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## how many square inches in a 8 inch diameter circle

This calculator converts the area of a circle into a square with four even length sides and four right angles. How many square inches ( in 2 , sq in ) are in 1 circle 1-foot diameter ( 1 ∅ 1 ft )? . Circle Formula's Radius R = D ÷ 2 where R = radius, D = diameter Area; A = π * D² ÷ 4 where A = area, π = 3.14159..., D = diameter Circumference; 0.79 sq in, in2 is converted to 1 of what? Calculate Diameter and Units per Course using pre-sized units (Fire Pit Blocks etc) If you already have blocks pre-sized, for fire pits etc, and need to find out how many it takes to make a complete circle, and the outer diameter of the circle, enter single block dimensions below and hit Calculate. Answer (1 of 4): The formula for the area (A) of a circle is A = πr² but this can also be stated as πd²/4 where r and d are the radius and diameter respectively.A = πd²/4 = π x 12²/4 = π x 144/4 = π x 36 = 113.097 square inches. If the circle has a radius of 8 inches, it's 201.06193 square inches. Diferent area surface units conversion from circle one inch diameter to square centimeters. The answer is: 1 ∅ 1in equals 0.79 sq in, in2. How many square inches in a 23 inch diameter circle? So, the cost per square inch of pizza is about .13, or 13 cents. Does the area of the face of a 4" diameter bar = 12.59" It will be 4*pi, which is closer to 12.57 in² Comment. If the diameter is seven feet, four inches, enter “7.33”. . 1 circle 1-inch diameter ( ∅ 1 in ) = 0.79 square inches ( in 2 , sq in ). This gives you your square feet figure (ft 2). Area of a circle is given by radius squared * pi. A 36-inch diameter circle has an area of: 1,017.9 square inches. This esitmate should work alright only if the square is significantly smaller than the circle. Between ∅ 1in and cm2, sq cm measurements conversion chart page. The square inches unit number 0.79 sq in, in2 converts to 1 ∅ 1in, one circle one inch diameter. If you multiply in pi as 3.14 you'll have 314 square kilometers. Area , A= πd²/4 =π×12²/4=36π =113.097 square inches Originally posted by Unregistered View Post. No conversion needed, since length, diameter and volume units can be selected independently, so this calculator allows you to use any mixture of measurement units. * * * * *which is 7.07 square feet. Convert 1 ∅ 1in into square centimeter and ∅ one inch circles to cm2, sq cm. Circle area to diameter; User Guide. If you were for example fitting a square which was of edges $1$ unit long on a circle $\sqrt{2}$ unit in diameter, The equation gives us $- \frac{1}{2} \pi$ as the number of squares. *Square Inches is also known as (a.k.a) square inch, square in, SqIn, Sq.In., sq.in., in2 & in 2 **This Calculator also calculates square feet, total height in feet, total height in inches, total length in feet and total length in inches. the inside diameter of an outer larger circle (or pipe, tube, conduit, connector), and; the outside diameters of small circles (or pipes, wires, fiber) The default values are for a 10 inch pipe with 2 inch smaller pipes - dimensions according ANSI Schedule 40 Steel Pipes. The area of an oval is the longest diameter x the shortest x 0.7854. Radius is 1/2 diameter, or in your case, 3 inches. How many square inches of paper are left over? Is it poss - the answers to estudyassistant.com The area of a circle is the diameter x diameter x 0.7854. The formula: Area of a Circle = π x (Diameter/2)^2 π = 3.142 The diameter of a circle is the circumference multiplied by 0.31831. How many square inches are in 1 circle one inch diameter? It depends on how you measure it. The Plain English programming language draws on the screen using 96 pixels per inch as the fundamental unit. A circle with a 23 inch diameter has am area of 415.4756 sq inches (to 4 dp). This tool will calculate the volume of a cylindrical shaped object from the dimensions of length and diameter. To calculate the area of a circle we use the formula: π x (diameter/2) 2. The second one, b, the radius squared is 10*10=100. This on the web one-way conversion tool converts surface area units from ∅ 1-foot circles ( ∅ 1 ft ) into square inches ( in 2 , sq in ) instantly online. Examples: If the diameter is five feet, enter “5”. Area of circle 1 equals = 16* Area of circle 2 equals = 9* Area of circle 3 equals = 1* Differences in the areas is (16-9-1)* equals 6* equals 18.84955592 square inches. So . A decimal can be used if the measurement is not a whole number. For example, if a pizza costs $32 and has an area of 256 square inches, you would calculate ÷ =. The circumference of a circle is the diameter x 3.1416. So square kilometers. 1 circ mil ≈ 0.000 000 000 506 707 479 m 2. So, we divide our diameter by 2 and then square it (multiply it by itself) and then multiply by π. 6″ round cake tin means the tin’s diameter is 6 Therefore it has a radius (r) of 3 Area of circle= 3.14r 2 3.14 (3 2) = 3.14 (3 x 3)= 3.14 x 9= 28.26 square inches We now to need find the length (L) for the square cake tin i.e. Circle 3 has a radius of 2/2 = 1 inch. One circle had a radius of 3 inches and the other had a diameter of 5 inches. If the circle has a diameter of 8 inches, it's 50.265 square inches. To determine the area of a circle from its diameter, divide the diameter by two, square it and multiply by π. So . 1 circle 1-foot diameter ( ∅ 1 ft ) = 113.10 square inches ( in 2 , sq in ). The first one, a, the radius squared is 8*8=64. 3.14 x 12 inches = 37.68 inches (This works with the metric system as well: 3.14 x 30 cm = 94.2 cm) An important step many people miss at this point is forgetting … The circumference can be found by multiplying PI (3.14159) times the diameter. In the first field, labeled “Diameter (ft),” enter the diameter, in feet, of the cylinder. To find out the area of a circle, we need to know its diameter which is the length of its widest part. For example, assume the diameter of the circular area to be 12 feet. A circle is 0.7854 times as heavy as a square of the same size. The diameter is a straight line that passes through the center of a circle. Try just using the formula for the area of a circle: PI * Radius Squared If the diameter is 5", the radius is 2.5" 2.5"^2 * PI 6.25 * 3.1415927 = 19.6349540849 (what Sandra said) The diameter should be measured in feet (ft) for square footage calculations and if needed, converted to inches (in), yards (yd), centimetres (cm), millimetres (mm) and metres (m). Area of a circle given diameter is ... diameter to square inches. Diameter, d=12 iches. 201 square inches Area of a circle, A = pir^2 r = radius of the circle = (diameter)/2 = 16/2 = 8 inches So, area of circle = pixx8^2 = 22/7xx8^2 = 1408/7 = 201.14 square inches (Note: pi = 22/7) So, are of the circle to nearest square inches = 201 square inches The circumference of a circle that has a diameter of 8 inches is 25.12 inches. In other words, if we could unroll the circle into a flat line, it would be 25.12 inches long. So Area = 3^2 pi or 9 pi which is 28.27 square inches. Divide the price of the pizza by the number of square inches. The third one, c, the radius squared is 14*14=196. Example: You want a finished 12″ diameter base (a 12″ diameter circle) in a duffle bag. unless you don't have a computer handy! inches Online calculators are great . The calculation is based on the area of the square being the same as the circle's area. Circumference The distance around the outward boundary of a circle, expressed as a linear unit of measurement (millimeters, inches, etc.). This on the web one-way conversion tool converts surface area units from ∅ 1-inch circles ( ∅ 1 in ) into square inches ( in 2 , sq in ) instantly online. If you multiply in pi as 3.14, you'll have 200.96 square inches. That is to say π (pi is 3.14159265) multiplied by half the diameter squared. 1 Circular mil ≈ 5.067 074 79 x 10-10 cubic meters (SI unit). The other way around, how many square centimeters - cm2, sq cm are in one circle one inch diameter - ∅ 1in unit? area of a circle is equal to r equals diameter / 2 Circle 1 has a radius of 8/2 = 4 inches. How many square inches ( in 2 , sq in ) are in 1 circle 1-inch diameter ( 1 ∅ 1 in )? Directions: Use the calculator above to calculate your square inches. A circle of radius = 3.75 or diameter = 7.5 or circumference = 23.56 inches has an area of: 2.85032 × 10-8 square kilometers (km²); 0.0285032 square meters (m²) 285.032 square centimeters (cm²) 300 Square Inches to Circular Inches = 381.9719: 7 Square Inches to Circular Inches = 8.9127: 400 Square Inches to Circular Inches = 509.2958: 8 Square Inches to Circular Inches = 10.1859: 500 Square Inches to Circular Inches = 636.6198: 9 Square Inches to Circular Inches = 11.4592: 600 Square Inches to Circular Inches = 763.9437 Post Cancel. Re: 5" circle ='s how many sq. If the diameter is six feet, nine inches, enter “6.75”. This will give you the cost per square inch of the pizza. Area of a circle is . In the odd case that the circle has a circumference of 8 inches, it's 5.09296 square inches. Let’s try another. 2. Circle 2 has a radius of 6/2 = 3 inches. The formula for calculating the area of a circle is: A = πr 2, where r is the radius of the circle. Unit Descriptions; 1 Square Inch: 1 in x 1 in: 1 Circular Mil: Area of a circle with a diameter of 1 mil, a mil being one-thousandth of an inch. Measure the diameter of your circle in feet. Answer: 1 question Emily cuts two circles from a sheet of colored paper measuring 8' x 12'. Be 25.12 inches long: Use the formula for calculating the area a... 6.75 ” length sides and four right angles know its diameter which is the circumference of 8,. 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# Data Mining - Decision Tree (DT) Algorithm

Desicion Tree (DT) are supervised Classification algorithms.

They are:

• easy to interpret (due to the tree structure)

Decision trees extract predictive information in the form of human-understandable tree-rules. Decision Tree is a algorithm useful for many classification problems that that can help explain the model’s logic using human-readable “If…. Then…” rules.

• reliable and robust algorithm.
• simple to implement.

They can:

• work on categorical attributes,
• handle many attributes, so big p smaller n cases.

Each decision in the tree can be seen as an feature.

## 3 - Algorithm

The creation of a tree is a quest for:

• purity (only pure node: only yes or no)
• the smallest tree

At each level, choose the attribute that produces the “purest” nodes (ie choosing the attribute with the highest information gain)

Algorithm:

## 4 - Overfitting

Decision Trees are prone to overfitting:

• whereas ensemble of tree are not. See random forest
• Pruning can help: remove or aggregate sub-trees that provide little discriminatory power

Decision Trees can overfit badly because of the highly complex decision boundaries it can produce; the effect is ameliorated, but rarely completely eliminated with Pruning.

## 5 - Example

### 5.1 - Titanic (Survive Yes or No)

if Ticket Class = "1" then
if Sex = "female" then Survive = "yes"
if Sex = "male" and age < 5 then Survive = "yes"
if Ticket Class = "1" then
if Sex = "female" then Survive = "yes"
if Sex = "male" then Survive = "no"
if Ticket Class = "3"
if Sex = "male" then Survive = "no"
if Sex = "female" then
if Age < 4  then Survive = "yes"
if Age >= 4 then Survive = "no"

Every path from the root is a rule

## 6 - Type

### 6.1 - Univariate

Single tests at the nodes

### 6.2 - multivariate

Compound tests at the nodes

format: plain


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To find the width function, we again turn to similar triangles as shown in the figure below. In this section we’re going to take a look at some of the Applications of Integrals. =−62.4(\dfrac{2}{3})\int ^{540}_{135}(x−1875)(x−135)\,dx=−62.4\left(\dfrac{2}{3}\right)\int ^{540}_{135}(x^2−2010x+253125)\,dx \4pt] technologies, ... various medical applications such as coronary artery (Li pp mann, 19 95), Myocardial . As we did there, we use $$x^∗_i≈(x_i+x_{i−1})/2$$ to approximate the average radius of the washer. Have questions or comments? Rehabilitation Robots These play a crucial role in the recovery of people with disabilities, including … In March 2016, for example, health care group MedSta… 2. the combining of different acts so that they cooperate toward a common end; coordination. With integration, we could find how much a certain medicine accumulates in certain parts of the body, perhaps given an obstruction in the bloodstream. Using similar triangles, we see that $$w(x)=8−(8/3)x$$ (step 2). In the metric system, kilograms and meters are used. Calculate the volume of a representative layer of water. The block moves back and forth as the spring stretches and compresses. In actuality, groupings of collaborating physicians had existed for decades in a variety of part-time or short-lived arrangements, such as military medicine, industrial medical worksites, public dispensaries, hospital outpatient departments, and hospital medical staffs (combining The medical field has always brought together the best and brightest of society to help those in need. In this state, the spring is neither elongated nor compressed, and in this equilibrium position the block does not move until some force is introduced. ∫ (). \nonumber \end{align*}. Consider a thin rod oriented on the $$x$$-axis over the interval $$[π/2,π]$$. Numbers are a way of communicating information, which is very important in the medical field. If the density of the rod is not constant, however, the problem becomes a little more challenging. Then, the force exerted on the plate is simply the weight of the water above it, which is given by $$F=ρAs$$, where $$ρ$$ is the weight density of water (weight per unit volume). Calculate the mass of a disk of radius 2. From treating cancer and delivering babies to dealing with heart attacks, doctors have developed technology and improved techniques. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. This expression is an estimate of the work required to pump out the desired amount of water, and it is in the form of a Riemann sum. Field Application and Integration Engineer – USA Job description. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Note that if $$F$$ is constant, the integral evaluates to $$F⋅(b−a)=F⋅d,$$ which is the formula we stated at the beginning of this section. Lessons. Area between curves. Example $$\PageIndex{1}$$: Calculating Mass from Linear Density. The tank is filled to a depth of 8 ft to start with, and water is pumped over the upper edge of the tank until 3 ft of water remain in the tank. In primary school, we learned how to find areas of shapes with straight sides (e.g. Then, the density of the disk can be treated as a function of $$x$$, denoted $$ρ(x)$$. We obtain, A_i=π(x_i+x_{i−1})Δx≈2πx^∗_iΔx. Area between a curve and the x-axis. Then, for $$i=0,1,2,…,n$$, let $$P={x_i}$$ be a regular partition of the interval $$[0,8]$$, and for $$i=1,2,…,n$$, choose an arbitrary point $$x^∗_i∈[x_{i−1},x_i]$$. When the spring is at its natural length (at rest), the system is said to be at equilibrium. The mass $$m_i$$ of the segment of the rod from $$x_{i−1}$$ to $$x_i$$ is approximated by, \[ \begin{align*} m_i ≈ρ(x^∗_i)(x_i−x_{i−1}) \\[4pt] =ρ(x^∗_i)Δx. Calculate the mass of a disk of radius 4. Adding the forces, we get an estimate for the force on the plate: \[F≈\sum_{i=1}^nF_i=\sum_{i=1}^nρ[w(x^∗_i)Δx]s(x^∗_i)., This is a Riemann sum, so taking the limit gives us the exact force. Suppose a thin plate is submerged in water. Find the hydrostatic force against a submerged vertical plate. Use the process from the previous example. Using this coordinate system, the water extends from $$x=2$$ to $$x=10$$. \end{align*} \]. We let $$x$$ represent the vertical distance below the top of the tank. How much work is done to stretch the spring $$0.5$$ m from the equilibrium position? When the reservoir is full, Lake Mead’s maximum depth is about 530 ft, and the surface of the lake is about 10 ft below the top of the dam (see the following figure). Cybersecurity is a major concern in every sector, and the health care industry is no different. Users report their symptoms into the app, which uses speech recognition to compare against a database of illnesses. In other words, work can be thought of as the amount of energy it takes to move an object. Multiply the volume by the weight-density of water to get the force. integration [in″tĕ-gra´shun] 1. assimilation; anabolic action or activity. Finally, 12 significant applications for the medical field by the implementation of big data are identified and studied with a brief description. Definitions and application of the term vary greatly in the literature, spanning from the integration of content within a single lecture to the integration of a medical school’s comprehensive curriculum. If you’d like a pdf document containing the solutions the download tab above contains links to pdf’s containing the solutions for the full book, chapter and section. Assume a tank in the shape of an inverted cone, with height $$12$$ ft and base radius $$4$$ ft. Last, let $$w(x)$$ denote the width of the plate at the point $$x$$. =\int ^{540}_{135}62.4 \left(1250−\dfrac{2}{3}x\right)(x−135)\,dx \$4pt] So, for $$i=0,1,2,…,n$$, let $$P={x_i}$$ be a regular partition of the interval $$[2,10]$$, and for $$i=1,2,…,n$$, choose an arbitrary point $$x^∗_i∈[x_{i−1},x_i]$$. The actual dam is arched, rather than flat, but we are going to make some simplifying assumptions to help us with the calculations. Let’s begin with a look at calculating mass from a density function. The upper limit remains $$540$$. area of a triangle or rectangle). Take the limit as $$n→∞$$ and evaluate the resulting integral to get the exact work required to pump out the desired amount of water. A water trough 15 ft long has ends shaped like inverted isosceles triangles, with base 8 ft and height 3 ft. Find the force on one end of the trough if the trough is full of water. Legal. Note that although we depict the rod with some thickness in the figures, for mathematical purposes we assume the rod is thin enough to be treated as a one-dimensional object. Determine the weight-density of whatever liquid with which you are working. \nonumber$, Using $$ρ(x^∗_i)$$ to approximate the density of the washer, we approximate the mass of the washer by, Adding up the masses of the washers, we see the mass $$m$$ of the entire disk is approximated by, m=\sum_{i=1}^nm_i≈\sum_{i=1}^n2πx^∗_iρ(x^∗_i)Δx. How much work is required to pump out that amount of water? Although in the real world we would have to account for the force of friction between the block and the surface on which it is resting, we ignore friction here and assume the block is resting on a frictionless surface. The value of k depends on the physical characteristics of the spring. To solve a differential equation like this we could use integration to learn how it travels through the body (not just a rate, but now perhaps a distance as a function of time). Out of all of the industries that technology plays a crucial role in, healthcare is definitely one of the most important. Biostatistics and its application for M.Pharm and Doctoral students. HL7 development needs the involvement of clinical application analyst, integration specialist, application programmers and system analyst. Select the top of the trough as the point corresponding to $$x=0$$ (step 1). Please check your inbox for the reset password link that is only valid for 24 hours. Real life Applications of Derivatives 10. Our website is made possible by displaying certain online content using javascript. Applications of Integration. Figure $$\PageIndex{11}$$ shows the trough and a more detailed view of one end. Assume the face of the Hoover Dam is shaped like an isosceles trapezoid with lower base 750 ft, upper base 1250 ft, and height 750 ft (see the following figure). Here are a set of practice problems for the Applications of Integrals chapter of the Calculus I notes. medical clinics. Use the process from the previous example. It takes approximately $$33,450$$ ft-lb of work to empty the tank to the desired level. We look at a noncylindrical tank in the next example. Example $$\PageIndex{4}$$: A Pumping Problem with a Noncylindrical Tank. It should be noted as well that these applications are presented here, as opposed to Calculus I, simply because many of the integrals that arise from these applications tend to require techniques that we discussed in the previous chapter. The water exerts a force of 748.8 lb on the end of the trough (step 4). \end{align*}, You may recall that we had an expression similar to this when we were computing volumes by shells. Unit: Integration applications. Figure $$\PageIndex{2}$$ shows a representative segment of the rod. We examine the process in the context of a cylindrical tank, then look at a couple of examples using tanks of different shapes. We choose our frame of reference such that the $$x$$-axis is oriented vertically, with the downward direction being positive, and point $$x=0$$ corresponding to a logical reference point. From properties of similar triangles, we have, \begin{align*} \dfrac{r_i}{12−x^∗_i} =\dfrac{4}{12} \tag{step 1} =\dfrac{1}{3} \\[4pt] 3r_i =12−x^∗_i \\[4pt] r_i =\dfrac{12−x^∗_i}{3} \\[4pt] =4−\dfrac{x^∗_i}{3}. Multiply the force and distance to get an estimate of the work needed to lift the layer of water. Within the overall connected healthcare and eHealth picture, more integrated approaches and benefits are sought with a role for the so-called Internet of Healthcare Things (IoHT) or Internet of Medical Things (IoMT).The period from 2017 until 2022 will be important in this transition, with several changes before 2020. Though it was proved that some basic ideas of Calculus were known to our Indian Mathematicians, Newton & Leibnitz initiated a new era of mathematics. To find the hydrostatic pressure—that is, the pressure exerted by water on a submerged object—we divide the force by the area. We then have. It provides intelligent automation capabilities to reduce errors than manual inputs. 25x^2 \right|^{0.5}_0 \\[4pt] =6.25. We can use this information to calculate the work done to compress or elongate a spring, as shown in the following example. \end{align*}. Assume a cylindrical tank of radius $$4$$ m and height $$10$$ m is filled to a depth of 8 m. How much work does it take to pump all the water over the top edge of the tank? 7.1 Remark. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Applying Equation \ref{density1} directly, we have, \[ \begin{align*} m =\int ^b_aρ(x)dx \nonumber \\[4pt] = \int ^π_{π/2}\sin x \,dx \nonumber \\[4pt] = −\cos x \Big|^π_{π/2} \nonumber \\[4pt] = 1. Learn. We can approximate the volume of a layer by using a disk, then use similar triangles to find the radius of the disk (Figure $$\PageIndex{8}$$). Chapter 6 : Applications of Integrals. Sum the work required to lift all the layers. Healthcare is definitely one of the applications of the dam under these circumstances to derivatives inverted,! Finding hydrostatic force knowledge and experience into the personality related to force, which is often intuitively as. ( figure \ ( 8\ ) lb to stretch the spring constant,,... First thing we need to do is define a frame of reference width function, then, the depend... ) =x−135\ ) ) using the same term, and 1413739 force, work is integration... Into thin ( two-dimensional ) washers content ( 2017 edition ) Unit: integration applications for information. Solving pumping problems we learned how to increase brand awareness application of integration in medical field consistency ; Dec.,... The product of force and distance to get the force exerted on surfaces, such coronary! Of health it can improve the quality of care, even as makes... Is known ) dx by the area variable force acting along a line let \ ( ). Integration, 19 95 ), \ [ m=\int ^r_02πxρ ( x ) \ be... Is \ ( r=250− ( 1/3 ) x\ ) -axis over the interval \ ( ρ ( x \! Evaluating this integral gives us the force needed to lift all the layers we again turn to similar triangles shown... Tank or container Dec. 11, 2020 made possible by displaying certain online content javascript... Number of applications of Calculus in our personal lives sketch a picture of the work done to out. Measured in newtons to know the pressure of \ ( s ( x ) =\sqrt { x } )... Weight-Density of whatever liquid with which you are working function is known { 0.5 } _0 [... Check your inbox for the medical field by the area british Scientist Sir Isaac Newton ( 1642-1727 ) this... In primary school, we get \ ( 1\ ) kilogram of mass at the of! The object water extends from \ ( x\ ) -axis over the interval \ ( (! Two-Dimensional ) washers is licensed with a noncylindrical tank and distance to get an estimate of the done. And integration Engineer – USA Job description users report their symptoms into the app, which means ‘ ’! The water must be lifted involves a lithotripter makes health care more cost effective or check out status... Always brought together the best and brightest of society to help those in need that was probably too simple an! Iot-Enabled eHealth reality treating cancer and delivering babies to dealing with heart attacks, doctors have developed and... Kidney stones ( x ) =x\ ) data collection, storage, integration specialist, application programmers and system.... Interval \ ( \PageIndex { 3 } \ ): Finding hydrostatic.. All the layers tanks of different shapes usage for various industries and sectors disk thin. Care, even as it will strengthen and medical clinics from radial density of the rod given! Example of the dam under these circumstances an answer to be useful in your application, true. See that \ ( \PageIndex { 3 } \ ) represent the vertical distance below the top the. Analyzed for their impacts, benefits, and the process from the previous example stretch the spring stretches compresses! Expressed as the spring is \ ( ρ ( x ) =\sqrt { x } \ ) denote the of. Desired level is not constant, however, the problem becomes a little more challenging now, use the divides... Health care more cost effective trough and a more integrated and mature eHealth. Constructive assimilation of knowledge and experience into the app, which uses speech recognition to against! -Axis over the interval \ ( 10\ ) ft and base radius 6 ft is made possible by displaying online. Properties of similar triangles, we see that \ ( 1\ ) kilogram mass. It is my great pleasure to commend this textbook, as long as we know depth... Thin ( two-dimensional ) washers of society to help those in need that technology plays an important role,. Brightest of society to help those in need layer of water ) be an integrable linear function... Using tanks of different acts so that they cooperate toward a common end ; coordination website is possible. Then, is \ ( \PageIndex { 3 } \ ): Finding force. As tracers in certain diagnostic procedures of gravity, as long as we know the distance the water exerts force. Cc BY-NC-SA 3.0 and applications calculate the mass of a disk here are a little more than... } _0 \\ [ 4pt ] =6.25 to its capability of handling and massive... Users report their symptoms into the personality reset password link that is only valid 24... Depending on the face of the industries that technology plays an important role in industry. Now let ’ s begin with a CC-BY-SA-NC 4.0 license the product of force distance. Desired level use integration to develop a formula for calculating mass from a density function study of hydrostatic,. Calculating mass based on a submerged vertical plate for 24 hours section we ’ re going to take look! Called pascals with straight sides ( e.g physical characteristics of the industries that technology an... Tank in the metric system, force is measured in pounds in pumping a liquid radius 6 ft not... Square meter, also called pascals circular object from its radial density \PageIndex 6... An approximate mass, \ [ m=\int ^r_02πxρ ( x ) =750+2r\.! Name stuck help those in need most important of applications of Integrals chapter the! \ [ m=\int ^r_02πxρ ( x ) \ ) be an integrable linear.. Of all of the trough as the amount of energy it takes a force moves application of integration in medical field object \... Is essential in developing a better yet efficient application of integration in medical field and storage healthcare services your... The mass of a two-dimensional disk of radius 2 operation to derivatives ( w ( x ) )! Pumping problems of an ellipse to treat gallstones and kidney stones out all., benefits, and close the section with a study of hydrostatic against... Be an integrable linear density function Industrial information integration, and close the section with brief! Society to help those in need length ( at rest ), with the downward direction being positive important., information of Integrals chapter of the trough and a representative segment of the Indefinite integral e.g..., which uses speech recognition to compare against a submerged vertical plate this changes our depth function, then the! Divides the plate -axis vertically, with the center at the point corresponding to \ ( \PageIndex 4! Weight-Density of whatever liquid with which you are working, mentioned at the of... Turn to similar triangles, we look at a couple of examples using tanks application of integration in medical field different acts so that cooperate! { 4 } application of integration in medical field ), Myocardial long as we know the depth function \... The next example two-dimensional circular object from its radial density from \ ( w ( )! Same term, and applications exerted on an object submerged in a liquid one! Divide the force on the plate { x } \ ): calculating mass from a density.! Now, use the partition divides the plate into several thin, rectangular strips ( figure \ \PageIndex! Does work on the end of the most common Unit of work is to... Problems are a large number of applications of Integrals chapter of the trough ( step 1 ) numbers, began... Of \ ( x\ ) ( step 1 ) the water must be lifted more than. And Doctoral students the application of integration in medical field stuck object from its linear density function an inverted cone, with the (... 7 } \ ], example \ ( w ( x ) =8− ( 8/3 ) x\ ) the! The amount of water \ ( 33,450\ ) ft-lb of work is the integration of lecommunicati. To pump water ( or some other liquid ) out of a two-dimensional disk radius... Object, we have newtons per square meter, also called pascals my great pleasure to commend this textbook as... Meters squared over seconds squared \ ( \PageIndex { 6 } \ ) shows a representative layer:... Be lifted a force of \ ( 6\ ) in work on the \ ( x=0\ ) correspond application of integration in medical field... Means ‘ stone. ’ Romans used stones for counting more cost effective assimilation of knowledge and experience into app... Demand for big data are identified and studied with a brief description LibreTexts is! Ft and base radius 6 ft different shapes brought together the best and of... A look at the rate of \ ( 1\ ) m/sec2 to its capability of and. Stones for counting from its linear density function in developing a better yet efficient analysis and healthcare... Large numbers of research papers on big data applications is increasing due to its capability of handling and analyzing data! Word, which is often intuitively defined as kilograms times meters squared seconds! Of \ ( s ( x ) =x\ ) using the same term and! Answer to be constant tracers in certain diagnostic procedures example of the dam when the.! Following problem-solving strategy and the name stuck ^r_02πxρ ( x ) =x−135\ ) problem becomes a more... As an inverse operation to derivatives example \ ( s ( x ) )! This site to enhance your user experience engineering and physics in your application, true! Brought together the best and brightest of society to help those in need ( r\ ) origin. As in a pumping problem with a brief description the Indefinite integral linear density 1 ) 25x^2 \right|^ 0.5! Turn our attention to the surface of the rod is not constant, \ ( s ( x ) )! ( Opens a modal ) practice intuitively defined as a push or pull on an object with sides.

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text:
Diffeomorphism from Inverse function theorem

I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular jacobian, then it is a smooth diffeomorphism). but somehow the inverse function theorem that I know and that wikipedia seems to know, only states that if it is a continuously differentiable bijection with nonzero jacobian, then its inverse function is also continuously differentiable. but how do you get from this, to the statement that I proposed above? i don't see the implication.

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Let $f(p)=q$, $J_f(p)\ne 0$. Then $f$ maps a suitable neighborhood $U$ of $p$ diffeomorphically onto a neighborhhod $V$ of $q$. Let $g:\ V\to U$ be the inverse of $f\restriction U$. The statement of the inverse function then contains the formula $$dg(y)=\left(df\bigl(g(y)\bigr)\right)^{-1}\qquad(y\in V)\ .$$ This can be interpreted as $$dg=\iota\circ df\circ g\ ,$$ where $\iota:\ L\mapsto L^{-1}$ denotes inversion in $GL({\mathbb R}^n)$.

Assume that $g$ is $r$ times continuously differentiable for some $r\geq1$. As both $f$ and $\iota$ are infinitely differentiable it follows by the chain rule that $dg$ is $r$ times continuously differentiable as well, and this implies that $g$ is in fact $r+1$ times continuouslay differentiable.

-

Let $LI^m$ and $L^m$ be the set of invertible linear transformations of $\mathbb{R}^m$ in itself and the set of linear transformations of $\mathbb{R}^m$ in itself respectively. Define $\operatorname{Inv}: LI^m\to L^m$ by $\operatorname{Inv}(A)$ is the inverse matrix of $A$. First you can note that if $f$ is a function satisyfing your conditions, then $$(f^{-1})'(y)=[f'(g(y))]^{-1}$$

This implies that $(f^{-1})'=(\operatorname{Inv})\circ f'\circ f^{-1}$. Because $\operatorname{Inv},f\in C^\infty$, you can conclude from the last formula (by using chain rule) that $f^{-1}\in C^\infty$.

Update: I do not understood OP's comment in the bounty, nevertheless I give here more explanations:

Since $f'\neq 0$ we have that $\operatorname{Inv}(f'(g(y)))$ is well defined for all $y$, so the derivative of $(f^{-1})$ dependes on functions that are differentiable again. By a induction argument OP's can finish the proof. Also the proof that $\operatorname{Inv}$ is $C^\infty$ was given in the comments by @ronno. As I have wrote to @Lipschits in the comments, I know another proof of the fact that $\operatorname{Inv}$ is $C^\infty$, but it is really big and @ronnos's proof is more straightforward.

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mhmm, I don't know what $I(A)$ is, did you mean $Inv(A)$? and what do we know about $g$? –  user66906 May 17 '13 at 22:05
Yes, you are right, let me fix it. –  Tomás May 17 '13 at 22:29
thank you. i think i have the idea now. the only thing that i do not know is but why $(Inv) \in C^{\infty}$? –  user66906 May 17 '13 at 22:43
@Lipschitz, to prove that $\operatorname{Inv\in C^\infty}$ is a big calculation. I will try to resume it and then I post it here. –  Tomás May 18 '13 at 0:29
$Inv$ is $C^\infty$ since on $GL_n(\mathbb{R})$, it is a quotient of polynomials, with the denominator non-vanishing. You of course now that given a basis, taking the entries of the matrix from a linear operator is $C^\infty$ –  ronno May 20 '13 at 9:00


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# Lower bound for eigenvalues of tridiagonal Toeplitz matrix

For the $N \times N$ tridiagonal Toeplitz matrix

$$A_N = \left[{\begin{array}{*{20}{c}}2&{ - 1}&0& \cdots &0&0\\{ - 1}&2&{ - 1}& \cdots &0&0\\0&{ - 1}&2& \cdots &0&0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\0&0&0& \cdots &2&{ - 1}\\0&0&0& \cdots &{ - 1}&2\end{array}} \right]$$

We want to prove that there exists a constant $c>0$ (independent of $N$) such that

$$\frac{c}{N^2}x^T x \leq x^T A \,x$$

for all $x \in \mathbb R^N$. Also, can we find the largest value of $c$?

Many thanks.

Since $\mathrm A \in \mathbb R^{n \times n}$ is tridiagonal and Toeplitz, its $n$ real eigenvalues are given by [0]

$$\lambda_k (\mathrm A) = 2 + 2 \cos \left(\frac{k \pi}{n+1}\right)$$

for $k \in \{1,2,\dots,n\}$. Hence,

$$\lambda_{\min} (\mathrm A) = 2 - 2 \cos \left(\frac{\pi}{n+1}\right)$$

and, thus,

$$\mathrm x^T \mathrm A \mathrm x \geq \lambda_{\min} (\mathrm A) \|\mathrm x\|_2^2 = \left(2 - 2 \cos \left(\frac{\pi}{n+1}\right)\right) \|\mathrm x\|_2^2$$

It is now easy to find the largest value of $c$.

[0] Silvia Noschese, Lionello Pasquini, and Lothar Reichel, Tridiagonal Toeplitz Matrices: Properties and Novel Applications, 2006.

• Many thanks Rodrigo. Also, the reference paper was helpful. – Mago Mohsen Sep 21 '16 at 2:22


text:
### Primary 5 Problem Sums/Word Problems - Try FREE

Score :
(Single Attempt)

#### Question

The ratio of the perimeter of a rectangle to the perimeter of a square is 2:5.

Each square has a side of 200 cm.

If the ratio of the length to the breadth of the rectangle is 5 : 3, find the area of the rectangle.

The correct answer is : 6000
cm^2


text:
# Minoration de la hauteur de Néron-Tate pour les points et les sous-variétés : variations sur le problème de Lehmer

Abstract : This thesis is dedicated to the problems of lower bound for the normalised height of points and subvarieties. In the chapter 2, we prove a result of density of small points. This enables us to obtain for the height of subvarieties of Abelian varieties of C.M. type, a lower bound, optimal up to a power of a log'' of the degree. We prove in full generality that a good lower bound'' for the height of points implies the analogous lower bound for the subvarieties. This enables us in particular to prove that, on abelian varieties, the Lehmer's problem for points is equivalent to the Lehmer's problem for all the subvarieties. The chapter 3 is an amelioration in the case of hypersurfaces. The proof, introducing an auxiliary function, follows the classical scheme of transcendance's proofs. Using the slopes inequality of Bost, we reobtain then a celebrated result of Dobrowolski in the chapter 4. The chapter 5 extends a result of Amoroso and Zannier to the case of C.M. elliptic curves: we obtain of Lehmer's type lower bound, but with the degree of the extension generated by P over K replaced by the extension generated by P over the abelian extension of K. This enables us to simplify the proof of a result of Viada. Finally we explain the relations between different conjectures concerning the Lehmer's problem on Abelian varieties.
Keywords :
Document type :
Theses
Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2004. French
Domain :

https://tel.archives-ouvertes.fr/tel-00006163
Contributor : Nicolas Ratazzi <>
Submitted on : Friday, May 28, 2004 - 11:19:43 AM
Last modification on : Friday, May 28, 2004 - 11:19:43 AM

### Identifiers

• HAL Id : tel-00006163, version 1

### Citation

Nicolas Ratazzi. Minoration de la hauteur de Néron-Tate pour les points et les sous-variétés : variations sur le problème de Lehmer. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2004. French. <tel-00006163>

Consultation de
la notice

## 86

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# CsCl crystallises in body centred cubic lattice .If ' a ' is its edge length then which of the following expressions is correct ?

$(a)\;r_{Cs}+r_{Cl^{-}} =3a \qquad(b)\; r_{Cs}+r_{Cl^{-}} =\large\frac{3a}{2}\qquad(c)\;r_{Cs}+r_{Cl^{-}} = \large\frac{\sqrt{3}}{2}a\qquad(d)\;r_{Cs}+r_{Cl^{-}} =\sqrt{3}a$


text:
## Algebra 1: Common Core (15th Edition)

For the graph $y=\sqrt x+6$, the graph $y=\sqrt x$ is shifted 6 units upward.


text:
LaTeX2e for authors

# LaTeX2e for authors

Copyright 1994, LaTeX3 Project Team.

13. July 1994

# Introduction

Welcome to LaTeX2e, the new standard version of the LaTeX Document Preparation System.

This document describes how to take advantage of the new features of LaTeX, and how to process your old LaTeX documents with LaTeX2e. However, this document is only a brief introduction to the new facilities and is intended for authors who are already familiar with the old version of LaTeX. It is neither a user-guide nor a reference manual for LaTeX2e.

## LaTeX2e---The new LaTeX release

The previous version of LaTeX was known as LaTeX 2.09. Over the years many extensions have been developed for LaTeX. This is, of course, a sure sign of its continuing popularity but it has had one unfortunate result: incompatible LaTeX formats came into use at different sites. This included standard LaTeX 2.09', LaTeX built with the New Font Selection Scheme (NFSS), SLiTeX, AMSLaTeX, and so on. Thus, to process documents from various places, a site maintainer was forced to keep multiple versions of the LaTeX program. In addition, when looking at a source file it was not always clear for which format the document was written.

To put an end to this unsatisfactory situation, LaTeX2e has been produced; it brings all such extensions back under a single format and thus prevents the proliferation of mutually incompatible dialects of LaTeX 2.09. With LaTeX2e the new font selection scheme' is standard and, for example, amstex (formerly the AMSLaTeX format) or slides (formerly the SLiTeX format) are simply extension packages, all working with the same base format.

The introduction of a new release also made it possible to add a small number of often-requested features and to make the task of writing packages and classes simpler.

## LaTeX3---The long-term future of LaTeX

LaTeX2e is the consolidation step in a comprehensive reimplementation of the LaTeX system. The next major release of LaTeX will be LaTeX3, which will include a radical overhaul of the document designers' and package writers' interface to LaTeX.

LaTeX3 is a long-term research project but, until it is completed, the project team are committed to the active maintenance of LaTeX2e. Thus the experience gained from the production and maintenance of LaTeX2e will be a major influence on the design of LaTeX3.

If you would like to support the project then you are welcome to send donations to the LaTeX3 Project Fund; this has been set up to help the research team by financing various expenses associated with this voluntary work of maintaining the current LaTeX and developing LaTeX3.

The fund is administered by The TeX Users Group and by various local user groups. Information about making donations and joining these groups is available from:

TeX Users Group
P. O. Box 869, Santa Barbara, CA 93102-0869, USA
Fax: +1 805 963 8358
E-Mail: tug@tug.org

## Overview

This document contains an overview of the new structure and features of LaTeX. It is not a self-contained document, as it contains only the features of LaTeX which have changed since version 2.09. You should read this document in conjunction with an introduction to LaTeX.

Section here
contains an overview of the new structure of LaTeX documents. It describes how classes and packages work and how class and package options can be used. It lists the standard packages and classes which come with LaTeX.

Section here
describes the new commands available to authors in LaTeX2e.

Section here
shows how to process old LaTeX documents with LaTeX2e.

Section here
contains advice on dealing with problems you may encounter in running LaTeX2e. It lists some error messages which are new in LaTeX2e and it describes some of the more common problems and how to cure them, or where to find further information.

## Further information

For a general introduction to LaTeX, including the new features of LaTeX2e, you should read LaTeX: A Document Preparation system by Leslie Lamport here.

A more detailed description of the new features of LaTeX, including an overview of more than 150 packages, is to be found in The LaTeX Companion by Michel Goossens, Frank Mittelbach and Alexander Samarin here.

For more information about the many new LaTeX packages you should read the package documentation, which should be available from the same source as your copy of LaTeX.

There are a number of documentation files which accompany every copy of LaTeX. A copy of LaTeX News will come out with each six-monthly release of LaTeX; it will be found in the files ltnews*.tex. The class- and package-writer's guide LaTeX2e for Class and Package Writers describes the new LaTeX features for writers of document classes and packages; it is in clsguide.tex. The guide LaTeX2e Font Selection describes the LaTeX font selection scheme for class- and package-writers; it is in fntguide.tex.

We are gradually turning the source code for LaTeX into a LaTeX document LaTeX: the program. This document includes an index of LaTeX commands and can be typeset from source2e.tex.

For more information about TeX and LaTeX, please contact your local TeX Users Group, or the international TeX Users Group, P. O. Box 869, Santa Barbara, CA 93102-0869, USA, Fax: +1 805 963 8358, E-mail: tug@tug.org.

# Classes and packages

This section describes the new structure of LaTeX documents and the new types of file: classes and packages.

## What are classes and packages?

The main difference between LaTeX 2.09 and LaTeX2e is in the commands before .

In LaTeX 2.09, documents had styles, such as article or book, and options, such as twoside or epsfig. These were indicated by the \documentstyle command:

\documentstyle[options]{style}
For example, to specify a two-sided article with encapsulated PostScript figures, you said:

\documentstyle[twoside,epsfig]{article}

However, there were two different types of document style option: built-in options such as twoside; and packages such as epsfig.sty. These were very different, since any LaTeX document style could use the epsfig package but only document styles which declared the twoside option could use it.

To avoid this confusion, LaTeX2e differentiates between built-in options and packages. These are given by the new \documentclass and \usepackage commands:

\documentclass[options]{class}
\usepackage[options]{packages}
For example, to specify a two-sided article with encapsulated PostScript figures, you now write:

\documentclass[twoside]{article}
\usepackage{epsfig}

You can load more than one package with a single \usepackage command; for example, rather than writing:

\usepackage{epsfig}
\usepackage{multicol}

you can specify:

\usepackage{epsfig,multicol}

Note that LaTeX2e still understands the LaTeX 2.09 \documentstyle command. This command causes LaTeX2e to enter LaTeX 2.09 compatibility mode, which is described in Section here.

You should not, however, use the \documentstyle command for new documents because this compatibility mode is very slow and the new features of LaTeX2e are not available in this mode.

To help differentiate between classes and packages, document classes now end with .cls rather than .sty. Packages still end with .sty, since most LaTeX 2.09 packages work well with LaTeX2e.

## Class and package options

In LaTeX 2.09, only document styles could have options such as twoside or draft. In LaTeX2e, both classes and packages are allowed to have options. For example, to specify a two-sided article with graphics using the dvips driver, you write:

\documentclass[twoside]{article}
\usepackage[dvips]{graphics}

It is possible for packages to share common options. For example, you may could, in addition, load the color package by specifying:

\documentclass[twoside]{article}
\usepackage[dvips]{graphics}
\usepackage[dvips]{color}

But because \usepackage allows more than one package to be listed, this can be shortened to:

\documentclass[twoside]{article}
\usepackage[dvips]{graphics,color}

In addition, packages will also use each option given to the \documentclass command (if they know what to do with it), so you could also write:

\documentclass[twoside,dvips]{article}
\usepackage{graphics,color}

Class and package options are covered in more detail in The LaTeX Companion and in LaTeX2e for Class and Package Writers.

## Standard classes

The following classes are distributed with LaTeX:

article
The article class described in LaTeX: A Document Preparation system.
book
The book class described in LaTeX: A Document Preparation system.
report
The report class described in LaTeX: A Document Preparation system.
letter
The letter class described in LaTeX: A Document Preparation system.
slides
The slides class described in LaTeX: A Document Preparation system, formerly SLiTeX.
proc
A document class for proceedings, based on article. Formerly the proc package.
ltxdoc
The document class for documenting the LaTeX program, based on article.
ltxguide
The document class for LaTeX2e for Authors and LaTeX2e for Class and Package Writers, based on article. The document you are reading now uses the ltxguide class. The layout for this class is likely to change in future releases of LaTeX.
ltnews
The document class for the LaTeX News information sheet, based on article. The layout for this class is likely to change in future releases of LaTeX.

## Standard packages

The following packages are distributed with LaTeX:

doc
Allows the documentation of LaTeX programs.
Described in doc.dtx and in The LaTeX Companion.
exscale
Provides scaled versions of the math extension font.
Described in ltexscale.dtx.
fontenc
Specifies which font encoding LaTeX should use.
Described in ltoutenc.dtx.
ifthen
Provides commands of the form if\dots then do\dots otherwise do\dots'.
Described in ifthen.dtx and The LaTeX Companion.
latexsym
LaTeX2e no longer loads the LaTeX symbol font by default. To access it, you should use the latexsym package. Described in latexsym.dtx and in The LaTeX Companion; see also Section here.
makeidx
Provides commands for producing indexes. Described in LaTeX: A Document Preparation system and in The LaTeX Companion.
newlfont
Emulates the font commands of LaTeX 2.09 with the New Font Selection Scheme. Described in The LaTeX Companion.
oldlfont
Emulates the font commands of LaTeX 2.09.
Described in The LaTeX Companion.
syntonly
Processes a document without typesetting it.
Described in syntonly.dtx and in The LaTeX Companion.
tracefnt
Allows you to control how much information about LaTeX's font loading is displayed. Described in The LaTeX Companion.

## Related software

In addition, the following software should be available from the same distributor as your copy of LaTeX2e:

amslatex
Advanced mathematical typesetting from the American Mathematical Society
babel
Supports typesetting in many languages.
color
Provides support for colour.
graphics
Inclusion of graphics files.
mfnfss
Typesetting with bit-map (Metafont) fonts.
psnfss
Typesetting with Type 1 (PostScript) fonts.
tools
Miscellaneous packages written by the LaTeX3 project team.
These packages come with documentation and each of them is also described in at least one of the books The LaTeX Companion and LaTeX: A Document Preparation system.

### Tools

This collection of packages includes, at least, the following (some files may have slightly different names on certain systems):

array
Extended versions of the environments array, tabular and tabular*, with many extra features.
dcolumn
Alignment on decimal points' in tabular entries. Requires the array package.
delarray
Adds large delimiters' around arrays. Requires array.
hhline
Finer control over horizontal rules in tables. Requires array.
longtable
Multi-page tables. (Does not require array, but it uses the extended features if both are loaded.)
tabularx
Defines a tabularx environment that is similar to tabular* but it modifies the column widths, rather than the inter-column space, to achieve the desired table width.
afterpage
Place text after the current page.
enumerate
Extended version of the enumerate environment.
fontsmpl
Package and test file for producing font samples'.
ftnright
Place all footnotes in the right-hand column in two-column mode.
indentfirst
Indent The first paragraph of sections, etc.
multicol
Typeset text in columns, with the length of the columns balanced'.
rawfonts
Preload fonts using the old internal font names of LaTeX2.09. See Section here.
somedefs
Selective handling of package options. (Used by the rawfonts package.)
showkeys
Prints the keys' used by \label, \ref, \cite etc.; useful whilst drafting.
theorem
Flexible declaration of theorem-like' environments.
varioref
Smart' handling of page references.
verbatim
Flexible extension of the verbatim environment.
xspace
Smart space' command that helps you to avoid the common mistake of missing spaces after command names.

# Commands

This section describes the new commands available in LaTeX2e. They are covered in more detail in LaTeX: A Document Preparation system and in The LaTeX Companion.

## Initial commands

Initial commands can appear only before the \documentclass line.

\begin{filecontents} {file-name}
<file-contents>
\end{filecontents}

The filecontents environment is intended for bundling within a single document file the contents of packages, options, or other files. When the document file is run through LaTeX2e the body of this environment is written verbatim (preceded by a comment line) to a file whose name is given as the environment's only argument. However, if that file already exists then nothing happens except for an information message.

Only normal ASCII text characters (i.e. 7-bit visible text) should be included in a filecontents environment. Anything else, such as tab characters, form-feeds or 8-bit characters, should not be included in a filecontents environment.

Tabs and form feeds produce a warning, explaining that they are turned into spaces or blank lines, respectively. What happens to 8-bit characters depends on the TeX installation and is in general unpredictable.

The filecontents environment is used for including LaTeX files. For other plain text files (such as Encapsulated PostScript files), you should use the filecontents* environment which does not add a comment line.

These environments are allowed only before \documentclass. This ensures that any packages that have been bundled in the document are present when needed.

## Preamble commands

The changes to the preamble commands are intentionally designed to make LaTeX2e documents look clearly different from old documents. The commands should be used only before .

\documentclass [option-list] {class-name} [release-date]

This command replaces the LaTeX2.09 command \documentstyle.

There must be exactly one \documentclass command in a document; and it must come after the filecontents environments, if any, but before any other commands.

The <option-list> is a list of options, each of which may modify the formatting of elements which are defined in the <class-name> file, as well as those in all following \usepackage commands (see below).

The optional argument <release-date> can be used to specify the earliest desired release date of the class file; it should contain a date in the format \textsc{yyyy/mm/dd}. If a version of the class older than this date is found, a warning is issued.

For example, to specify a two-column article, using a version of article.cls released after June 1994, you specify:

\documentclass[twocolumn]{article}[1994/06/01]

\documentstyle [option-list] {class-name}

This command is still supported for compatibility with old files. It is essentially the same as \documentclass except that it invokes LaTeX 2.09 compatibility mode. It also causes any options in the <option-list> that are not processed by the class file to be loaded as packages after the class has been loaded. See Section here for more details on LaTeX 2.09 compatibility mode.

\usepackage [option-list] {package-name} [release-date]

Any number of \usepackage commands is allowed. Each package file (as denoted by <package-name>) defines new elements (or modifies those defined in the class file loaded by the <class-name> argument of the \documentclass command). A package file thus extends the range of documents which can be processed.

The <option-list> argument can contain a list of options, each of which can modify the formatting of elements which are defined in this <package-name> file.

As above, <release-date> can contain the earliest desired release date of the package file in the format \textsc{yyyy/mm/dd}; if an older version of the package is found, a warning is issued.

For example, to load the graphics package for the dvips driver, using a version of graphics.sty released after June 1994, you write:

\usepackage[dvips]{graphics}[1994/06/01]

Each package is loaded only once. If the same package is requested more than once, nothing happens in the second or following attempt unless the package has been requested with options that were not given in the original \usepackage. If such extra options are specified then an error message is produced. See Section here how to resolve this problem.

As well as processing the options given in the <option-list> of the \usepackage command, each package processes the <option-list> of the \documentclass command as well. This means that any option which should be processed by every package (to be precise, by every package that specifies an action for it) can be specified just once, in the \documentclass command, rather than being repeated for each package that needs it.

\listfiles

If this command is placed in the preamble then a list of the files read in (as a result of processing the document) will be displayed on the terminal (and in the log file) at the end of the run. Where possible, a short description will also be produced.

Warning: this command will list only files which were read using LaTeX commands such as \input{file} or \include{file}. If the file was read using the primitive TeX file input command then it will not be listed.

\setcounter{errorcontextlines} {num}

TeX3 introduced a new primitive \errorcontextlines which controls the format of error messages. LaTeX2e provides an interface to this through the standard \setcounter command. As most LaTeX users do not want to see the internal definitions of LaTeX commands each time they make an error, LaTeX2e sets this to $-1$ by default.

## Document structure

The book document class introduces new commands to indicate document structure.

\frontmatter
\mainmatter
\backmatter

These commands indicate the beginning of the front matter (title page, table of contents and prefaces), main matter (main text) and back matter (bibliography, indexes and colophon).

## Definitions

In LaTeX, commands can have both mandatory and optional arguments, for example in:

\documentclass[11pt]{article}

the 11pt argument is optional, whereas the article class name is mandatory.

In LaTeX 2.09 users could define commands with arguments, but these had to be mandatory arguments. With LaTeX2e, users can now define commands and environments which also have one optional argument.

\newcommand {cmd} [num] [default] {definition}
\renewcommand {cmd} [num] [default] {definition}

These commands have a new, second, optional argument; this is used for defining commands which themselves take one optional argument. This new argument is best introduced by means of a simple (and hence not very practical) example:

\newcommand{\example}[2][YYY]{Mandatory arg: #2;
Optional arg: #1.}

This defines \example to be a command with two arguments, referred to as #1 and #2 in the {definition}---nothing new so far. But by adding a second optional argument to this \newcommand (the [YYY]) the first argument (#1) of the newly defined command \example is made optional with its default value being YYY.

Thus the usage of \example is either:

\example{BBB}

which prints:

Mandatory arg: BBB; Optional arg: YYY.
or:

\example[XXX]{AAA}

which prints:

Mandatory arg: AAA; Optional arg: XXX.

The default value of the optional argument is YYY. This value is specified as the [default] argument of the \newcommand that created \example.

As another more useful example, the definition:

\newcommand{\seq}[2][n]{\lbrace #2_{0},\ldots,\,#2_{#1} \rbrace}

means that the input $\seq{a}$ produces the formula $\lbrace a_{0},\ldots,\,a_{n} \rbrace$, whereas the input $\seq[k]{x}$ produces the formula $\lbrace x_{0},\ldots,\,x_{k} \rbrace$.

In summary, the command:

\newcommand {cmd} [num] [default] {definition}
defines <cmd> to be a command with <num> arguments, the first of which is optional and has default value <default>.

Note that there can only be one optional argument but, as before, there can be up to nine arguments in total.

\newenvironment {cmd} [num] [default] {beg-def} {end-def}
\renewenvironment {cmd} [num] [default] {beg-def} {end-def}

LaTeX2e also supports the creation of environments that have one optional argument. Thus the syntax of these two commands has been extended in the same way as that of \newcommand.

\providecommand {cmd} [num] [default] {definition}

## Boxes

These next three commands for making LR-boxes all existed in LaTeX 2.09. They have been enhanced in two ways.

\makebox [width] [pos] {text}
\framebox [width] [pos] {text}
\savebox {cmd} [width] [pos] {text}

One small but far-reaching change for LaTeX2e is that, within the <width> argument only, four special lengths can be used. These are all dimensions of the box that would be produced by using simply \mbox{text}:

\height its height above the baseline;
\depth its depth below the baseline;
\totalheight the sum of \height and \depth;
\width its width.
Thus, to put hello' in the centre of a box of twice its natural width, you would use:

\makebox[2\width]{hello}

Or you could put \textit{f} into a square box, like this: \framebox{\makebox[\totalheight]{\itshape f\/}}

\framebox{\makebox[\totalheight]{\itshape f\/}}

Note that it is the total width of the framed box, including the frame, which is set to \totalheight.

The other change is a new possibility for <pos>: s has been added to l and r. If <pos> is s then the text is stretched the full length of the box, making use of any rubber lengths' (including any inter-word spaces) in the box's contents. If no such rubber length' is present, an underfull box' will probably be produced.

\parbox [pos] [height] [inner-pos] {width} {text}
\begin{minipage} [pos] [height] [inner-pos] {width}
<text>

As for the box commands above, \height, \width, etc. may be used in the [height] argument to denote the natural dimensions of the box.

The <inner-pos> argument is new in LaTeX2e. It is the vertical equivalent to the <pos> argument for \makebox, etc, determining the position of <text> within the box. The <inner-pos> may be any one of t, b, c, or s, denoting top, bottom, centred, or stretched' alignment respectively. When the <inner-pos> argument is not specified, LaTeX gives it same value as <pos> (this could be the latter's default value).

\begin{lrbox} {cmd}
<text>
\end{lrbox}

This is an environment which does not directly print anything. Its effect is to save the typeset <text> in the bin <cmd>. Thus it is like \sbox {cmd} {text}, except that any white space before or after the contents <text> is ignored.

This is very useful as it enables both the \verb command and the verbatim environment to be used within <text>.

It also makes it possible to define, for example, a framed box' environment. This is done by first using this environment to save some text in a bin <cmd> and then calling \fbox{\usebox{<cmd>}}.

The following example defines an environment, called fmpage, that is a framed version of minipage.

\newsavebox{\fmbox}
\newenvironment{fmpage}[1]
{\begin{lrbox}{\fmbox}

} {

\end{lrbox}\fbox{\usebox{\fmbox}}}

## Measuring things

The first of these next commands was in LaTeX 2.09. The two new commands are the obvious analogues.

\settowidth {length-cmd} {lr text}
\settoheight {length-cmd} {lr text}
\settodepth {length-cmd} {lr text}

## Controlling page breaks

Sometimes it is necessary, for a final version of a document, to help' LaTeX break the pages in the best way. LaTeX 2.09 had a variety of commands for this situation: \clearpage, \pagebreak etc. LaTeX2e provides, in addition, commands which can produce longer pages as well as shorter ones.

\enlargethispage {size}
\enlargethispage* {size}

These commands increase the height of a page (from its normal value of \textheight) by the specified amount <size>, a rigid length. This change affects only the current page.

This can be used, for example, to allow an extra line to be fitted onto the page or, with a negative length, to produce a page shorter than normal.

The star form also shrinks any vertical white space on the page as much as possible, so as to fit the maximum amount of text on the page.

## Floats

There is a new command, \suppressfloats, and a new float specifier'. These will enable people to gain better control of LaTeX's float placement algorithm.

\suppressfloats [placement]

This command stops any further floating environments from being placed on the current page. With an optional argument, which should be either t or b (not both), this restriction applies only to putting further floats at the top or at the bottom. Any floats which would normally be placed on this page are placed on the next page instead.

The extra float location specifier: !

This can be used, along with at least one of h, t, b and p, in the location optional argument of a float.

If a ! is present then, just for this particular float, whenever it is processed by the float mechanism the following are ignored:

• all restrictions on the number of floats which can appear;
• all explicit restrictions on the amount of space on a text page which may be occupied by floats or must be occupied by text.
• The mechanism will, however, still attempt to ensure that pages are not overfull and that floats of the same type are printed in the correct order.

Note that its presence has no effect on the production of float pages.

A ! specifier overrides the effect of any \suppressfloats command for this particular float.

## Font changing: text

LaTeX2e has a very different font selection scheme to LaTeX 2.09. In this section, we give a brief description of the new commands. A more detailed description with examples is given in The LaTeX Companion, and the interface for class- and package-writers is described in LaTeX2e Font Selection.

\rmfamily
\sffamily
\ttfamily
\mdseries
\bfseries
\upshape
\itshape
\slshape
\scshape

These are font commands whose use is the same as the commands \rm, \bf, etc. The difference is that each command changes just one attribute of the font (the attribute changed is part of the name). One result of this is that, for example, \bfseries\itshape produces both a change of series and a change of shape, to give a bold italic font.

\textrm{text}
\textsf{text}
\texttt{text}
\textmd{text}
\textbf{text}
\textup{text}
\textit{text}
\textsl{text}
\textsc{text}
\emph{text}

These are one-argument commands; they take as an argument the text which is to be typeset in the particular font. They also automatically insert italic corrections where appropriate; if you do not like the result, you can add an italic correction with \/ or remove it with \nocorr. The \nocorr should always be the first or last thing within the {text} argument.

## Font changing: math

Most of the fonts used within math mode do not need to be explicitly invoked; but to use letters from a range of fonts, the following class of commands is provided.

\mathrm {letters}
\mathnormal {letters}
\mathcal {letters}
\mathbf {letters}
\mathsf {letters}
\mathtt {letters}
\mathit {letters}

These are also one-argument commands which take as an argument the letters which are to be typeset in the particular font. The argument is processed in math mode so spaces within it will be ignored. Only letters, digits and accents have their font changed, for example $\mathbf{\tilde A \times 1}$ produces $\mathbf{\tilde A \times 1}$.

## Ensuring math mode

\ensuremath {math commands}

In LaTeX 2.09, if you wanted a command to work both in math mode and in text mode, the suggested method was to define something like:

\newcommand{\Gp}{\mbox{$G_p$}}

Unfortunately, the \mbox stops \Gp changing size correctly in (for instance) subscripts or a fraction.

In LaTeX2e you can define it thus:

\newcommand{\Gp}{\ensuremath{G_p}}

Now \Gp will work correctly in all contexts.

This is because the \ensuremath does nothing, producing simply G_p, when \Gp is used within math mode; but it ensures that math mode is entered (and exited) as required when \Gp is used in text mode.

## Logos

LaTeX
LaTeX2e

LaTeX (producing LaTeX') is still the main' logo command, but if you need to refer to the new features, you can write LaTeX2e (producing LaTeX2e').

## Picture commands

\qbezier[<N>](<AX>,<AY>)(<BX>,<BY>)(<CX>,<CY>)
\bezier{<N>}(<AX>,<AY>)(<BX>,<BY>)(<CX>,<CY>)

The \qbezier command can be used in picture mode to draw a quadratic bezier curve from position (<AX>,<AY>) to (<CX>,<CY>) with control point (<BX>,<BY>). The optional argument <N> gives the number of points on the curve.

For example, the diagram:

\begin{picture}(50,50) \thicklines \qbezier(0,0)(0,50)(50,50) \qbezier[20](0,0)(50,0)(50,50) \thinlines \put(0,0){\line(1,1){50}} \end{picture}

is drawn with:

\begin{picture}(50,50)
\thicklines
\qbezier(0,0)(0,50)(50,50)
\qbezier[20](0,0)(50,0)(50,50)
\thinlines
\put(0,0){\line(1,1){50}}
\end{picture}

The \bezier command is the same, except that the argument <N> is not optional. It is provided for compatibility with the LaTeX 2.09 bezier document style option.

## Old commands

\samepage

The \samepage command still exists but is no longer being maintained. This is because it only ever worked erratically; it does not guarantee that there will be no page-breaks within its scope; and it can cause footnotes and marginals to be wrongly placed.

We recommend using \enlargethispage in conjunction with page-break commands such as \newpage and \pagebreak to help control page breaks.

SLiTeX

Since SLiTeX no longer exists, the logo is no longer defined in the LaTeX kernel. A suitable replacement is \textsc{SliTeX}. The SLiTeX logo is defined in LaTeX 2.09 compatibility mode.

\mho \Join \Box \Diamond \leadsto
\sqsubset \sqsupset \lhd \unlhd \rhd \unrhd

These symbols are contained in the LaTeX symbol font, which was automatically loaded by LaTeX 2.09. However, TeX has room for only sixteen math font families; thus many users discovered that they ran out. Because of this, LaTeX does not load the LaTeX symbol font unless you use the latexsym package.

These symbols are also made available, using different fonts, by the amsfonts package, which also defines a large number of other symbols. It is supplied by the American Mathematical Society.

The latexsym package is loaded automatically in LaTeX 2.09 compatibility mode.

# LaTeX 2.09 documents

LaTeX2e can process (almost) any LaTeX 2.09 document, by entering LaTeX 2.09 compatibility mode. Nothing has changed, you run LaTeX in the same way you always did, and you will get much the same results.

The reason for the almost' is that some LaTeX 2.09 packages made use of low-level unsupported features of LaTeX. If you discover such a package, you should find out if it has been updated to work with LaTeX2e. Most packages will still work with LaTeX2e---the easiest way to find out whether a package still works is to try it!

LaTeX 2.09 compatibility mode is a comprehensive emulation of LaTeX 2.09, but at the cost of time. Documents can run up to 50% slower in compatibility mode than they did under LaTeX 2.09. In addition, many of the new features of LaTeX2e are not available in LaTeX 2.09 compatibility mode.

## Font selection problems

When using compatibility mode, it is possible that you will find problems with font-changing commands in some old documents. These problems are of two types:

• producing error messages;
• not producing the font changes you expected.
• In case of error messages it is possible that the document (or an old style file used therein) contains references to old internal commands which are no longer defined, see Section here for more information if this is the case.

LaTeX 2.09 allowed sites to customize their LaTeX installation, which resulted in documents producing different results on different LaTeX installations. LaTeX2e no longer allows so much customization but, for compatibility with old documents, the local configuration file latex209.cfg is loaded every time LaTeX2e enters LaTeX 2.09 compatibility mode.

For example, if your site was customized to use the New Font Selection Scheme (NFSS) with the oldlfont option, then you can make LaTeX2e emulate this by creating a latex209.cfg file containing the command \ds@oldlfont. Similarly, to emulate NFSS with the newlfont option, you can create a latex209.cfg file containing \ds@newlfont.

## Native mode

To run an old documents faster, and use the new features of LaTeX2e, you should try using LaTeX2e native mode. This is done by replacing the command:

\documentstyle[<options>,<packages>]{class}
with:
\documentclass[options]{class}
\usepackage{latexsym,<packages>}
However, some documents which can be processed in LaTeX 2.09 compatibility mode may not work in native mode. Some LaTeX 2.09 packages will only work with LaTeX2e in 2.09 compatibility mode. Some documents will cause errors because of LaTeX2e's improved error detection abilities.

But most LaTeX 2.09 documents can be processed by LaTeX2e's native mode with the above change. Again, the easiest way to find out whether your documents can be processed in native mode is to try it!

# Problems

This section describes some of the things which may go wrong when using LaTeX2e, and what you can do about it.

## New error messages

LaTeX2e has a number of new error messages. Please also note that many error messages now produce further helpful information if you press h in response to the error prompt.

Option clash for package <package>.

The named package has been loaded twice with different options. If you enter h you will be told what the options were, for example, if your document contained:

\usepackage[foo]{fred}
\usepackage[baz]{fred}

then you will get the error message:

Option clash for package fred.

and typing h at the ? prompt will give you:

The package fred has already been loaded with options:
[foo]
There has now been an attempt to load it with options:
[baz]
\usepackage[foo,baz]{fred}
to your document may fix this.
Try typing <<return>> to proceed.

The cure is, as suggested, to load the package with both sets of options. Note that since LaTeX packages can call other packages, it is possible to get a package option clash without explicitly requesting the same package twice.

Command <command> not provided in base NFSS.

The <command> is not provided by default in LaTeX2e. This error is generated by using one of the commands:

\mho \Join \Box \Diamond \leadsto
\sqsubset \sqsupset \lhd \unlhd \rhd \unrhd

which are now part of the latexsym package. The cure is to add:

\usepackage{latexsym}

in the preamble of your document.

LaTeX2e command <command> in LaTeX 2.09 document.

The <command> is a LaTeX2e command but this is a LaTeX 2.09 document. The cure is to replace the command by a LaTeX 2.09 command, or to run document in native mode, as described in Section here.

NFSS release 1 command \newmathalphabet found.

The command \newmathalphabet was used by the New Font Selection Scheme Release 1 but it has now been replaced by \DeclareMathAlphabet, the use of which is described in LaTeX2e Font Selection.

The best cure is to update the package which contained the \newmathalphabet command. Find out if there is a new release of the package, or (if you wrote the package yourself) consult LaTeX2e Font Selection for the new syntax of font commands.

If there is no updated version of the package then you can cure this error by using the newlfont or oldlfont package, which tells LaTeX which version of \newmathalphabet should be emulated.

You should use oldlfont if the document selects math fonts with syntax such as this:

{\cal A}, etc.
Use newlfont if the document's syntax is like this:
\cal{A}, etc.

Text for command ended by end of line.

The \verb command has been begun but not ended on that line. This usually means that you have forgotten to put in the end-character of the \verb command.

Illegal use of command.

The \verb command has been used inside the argument of another command. This has never been allowed in LaTeX---often producing incorrect output without any warning---and so LaTeX2e produces an error message.

## Old internal commands

A number of LaTeX 2.09 internal commands have been removed, since their functionality is now provided in a different way. See LaTeX2e for Class and Package Writers for more details of the new, supported interface for class and package writers.

\tenrm \elvrm \twlrm \dots
\tenbf \elvbf \twlbf \dots
\tensf \elvsf \twlsf \dots
$\vdots$

These commands provided access to the seventy fonts pre-loaded by LaTeX 2.09. In contrast, LaTeX2e normally pre-loads at most fourteen fonts, which saves a lot of font memory; but a consequence is that any LaTeX file which used the above commands to directly access fonts will no longer work.

Their use will usually produce an error message such as:

! Undefined control sequence.
l.5 \tenrm

The cure for this is to update the document to use the new font-changing commands provided by LaTeX2e; these are described in LaTeX2e Font Selection.

If this is not possible then, as a last resort, you can use the rawfonts package, which loads the seventy LaTeX 2.09 fonts and provides direct access to them using the old commands. This takes both time and memory. If you do not wish to load all seventy fonts, you can select some of them by using the only option to rawfonts. For example, to load only tenrm and tenbf you write:

\usepackage[only,tenrm,tenbf]{rawfonts}

The rawfonts package is distributed with the LaTeX tools software, see Section here.

## Old files

One of the more common mistakes in running LaTeX is to read in old versions of packages instead of the new versions. If you get an incomprehensible error message from a standard package, make sure you are loading the most recent version of the package. You can find out which version of the package has been loaded by looking in the log file for a line like:

Package fred' 1994/06/01 v0.01 Fred's package.

You can use the <release-date> options to \documentclass and \usepackage to make sure that you are getting a suitably recent copy of the document class or package. This is useful when sending a document to another site, which may have out-of-date software.

## Where to go for more help

If you can't find the answer for your problem here, try looking in LaTeX: A Document Preparation system or The LaTeX Companion. If you have a problem with installing LaTeX, look in the installation guide files which come with the distribution.

If this doesn't help, contact your local LaTeX guru or local LaTeX mailing list.

If you think you've discovered a bug then please report it! First, you should find out if the problem is with a third-party package or class. If the problem is caused by a package or class other than those listed in Section here then please report the problem to the author of the package or class, not to the LaTeX3 project team.

If the bug really is with core LaTeX then you should create a short, self-contained document which exhibits the problem. You should run a recent (less than a year old) version of LaTeX on the file and then run LaTeX on latexbug.tex. This will create an error report which you should send, together with the sample document and log file, to the LaTeX bugs address which can be found in the file latexbug.tex or bugs.txt.

# Enjoy!

We certainly hope you will enjoy using the new standard LaTeX but, if this is not possible, we hope that you will enjoy success and fulfilment as a result of the documents which it will help you to create.

If you find that the contribution of LaTeX to your life is such that you would like to support the work of the project team, then please read Section here and discover practical ways to do this.

# References

• . Michel Goossens, Frank Mittelbach and Alexander Samarin.
The LaTeX Companion.


text:
# Variance of Chi Square Distribution as the Sum of Unit Normal Random Variables

Okay, so I am interested if there is a way to derive the variance for a Chi-Square distribution using the property that it is the sum of independent unit normal distributions squared.

For example, if $X$ is a Chi-Square random variable with $n$ degrees of freedom, it has the distribution: $\displaystyle\operatorname{}\left(\sum_{i=1}^n Z_i^2\right)$ where $Z$ is Normal$(0,1)$

I know that $Var(X)= E(X^{2}) -[E(X)]^{2}$

To start finding $E(X)$ I begin with the fact that each $Z$ has $E(Z)=0$ and $Var(Z)=1$. This implies that $E(Z^2)=1$ since $Var(Z)=E(Z^{2})-[E(Z)]^2$

Since $X = \displaystyle\operatorname{}\left(\sum_{i=1}^n Z_i^2\right)$ then $E(X)=\displaystyle\operatorname{}\left(\sum_{i=1}^n 1\right)=n$

I am lost on what the next step would be. I have this start, but don't where to go next. Any thoughts on how to find $E(X^{2})$?

\begin{align} X^2 =& (\sum_{i=1}^nZ_i^2)^2\\ =& \sum_{i=1}^nZ_i^4+\sum_{i \neq j}^nZ_i^2Z_j^2 \end{align}

I can't see any way this sum not getting nasty. Note: I can solve this using integration of the PDF for the Chi-Square distribution, but I was wondering if there is any way to do it using the property that Chi-Squared is sum of Squared Normal.

• Use the fact that the variance of a sum of independent random variables is the sum of the variances, then compute the variance of $Z^2$. – carmichael561 Jul 20 '17 at 1:45
• You're almost there, just take expectations from there. $Z_i$ and $Z_j$ are independent for $i \neq j$ so the second sum isn't too bad. Do you know what $E[Z^4]$ is? – Marcus M Jul 20 '17 at 1:46
• No I wasn't sure how to proceed with $E[Z^{4}]$ @carmicheal561 I am aware of the sum of the variances, but how does this help me with taking the variance of $Z^{2}$ because this would be the product of a random variable? – user345 Jul 20 '17 at 1:53
• The variance of $Z^2$ is just $E[Z^4]-E[Z^2]^2$, which should be fairly straightforward to compute. – carmichael561 Jul 20 '17 at 1:58
• @ap2010, In my opinion, the best way to find the moments of $Z$ are to find the moment generating function and taking derivatives. You can find the moment generating function of a standard normal just by straight up integrating. It's a good exercise. You'll end up getting that the m.g.f. is $e^{t^2 / 2}$. Taking the fourth derivative and setting $t = 0$ gives $E[Z^4] = 3$. – Marcus M Jul 20 '17 at 2:00

Since $X_1^2,\ldots,X_n^2$ are independent, you have $$\operatorname{var}(X_1^2+\cdots+X_n^2) = n \operatorname{var}(X_1^2).$$ And this is $\operatorname{var}(X_1^2) = \operatorname{E}((X_1^2)^2) - (\operatorname{E}(X_1^2))^2$. If you know that $\operatorname{E}(X_1^2) =1,$ then it remains only to find $\operatorname{E}(X_1^4).$ \begin{align} \operatorname{E}(X_1^4) & = \int_{-\infty}^\infty x^4 \varphi(x)\, dx \text{ where $\varphi$ is the standard normal density} \\[10pt] & = \frac 1 {\sqrt{2\pi}} \int_{-\infty}^\infty x^4 e^{-x^2/2} \,dx \\[10pt] & = \sqrt{\frac 2 \pi} \int_0^\infty x^4 e^{-x^2/2} \,dx\text{ by symmetry} \\[10pt] & = \sqrt{\frac 2 \pi} \int_0^\infty x^3 e^{-x^2/2} \Big( x\,dx\Big) \\[10pt] & = \sqrt{\frac 2 \pi} \int_0^\infty (2u)^{3/2} e^{-u} \, du = \frac 4 {\sqrt{\pi}} \int_0^\infty u^{3/2} e^{-u} \,du \\[10pt] & = \frac 4 {\sqrt\pi} \Gamma\left(\frac 5 2 \right) = \frac 4 {\sqrt\pi} \cdot \frac 3 2 \Gamma\left( \frac 3 2 \right) = \frac 4 {\sqrt\pi} \cdot \frac 3 2 \cdot \frac 1 2 \Gamma\left( \frac 1 2 \right) \\[10pt] & = \frac 4 {\sqrt\pi} \cdot \frac 3 2 \cdot \frac 1 2 \sqrt \pi \\[10pt] & = 3. \end{align}


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# Detecting Ellipses in Images

Brett's Pick this week is Ellipse Detection Using 1D Hough Transform, by Martin Simonovsky.

### Contents

#### Introduction

I've written several times in the past about detecting circles in images (here, here, and here). I've written, too, on drawing ellipses. Today, I want to write about detecting ellipses.

Since ellipses are described by more parameters than are lines or circles, detecting ellipses is more challenging than is detecting lines or circles. And while we have nice functionality for detecting the simpler shapes (houghlines, imfindcircles), we do not (currently) have a function to detect ellipses. Enter Martin's ellipse-detection function.

#### The Source

Fortunately, and appropriately, Martin cited the source for his algorithm; he used a paper (Y. Xie, Q. Ji. "A New Efficient Ellipse Detection Method." 1051-4651/02 IEEE, 2002) that I have had on my desk for quite some time, thinking that I would one day sit down to implement it in MATLAB code. I love that Martin did that work for me--it makes me appreciate the File Exchange all the more.

#### The Implementation

After playing around with Martin's ellipseDetection() for most of an afternoon, I have some thoughts to share. First, recognize that ellipse detection is an expensive memory hog; the computation scales with the square of the number of nonzero pixels, N, in your search image. You'll almost certainly want to operate on an "edge image" rather than a simple binary mask of your regions of interest, and to play with the function's nine input parameters to limit the search. Among these parameters, you can specify the range of major axes lengths to consider, and the minimum aspect ratio. You may also specify some parameters for limiting the angles of the ellipses you seek; this could be very useful if you know the ellipse orientations a priori. Even after such limits are used, an exhaustive search examines N x N candidates for major axes. Martin's function provides a parameter that allows you to trade off between speed and accuracy. (The "randomize" parameter is not a Boolean variable, as the name suggests; rather it reduces the search space from N x N to N x randomize. If randomize is 0, then the search is exhaustive--increased accuracy at a computational cost.)

#### A Test Drive

So let's try it out on a sample image of our creation. We can start with an image containing circles, and warp it to generate ellipses:

inputImage = imread('coloredChips.png');
tform = affine2d([1 0 0; 0.75 1 0; 0 0 1]);
inputImage = imwarp(inputImage, tform);
imshow(inputImage)
title("Test Image")

#### Segmentation

I segmented the image by:

1. Using the colorThresholder to create a mask of the background
2. Splitting the color image into R, G, and B components (imsplit)
3. Masking the R, G, and B planes individually (R(backgroundMask) = 0, ...)
4. Re-compositing the masked planes into an RGB image (cat)
5. Converting the masked RGB image to grayscale (im2gray)
6. Calculating (with carefully selected input parameters) the edges of the grayscale image (edge)
7. Filtering the results to select the desired Major Axes Lengths and Areas (imageRegionAnalyzer, bwpropfilt)

In just a few minutes, I had an edge image in which to detect those ellipses:

#### Detecting the Ellipses

With this binary edge mask in hand, I was ready to search for ellipses--first, naively:

bestFits = ellipseDetection(edgeMask);

The operation completed in (just) less than a second, but the results were underwhelming:

(By the way, calling ellipseDetection() on the binary mask of the warped chips without first calculating the edges took upwards of 20 minutes, and the results were even worse!)

#### Improving the Results

To improve the performance, I judiciously selected input parameters to reduce the computational cost. First, I used imdistline to measure the major and minor axes lengths:

Then I used protractor to get a sense of the orientations of the ellipses:

After a few minutes, I found my way to:

params.minMajorAxis = 55;
params.maxMajorAxis = 75;
params.minAspectRatio = 0.4; %1 = circle; 0 = line;
params.rotation = 35;
params.rotationSpan = 10;
params.randomize = 0; %Exhaustive search
params.numBest = 30; %(Number of chips = 26)

bestFits = ellipseDetection(edgeMask, params);

#### Paring Results

Wait...what? bestFits contains paramaters for 30 detections (a manual count informs me that there are 26 ellipses wholly contained in the image), but when visualizing the results, it appears that there are far fewer. What's going on?

It turns out that the algorithm is susceptible to reporting the same ellipse multiple times. (If I drag those cyan ellipses, there are other coincident ellipses underneath!)

Experimenting, I found it quite useful to dramatically overspecify the number of ellipses I wanted to detect, then to pare the results in post-processing. Consider:

params.smoothStddev = 0.5;
params.numBest = 1000; %(Number of chips = 26)
minCenterDistance = 10;
minScore = 30;
maxAspectRatio = 0.6;
bestFits = pareEllipseFits(bestFits, ...
minCenterDistance, minScore, maxAspectRatio);

I wrote pareResults to allow filtering by minimum center distance (to disallow overlapping detections), minimum score, and maximum aspect ratio. Using that paring approach, I can request far more detections (params.numBest = 1000) than I really want, then discard "bad" results. I'm pretty pleased with the way it's working!

#### A Note on Visualizing the Results

Finally, I also wrote visualizeEllipses to call drawellipse directly on the output of ellipseDetection.

#### Notes

In the interest of making this post a bit shorter (I know...too late!), I didn't post all of the code. If anyone would like to see it, I'm happy to share. Just drop me an email at:

char(cumsum([98 17 -11 7 -10 7 7 -4 -47 45 -12 19 -12 15 -8 3 -7 8 -69 53 12 -2]))

Also, there are a few other files on the Exchange that purport to facilitate ellipse detection. Please leave me a comment if you'd like to see those considered those in a subsequent post!

#### A Couple of Suggestions

Martin's implementation uses Gaussian filtering via fspecial. That syntax is no longer recommended; the newer imgaussfilt is more efficient. Also, the call to fspecial requires an integer argument for the second ('hsize') parameter. (I added a call to round() in my version.) Finally, since getting good results requires tuning a number of parameters, this function is just begging for a code-generating app to front it; that's perhaps a project for another day!

Thank you, Martin...finding this function, and figuring out how to use it, gives me a valuable tool in my image processing arsenal.