the expression for the electrostatic energy between the $$\ce{M^{2+}}$$ with the 12 $$\ce{M^{2+}}$$ surrounding it is:

$$E_{2} = 12 \frac{2\cdot 2 e^2}{4πε_0 \sqrt{2}r}$$

Etc. You can do another $$\pu{E_{i}}$$s of other sets of further ions surrounding the $$\ce{M^{2+}}$$ ion.

Adding all of this electrostatic energy equations together we get the sum of all the electrostatic energy on one particular ion:

$$E_\mathrm{all} = \frac{2\cdot (-2) \cdot e^2}{4πε_0 r} \cdot \pu{(6 - \frac{12}{\sqrt{2}}+...)}$$

The sum of $$(6 - \frac{12}{\sqrt{2}}+...)$$ in the equation is the Madelung constant. Since those numbers derive from the geometry and coordination number of the lattice, it is dependent on the crystal system rather than the nature of the ions, as you might have learned in textbooks.

Finally, combining this with the account for the electron-electron and nucleus-nucleus repulsions between the ions and multiply with the Avogadro constant since we are using $$\pu{kJ/mol}$$, we arrive at the Born-Lande equation:

$$E_{lattice} = \frac{N_A \cdot z_1 \cdot z_2 \cdot e^2 \cdot M}{4πε_0 r} \cdot ( 1-\frac{1}{d})$$

As you can see, since the lattice energy involves the interaction forces of the ions, having too much repulsion would decrease it. The benefit of increased lattice energy from this factor is more than enough to compensate for the increased ion distance.

References:

1. The Strange Case of the Alkali Oxides. (2021, June 7). Retrieved June 11, 2021, from https://chem.libretexts.org/@go/page/189354

2. Energetics of Crystalline Solids- The Ionic Model. (2021, June 7). Retrieved June 11, 2021, from https://chem.libretexts.org/@go/page/183348


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# Short put prices different strikes

I was looking at Robinhood and can't find a rational reason for a put price for the $$\\\117$$ strike to be higher than both the $$\\\116$$ and the $$\\\119$$ strike.

• What are the buy prices for these puts? Apr 20 at 19:37
• On the right. The orange and green values. Apr 20 at 21:07
• I meant the corresponding values for "Buy Puts" Apr 21 at 4:57
• They are the same !!! Apr 21 at 12:40
• Precisely ! Seems to good to be true. Apr 21 at 12:53


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# If I have a list of numbered questions, how can I align my questions to the right of the number?

I'm editing a paper with a bunch of questions numbered from 1 to 10. In the example that my professor gave me, he had it formatted so that the number of the question would be listed, and then the question itself would be aligned with it's own margin next to the number. I know in Microsoft Word, for example, it aligns itself automatically, but I don't know how to do that in TeXStudio.

Is there some command that I don't know about?

-
have you tried enumerate environment? – mythealias Sep 20 '12 at 23:19

## 1 Answer

A simple enumerate environment seems to be what you need (or do you want some other alignment for the labels?):

\documentclass{article}

\begin{document}

\begin{enumerate}
\item Question 1.
\item Question 2.
\item Question 3.
\item Question 4.
\item Question 5.
\item Question 6.
\item Question 7.
\item Question 8.
\item Question 9.
\item Question 10.
\end{enumerate}

\end{document}

-


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# yael stone siblings

The eigenvectors of P span the whole space (but this is not true for every matrix). (1) Geometrically, one thinks of a vector whose direction is unchanged by the action of A, but whose magnitude is multiplied by λ. If λ = 1, the vector remains unchanged (unaffected by the transformation). Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … T ( v ) = λ v. where λ is a scalar in the field F, known as the eigenvalue, characteristic value, or characteristic root associated with the eigenvector v. Let’s see how the equation works for the first case we saw where we scaled a square by a factor of 2 along y axis where the red vector and green vector were the eigenvectors. whereby λ and v satisfy (1), which implies λ is an eigenvalue of A. Both Theorems 1.1 and 1.2 describe the situation that a nontrivial solution branch bifurcates from a trivial solution curve. 2 Fact 2 shows that the eigenvalues of a n×n matrix A can be found if you can ﬁnd all the roots of the characteristic polynomial of A. The ﬁrst column of A is the combination x1 C . We state the same as a theorem: Theorem 7.1.2 Let A be an n × n matrix and λ is an eigenvalue of A. In such a case, Q(A,λ)has r= degQ(A,λ)eigenvalues λi, i= 1:r corresponding to rhomogeneous eigenvalues (λi,1), i= 1:r. The other homoge-neous eigenvalue is (1,0)with multiplicity mn−r. 6.1Introductiontoeigenvalues 6-1 Motivations •Thestatic systemproblemofAx =b hasnowbeensolved,e.g.,byGauss-JordanmethodorCramer’srule. Eigenvalues so obtained are usually denoted by λ 1 \lambda_{1} λ 1 , λ 2 \lambda_{2} λ 2 , …. :2/x2 D:6:4 C:2:2: (1) 6.1. The eigenvalue equation can also be stated as: Let A be an n×n matrix. An application A = 10.5 0.51 Given , what happens to as ? Combining these two equations, you can obtain λ2 1 = −1 or the two eigenvalues are equal to ± √ −1=±i,whereirepresents thesquarerootof−1. If x is an eigenvector of the linear transformation A with eigenvalue λ, then any vector y = αx is also an eigenvector of A with the same eigenvalue. Then the set E(λ) = {0}∪{x : x is an eigenvector corresponding to λ} x. remains unchanged, I. x = x, is defined as identity transformation. v; Where v is an n-by-1 non-zero vector and λ is a scalar factor. Introduction to Eigenvalues 285 Multiplying by A gives . Observation: det (A – λI) = 0 expands into a kth degree polynomial equation in the unknown λ called the characteristic equation. This illustrates several points about complex eigenvalues 1. This eigenvalue is called an inﬁnite eigenvalue. Eigenvalues and Eigenvectors Po-Ning Chen, Professor Department of Electrical and Computer Engineering National Chiao Tung University Hsin Chu, Taiwan 30010, R.O.C. detQ(A,λ)has degree less than or equal to mnand degQ(A,λ)


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# HistogramVectorPostprocessor

## Short Description

Compute a histogram for each column of a VectorPostprocessor

## Description

Used to compute the histogram for all columns of another VectorPostprocessor (VPP). The only inputs are the other VPP (vpp) and the number of bins to use in the histogram (num_bins).

This will actually generate three columns for each column in the original VPP:

• column_name: The histogram data

• column_name_lower: The lower bound for each bin

• column_name_upper: The upper bound for each bin

## Plotting

MOOSE comes with built-in plotting capabilities that can help plot the output of a HistogramVectorPostprocessor. These plotting capabilities are part of the Chigger suite of visualization tools located in the moose/python directory. To use them you must add the full path of your moose/python directory to the environment variable called $PYTHONPATH using something like:  export PYTHONPATH=/full/path/to/moose/python:$PYTHONPATH

Once that is completed a script such as the following will plot your data:

import matplotlib.pyplot as plt
import mooseutils

# Create Figure and Axes
figure = plt.figure(facecolor='white')

# Grab upper and lower bin bounds
lower = data('value_lower')
upper = data('value_upper')

# Compute the midpoint and width of each bin
mid = (lower + upper) / 2.0
width = upper - lower

# Grab the data to be plotted
y = data('value')

# Plot everything
axes0.bar(mid, y, width=width)

# Show the plot and save it
plt.show()
figure.savefig("output.pdf")

## Input Parameters

• vppThe VectorPostprocessor to compute histogram of

C++ Type:VectorPostprocessorName

Options:

Description:The VectorPostprocessor to compute histogram of

• num_binsThe number of bins for the histograms

C++ Type:unsigned int

Options:

Description:The number of bins for the histograms

### Required Parameters

• execute_onTIMESTEP_ENDThe list of flag(s) indicating when this object should be executed, the available options include NONE, INITIAL, LINEAR, NONLINEAR, TIMESTEP_END, TIMESTEP_BEGIN, FINAL, CUSTOM.

Default:TIMESTEP_END

C++ Type:ExecFlagEnum

Options:NONE INITIAL LINEAR NONLINEAR TIMESTEP_END TIMESTEP_BEGIN FINAL CUSTOM

Description:The list of flag(s) indicating when this object should be executed, the available options include NONE, INITIAL, LINEAR, NONLINEAR, TIMESTEP_END, TIMESTEP_BEGIN, FINAL, CUSTOM.

• contains_complete_historyFalseSet this flag to indicate that the values in all vectors declared by this VPP represent a time history (e.g. with each invocation, new values are added and old values are never removed). This changes the output so that only a single file is output and updated with each invocation

Default:False

C++ Type:bool

Options:

Description:Set this flag to indicate that the values in all vectors declared by this VPP represent a time history (e.g. with each invocation, new values are added and old values are never removed). This changes the output so that only a single file is output and updated with each invocation

### Optional Parameters

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Options:

Description:Set the enabled status of the MooseObject.

• use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Default:False

C++ Type:bool

Options:

Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

• outputsVector of output names were you would like to restrict the output of variables(s) associated with this object

C++ Type:std::vector

Options:

Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object

• control_tagsAdds user-defined labels for accessing object parameters via control logic.

C++ Type:std::vector

Options:

Description:Adds user-defined labels for accessing object parameters via control logic.

• allow_duplicate_execution_on_initialFalseIn the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).

Default:False

C++ Type:bool

Options:

Description:In the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).

• force_preauxFalseForces the GeneralUserObject to be executed in PREAUX

Default:False

C++ Type:bool

Options:

Description:Forces the GeneralUserObject to be executed in PREAUX


text:
# Math Help - Help a LaTeX Newb Please.

1. ## Help a LaTeX Newb Please.

Hi everyone. I feel very stupid for asking this. Ok, I have dowloaded MiKTeX and installed it. Now, do I need to download LaTeX as well? If I want to create documents do I need a tex editor like winEDT? Help! I am lost and very confused

Thanks for any help.

2. Originally Posted by qspeechc
Hi everyone. I feel very stupid for asking this. Ok, I have dowloaded MiKTeX and installed it. Now, do I need to download LaTeX as well? If I want to create documents do I need a tex editor like winEDT? Help! I am lost and very confused

Thanks for any help.
You don't need to, if you know what you are doing you can use any text editor and the tools provided by MiKTeX, but you don't want to do that, so yes you will need a TeX editor of some kind. It also a good idea to make sure you have Ghostscript and Ghostview installed.

RonL

3. Thanks for the help. Would WinShell count as a text editor (I have downloaded it)? Would notepad?
And what are Ghostscript and Ghostview, why do I need them, and how do I know if my MiKTeX has it?
Thank you for the help.

4. Originally Posted by qspeechc
Thanks for the help. Would WinShell count as a text editor (I have downloaded it)? Would notepad?
And what are Ghostscript and Ghostview, why do I need them, and how do I know if my MiKTeX has it?
Thank you for the help.
All the bits are seperate WinShell is a front end for MiKTeX, that is it is an editor and integrates the main MiKTeX tools in its interface.

The attachment shows a screen shot of WinShell, notice the button bar, there are buttons for invoking 7 tools the first compiles to DVI, the second something to do with bibtex, DVI viewer, compile to postscript, postscript viewer, compile to pdf, and finally pdfviewer.

The installation sequence for WinShell is something like MiKTeX, Ghostscript/Ghostview, WinShell (I assume you already have Acrobat reader installed (but its not essential anyway).

RonL