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# zbMATH — the first resource for mathematics

Kernel smoothers: an overview of curve estimators for the first graduate course in nonparametric statistics. (English) Zbl 1127.62330
Summary: An introduction to nonparametric regression is accomplished with selected real data sets, statistical graphics and simulations from known functions. It is pedagogically effective for many to have some initial intuition about what the techniques are and why they work. Visual displays of small examples along with the plots of several types of smoothers are a good beginning. Some students benefit from a brief historical development of the topic, provided that they are familiar with other methodology, such as linear regression. Ultimately, one must engage the formulas for some of the linear curve estimators. These mathematical expressions for local smoothers are more easily understood after the student has seen a graph and a description of what the procedure is actually doing. In this article there are several such figures. These are mostly scatterplots of a single response against one predictor. Kernel smoothers have series expansions for bias and variance. The leading terms of those expansions yield approximate expressions for asymptotic mean squared error. In turn these provide one criterion for selection of the bandwidth. This choice of a smoothing parameter is done a rich variety of ways in practice. The final sections cover alternative approaches and extensions. The survey is supplemented with citations to some excellent books and articles. These provide the student with an entry into the literature, which is rapidly developing in traditional print media as well as on line.

##### MSC:
 62G08 Nonparametric regression and quantile regression
##### Software:
ElemStatLearn; fda (R); KernSmooth; SemiPar
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##### References:
 [1] Altman, N. S. (1992). An introduction to kernel and nearest-neighbor nonparametric regression. Amer. Statist. 46 175–185. [2] Bartlett, M. S. (1963). Statistical estimation of density functions. Sankhyā Ser. A 25 245–254. · Zbl 0129.32302 [3] Benedetti, J. K. (1977). On the nonparametric estimation of regression functions. J. Roy. Statist. Soc. Ser. B 39 248–253. · Zbl 0367.62088 [4] Chu, C.-K. and Marron, J. S. (1991). Choosing a kernel regression estimator (with discussion). Statist. Sci. 6 404–436. JSTOR: · Zbl 0955.62561 [5] Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 74 829–836. · Zbl 0423.62029 [6] Epanechnikov, V. A. (1969). Nonparametric estimation of a multivariate probability density. Theory Probab. Appl. 14 153–158. · Zbl 0175.17101 [7] Eubank, R. L. (1999). Nonparametric Regression and Spline Smoothing , 2nd ed. Dekker, New York. · Zbl 0936.62044 [8] Fan, J. (1992). Design-adaptive nonparametric regression. J. Amer. Statist. Assoc. 87 998–1004. · Zbl 0850.62354 [9] Fan, J. and Gijbels, I. (1995). Data-driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaptation. J. Roy. Statist. Soc. Ser. B 57 371–394. · Zbl 0813.62033 [10] Fan, J. and Gijbels, I. (1996). Local Polynomial Modeling and Its Applications . Chapman and Hall, London. · Zbl 0873.62037 [11] Gasser, T. and Müller, H.-G. (1979). Kernel estimation of regression functions. Smoothing Techniques for Curve Estimation . Lecture Notes in Math. 757 23–68. Springer, Heidelberg. · Zbl 0418.62033 [12] Gerard, P. D. and Schucany, W. R. (1997). Locating exotherms in differential thermal analysis with nonparametric regression. J. Agric. Biol. Environ. Stat. 2 255–268. [13] Hart, J. D. (1997). Nonparametric Smoothing and Lack-of-Fit Tests . Springer, New York. · Zbl 0886.62043 [14] Hart, J. D. and Lee, C.-L. (2005). Robustness of one-sided cross-validation to autocorrelation. J. Multivariate Anal. 92 77–96. · Zbl 1065.62068 [15] Hastie, T., Tibshirani, R. and Friedman, J. (2001). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York. · Zbl 0973.62007 [16] Hodges, J. L., Jr. and Lehmann, E. L. (1956). The efficiency of some nonparametric competitors of the $$t$$-test. Ann. Math. Statist. 27 324–335. · Zbl 0075.29206 [17] Hurvich, C. M., Simonoff, J. S. and Tsai, C.-L. (1998). Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 271–293. · Zbl 0909.62039 [18] Hurvich, C. M. and Tsai, C.-L. (1989). Regression and time series model selection in small samples. Biometrika 76 297–307. · Zbl 0669.62085 [19] Jia, A. and Schucany, W. R. (2004). Recursive partitioning for kernel smoothers: A tree-based approach for estimating variable bandwidths in local linear regression. Unpublished manuscript. [20] Lin, X., Wang, N., Welsh, A. H. and Carroll, R. J. (2004). Equivalent kernels of smoothing splines in nonparametric regression for clustered/longitudinal data. Biometrika 91 177–193. · Zbl 1132.62321 [21] Loader, C. R. (1996). Change point estimation using nonparametric regression. Ann. Statist. 24 1667–1678. · Zbl 0867.62033 [22] Loader, C. R. (1999). Local Regression and Likelihood . Springer, New York. · Zbl 0929.62046 [23] Müller, H.-G. (1987). Weighted local regression and kernel methods for nonparametric curve fitting. J. Amer. Statist. Assoc. 82 231–238. · Zbl 0632.62039 [24] Müller, H.-G. (1992). Change-points in nonparametric regression analysis. Ann. Statist. 20 737–761. JSTOR: · Zbl 0783.62032 [25] Nadaraya, E. A. (1965). On nonparametric estimates of density functions and regression curves. Theory Probab. Appl. 10 186–190. · Zbl 0134.36302 [26] Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065–1076. · Zbl 0116.11302 [27] Pitblado, J. (2000). Estimating partially variable bandwidths in local linear regression using an information criterion. Ph.D. dissertation, Dept. Statistical Science, Southern Methodist Univ. [28] Priestley, M. B. and Chao, M. T. (1972). Nonparametric function fitting. J. Roy. Statist. Soc. Ser. B 34 385–392. · Zbl 0263.62044 [29] Ramsay, J. O. and Silverman, B. W. (1997). Functional Data Analysis . Springer, New York. · Zbl 0882.62002 [30] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 832–837. · Zbl 0073.14602 [31] Ruppert, D. (1997). Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation. J. Amer. Statist. Assoc. 92 1049–1062. · Zbl 1067.62531 [32] Ruppert, D., Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression . Cambridge Univ. Press. · Zbl 1038.62042 [33] Schlee, W. (1988). Regressograms. Encyclopedia of Statistical Sciences 8 1–3. Wiley, New York. [34] Scott, D. W. (1992). Multivariate Density Estimation . Wiley, New York. · Zbl 0850.62006 [35] Sheather, S. (2005). Density estimation. Statist. Sci. 19 588–597. · Zbl 1100.62558 [36] Signorini, D. F. and Jones, M. C. (2004). Kernel estimators for univariate binary regression. J. Amer. Statist. Assoc. 99 119–126. · Zbl 1089.62506 [37] Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis . Chapman and Hall, London. · Zbl 0617.62042 [38] Spencer, J. (1904). On the graduation of rates of sickness and mortality. J. Institute of Actuaries 38 334–343. [39] Stone, C. J. (1977). Consistent nonparametric regression (with discussion). Ann. Statist. 5 595–645. JSTOR: · Zbl 0366.62051 [40] Tukey, J. W. (1961). Curves as parameters and touch estimation. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1 681–694. Univ. California Press, Berkeley. · Zbl 0105.12304 [41] Ullah, A. (1985). Specification analysis of econometric models. J. Quantitative Economics 1 187–209. [42] Wahba, G. and Wold, S. (1975). A completely automatic French curve. Comm. Statist. 4 1–17. · Zbl 0305.62043 [43] Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing . Chapman and Hall, London. · Zbl 0854.62043 [44] Watson, G. S. (1964). Smooth regression analysis. Sankhyā Ser. A 26 359–372. · Zbl 0137.13002 [45] Welsh, A. H., Lin, X. and Carroll, R. J. (2002). Marginal longitudinal nonparametric regression: Locality and efficiency of spline and kernel methods. J. Amer. Statist. Assoc. 97 482–493. · Zbl 1073.62529
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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# $Y \subset X$ where $X$ and $Y$ are connected but $X-Y$ is separated by $A$ and $B$

It is known that $Y \cup A$ and $Y \cup B$ are connected given that $X$ is a connected space, $Y$ is a connected subspace of $X$, and that $X-Y$ is separated by $A$ and $B$. Is it true that $A$ and $B$ are connected as well?

No, this is false. Let $X \subset \Bbb{R}^2$ be the union of two closed discs $D_1, D_2$ whose boundaries intersect in a single point $p$. Let $L$ be the line connecting $p$ to the centers of both discs, and $Y = X \cap L$. If you let $A = D_1 /Y$, $B = D_2/Y$ then $Y$, $Y \cup A$ and $Y \cup B$ are all connected and $X/Y$ is separated by $A$ and $B$, but both $A$ and $B$ are not connected.

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# HatHoseHook x 1000

So, things going on today.

This is the 1000th post, for starters. Should have posted it yesterday, but failed. Heck, I only just made today

### Teel

Author, artist, romantic, insomniac, exorcist, creative visionary, lover, and all-around-crazy-person.

## 4 thoughts on “HatHoseHook x 1000”

1. Teel says:

I took the easy way out, and posted to 445000. Which means the average post must have 445 words, right?

Sure.

2. Teel says:

I took the easy way out, and posted to 445000. Which means the average post must have 445 words, right?

Sure.

3. Teel says:


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# Search

You can find the results of your search below.

## Fulltext results:

mark_twain @air
llection of curiosities in Heidelberg Castle, and one day I surprised the keeper of it with my German. ... mless, and so slippery and elusive to the grasp. One is washed about in it, hither and thither, in the... be, my experience. Every time I think I have got one of these four confusing "cases" where I am master... mething -- that is, resting (which is one of the German grammar's ideas of doing something)
quotes @dob
al Indestrielller und Politiker 1916 - 2001 “If one would give me six lines written by the hand of th... includes economic reality..... "Bigamy is having one wife too many. Monogamy is the same." - Oscar Wil... n happiness d. "If the whole human race lay in one grave, the epitaph on its headstone might well be... learned this, at least, by my experiment; that if one advances confidently in the direction of his dre
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avily braided white cap pulled down rakishly over one cold gray eye. "We can't make it, sir. It's spoil... d. "The old man will get us through" they said to one another. "The Old Man ain't afraid of Hell!" . . ... "You're tensed up again," said Mrs. Mitty. "It's one of your days. I wish you'd let Dr. Renshaw look y... r is giving way!" shouted an intern. "There is no one in the East who knows how to fix it!" "Quiet, man
bayesian_statistics @blog
lt for your total sample. [[wp>Bayes Theorem]] is one tool to look at this and it yields some unexpecte... $$where the event will be clearly decided, i.e. one of the options must occur$$Occurance + Non\mbox... the probability of getting 3 heads and 7 tails if one flips a fair coin 10 times. You do $n=10$ trials... or large $n$ and for $p$ not too close to zero or one.  f(x| \mu, \sigma^2) = \frac{1}{\sqrt{2 \pi \
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ink points to google]]. Email addresses like this one: <andi@splitbrain.org> are recognized, too. Do... k points to google]]. Email addresses like this one: <andi@splitbrain.org> are recognized, too. ====... ll automatically add them as alternatives so that one of the three files is understood by your browser.... s) by adding ''%%:::%%'' into the cells below the one to which they should connect. ^ Heading 1 ^
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welcome @wiki


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# Prove that $BXOY$ is cyclic with spiral similarity over a midpoint

Let $$O$$ be the circumcenter of triangle $$ABC$$. Line $$CO$$ intersects the altitude from $$A$$ at point $$K$$. Let $$P,M$$ be the midpoints of $$AK$$, $$AC$$ respectively. If $$PO$$ intersects $$BC$$ at $$Y$$, and the circumcircle of triangle $$BCM$$ meets $$AB$$ at $$X$$, prove that $$BXOY$$ is cyclic.

My Progress: Since this was from a spiral similarity handout, I am using spiral similarity.

Note that $$XM$$ is anti parallel to $$BC$$. Now since $$AK$$ and $$AO$$ are isogonals and $$AK\perp BO \implies AO \perp XM$$.

Now define $$AO\cap XM=U$$ , $$M'$$=reflection of $$O$$ wrt $$M$$ , $$A'$$= antipode of $$A$$ wrt $$(ABC)$$ , $$W=AC\cap A'K$$ , $$Z=A'M' \cap (ABC) .$$

I observed that there is a spiral similarity (say $$\gamma$$ ) centred at $$X$$ taking $$M'M$$ to $$AK$$ , and hence $$O$$ to $$P$$ . So I am trying to prove that observation. Here is what I got.

• $$ACA'Z$$ is a rectangle : as $$A-O-A'$$ and $$Z-O-C$$ are collinear
• $$MM'A'C$$ is a rectangle : as $$M$$ and $$M'$$ are midpoint of $$AC$$ and $$ZA'$$
• $$MUM'A'C$$ is cyclic : Note that $$\angle MUA'=\angle MM'A'=90$$

So by the above claims , we get $$\angle XAK =\angle MAO=\angle OA'M'= \angle UA'M'= \angle UMO=\angle XMM'$$.

Now after this I want to prove

$$\angle MXM'= \angle AXK$$

After this we will be done, because then we will have $$\gamma : M'M \rightarrow KA \implies \gamma : O \rightarrow P \implies \Delta POX \sim \Delta AMX \implies \angle POX = \angle AMX=\angle ABC$$

So, as you have already shown that $$\angle XAK = \angle XMM'$$, its enough to show that $$\angle AXP =\angle MXO$$ (Notice that instead of $$\gamma: MM'\mapsto AK$$, I'm trying to show $$\gamma: MO\mapsto AP$$ and the reason for this is because as you observed $$XM\perp AO$$, showing angles $$AXP$$ and $$MXO$$ same would be equivalent of showing center of $$\odot(AXO)$$ lies on $$XP$$ and thus, this is a more natural way of approaching the problem.)
Note that as $$\angle AUM = \angle AMO =90,$$ we get, $$\angle UMO =\angle OAM=\angle OCA\overset{Thales'}{=}\angle AMP$$. Further note that $$\{AP,AO\}$$ are isogonal lines with respect to $$\triangle XAM$$ and by the fact that $$\angle UMO =\angle AMP$$ we conclude that points $$\{P,O\}$$ are isogonal with respect to $$\triangle XAM$$ and thus, $$\angle PXA =\angle OXM$$ and we are done!$$\tag*{\blacksquare ::&}$$


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# JEE Advanced 2023 Revision Notes for Optics

Last updated date: 27th Mar 2023
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The study of light rays and their natural features is quite fascinating. It is covered in the chapter on optics along with the concepts of laws of reflection, optical rays, optical mediums, mirrors, lenses, etc. This chapter will explain how a light ray follows a rectilinear path and deviates from its path when it passes through a transparent solid medium. This chapter will also cover problems related to the focal length of mirrors and lenses. To understand these concepts easily, refer to the JEE Advanced Physics Optics revision notes prepared by the subject matter experts of Vedantu.

These revision notes will enable students to revise the chapter faster and recall what they have studied. The concise format of these notes will help you understand the meaning of the scientific terms, their derivations, and descriptions. Add these notes to your Optics study material and prepare this chapter excellently.

Competitive Exams after 12th Science

## Access JEE Advanced Revision Notes Physics Optics

### Light

• Light is a type of energy that causes our eyes to experience visual sensations.

• Light is a type of electromagnetic wave that travels in a transverse direction.

• The speed of light in vacuum is $3 \times 10^{8} \mathrm{~m} / \mathrm{s}$.

### Reflection of Light

• Reflection of light is the phenomenon of light rays bouncing back in the same medium after striking a smooth surface.

Reflection

### There are Two Types of Reflection

1. Regular Reflection: when light reflects off a very smooth surface and forms a clear image, this is referred to as regular reflection.

2. Irregular Reflection: when a parallel light ray beam strikes a rough surface, the reflected rays scatter in various directions. This is referred to as irregular or diffuse reflection. Diffuse reflection refers to reflection from a rough surface.

Different Types of Reflection

### Laws of Reflection

• There are Two Laws of Reflection

1. The angle of incidence and reflection are always equal.

$\angle i=\angle r$

1. At the point of incidence, the incident ray, reflected ray, and normal are all lie in the same plane.

### Mirror

• A mirror is a polished glass surface onto which almost all light is reflected. The mirror is called a plane mirror if the reflecting surface is plane, and a spherical mirror if the reflecting surface is spherical.

### Image

• Image is an optical appearance produced when light rays coming from an object are reflected or refracted from a mirror ( or appearing to  meet ).

• There are Two Types of Image

1. Real Image: When light rays from one point meet at another point after reflection, the image formed is known as a real image.

2. Virtual Image: The image formed is called a virtual image if the light rays coming from a point do not actually meet after reflection or appear to come from another point.

### Reflection From Plane Mirror

• The size of the image in a plane mirror is always equal to the size of the object, and it is virtual.

• In a plane mirror, the image appears to be as far behind the mirror as the object in front of it.

### Important Points Related to Simple Mirror are Given Below

• When an object moves a distance towards or away from a mirror, its image moves a distance towards or away from the mirror.

• The minimum mirror size required to see the full image of an observer is half the observer's height.

• When the plane mirror is rotated in the plane of incidence by an angle $\theta$, then the reflected ray rotates by an angle $2 \theta .$

• The focal length of a plane mirror is infinite, implying that its power is nil. The plane mirror produces a linear magnification of 1.

• If  two plane mirrors are kept facing each other at an angle $\theta$ and an object is placed between them, then number of images, $n=\dfrac{360^{\circ}}{\theta}-1$, if $\dfrac{360^{\circ}}{\theta}$ is even or the object lies symmetrically. Number of images, $n=\left(\dfrac{360^{\circ}}{\theta}\right)$, if $\dfrac{360^{\circ}}{\theta}$ is odd or the object lies asymmetrically.

• There are an infinite number of images when two plane mirrors are parallel to each other.

• The plane mirror is used in kaleidoscopes, periscopes, and looking glasses, among other things.

### Spherical Mirror

• The mirror is referred to as a spherical mirror if the reflecting surface is curved inwards or outwards.

• Spherical Mirrors are of Two Types

1. Concave Mirror: Concave mirrors are spherical mirrors with an inwardly curved reflecting surface. Because a beam of light converges after reflection from its surface, it is also known as a convergent mirror. e.g. A concave mirror can be thought of as the inner curved surface of a shining spoon.

Concave Mirror

1. Convex Mirror: Convex mirrors are spherical mirrors with an outwardly curved reflecting surface. Because a beam of light diverges after reflection from its surface, it is also known as a divergent mirror. e.g. A shining spoon's outer curved surface can be treated or thought of as a convex mirror.

### Some Definitions Related to Spherical Mirrors

1. Centre of Curvature: The centre of curvature of a spherical mirror is the centre of the sphere from which it is separated. In the above figures, it is marked by $C$.

2. Radius of Curvature: A spherical mirror's radius of curvature is the radius of the sphere from which the mirror is separated. In the above figure, it is shown by $R$.

3. Pole: The spherical mirror's pole is the midpoint of its reflecting surface. In the figure, it is shown by $P$.

4. Principal Axis: The line connecting the pole and the centre of curvature is the principal axis of a spherical mirror. In the figure, PC is the principal axis.

5. Aperture of Mirror: The aperture of a mirror refers to the portion of the reflecting surface that can be exposed to incident light.

6. Principal Focus or Focus: After reflection, the light rays coming parallel to the principal axis actually meet or appear to be coming from this point on the principal axis of the mirror. It is represented by $F$. The focus of a concave mirror is in front of the mirror, whereas the focus of a convex mirror is behind the mirror. A concave mirror's focus is real, whereas a convex mirror's focus is virtual.

7. Focal Length: The focal length of a spherical mirror is the distance between the pole and the focus. It is represented by $f$.

(i) Plane mirrors have an infinite focal length.

(ii) The curvature radius of a plane mirror is infinite.

### Relation Between Focal Length and Radius of Curvature of Spherical Mirror

• When the aperture of the mirror is small, then the following relation is given below between focal length $(f)$ and radius of curvature$(R)$. i.e. $f=\dfrac{R}{2}$

### Sign Convention for Reflection by Spherical Mirrors

• We will use a set of sign conventions called the new Cartesian sign convention based on Cartesian coordinates when dealing with light reflection by spherical mirrors.

• The mirror's pole (P) is used as the origin in this convention. The X-axis of the coordinate system is taken as the principal axis of the mirror.

Sign Convention for Reflection by Spherical Mirrors

• The Conventions are as Follows

• Always place the object to the left of the mirror.

• The pole of the mirror is used to measure all distances parallel to the principal axis (X-axis).

• Negative distances to the left of the pole (-ve X-axis). Positive distances are those to the right of the pole (+ve X-axis).

• Positive distances are those measured perpendicularly above the principal axis (along the +Y-axis).

• Negative distances are those measured perpendicularly below the principal axis (along the -Y-axis).

### Image Formation by Spherical Mirrors

• There are two types of image formation by spherical mirrors.

1. Image Formation by a Concave Mirror

• The ray diagram, as well as the position and nature of the image formed by a concave mirror, is shown in the table below for various positions of the object.

### Formation of Image by Concave Mirror for Different Positions of Object

 Position of Object Ray Diagram Position of image Nature and size of image At infinity At focus orin the focalplane Real, inverted, extremelydiminishedpoint size Beyond the centre of curvature but at finite distance Betweenlocus andthe centre ofcurvature Real, inverted and diminished At the centre of curvature At the centre of curvature Real, inverted and equal to the object Between locus and centre of curvature Beyond thecentre ofcurvature Real, inverted and bigger than object At the focus At infinity Real, inverted and extremely magnified Between thepole and locus Behind the mirror Virtual, erect and magnified

(ii) Image Formation by a Convex Mirror

• There are two positions of the object that are considered when studying the image formed by a convex mirror. The first is when the object is infinitely far away from the mirror, and the second is when the object is finitely far away from the mirror.

• The ray diagrams, as well as the position and nature of the image, are shown in the table below.

• For the above two positions of the object, a convex mirror was used.

### Formation of Image by Convex:

Mirror for Different Positions of Object

 Position ofObject Ray Diagram Position of Image Nature andSize ofImage At infinity At the focus.behind themirror Virtual, erect and extremely diminished Between infinity and the pole of the mirror (i.e. at finite distance) Between the locus and the pole, behind the mirror Virtual, erectanddiminished

### Mirror Formula

• In a spherical mirror, the object distance $(u)$ is the distance between the object and its pole. The image distance $(v)$ is the distance between the image and the mirror's pole. We know that the relation between quantities $(u, v$ and $f)$, i.e. mirror formula, $\dfrac{1}{v}+\dfrac{1}{u}=\dfrac{1}{f}$

• Here, u, v and $f$ are to be used according to their new Cartesian sign convention.

### Linear Magnification

• Linear magnification is defined as the ratio of image height to object height. It is represented by $(m)$.

• i.e. $m=\dfrac{\text { height of image }(I)}{\text { height of object }(O)}$

• Or Linear magnification is the ratio of image distance to object distance.$\text{ i}\text{.e}\text{. }m=\dfrac{\text{ image distance }(v)}{\text{ object distance }(u)}=\dfrac{-v}{u}$

• The plane mirror produces a linear magnification of 1.

### Identification of Mirrors

• The nature of the mirror can be determined by observing the image produced by it at various positions of the object.

1. The mirror is called a plane mirror if the image formed by it is the same size as the object at different positions.

2. When the image formed by a mirror is diminished for all object positions, the mirror is said to be convex.

3. A concave mirror is one in which the image formed behind the mirror is longer than the object.

### Refraction of Light

• Refraction of light is the change in the path of a light ray as it travels from one medium to another.

• If light travels from a rarer medium to a denser one, it bends towards the normal $(i>r)$ and when it travels from a denser medium to a rarer one, it bends away from the normal $(i<r)$.

Refraction of Light

• Here , $i=$ angle of incidence And $r=$ angle of refraction

• The optically rarer medium is one in which the speed of light is greater, while the optically denser medium is one in which the speed of light is lower.

• When a ray of light is refracted, its velocity and wavelength change, but the frequency remains constant.

• i.e. $n_{1}=n_{2}$ or $\dfrac{v_{1}}{\lambda_{1}}=\dfrac{v_{2}}{\lambda_{2}}$

### Laws of Refraction

The following laws govern the laws of refraction.

(i) At the point of incidence, the incident ray, the normal to the surface of separation, and the refracted ray all lie in the same plane.

(ii) The ratio of sine of angle of incidence to the sine of angle of refraction is constant for light rays of the same color passing through two medium, i.e. $\dfrac{\sin i}{\sin r}={ }_{1} \mu_{2}=\dfrac{\mu_{2}}{\mu_{1}} \text { [constant] }$ for same pair of media.

• Here, ${ }_{1} \mu_{2}$ is called the refractive index of the second medium with respect to the first medium. This law is known as Snell's law.

### Refractive Index

• Refractive index of a medium,$\mu=\dfrac{c}{v}=\dfrac{\lambda_{\text {air }}}{\lambda_{\text {medium }}}=\dfrac{\text { speed of light in vacuum }}{\text { speed of light in the medium }}$

• Here, $c$=speed of light in vacuum $=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$

• The refractive index of two mediums is determined by the nature of the two mediums, the colour of the light ray, and the temperature of the two mediums.

• The refractive index of a medium decreases as its temperature rises.

• The optical property of a medium is its refractive index.

• All mediums' refractive indices are measured in relation to air.

• Diamonds have the highest refractive index.

• If a light ray travels from vacuum to a medium, the absolute refractive index of that medium is called refractive index.

• ${ }_{1} \mu_{2}$ $=\dfrac{\text { velocity of light in first medium }}{\text { velocity of light in second medium }}$

• ${ }_{1} \mu_{2}$ $=\dfrac{\text { wavelength }(\lambda) \text { in first medium }}{\text { wavelength }(\lambda) \text { in second medium }}$

• Note For glass/water pair, ${{\mu }_{g}}=\dfrac{_{a}{{\mu }_{g}}}{_{a}{{\mu }_{w}}}$.

### Refraction Through a Rectangular Glass Slab

• When a light ray passes through a glass slab, the emergent ray is parallel to the incident ray but slightly offset to the side.

• Refraction occurs twice in this case, once when the ray enters the glass slab from the air and again when it exits the glass slab to the air.

• The angle between the incident and emergent rays is perpendicular, $d=t\left[1-\dfrac{\cos i}{\sqrt{\mu^{2}-\sin ^{2} i}}\right] \sin i$

• This value is equal to the lateral shift of the ray.

Refraction Through a Rectangular Glass Slab

• Here, i = angle of incidence, $r=$ angle of refraction And $e=$ angle of emergence.

### Critical Angle

• When a light ray travels from a denser medium to a rarer medium, the angle of refraction increases as the angle of incidence increases, so the critical angle is the angle of incidence at which the angle of refraction equals 90 degree.

• $\sin C=\dfrac{\mu_{R}}{\mu_{D}} \Rightarrow C=\sin ^{-1}\left(\dfrac{\mu_{R}}{\mu_{D}}\right)=\sin ^{-1}\left(\dfrac{1}{{ }_{r} \mu_{d}}\right)$

### Total Internal Reflection

• The ray is reflected back into the denser medium if the angle of incidence in the denser medium is greater than the critical angle (c).

• This phenomenon is known as total internal reflection.

• For total internal reflection, it is necessary that

• From a denser medium, light rays should travel to a rarer medium.

• In a denser medium, the angle of incidence should be greater than the critical angle.

### Some Phenomena Based on Total Internal Reflection (TIR)

1. A diamond's brilliance is due to total internal reflection.

2. A fish can't see the entire surface from inside the water, it sees only a circular path of radius r of light. The expression of $r=\dfrac{h}{\sqrt{\mu^{2}-1}}$

3. The phenomenon of mirage occurs in the desert due to total internal reflection.

4. Due to total internal reflection, the air bubbles in glass paper weight appear silvery white.

5. Due to total internal reflection, a blackened test tube dipped in water from the outside appears silvery white.

6.  As the temperature rises, so does the critical angle.

### Refraction From Spherical Surfaces

There are two types of refraction at a spherical surface

1. Convex

2. Concave

• The refraction formula for both surfaces is $\dfrac{_{1}{{\mu }_{2}}}{v}-\dfrac{1}{u}=\dfrac{_{1}{{\mu }_{2}}-1}{R}$

• $\mu_{2}$ is the refractive index of the second medium with respect to the first. If $\mu_{1}$ and $\mu_{2}$ are refractive indices of first and second medium with respect to air, then $\dfrac{\mu_{2}}{v}-\dfrac{\mu_{1}}{u}=\dfrac{\mu_{2}-\mu_{1}}{R}$

### Lens

• A lens is a refracting medium piece that is bounded by two surfaces, at least one of which is curved. The most common lenses are spherical lenses, which have either both spherical surfaces or one spherical and one plane surface.

### There are Two Types of Lenses:

Convex or Converging Lens

• Convex lens is a transparent medium that is bound by two bulging surfaces. There are three types of convex lenses (as shown below)

Convex or Converging Lens

• When a plano-convex lens' plane surface is silver polished, it behaves like a concave mirror with a focal length half that of the initial value.

### Concave or Divergent Lens

• Concave lens is a transparent medium bounded by two hollow surfaces.

Concave or Divergent Lens

### Some Definitions Related to Lenses

• Optical Center: The optical centre of a lens is its central point. It is symbolised by the letter O. The optical centre is a point inside or outside the lens where incident rays refract without deviating from their path.

• Principal focus Lens has two principal foci.

1. First Principal Focus: It's a point on the lens's principal axis where the rays that originate from or are directed to become parallel to the axis after refraction.

First Principal Focus

1. Second Principal Focus: After refraction from the lens, it is the point on the principal axis where rays coming parallel to the principal axis converge on the opposite side of the lens (convex) or appear to meet on the same side of the lens (concave).

Second Principal Focus

Convex lenses have real foci, whereas concave lenses have virtual foci.

1. Centre of Curvature: The centres of curvature of the lens are the centres of the two imaginary spheres of which the lens is a part. With respect to its two curved surfaces, a lens has two centres of curvature.

2.  Radii of Curvature: The radii of curvature of the lens are the radii of the two imaginary spheres of which the lens is a part. R is the symbol for it. A lens has two curvature radii. These could be equal or not.

### Image Formation by Lenses

• Formation of Image by a Convex Lens

• The ray diagrams, as well as the position and nature of the image formed by a convex lens, are shown in the table below for various positions of the object.

### Formation of Image by Convex Lens for Different Positions of Object

 Case 1 - Object is Placed at infinity Case 2 - Object is beyond 2F Case 3 - Object is at 2F Case 4 - Object is between F and 2F Case 5 - Object is between O and F

### Formation of Image by a Concave Lens

• There are two positions of objects for studying image formation by concave lens.

• We always get erect images, diminished images, and virtual images when using a concave lens. If the object is very far away, the images formed by lenses will be greatly reduced.

• The ray diagrams, as well as the position and nature of the image formed by the concave lens for the above two positions of the object, are shown in the table below.

Formation of Image by Concave Lens for Different Positions of Object

 Position of Object Position of Image Size of Image Nature of Image Ray Diagram At infinity At focus on same side of lens as an object Highly diminished Virtual and erect At finite distance Between focus and optical centre on the same size of lens as an object Diminished Virtual and erect

### Important Formula Related to Lens

• Lens Formula: For both convex and concave lenses, the formula connecting the object and image distance is $\dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u} .$ Here, f=focal length of lens, $u=$ Object distance and $v=$ image distance

• Power of Lens: The power of a lens refers to its ability to converge or diverge light rays. The reciprocal of focal length is used to calculate it. Power, $\quad P=\dfrac{1}{f \text { (in metre) }} \Rightarrow P=\dfrac{100}{f \text { (in cm) }}$. Its unit is diopter (D). Power of a plane mirror is zero.

• Newton's Formula for Lenses: The lens formula is given by Newton. $f^{2}=a_{1} a_{2} \quad \Rightarrow \quad f=\sqrt{a_{1} a_{2}}$ Here, $a_{1}=$ distance of object from first focus and $a_{2}=$ distance of object from second focus

### Magnification Produced by Lens

1. The linear magnification of a lens is given by

Linear Magnification

1.  Areal magnification of lens Areal magnification $\left(m_{s}\right)$ is given by$m_{s}=\dfrac{\text { area of image }}{\text { area of object }}=\dfrac{A_{i}}{A_{o}}=m^{2}=\left(\dfrac{f}{f+u}\right)^{2}$

2. Relationship between object speed and image $v_{i}=\left(\dfrac{f}{f+u}\right) v_{a}$

### Some Phenomenon Related to Lens

1. Lens Immersed in a Liquid If $_{a}{\mu }_{g}$ is the refractive index of lens and ${\mu}_l$ refractive index of liquid, then the total length of lens is $\dfrac{1}{f}=\left( _{l}{{\mu }_{g}}-1 \right)\left( \dfrac{1}{{{R}_{1}}}-\dfrac{1}{{{R}_{2}}} \right)$

2. Cutting of a Lens

1. If a focal length symmetrical convex lens is cut into two parts along its optical axis, each part (a plano-convex lens) has a focal length of 2 f. If the two parts are joined as shown in the figure, however, then the focal length of the combination is again $f$.

Cutting of a Lens

1. If a symmetrical convex lens with a focal length of f is divided into two parts along the principal axis, the focal length of each part remains constant at f. The focal length of the combination if these two parts are joined with curved ends on one side is $\dfrac{f}{2}$. But on joining two parts in opposite senses the net focal length becomes infinite (or net power $=0$ ).

### Combination of Thin Lenses

1. If two or more lenses are in contact, the combination's equivalent focal length is

$\dfrac{1}{f}=\dfrac{1}{f_{1}}+\dfrac{1}{f_{2}}+\ldots=\sum_{i=1}^{n} \dfrac{1}{f_{i}}$

Power of combination,

$P=P_{1}+P_{2}+\ldots=\sum_{i=1}^{n} P_{i}$

1. If two lenses of focal lengths $f_1$ and $f_2$ are separated by a distance $x$, then its equivalent focal length,

$\dfrac{1}{F}=\dfrac{1}{f_{1}}+\dfrac{1}{f_{2}}-\dfrac{x}{f_{1} f_{2}}$

Power of combination,

$P=P_{1}+P_{2}-x P_{1} P_{2}$

### Prism

• A prism is a transparent refracting medium that is defined by two plane surfaces that are angled at an angle.

### Angle of Deviation

• The angle formed by the incident rays.

• And emergent ray (e) is called angle of deviation $(\delta)$

Prism

### Dispersion of Light

• When white light strikes a glass prism, it splits into its seven colour components according to the VIBGYOR sequence.

• Dispersion of white light is the term for this phenomenon.

Dispersion of Light

• The refractive index of glass is highest for violet light and lowest for red light.

• As a result, the violet colour of white light (sunlight) deviated the most, while the red colour deviated the least.

### Angular Dispersion

• It refers to the angle between the two extreme rays.

• Angular dispersion, $\theta=\delta_{V}-\delta_{R}=\left(\mu_{V}-\mu_{R}\right) A$

### Dispersive Power

• The ratio of angular dispersion to the mean deviation suffered by a light beam is used to determine the dispersive power of a prism material.

• Dispersive power, $\omega=\dfrac{\delta_{V}-\delta_{R}}{\delta}=\dfrac{\mu_{V}-\mu_{R}}{\mu-1}$ Here, $\mu$ Is the mean value of the refractive index of prism?

• The dispersive power of a prism is determined solely by its material and is unaffected by the prism's angle, angle of incidence, or size.

• Dispersive power is a non-dimensional and unitless term.

• A flint glass prism has more dispersive power than a crown glass prism.

### Dispersion without Deviation

(Direct Vision Prism)

1. To achieve dispersion without mean deviation, we combine two prisms made of different materials.

Direct Vision Prism

$A^{\prime}=\left(\dfrac{\mu-1}{\mu^{\prime}-1}\right) A$

### Deviation without Dispersion (Achromatic Prism)

1. We use a combination of two prisms made of different materials to produce deviation without dispersion.

$A^{\prime}=\dfrac{\left[\mu_{V}-\mu_{R}\right]}{\left[\mu_{V}^{\prime}-\mu_{R}^{\prime}\right]} A$

1. Net dispersion caused $=\left(\mu_{V}-\mu_{R}\right) A+\left(\mu_{V}^{\prime}-\mu_{R}^{\prime}\right) A^{\prime}$ $=(\mu-1) A\left(\omega-\omega^{\prime}\right)=\delta\left(\omega-\omega^{\prime}\right)$

2. Resultant deviation produced $=\delta\left[1-\dfrac{\omega}{\omega^{\prime}}\right]$

### Scattering of Light

• When light passes through a medium in which particles of the order of the wavelength of light are suspended, light deviates in different directions when it strikes these particles.

• Scattering of light is the term for this phenomenon. As a result, red light scatters the least (wavelength is highest) and violet light scatters the most (wavelength is least).

• Danger signals are red in colour because red light scatters the least and can thus be seen from a long distance.

• The sky appears blue because the blue colour of light scatters the most in sunlight, and this scattered blue light enters our eyes, causing the sky to appear blue.

• Clouds appear white because they are made up of large water droplets and dust particles that scatter all light colours equally. White is the result of combining all seven colours of sunlight.

• Because there is no atmosphere in space, no light scattering occurs, the sky appears black to astronauts from space.

### Human Eye

• One of the most important and sensitive sense organs in the human body is the eye.

• It allows us to appreciate the beautiful world and colours that surround us.

• It is made up of a lens made up of living tissues.

### A Human Eye has the Following Main Parts

• Cornea: It's the spherical transparent membrane that covers the front of the eye. This membrane allows light to enter the eye.

• Crystalline Lens: A convex lens, similar to a jelly made of proteins, is made of a transparent, soft, and flexible material.

• Iris: Between the cornea and the lens, there is a dark muscular diaphragm. It regulates the pupil's size.

• Pupil: It is a small opening between the iris and the cornea that allows light to enter the eye.

• Ciliary Muscles: They help to keep the lens in place and modify the curvature of the lens.

• Retina: The image is formed on the light-sensitive surface of the eye. It is made up of rods and cones that are sensitive to light.

• Optic Nerve: It connects the retina to the brain and transmits visual information.

• Sclera: It's an opaque, fibrous, protective layer of the eye that's made up of collagen and elastic fibre. The white of the eye is another name for it.

• Blind Spot: The optic nerve leaves the eye at this location. Because it lacks rods and cones, any image formed here is not sent to the brain.

• Aqueous Humour: The aqueous humour is a transparent liquid that fills the space behind the cornea.

• Vitreous Humour: Another liquid called vitreous humour fills the space between the eye lens and the retina.

• Accommodation of Eye: It is the ability of the eye lens to change its focal length in order to form sharp images of objects at various distances from the eye on the retina.

• Range of Vision: It is the distance between the eye's near and far points.For normal eye, the range of vision is 25cm to infinity.

• Near Point: It is the distance between an object and the human eye at which a sharp image is formed on the retina.

Human Eye

### Defects of Vision

1. Myopia or Short Sightedness: It is an eye defect that causes a person to be able to see nearby objects clearly but not far away objects. The far point of the eye shifts from infinity to a closer distance in this defect. A concave lens of appropriate power can be used to remove the flaw.

2. Hypermetropia or Long Sightedness: It is an eye defect that causes a person to be able to see far away objects clearly but not nearby objects.

3. Presbyopia: It is found in the elderly. This flaw makes it difficult to read comfortably and clearly. Bifocal lenses can be used to correct this flaw. The near point of the eye shifts away from the eye in this defect. A convex lens of appropriate power can be used to remove the flaw.

4. Astigmatism: A person with this defect is unable to focus on both horizontal and vertical lines at the same time. This flaw can be corrected with the use of suitable cylindrical lenses.

5. Colour Blindness: A person with this defect is unable to distinguish between a limited number of colours. The absence of cone cells sensitive to only a few colours is the cause of this defect. This flaw is irreversible.

6. Cataract: An opaque, white membrane forms on the cornea in this defect, causing a person to lose vision power partially or completely. This defect can be corrected by surgically removing the membrane. Optical instruments are developed and used by applying generalised rules of optics.

### Optical Instruments

• An optical instrument is a device made up of a specific set of mirrors, prisms, and lenses.

• The principles of operation of optical instruments are based on the laws of light reflection and refraction.

### Microscope

• It's an optical device that magnifies a small nearby object and thus increases the visual angle occupied by the image at the eye, making the object appear larger and more distinct.

### Simple Microscope

• A simple microscope consists of a convex lens with a short focal length fixed in a frame with a handle.

Simple Microscope

### Magnification of Simple Microscope

• When the final image is formed at a distance equal to or greater than the distinct vision distance,$M=1+\dfrac{D}{f}$

• For relaxed eye, $M=\dfrac{D}{f}$ where, $D=$ least distance of distinct vision

### Compound Microscope

• It is made up of two coaxially arranged converging lenses. The one facing the object is known as the objective, while the one closest to the eye is known as the eye piece.

• The objective's aperture and focal length are smaller than those of the eyepiece.

Compound Microscope

### Magnification of Compound Microscope

• When the final image is formed at the least distance of distinct vision, $m=-\dfrac{v_{0}}{u_{0}}\left(1+\dfrac{D}{f_{e}}\right)$

• For the image formed at infinity, $u_{e}=f_{e}$and $m=-\dfrac{v_{0}}{u_{0}} \cdot \dfrac{D}{f_{e}}$

• By making the focal length of the objective small, the magnifying power can be increased.

### Telescope

• A telescope is an optical instrument that increases the visual angle at the eye by forming an image of a distant object at the shortest distance of distinct vision, allowing the object to appear larger and more distinct.

### Astronomical Telescope

• It is made up of two coaxially positioned converging lenses. The objective is the lens that faces the distant object and has a large aperture and focal length.

• The other is known as the eyepiece because the eye is placed against it. The eyepiece tube slides inside the objective tube, allowing the distance between the objective and the eyepiece to be adjusted.

### Magnification of Astronomical Telescope

• For relaxed eye, $M_{\infty}=-\dfrac{f_{o}}{f_{e}}$ In this position, length of telescope, $L_{\infty}=f_{o}+f_{e}$

• When final image is formed at least distance of distinct vision, $M_{D}=-\dfrac{f_{o}}{f_{e}}\left(1+\dfrac{f_{e}}{D}\right)$ Length of telescope, $L_{D}=f_{o}+f_{e}$ Here, $f_{o}=$ focal length of objective lens and $f_{e}=\text { focal length of eyepiece }$.

• It is used to observe distinct images of celestial bodies such as stars, planets, and so on.

• Two convex lenses with unequal focal lengths make up a refracting telescope.

Astronomical Telescope

### Critical Angle:

The critical angle (C) is the angle of incidence in the denser medium for which the angle of refraction in the rarer medium is 90°.

$C = {\sin ^{ - 1}}\left( {\dfrac{1}{n}} \right)$

### Lens’s Maker’s Formula:

$\dfrac{1}{f} = (n - 1)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)$ , where n is the refractive index of the material of the lens with respect to the outer medium, R1 is the radius of curvature of the surface facing the object and R2 that of the other surface.

### Prism Formula:

$n = \dfrac{{\sin \left( {\dfrac{{A + \delta }}{2}} \right)}}{{\sin \left( {\dfrac{A}{2}} \right)}}$, where $\delta$ is the minimum deviation and A is the angle of the prism.

### Wave Optics:

• Wave optics is based on the wave theory of light put forward by Huygen.

• According to wave theory, light is a form of energy which travels through a medium in the form of transverse waves.

• Light is a form of energy.

• Light travels even in a vacuum. It does not require a material medium for its propagation.

### Newton's Corpuscular Theory:

• Light consists of extremely small, invisible elastic particles travelling in a vacuum with a speed of 3 × 108 m/s.

• The theory could explain reflection and refraction.

• It could not explain interference, diffraction, polarisation, photoelectric effect and Compton effect. The theory failed as it could not explain why light travels faster in a rarer medium than in a denser medium.

### Huygen's Wave Theory:

• Light travels in a medium in the form of a wavefront where a wavefront is the locus of all the particles vibrating in the same phase.

• When the source of light is a point source, the wavefront is spherical. Amplitude is inversely proportional to distance.

• When the source of light is linear, the wavefront is cylindrical.

• At a very large distance from the source (or an extended large source), a portion of spherical or cylindrical wavefront appears to the plane. In this case amplitude and intensity remains constant.

• A wavefront travels parallel to itself and perpendicular to the rays.

• Wavefront always travels in the forward direction of the medium.

• In an anisotropic medium, a point source gives rise to an elliptic wavefront.

• Huygen's wave theory could explain reflection, refraction, interference and diffraction of light.

• It failed to explain polarisation of light. It assumed the light waves to be longitudinal and hence failed to explain polarisation. It also could not explain the backward propagation of light, photoelectric effect, Compton effect and Raman effect.

### Maxwell's Electromagnetic Theory:

• The electromagnetic waves are transverse waves which require no material medium for propagation. They can travel through vacuum.

• Due to transverse nature, light waves undergo polarisation.

• In these waves, the electric field vector and magnetic field vector vibrate in the same phase along mutually perpendicular directions.

• The electric field vector and magnetic field vector are perpendicular to the direction of velocity of light.

• It explained the phenomenon of reflection, refraction, interference, diffraction and polarisation.

• It failed to explain the phenomenon of photoelectric effect, Compton effect and Raman effect.

### Max Planck's Quantum Theory:

• Light consists of packets of energy known as quanta or photons.

• The photons are emitted discretely and not continuously.

• The velocity of a photon is equal to that of light.

• The rest mass of the photon = zero.

• Photons are electrically neutral.

• Quantum theory could explain the photoelectric effect, Compton effect and Raman effect.

• Quantum theory failed to explain interference, diffraction and polarisation of light.

### De-Broglie's Dual Theory:

• Light behaves both as a corpuscle and also as waves.

• The corpuscular and wave aspects are like the two faces of a coin which can never be separated. Light does not exhibit both the aspects simultaneously. In some phenomena it behaves like transverse waves (as in interference, diffraction and polarisation) and in other phenomena it behaves like particles (as in photoelectric effect, Compton effect and Raman effect). This dual theory explains all the phenomena and effects of light.

### Huygen's Principle:

• Every point on a given wavefront (called primary wave front) acts as a fresh source of new disturbance, called secondary wavelets.

• The secondary wavelets travel in all directions with the speed of light in the medium.

• A surface touching these secondary wavelets tangentially in the forward direction at any instant gives the new secondary wavefront at that instant.

### Interference of Light

• When two light waves of exactly equal frequency have a phase difference which is constant with respect to time travel in the same direction and overlap each other then the intensity is not uniform in space. The phenomenon of nonuniform distribution of energy in the medium due to superposition of two such light waves is called interference of light.

• The interference pattern in which the position of maxima and minima of intensity of light remain fixed all along on the screen is called sustained or permanent interference pattern.

• At the points where resultant intensity is maximum, interference is said to be constructive.

• At the points where the resultant intensity is minimum, interference is said to be destructive.

• The condition for constructive interference is that path difference between two waves should be zero or an integral multiple of full wavelength.

• The condition for destructive interference is that path difference between two waves should be an odd integral multiple of half the wavelength.

• Newton's rings are formed as a result of interference between the light waves reflected from the upper and a lower surface of the air film.

• In the interference of the light waves, we obtain alternate bright and dark bands of light, called interference fringes.

• Fringe width: The distance between the centres of two consecutive bright or dark fringes is called the fringe width.

• The angular fringe width is given by θ = λ/d, where λ is the wavelength of light, d is the distance between two coherent sources.

### Coherent Source:

Coherent sources are those which emit continuous light waves of same amplitude, same wavelength/frequency in the same phase or having a constant phase difference. Two independent sources can never be coherent. They are produced from a single source of light.

### Thomas Young's Double Slit Experiment:

• In this experiment, two points of the same wavefront are used as two coherent sources.

• Interference fringes obtained in Young's experiment consist of alternate bright and dark bands.

• All bright fringes have the same intensity and all dark fringes are perfectly dark.

• Bright fringes are due to constructive interference. Dark fringes are due to destructive interference.

• The central fringe is bright with monochromatic light. It is achromatic/white with white light.

• For nth bright fringe, path difference = $\dfrac{{xd}}{D}$ = .

• For nth bright fringe, path difference = $\dfrac{{xd}}{D}$  = $\dfrac{{2n - 1}}{2}\lambda$.

• Fringe width = $\dfrac{{\lambda D}}{d}$

n = order of fringe

x = distance of nth bright fringe from centre

d = distance between two coherent sources

D = distance between screen and slit/source

λ = wavelength of light

### Diffraction of Light:

The act of bending light around corners such that it spreads out and illuminates regions, where a shadow is anticipated, is known as diffraction of light. Since both occur simultaneously, it is challenging to distinguish between diffraction and interference. The diffraction of light is what causes the silver lining we see in the sky. A silver lining appears in the sky when the sunlight penetrates or strikes the cloud.

In Young's experiment, a single narrow slit in place of the double slit produced a broad pattern with a bright patch in the middle. Dark and bright sections alternate on either side of the centre. As one moves out from the centre, the intensity decreases. In this post, we go into great detail about the diffraction of light through a single slit.

### Single Slit Diffraction

A bright and dark band appears on the screen when a monochromatic light beam strikes a single slit and is diffracted by the slit.

• Maxima and minimum are other names for the bright pattern and the black band, respectively.

• The intensity of light is at its highest point during maxima and its lowest point at minima.

Single Slit Diffraction

The width of the maxima is given by:

$a \sin \theta=n \lambda$

Where $\lambda$ is the wavelength of the light, $n$ is an integer value, a is slit width and $D$ is the distance of the screen from the slit.

Here if $\theta$ is very small (less than $\left.30^{\circ}\right) \sin \theta$ will be approximately equal to $\theta$.

Thus, for $m^{\text {th }}$ minima, the equation can be expressed as

$a \theta=m \lambda_m=\pm 1, \pm 2, \pm 3 \ldots$

$\text { I.e., } \theta=m \lambda / a, a \theta=m \lambda_m=\pm 1, \pm 2, \pm 3 \ldots$

The intensity of single-slit diffraction at some $m^{\text {th }}$ minima

### Doppler's Effect in Light and Polarisation of Light:

• The apparent shift in the frequency of the light that the observer perceives as a result of relative motion between the source of the light and the observer is known as the Doppler effect of light.

• However, the equations for the Doppler shift for sound waves vary significantly depending on whether the source, the observer, or the air is moving.

• Light doesn't need a medium to travel, and the observer and source relative speeds are the sole factors affecting the Doppler shift for light travelling in a vacuum.

• The frequency that the observer receives will be lower than the frequency that the light source transmits as it moves away from the observer.

• The visible light spectrum shifts towards the red end as a result. It's known as the redshift in astronomy.

• The frequency perceived by the observer will be higher than the frequency broadcast by the source when the light source advances in that direction.

• The visible light spectrum shifts in favour of the high-frequency region as a result. Scientists refer to it as the blue shift.

### Polarisation of Light:

• A characteristic of turning waves that demonstrates the geometrical blossoming of the oscillations is polarisation of light.

• A turning wave oscillates at an angle of 90 degrees to the wave's velocity. The two waves of plane polarised light have the same mode of vibration for all waves.

• Moving across space, electric and magnetic fields interact to form light. Light waves vibrate electrically and magnetically in opposition to one another. The magnetic and electric fields move perpendicular to one another in different directions. As a result, the direction of motion is perpendicular to both the plane occupied by the electric field and the plane of the magnetic field that is perpendicular to it.

• These magnetic and electric vibrations can take place on various planes. Unpolarized light is a type of light wave that vibrates in more than one plane. The sun, a lamp, and a tube light are all examples of unpolarized light sources.

• The propagation direction is constant, however, the planes on which the amplitude occurs are changing.

Polarised waves are another type of wave. Light waves that vibrate only in one plane are known as polarised waves. The waves that make up plane polarised light all have the same direction of vibration. Polarisation is the process through which non-polarized light is converted into polarised light.

### Formula Chart:

1. Relation between focal length and radius of curvature of a mirror/lens, $\quad f=R$

2. Mirror formula: $\quad \dfrac{1}{\mathrm{f}}=\dfrac{1}{\mathrm{v}}+\dfrac{1}{\mathrm{u}}$

3. Magnification produced by a mirror:

$m=-\dfrac{v}{u}=-\dfrac{f}{u-f}$

4. Snell's law: $\quad \dfrac{\sin i}{\sin r}={ }^{1} n_{2}=\dfrac{n_{2}}{n_{1}}$

5. ${ }^{2}{ }_{1}=\dfrac{1}{1_{n 2}}$

6. $\mathrm{n}=\dfrac{\mathrm{c}}{\mathrm{v}}=\dfrac{\text { speed of light in vacuum }}{\text { speed of light in a medium }}=\dfrac{\lambda_{\text {air }}}{\lambda_{\text {medium }}}$

7. If object is in medium of refractive index $\mathrm{n}$, then $\mathrm{n}=\dfrac{\text { real depth }}{\text { apparent depth }}=\dfrac{t}{t_{a p p}}$

8. Apparent shift, $x=t-\dfrac{t}{n}=t\left(1-\dfrac{1}{n}\right)$

9. Critical angle for total internal reflection: $\quad \sin \mathrm{C}=\dfrac{1}{r^{n} \mathrm{~d}}=\dfrac{1}{n}$

10. Refraction at spherical (convex) surface: For object in rarer medium and real image in denser medium, the formula is $\dfrac{n_{2}}{v}-\dfrac{n_{1}}{u}=\dfrac{n_{2}-n_{1}}{R}$ where $n_{2} \& n_{1}$ are the refractive indices of denser and rarer media.

11. Lens formula:

$\dfrac{1}{\mathrm{f}}=\dfrac{1}{\mathrm{v}}-\dfrac{1}{\mathrm{u}}$

12. Linear magnification produced by a lens: $m=\dfrac{\mathrm{I}}{0}=\dfrac{\mathrm{v}}{\mathrm{u}}$

13. Lens maker's formula : $\dfrac{1}{\mathrm{f}}=\dfrac{1}{\mathrm{v}}-\dfrac{1}{\mathrm{u}}=\left({ }^{a} n_{\mathrm{g}}-1\right)\left[\dfrac{1}{\mathrm{R}_{1}}-\dfrac{1}{\mathrm{R}_{2}}\right]=(\mathrm{n}-1)\left[\dfrac{1}{\mathrm{R}_{1}}-\dfrac{1}{\mathrm{R}_{2}}\right]$

14. Power of a lens: $P=\dfrac{1}{f}$ diopter

( $f$ is in metre)

15. Lenses in contact: $\dfrac{1}{\mathrm{f}}=\dfrac{1}{\mathrm{f}_{1}}+\dfrac{1}{\mathrm{f}_{2}} \quad$ or $\quad \mathrm{P}=\mathrm{P}_{1}+\mathrm{P}_{2}$

Example 1: A glass tumbler having an inner depth of $17.5 \mathrm{~cm}$ is kept on a table. A student starts pouring water $(\mu=4 / 3)$ into it while looking at the surface of water from the above. When he feels that the tumbler is half filled, he stops pouring water. Up to what height, the tumbler is actually filled?

(A) $11.7 \mathrm{~cm}$

(B) $10 \mathrm{~cm}$

(C) $7.5 \mathrm{~cm}$

(D) $8.75 \mathrm{~cm}$

Explanation:

Glass Tumbler

Consider the actual height of the tumbler be $\mathrm{H}$.

The refractive index of the water, $\mu=4 / 3$

The refractive index of the air, $\mu=1$

As we know that,

$\mu_{\text {water }}=\dfrac{H_{\text {real }}}{H_{\text {apparent }}} \Rightarrow \dfrac{4}{3}=\dfrac{H}{H_{\text {apparent }}}$

$\Rightarrow H_{\text {apparent }}=\dfrac{3 H}{4}$

Height of air observed by observer $=17.5-\mathrm{H}$

According to the question, both heights observed by observers are the same.

$\dfrac{3 H}{4}=17.5-\mathrm{H}$

$\Rightarrow \mathrm{H}=10 \mathrm{~cm}$

Hence, the correct answer is option B.

Example 2: Two plane mirrors $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ are at right angles to each other shown. A point source 'P' is placed at 'a' and '2a' meters away from $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ respectively. The shortest distance between the images thus formed is : (Take $\sqrt{5}=2.3)$

Two Plane Mirror

(A) $3 a$

(B )$4.6 a$

(C) $2.3 a$

(D) $2 \sqrt{10} a$

Explanation:

Plane Mirror Construction

Shortest distance is $2 \mathrm{a}$ between $\mathrm{I}_{1} \& \mathrm{I}_{3}$

But answer given is for $\mathrm{I}_{1} \& \mathrm{I}_{2}$

$=\sqrt{(4 a)^{2}+(2 a)^{2}}$

$=a \sqrt{20}$

$=4.47 \mathrm{a}$

Hence, the correct answer is option B.

### Importance of Class 12 Physics Chapter Optics

You have studied light rays, their reflection, refraction, and the formulae related to all these scientific principles. In this chapter on optics, you will be introduced to the advanced concepts along with what you have studied in the previous classes.

The fundamental concepts of ray optics and the mechanism of optical instruments will be taught in this chapter. Students will be introduced to the advanced concepts related to rectilinear propagation of light, the reflection of light on a plane and curved surfaces, refraction of light in transparent mediums, and linear magnification of images.

They will also study lenses and how different types of lenses are used in the optical instruments used these days. The scientific terms related to the lenses and spherical mirrors will also be explained with proper terminologies to represent them. The formulae used to determine the value of these terms will also be explained.

This chapter is very important for the preparation for the JEE Advanced exam. Conceptual questions from this chapter appear in the exam. Preparing this chapter will help you understand the basic and advanced principles of optics and score good marks in the competitive exams.

## Benefits of Vedantu’s Optics Revision Notes for JEE Advanced

• Optics notes are compiled by the experts of Vedantu to offer a simpler version of the crucial principles of this chapter. You will learn these scientific principles faster using these revision notes and keep yourself ahead of your class.

• These can be used for revising this chapter faster. The concise presentation of all the sections of this chapter will help you reduce your revision time. In fact, you can also recall the concepts, formulae, and derivations you have studied in this chapter by following the organised presentation of the notes.

• Use these notes at your convenience and make your study sessions much better. You can download and refer to these notes at your convenience. Simplify your knowledge development process and quickly find out the answers to the questions coming in the competitive exams.

• Follow the sample questions in these notes to evaluate your preparation level. Find out where you need to concentrate more and fortify your conceptual foundation for this chapter.

Get the free PDF version of these notes and complete your study material for optics. These notes are compiled to offer a better platform for understanding, studying, revising, and recalling the fundamental concepts of optics for the JEE Advanced exam. Follow how the experts have simply explained these concepts and escalate your answering skills to score more in the exam.

## FAQs on JEE Advanced 2023 Revision Notes for Optics

1. What is a lens?

A lens is an optical instrument made of glass or any transparent material used to divert light rays.

2. What is a spherical mirror?

A spherical mirror is a section of a sphere that has a reflecting surface inside or outside the curvature.

3. What is a focus in a lens?

A focus is a point on the principal axis of a lens where incident parallel rays meet.

4. What happens to the light rays emerging from the focal point?

Light rays emerging from the focus will become parallel to each other after refraction in a lens.


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## Central-Difference Formula

Good day everyone.

I'm so stuck with this homework in numerical methods. I'm supposed to derive an equation using Taylor expansion. The details are below:

Use the Taylor expansions for $f(x+h), f(x-h), f(x+2h)$ and $f(x-2h)$ and derive the central-difference formula:

$f^{(4)}(x)=\frac{f(x+2h)-4f(x+h)+6f(x)-4f(x-h)+f(x-2h)}{h^4}$

I know I'm supposed to start with these four equations,

$f(x+h)=f(x)+f'(x)h+\frac{f^{(2)}(x)h^2}{2!} + \frac{f^{(3)}(x)h^3}{3!} + \frac{f^{(4)}(x)h^4}{4!} + \frac{f^{(5)}(x)h^5}{5!}$
$f(x+h)=f(x)-f'(x)h+\frac{f^{(2)}(x)h^2}{2!} -\frac{f^{(3)}(x)h^3}{3!} + \frac{f^{(4)}(x)h^4}{4!} - \frac{f^{(5)}(x)h^5}{5!}$

$f(x+2h)=f(x)+f'(x)2h+\frac{f^{(2)}(x)4h^2}{2!} + \frac{f^{(3)}(x)8h^3}{3!} + \frac{f^{(4)}(x)16h^4}{4!} + \frac{f^{(5)}(x)32h^5}{5!}$
$f(x-2h)=f(x)-f'(x)2h+\frac{f^{(2)}(x)4h^2}{2!} - \frac{f^{(3)}(x)8h^3}{3!} + \frac{f^{(4)}(x)16h^4}{4!} - \frac{f^{(5)}(x)32h^5}{5!}$

After these, I can't move on. HELP!


text:
?

Free Version
Moderate

# Point Reflection - P'$(-2,5)$

ACTMAT-V4OCGK

P'$(-2,5)$ was reflected about the line $y = -x.$

What are the coordinates of pre-image point P?

A

$(-5,2)$

B

$(5,-2)$

C

$(2,5)$

D

$(2,-5)$

E

$(5,2)$


text:
Mechanics Homework 33-1 1035 - Bolt Cutter Name_______________ Mr. Haynes Date_______________

Problem 33-1 Determine the force that the bolt cutters exert on the bolt at C when 100 N vertical forces are applied at the handles.


text:
2019
Том 71
№ 11

# On the analyticity of solutions of $\overrightarrow{2b}$-parabolic systems

Abstract

It is proved that if the coefficients of a $\overrightarrow{2b}$ -parabolic system admit analytic extension to a complex region in the space variables, then the fundamental matrix of solutions of the Cauchy problem and regular solutions of the system also possess the same property.

English version (Springer): Ukrainian Mathematical Journal 58 (2006), no. 2, pp 178-188.

Citation Example: Ivasyshen S. D., Kondur O. S. On the analyticity of solutions of $\overrightarrow{2b}$-parabolic systems // Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 160-167.

Full text


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### 一、Data Structures (50%)

1. (20%)

Given a red-black tree in the following figure.

After one node with value 778 is inserted, the resulting red-black tree is shown as follows.

(a) Please describe the operation procedures for this insertion.
(b) Please indicate the color and value of node A and node B, respectively.

(a) 略
(b) node A: black, 614. node B: red, 419.

2. (10%) Given the expression, (x+y)*w+u/(v+x*w)+z, Please show the content in the stack after the operand v is read in postfix transformation.

3. (20%)

(a) Please finish the lost code for the choose function in the following Dijkstra shortest path implementation for a graph without negative-weight edges.

$$程式碼請看原檔$$

(b) In a directed graph without a cycle of negative length but with a negative-length edge, we can implement Bellman-Ford algorithm as follows to compute shortest paths. Please fill in the lost code.

$$程式碼請看原檔$$

(a)

.wp-block-code{border:0;padding:0}.wp-block-code>div{overflow:auto}.shcb-language{border:0;clip:rect(1px,1px,1px,1px);-webkit-clip-path:inset(50%);clip-path:inset(50%);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;word-wrap:normal;word-break:normal}.hljs{box-sizing:border-box}.hljs.shcb-code-table{display:table;width:100%}.hljs.shcb-code-table>.shcb-loc{color:inherit;display:table-row;width:100%}.hljs.shcb-code-table .shcb-loc>span{display:table-cell}.wp-block-code code.hljs:not(.shcb-wrap-lines){white-space:pre}.wp-block-code code.hljs.shcb-wrap-lines{white-space:pre-wrap}.hljs.shcb-line-numbers{border-spacing:0;counter-reset:line}.hljs.shcb-line-numbers>.shcb-loc{counter-increment:line}.hljs.shcb-line-numbers .shcb-loc>span{padding-left:.75em}.hljs.shcb-line-numbers .shcb-loc::before{border-right:1px solid #ddd;content:counter(line);display:table-cell;padding:0 .75em;text-align:right;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;white-space:nowrap;width:1%}for (i = 0; i < n; i++)
{
if (distance[i] < min && !found[i])
{
min = distance[i];
minpos = i;
}
}
Code language: C++ (cpp)

(b)

for (int i = 0; i < n; i++)
for (int u = 0; u < n; u++)
Code language: C++ (cpp)

### 二、Algorithms (50%)

1. (10%) Prove or disprove: The single-source shortest paths problem can be solved in linear time in directed acyclic graphs.

(1) 找Topological sort.
(2) 從sort的順序由前到後作Bellman-Ford algorithm的Relax function.

2. (15%) The matrix-chain multiplication problem can be stated as follows: Given a chain $(A_1, \, A_2,…, \, A_n)$ of $n$ matrices, where for $i = 1, \, 2,…, \, n,$ matrix $A_i$ has dimension $p_{i-1} \times p_i,$ fully parenthesize the product $A_1A_2…A_n$ in a way that minimizes the number of scalar multiplications. Suppose that you have 6 matrices: $A_1$ has dimension $30 \times 35, \, A_2$ has dimension $35 \times 15, A_3$ has dimension $15 \times 5, \, A_4$ has dimension $5 \times 10, \, A_5$ has dimension $10 \times 20, \, A_6$ has dimension $20 \times 30.$ Please calculate the minimum number of scalar multiplications.

3. (10%) Give asymptotic upper and lower bounds for $T(n) = 2T(\dfrac{n}{4}) + \sqrt{n}.$ Assume that $T(n)$ is constant for $n \leq 2.$ Make your bounds as tight as possible.

4. (15%) Consider the problem of finding the 5-vector $x = (x_i)$ that satisfies

$$\begin{pmatrix} 1 & -1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & -1\\ 0 & 1 & 0 & 0 & -1\\ -1 & 0 & 1 & 0 & 0\\ -1 & 0 & 0 & 1 & 0\\ 0 & 0 & -1 & 1 & 0\\ 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 0 & -1 & 1\\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3\\ x_4\\ x_5 \end{pmatrix} \leq \begin{pmatrix} 0\\ -1\\ 1\\ 5\\ -4\\ -1\\ -3\\ -3 \end{pmatrix}$$

Determine whether there exists a solution or there is no solution.


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# How should I remove the title from each slide in beamer?

I am preparing for a presentation using the beamer. Here is the code in the preamble.

\documentclass[9pt]{beamer}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usetheme{Goettingen}
\usecolortheme{whale}
\usepackage{verbatim}

\usebeamerfont{footline}%
\usebeamercolor[fg]{footline}%
\hspace{1em}%
}

\title{\texttt{\LARGE My Mathematical journey}  }
\institute{Institute of Mathematical Studies}
\author{Tom Scott}
\date{November 2021}

I did not write the above code but I adopted it from the examples given in the LaTeX editor and I understand it to some extent. But

I am not able to get rid of the title and author name which comes on every page in the right corner. So what should I add/modify in the above code?

Adding an empty optional argument to \author and \title as in \author[]{Tom Scott} should get you closer to the expected output:

Output of the original code output of the modified code

\documentclass[9pt]{beamer}
\usetheme{Goettingen}
\usecolortheme{whale}

\usebeamerfont{footline}%
\usebeamercolor[fg]{footline}%
\hspace{1em}%
}

\title[]{\texttt{\LARGE My Mathematical journey}}
\institute{Institute of Mathematical Studies}
\author[]{Tom Scott}
\date{November 2021}

\begin{document}
\maketitle

\section{section}

\begin{frame}
\frametitle{frame title}
contents
\end{frame}

\section{another section}
\begin{frame}
\frametitle{frame title}
contents
\end{frame}

\section{last section}
\begin{frame}
\frametitle{frame title}
contents
\end{frame}
\end{document}


text:
# How can log link still allow negative predicted values?

#### samuelle

##### New Member
I have been attempting to get a meaningful outcome of my data for months, and could really do with some help where to go next. I am attempting to fit a model based on past cost data.
y = Total ($/ha/year) (all positive, min = 0.02, max = 2773.05) x1 = Area (ha) x2 = Continent (factor) x3 = GDP ($/yr)
x3 Population density
x4 = Method (factor)
x5 = Type (factor)

In initial scatter graphs Total and Area were heavily skewed with a long right tail. I have so far attempted a linear regression with no logged variables, area logged, total logged and both logged, a GLM with log link, and a Gaussian GLM, and I consistently get negative predicted values. When I did log(Total+1) and log(Area+1) the fitted vs. residual graph appeared to have no negative fitted values but when I was using the predict function, one of my outputs was negative. Hat values seem high so perhaps some leverage is occurring. Does anyone have any ideas? Thanks.


text:
History of the Normal Basis Theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:53:38Z http://mathoverflow.net/feeds/question/35577 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35577/history-of-the-normal-basis-theorem History of the Normal Basis Theorem Anthony Pulido 2010-08-14T13:31:54Z 2010-08-16T03:16:56Z
The Normal Basis Theorem: If $E/F$ is a finite Galois extension, then there exists $a \in E$ such that the orbit of $a$ under the action of $\mathrm{Gal}(E/F)$ is a basis for $E$ as a vector space over $F.$

Who discovered this?

I've looked through the collected works of Frobenius and Dedekind, which are the earliest works I've seen referring to it, but it looks like the theorem led Dedekind to what is called the group determinant, and he doesn't give a reference. (p. 433 of Dedekind's Gesammelte Werke, v. 2, via Curtis's Pioneers of Representation Theory. See KConrad's answer below.) Among others, I've also looked at some of the correspondence of Hasse and Noether. The works are in German, which is second language to me, so it's possible I missed something. Needless to say, I've searched using Google to no avail. If anyone knows something, I'd be very grateful.
 http://mathoverflow.net/questions/35577/history-of-the-normal-basis-theorem/35578#35578 Answer by KConrad for History of the Normal Basis Theorem KConrad 2010-08-14T14:18:54Z 2010-08-15T02:51:08Z
The cached page

http://webcache.googleusercontent.com/search?q=cache:q5q43iNq1SQJ:siba2.unile.it/ese/issues/1/690/Notematv27n1p5.ps+normal+basis+theorem&cd=3&hl=en&ct=clnk&gl=us&client=safari

gives some information: Eisenstein conjectured it in 1850 for extensions of finite fields and Hensel gave a proof for finite fields in 1888. Dedekind used such bases in number fields in his work on the discriminant in 1880, but he had no general proof. (See the quote by Dedekind on the bottom of page 51 of Curtis's "Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer".) In 1932 Noether gave a proof for some infinite fields while Deuring gave a uniform proof for all fields (also in 1932).

In Narkiewicz's "Elementary and Analytic Theory of Algebraic Numbers" (3rd ed.) he writes on the bottom of p. 186 that the normal basis theorem is due to Noether, but as usual the history is slightly more complicated.


text:
# Objective Questions (MCQ)

Question:

Objective Questions (MCQ)

The roots of the quadratic equation $2 x^{2}-x-6=0$ are

(a) $-2, \frac{3}{2}$

(b) $2, \frac{-3}{2}$

(c) $-2, \frac{-3}{2}$

(d) $2, \frac{3}{2}$

Solution:

The given quadratic equation is $2 x^{2}-x-6=0$.

$2 x^{2}-x-6=0$

$\Rightarrow 2 x^{2}-4 x+3 x-6=0$

$\Rightarrow 2 x(x-2)+3(x-2)=0$

$\Rightarrow(x-2)(2 x+3)=0$

$\Rightarrow x-2=0$ or $2 x+3=0$

$\Rightarrow x=2$ or $x=\frac{-3}{2}$

Thus, the roots of the given equation are 2 and $\frac{-3}{2}$.

Hence, the correct answer is option B.


text:
## The Annals of Probability

### On Limiting Distributions of Intermediate Order Statistics from Stationary Sequences

#### Abstract

Let $X_1, X_2, \cdots$, be a sequence of random variables and write $X^{(n)}_k$ for the $k$th largest among $X_1, X_2, \cdots, X_n$. If $\{k_n\}$ is a sequence of integers such that $k_n \rightarrow \infty, k_n/n \rightarrow 0$, the sequence $\{X^{(n)}_{k_n}\}$ is referred to as the sequence of intermediate order statistics corresponding to the intermediate rank sequence $\{k_n\}$. The possible limiting distributions for $X^{(n)}_{k_n}$ have been characterized (under mild restrictions) by various authors when the random variables $X_1, X_2, \cdots$ are independent and identically distributed. In this paper we consider the case when the $\{X_n\}$ form a stationary sequence and obtain a natural dependence restriction under which the "classical" limits still apply. It is shown in particular that the general dependence restriction applies to normal sequences when the covariance sequence $\{r_n\}$ converges to zero as fast as an appropriate power $n^{-\rho}$ as $n \rightarrow \infty$.

#### Article information

Source
Ann. Probab., Volume 10, Number 3 (1982), 653-662.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993774

Digital Object Identifier
doi:10.1214/aop/1176993774

Mathematical Reviews number (MathSciNet)
MR659535

Zentralblatt MATH identifier
0487.62015

JSTOR


text:
# Reference request: Reidemeister type moves for immersed curves on surfaces

Preliminaries

Let $$\Gamma$$ be a closed $$1$$-manifold (i.e. a union of finitely many circles) and let $$\Sigma$$ be a closed $$2$$-manifold (i.e. a surface). I'll adopt the following terminology.

Definition 1. A clean immersion $$\iota:\Gamma \to \Sigma$$ is an immersion such that

• $$\iota$$ restricts to an embedding on each component of $$\Gamma$$ and
• the self-intersections of the image consist of transverse double points.

Reidemeister-Type Moves For Clean Immersions

Consider the following two analogues of the Reidemeister moves. The first is an analogue of Reidemeister 2.

Definition 2. A two-point move on $$\iota$$ is any modification of $$\iota$$ to a new clean immersion $$\iota'$$ as follows. Let $$D \subset \Sigma$$ be a disk such that $$(D,D \cap \iota(\Gamma))$$ is diffeomorphic to either $$(D_+,\gamma_+ \cup \eta_+)$$ or $$(D_-,\gamma_-\cup\eta_-)$$, which are given by the pictures

Then, swap $$(D_+,\gamma_+\cup\eta_+)$$ for $$(D_-,\gamma_-\cup\eta_-)$$ (or visa-versa).

The second is an analogue of Reidemeister 3.

Definition 3. A three-point move on $$\iota$$ is any modification of $$\iota$$ to a new clean immersion $$\iota'$$ as follows. Let $$D \subset \Sigma$$ be a disk such that $$(D,D \cap \iota(\Gamma))$$ is diffeomorphic to either $$(D_+,\gamma_+ \cup \eta_+ \cup \xi_+)$$ or $$(D_-,\gamma_-\cup\eta_-\cup\xi_-)$$, whcih are given by the pictures

Then, swap $$(D_+,\gamma_+ \cup\eta_+\cup\xi_+)$$ for $$(D_-,\gamma_-\cup\iota_-\cup\xi_-)$$ (or visa-versa).

A Reidemeister-Type Theorem

I believe the following result is true of homotopies of clean embeddings.

Proposition. If $$\iota_0:\Gamma \to \Sigma$$ and $$\iota_1:\Gamma \to \Sigma$$ are homotopic, then they are related by a sequence of two and three point moves.

Here is a sketch of the argument that I have in mind.

Proof: Pick an smooth homotopy $$\iota:\Gamma \times I \to \Sigma$$. We can choose this isotopy so that it is an isotopy embeddings on each component. Now consider the trace $$I:\Gamma \times I \to \Gamma \times I$$ given by $$I(s,t) = (\iota_t(s),t)$$.

By Papakyriakopoulos (?) we can perturb $$I$$ away from a neighborhood of the boundary of $$\Gamma \times I$$ to a map $$I'$$ with only arcs of double points, isolated triple points and isolated branch points, and where the double point arcs must have boundary on the branch points. Also, we can pick $$I'$$ so that the components of $$\Gamma$$ are still embedded. In particular, there can be no branch points and so the double point arcs must be closed curves.

We can perturb again to an embedding $$I''$$ where the double point circles so that they are transverse to the projection map to $$[0,1]$$. Then these represent (forward and backward) two-point moves. In a similar manner, the triple points correspond to the triple-point moves.

Main Questions

My issue is that I can't for the life of me find a reference for Proposotion 1! Also I think it would be ideal for the particular project I'm working on to not include a fleshed out version of the proof I discussed above.

Question 1. Where can I find a reference for Proposition 1? Or a statement from which I can conclude Proposition 1 as a quick Corollary?

Also, just to be sure...

Question 2. Proposition 1 is correct, right? Or am I just totally off base?

This isn't true as stated. What is true is that if $$\gamma$$ and $$\gamma'$$ two homotopic curves that are in minimal position (i.e. each contains the minimal number of self-intersections in their homotopy class), then $$\gamma$$ and $$\gamma'$$ can be connected by a sequence of isotopies and Reidemeister III moves. This was proved in

J. Hass and P. Scott, Shortening curves on surfaces, Topology 33 (1994), no. 1, 25–43.

This paper does not state the precise result I describe above, but see

J. M. Paterson, A combinatorial algorithm for immersed loops in surfaces, Topology Appl. 123 (2002), no. 2, 205–234.

for an alternate proof that states things as I did.

What is needed, then, is a set of moves that take a curve and reduce it to minimal position. Unfortunately, it is not true that you can do this with Reidemeister I and II moves (I am assuming that you meant to include Reidemeister I, by the way), even if you make the obvious modification to allow the curve to pass through the "modification region" in several other arcs that are not involved in the moves. What is needed are moves that eliminate "singular monogons" and "singular bigons" (which can't be assumed to be embedded, as in the usual Reidemeister moves). See the paper

J. Hass and P. Scott, Intersections of curves on surfaces, Israel J. Math. 51 (1985), no. 1-2, 90–120.

for a precise statement and proof (plus examples showing why this is complicated).

• In the original definition of clean immersion, each component curve was embedded, (and hence in minimal position). I suspect that's enough to make the Proposition in the question correct. – KSackel Sep 16 '19 at 3:38
• @KSackel: Ah, I didn't catch that. It's a long post. I will leave this answer up even though it doesn't exactly answer the question since I think the references in it will be useful to the OP, and also to make this useful to people who find this question via google searches. – Andy Putman Sep 16 '19 at 3:51
• Hi @AndyPutman, thanks for the answer! The result I'm asking for seems to be provable pretty quickly from the result about minimally intersecting curves plus a result about intersecting pairs of embedded curves in the 2nd Haas-Scott paper. Maybe I'll write that answer below this one. – Julian Chaidez Sep 16 '19 at 5:20

Ok, here's a satisfactory answer to my question using the papers that Andy posted (I'm just posting this for clarity). As Andy pointed out, the result is not true if the components of $$\Gamma$$ are allowed to not be embedded.

Proof: First assume that $$\iota_0$$ and $$\iota_1$$ be homotopic clean immersions with the minimal intersection property. That is, $$\iota_0(\Gamma)$$ and $$\iota_1(\Gamma)$$ have the minimal possible number of transverse double points in their homotopy class. In Paterson's paper [1], they prove that any minimal two such minimal transversely self-intersecting immersions are ambiently isotopic after a sequence of three-point moves (see p. 231-232 and Lemma 3.4).

Thus it suffices to show that $$\iota_0$$ and $$\iota_1$$ can be isotoped to minimal clean immersions $$\iota_0'$$ and $$\iota_1'$$ through two-point moves. This is implied immediately by Lemma 3.1 of [2], which states that if $$\iota_0$$ (for instance) does not minimize intersections, then there is an inner-most $$2$$-gon (i.e. a copy of $$(D_+,\gamma_+ \cup \eta_+)$$ as above) on which one can perform a 2-point move to decrease the self-intersections by $$1$$.

[1] Paterson, J. M., A combinatorial algorithm for immersed loops in surfaces, Topology Appl. 123, No. 2, 205-234 (2002). ZBL1025.57025.

[2] Hass, Joel; Scott, Peter, Intersections of curves on surfaces, Isr. J. Math. 51, 90-120 (1985). ZBL0576.57009.


text:
If you find any mistakes, please make a comment! Thank you.

## The diagonal subset of the Cartesian square of a ring is a subring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.18

Let $R$ be a ring. Prove that $\Delta(R) = \{(r,r) \ |\ r \in R \}$ is a subring of $R \times R$.

Solution: In this previous exercise, we saw that $\Delta(R)$ is an additive subgroup of $R \times R$; thus it suffices to show closure under multiplication. To that end, if $r,s \in R$, then $$(r,r)(s,s) = (rs,rs) \in \Delta(R).$$


text:
# What class of problem is this?

If I have a lot of input output pairs as training data

<float Xi, float Yi>

and I have a parametrized approximation function (I know the function algorithm, but not the values of the many many parameters it contains) which shall approximate the process by which the original data pairs were generated. The function takes two input values:

// c is a precomputed classifier for x and can have values from 0 to 255, so there can be up to 256 different classes
y = f(float x, int c)

the hidden parameters of the function are some big lookup tables (a lot of free parameters, but still much fewer than the amount of data points in the training data)

Now I want to fit all the hidden parameters that f contains AND compute for each Xi a ci, such that for the fitted function the error over all i of Yi - f(Xi, ci) is minimized

So, using some algorithm I want to fit the parameters of f and also classify the inputs Xi so that f(Xi, ci) aproximates Yi

How is this kind of problem called and what kind of algorithm is used to solve it?

I assume it's possible to initialize all hidden parameters as well as all ci with random values and then somehow use back propagation of the error to iteratively find parameters and ci such that the function works well.

What I don't know is whether this is a well known class of problem and I just don't know the name of it, so I'm asking for pointers.

Or maybe in other words: I have a function that has a certain layout (for performance reasons) which I want to use to approximate and interpolate my training data, I want to tune the parameters of this function such that it approximates the original data well. since the data points fall into some 'categories', I want to pre-classify the data-points to make it easier for the function to do its job. What kind of algorithm do I use to find the function's parameters and to pre-classify the input?


text:
# NAG FL Interfacee04rgf (handle_set_nlnobj)

## ▸▿ Contents

Settings help

FL Name Style:

FL Specification Language:

## 1Purpose

e04rgf is a part of the NAG optimization modelling suite and declares the objective function of the problem as a nonlinear function with a particular gradient sparsity structure.

## 2Specification

Fortran Interface
 Subroutine e04rgf (
 Integer, Intent (In) :: nnzfd, idxfd(nnzfd) Integer, Intent (Inout) :: ifail Type (c_ptr), Intent (In) :: handle
#include <nag.h>
 void e04rgf_ (void **handle, const Integer *nnzfd, const Integer idxfd[], Integer *ifail)
The routine may be called by the names e04rgf or nagf_opt_handle_set_nlnobj.

## 3Description

After the handle has been initialized (e.g., e04raf has been called), e04rgf may be used to declare the objective function of the problem as a nonlinear function and define the sparsity pattern (list of nonzero elements) of its gradient. If the objective function has already been defined, it will be overwritten and its Hessian (or the Hessian of the Lagrangian) will be removed. If e04rgf is called with no nonzeroes in the sparsity pattern, any existing objective function is removed, no new one is added and the problem will be solved as a feasible point problem.
This objective function will typically be used for nonlinear programming problems (NLP) of the kind:
 $minimize x∈ℝn f(x) (a) subject to lg≤g(x)≤ug, (b) lB≤Bx≤uB. (c) lx≤x≤ux (d)$ (1)
The values of the nonlinear objective function $f\left(x\right)$ and the nonzero values of its gradient $\frac{\partial f}{\partial {x}_{i}}$ (matching the sparsity pattern) evaluated at particular points in the decision variable space will be communicated to the NLP solver by user-supplied functions (e.g., objfun and objgrd for e04stf). See Section 3.1 in the E04 Chapter Introduction for more details about the NAG optimization modelling suite.

None.

## 5Arguments

1: $\mathbf{handle}$Type (c_ptr) Input
On entry: the handle to the problem. It needs to be initialized (e.g., by e04raf) and must not be changed between calls to the NAG optimization modelling suite.
2: $\mathbf{nnzfd}$Integer Input
On entry: the number of nonzero elements in the sparse gradient vector of the objective function.
Constraint: ${\mathbf{nnzfd}}\ge 0$.
3: $\mathbf{idxfd}\left({\mathbf{nnzfd}}\right)$Integer array Input
On entry: the one-based indices of the nonzero elements of the sparse gradient vector. The indices must be stored in ascending order. Note that $n$ is the current number of decision variables in the model.
If ${\mathbf{nnzfd}}=0$, the objective is assumed to be zero and the array idxfd will not be referenced.
Constraints:
• $1\le {\mathbf{idxfd}}\left(\mathit{i}\right)\le n$, for $\mathit{i}=1,2,\dots ,{\mathbf{nnzfd}}$;
• ${\mathbf{idxfd}}\left(\mathit{i}\right)<{\mathbf{idxfd}}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nnzfd}}-1$.
4: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $-1$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
The supplied handle does not define a valid handle to the data structure for the NAG optimization modelling suite. It has not been properly initialized or it has been corrupted.
${\mathbf{ifail}}=2$
The problem cannot be modified right now, the solver is running.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{nnzfd}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nnzfd}}\ge 0$.
${\mathbf{ifail}}=7$
On entry, $i=⟨\mathit{\text{value}}⟩$, ${\mathbf{idxfd}}\left(i\right)=⟨\mathit{\text{value}}⟩$ and ${\mathbf{idxfd}}\left(i+1\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{idxfd}}\left(\mathit{i}\right)<{\mathbf{idxfd}}\left(i+1\right)$ (ascending order).
On entry, $i=⟨\mathit{\text{value}}⟩$, ${\mathbf{idxfd}}\left(i\right)=⟨\mathit{\text{value}}⟩$ and $n=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{idxfd}}\left(i\right)\le n$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

Not applicable.

## 8Parallelism and Performance

e04rgf is not threaded in any implementation.


text:
Lemma 33.42.2. Let $X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. If

1. $\mathcal{O}_{X, x} = \mathcal{O}_{S, s}$,

2. $X$ is reduced,

3. $X \to S$ is of finite type, and

4. $S$ has finitely many irreducible components,

then there exists an open neighbourhood $U$ of $x$ such that $f|_ U$ is an open immersion.

Proof. We may remove the (finitely many) irreducible components of $S$ which do not contain $s$. We may replace $S$ by an affine open neighbourhood of $s$. We may replace $X$ by an affine open neighbourhood of $x$. Say $S = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(B)$. Let $\mathfrak q \subset B$, resp. $\mathfrak p \subset A$ be the prime ideal corresponding to $x$, resp. $s$. As $A$ is a reduced and all of the minimal primes of $A$ are contained in $\mathfrak p$ we see that $A \subset A_\mathfrak p$. As $X \to S$ is of finite type, $B$ is of finite type over $A$. Let $b_1, \ldots , b_ n \in B$ be elements which generate $B$ over $A$ Since $A_\mathfrak p = B_\mathfrak q$ we can find $f \in A$, $f \not\in \mathfrak p$ and $a_ i \in A$ such that $b_ i$ and $a_ i/f$ have the same image in $B_\mathfrak q$. Thus we can find $g \in B$, $g \not\in \mathfrak q$ such that $g(fb_ i - a_ i) = 0$ in $B$. It follows that the image of $A_ f \to B_{fg}$ contains the images of $b_1, \ldots , b_ n$, in particular also the image of $g$. Choose $n \geq 0$ and $f' \in A$ such that $f'/f^ n$ maps to the image of $g$ in $B_{fg}$. Since $A_\mathfrak p = B_\mathfrak q$ we see that $f' \not\in \mathfrak p$. We conclude that $A_{ff'} \to B_{fg}$ is surjective. Finally, as $A_{ff'} \subset A_\mathfrak p = B_\mathfrak q$ (see above) the map $A_{ff'} \to B_{fg}$ is injective, hence an isomorphism. $\square$

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text:
# Macros in c++ with example

Macros can be implemented using #define. When the call to the macro occurs, the code of the macro will substitute in the source code before compilation.
Syntax of the macro is

NOTE: Do not leave the space between the function and the parenthesis. The parenthesis should start immediately after the macro name.
Below is a small example:

The disadvantages of the macros are they do not perform any type of checking.
But Macros directly substitute their code, they reduce the overhead of a function call.
Let us see the below example that helps the use of the macros

Output is:

One of the common mistakes we make when we use Macro is to forget what Macro is supposed to do. In the following example, if we miss parenthesis around it, it will give us unexpected results.

The macro should be like below

To overcome the problems that are come with the macros, the only solution is inline functions