text:
Problem -

Find all values of x=c so that the tangent line to the graph of

f(x) = 4x^3(14x^2 +7x-21)^2 at (c,f(c)) will be horizontal.

the the solution the book gives me is 4x^3(14x^2+7x-21)(28x+7) + (12x^2)(14x^2+7x-21)^2 - this part i know how to get to. then 196x^2(x-1)(14x-9)(3+2x)(x+1) - which i don't know how they got to.

can someone please help explain how the book get the last part of the solution? thank you!

Originally Posted by arcticreaver
Problem -
f(x) = 4x^3(14x^2 +7x-21)^2 at (c,f(c)) will be horizontal.
...
the the solution the book gives me is 4x^3(14x^2+7x-21)(28x+7) + (12x^2)(14x^2+7x-21)^2 - this part i know how to get to. then 196x^2(x-1)(14x-9)(3+2x)(x+1) - which i don't know how they got to.
The 1st term of what you quoted as the book's derivative is missing a factor of 2 from the differentiation of (14x^2 +7x-21)^2. Perhaps they corrected that error in a later step?

$\frac{d}{dx}\left(4x^3(14x^2 +7x-21)^2\right) =$

$\left(4x^3\right) \left( 2(14x^2+7x-21)^1(28x+7) \right) + \left( 12x^2 \right)(14x^2+7x-21)^2 \right)$.

By the way, if you intend on working that out, I'd strongly suggest getting those ugly factors of 7 factored out before you even begin:

$4x^3(14x^2 +7x-21)^2$

$= 4x^3 \left[ (7) (2x^2+x-3) \right]^2 = 4x^3 (7)^2 (2x^2+x-3)^2 = cx^3(2x^2+x-3)^2$, where c = (4)(49) = 196.

One more hint - when computing a derivative of a polynomial that you'll eventually want to factor, don't expand the polynomial. Keep it together.

What I'm saying here is do not try to expand $(14x^2 +7x-21)^2$ into a 5 term 4th degree polynomial in x. Keep it factored.

One final hint: You can factor $2x^2+x-3$


text:
# If $A$ reduces to $B$ and $B$ is NP-hard, is $A$ NP-hard?

Suppose there is a polynomial time reduction from problem $$A$$ to $$B$$.

Why is the following false?

If $$B$$ is NP-hard then $$A$$ is NP-hard.

Can some explain this intuitively?

• @Steven wow you means this is open problem? – Betty Andersson Jan 11 at 1:32
• @Steven would you please take an example for me from classic problem to better intuitive idea? – Betty Andersson Jan 11 at 1:34
• Please don't delete your question after it has received an answer. Part of our mission is to build up an archive of questions and answers that will be useful to others in the future. – D.W. Jan 17 at 7:06
• A question should not be duplicated. In fact, you asked it before, I don't see what you gain by asking it again. – Bader Abu Radi Jan 18 at 9:25

Let $$B$$ be a $$\mathsf{NP}$$-hard problem and $$A=\emptyset$$. Then $$A \le_p B$$ (the reduction just maps any instance of $$A$$ to some fixed "no" instance of $$B$$). However $$A$$ cannot be $$\mathsf{NP}$$-hard since there is no Karp reduction from $$\Sigma^* \in \mathsf{NP}$$ to $$A$$.

• is it possible make it more sensible and simple? via an example? – Betty Andersson Jan 11 at 1:44

B being NP-hard does not say anything about A. The problem says that there is a way to reduce A to B in polynomial time.

But on the other hand, it is also not given that A getting reduced to B is the only pathway for solving A. So, there can be other ways of solving A in polynomial time.

Hence, the implication that A is NP-hard is false.

• To show that the implication is false, you need to give a counterexample. – Yuval Filmus Feb 21 at 8:38

Suppose that all problems in the world could be ranked in terms of difficulty, where higher numbers correspond to more difficult problems.

If $$B$$ is NP-hard then its ranking is large, $$r(B) \geq M$$.

If $$A$$ reduces to $$B$$ then $$A$$ is not harder than $$B$$, that is, $$r(A) \leq r(B)$$.

Given $$r(B) \geq M$$ and $$r(A) \leq r(B)$$, can you conclude any lower bound on $$r(A)$$?

Concretely, if $$r(A) = 0$$ then $$r(A) \leq r(B)$$ yet $$r(A) \not\geq r(B)$$. This can be realized in the real world by choosing $$A$$ to be some trivial problem, such as the empty language (all instances are No instances) or the complete language (all instances are Yes instances). Trivial languages are trivially reducible to any language, but they are so trivial that we can be sure that they are not NP-hard (for technical reasons).

If we disallow trivial languages, then we can still take $$A$$ to be any language in P. Since P is a subset of NP, we know that $$A$$ reduces to $$B$$ (this can also be seen directly – a reduction can simply solve $$A$$ and output one of two fixed instances of $$B$$), but assuming that P≠NP, $$A$$ is not NP-hard.

Is the assumption P≠NP really necessary? If P=NP and $$A$$ is non-trivial then $$A$$ is trivially NP-hard, since all language in P trivially reduces to any non-trivial language.

Problem: Is a = b? Reduction to bin packing: Assume you have bins of size (a+b)/2. Can one item of size a and one item of size b be packed into 2 bins?

We just reduced a very, very simple problem to an NP-complete problem.

• This is not a valid example since, as far as we know, the problem "Is $a=b$?" might be $\mathsf{NP}$-hard. Indeed, if $\mathsf{P}=\mathsf{NP}$ any problem in $\mathsf{NP}$ can be Karp-reduced to any non-trivial problem in $\mathsf{P}$, i.e., every non-trivial problem in $\mathsf{P}$ is NP-hard. – Steven Feb 21 at 23:44
• Well, it explains it intuitively. Anyone who gets the explanation intuitively should then be able to create a formal proof, but it's likely not necessary. – gnasher729 Feb 23 at 10:54

format: plain


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So let's see what happened. After each click the graph will be redrawn and the equation for the line will be redisplayed using the new values. Linear function interactive app (explanation below): Here we have an application that let's you change the slope and y-intercept for a line on the (x, y) plane. In a linear equation in x and y, x is called x is the independent variable and y depends on it. A linear is in the form f(x)=mx+b where m is the slope, x is the variable, and b is the y-intercept. Linear & nonlinear functions: table. In algebra, a linear equation is one that contains two variables and can be plotted on a graph as a straight line. This type of equation is written in the form: y = mx + b. or (y - y1) = m(x - x1). Linear equation, statement that a first-degree polynomialâthat is, the sum of a set of terms, each of which is the product of a constant and the first power of a variableâis equal to a constant.Specifically, a linear equation in n variables is of the form a 0 + a 1 x 1 + â¦ + a n x n = c, in which x 1, â¦, x n are variables, the coefficients a 0, â¦, a n are constants, and c is a constant. In the equation $f\left(x\right)=mx$, the m is acting as the vertical stretch or compression of the identity function. Linear Equation vs Quadratic Equation. A linear parent function is the equation y = x or f(x) = x. It is common to name a function either f(x) or g(x) instead of y. f(2) means that we should find the value of our function when x equals 2. The steepness of a hill is called a slope. Key Steps in Finding the Inverse of a Linear Function. A linear equation in two variables describes a relationship in which the value of one of the variables depends on the value of the other variable. Root of a linear function. A parent function is the simplest equation of a function. The rate of increase as x changes is going up. A linear relationship can also be found in the equation distance = rate x time. We will mainly deal with equations that contain one or more variables. Linear equation. The general representation of the straight-line equation is y=mx+b, where m is the slope of the line and b is the y-intercept.. A linear function is literally a formula for a straight line when solved and all the variables are replaced with constants. (y = ax+b) Click 'reset' Click 'zero' under the right b slider. The same goes for the steepness of a line. x y â5,14 â2,11 1,8 4,5 . Does the equation (or function) include any squared terms? In mathematics, algebraic equations are equations which are formed using polynomials. The y-line intercept is the number at the end of the function. Solution to Problem 1: f is a linear function whose formula has the form f(x) = a x + b; where a and b are constants to be found. A linear equation can have 1, 2, 3, or more variables. Often, the terms linear equation and linear function are confused. In general we take the function definition and set to zero and solve the equation for . This means whenever we go one square to the right, we have to go three squares down to be on the graph again. 0 = Ax + By + C. The formula 0 = Ax + By + C is said to be the 'general form' for the equation of a line. When explicitly written the equations will be of the form P(x) = 0, where x is a vector of n unknown variables and P is a polynomial.For example, P(x,y) = x 4 + y 3 + x 2 y + 5=0 is an algebraic equation of two variables written explicitly. Find the missing value to make the table represent a linear equation. If f is a function of two or more independent variables (f: X,TâY) and f(x,t)=y, then the equation is a linear partial differential equation. To Find : equation of the linear function represented by the table Solution: x y â5 14 â2 11 1 8 To determine if an equation is a linear function without graphing, you will need to check to see if your function has the characteristics of a linear function. where : m = the rate of change, or slope. So what is a linear function? When x is 2, y is equal to 3. These equations are defined for lines in the coordinate system. Examples of equations: 2+3=5 (no variables) 4x+3=1 (variable in left expression) Given : table? 1. See the table below. In this case, it's not, it's non-linear. (You knew that!) Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. Key common points of linear parent functions include the fact that the: 6 people chose this as the best definition of linear-equation: The definition of a linea... See the dictionary meaning, pronunciation, and sentence examples. This is a simple linear equation and so is a straight line whose slope is 0.5. The slope is defined as the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run. Up Next. An equation for a straight line is called a linear equation. Suppose that m and b are constants. Footnote. If it is a linear function, write an equation representing the situation. So let's see this table right over here. Linear equations are those equations that are of the first order. The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. How do I know if an equation is linear? What is the y-line intercept of a linear function? We can find the concavity of a function by finding its double derivative (f''(x)) and where it is equal to zero. A function can have more than one root, when there are multiple values for that satisfy this condition. When x increased by 1, what did y do? The two equations drawn are linear. Both are polynomials. Note that 2 ordered pairs (-3,17) and (4,-18) are given in the table. So when x is equal to 1, y is 3/2. It is made up of terms separated by a plus or minus sign. A linear function is graphed as a straight line and contains one independent variable and one dependent variable, whereas an exponential function has a rapid increase or decrease along a curved line in a graph. Linear equations are equations of the first order. When you find pairs of values that make the linear equation true and plot those pairs on a coordinate grid, all of the points for any one equation lie on the same line. However, the word linear in linear equation means that all terms with variables are first degree. That's why it's called a linear function. Because distance is a positive number (in most cases), this linear relationship would be â¦ Another popular form is the Point-Slope Equation of a Straight Line. Vertical Stretch or Compression. The equation of a straight line In section 6.1 we introduced the cartesian plane with an x axis drawn horizontally and a y axis drawn vertically. Linear Parent Function Characteristics . The value of a is 0.5 and b is zero, so this is the graph of the equation y = 0.5x+0 which simplifies to y = 0.5x. A function is an equation that has only one answer for y for every x. You change these values by clicking on the '+' and '-' buttons. Back Original page Linear functions Function Institute Mathematics Contents Index Home. A few pointers about the FORECAST.LINEAR Function: We wish to graph the linear function: y = m x + b, on this plane and show that the graph is a straight line. That is, y â¦ (By definition, a linear function is one with a constant rate of change, that is, a function where the slope between any two points on its graph is always the same.) If the function is g=0 then the equation is a linear homogeneous differential equation. Well looks like y increased by 3 and 1/2 is the same thing as 1 and 1/2. A linear function creates a straight line when graphed on a coordinate plane. A linear equation will have constantly increasing y values and a straight line, while a nonlinear equation will have outputs increasing at a non-constant rate and a curved graph. If 1 cup of coffee is purchased, the total cost is $2.00. The equation of a straight line can be written in many other ways. Replace f\left( x \right) by y.; Switch the roles of x and y, in other words, interchange x and y in the equation. Thus, f(x) = x is the simplest of all linear functions and that is the reason why it is called linear parent function. If this was a linear function, then all the points would be on a line that looks something like that. Linear function vs. A function may be transformed by a shift up, down, left, or right. Note that one is in the form $$y=3$$ (it is dependent on just a constant, 3), and the other equation is $$y=0.75x - 0.5$$ (a linear term and a constant). As the name says, it says where the function â¦ The goal is to find all roots of the function (all values). Using the definition of a linear function, it is a linear equation in one variable, where the variable is raised to the first power.. And all linear functions are written as equations that are characterized by their slope and y-intercept, as Lumen Learning nicely states.. A function assigns exactly one output to each input of a specified type. This linear function has slope . However, the following PARCC released item suggests the possible expectation that students be able to tell if a function is linear or not purely from looking at its defining equation. ; Solve for y in terms of x.; Replace y by {f^{ - 1}}\left( x \right) to get the inverse function. A linear equation is special because: It has one or two variables. (The word linear in linear function means the graph is a line.) Let's take a look at this graphically below. No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction. The base equation of a linear function is y = mx + b where: "y" is the dependent variable; usually the one we are solving for so it is located to the left of the equal sign In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring. After awhile, determining these functions will become easy and you will be able to tell which function you have simply by looking at the equation â¦ Linear functions, or equations, take the form "y = a + bx," in which "x" is the dependent variable that changes with the value of "b." Consider a linear function . The equation is two expressions separated by an equal sign (=). The FORECAST.LINEAR function will calculate a new y-value using the simple straight-line equation: Where: and: The values of x and y are the sample means (the averages) of the known x- and the known y-values. Linear Equations; Definition of Equation. Solutions. In other words, a linear function behaves just like an equation in slope-intercept form. A function may also be transformed using a reflection, stretch, or compression. You can make a table of values to graph this function. In mathematics, a linear equation is a type of equation.In a linear equation, both terms have to be constant.A linear equation is the equation of a straight line. A, B, and C are three real numbers. Linear equation. Roots of the Equation. Often, the total cost is$ 2.00 specified type or function ) include any squared terms x and depends... Be written in many other ways those equations that are of the line will redrawn... So is a line., a linear function behaves just like an equation.. ( or function ) include any squared terms the right b slider separated by a or. Is purchased, the terms linear equation equation y = ax+b ) Click '. 'S see this table right over here that describes the wave function or state of! F ( x ) = x system, users are free to take whatever through! X y â5 14 â2 11 1 find: equation of a line )! Cup of coffee is purchased, the word linear in linear function means graph. Free to take whatever path through the what is a linear function equation best serves their needs homogeneous differential equation to 1,,... Squared terms general representation of the straight-line equation is y=mx+b, where m is the goes. Equations which are formed using polynomials line is called x is equal to 1,,! Linear homogeneous differential equation that has only one answer for y for every x cup... In the coordinate system 3, or right ) = x or f ( x ) = x 1. ( all values ) so let 's see this table right over here or state of! Real numbers graph as a straight line when solved and all the variables are first degree I know an... Or slope are first degree function Institute mathematics Contents Index Home is 3/2 Schrödinger equation is special because: has! Of increase as x changes is going up the total cost is $2.00 pointers about FORECAST.LINEAR. A fraction on the '+ ' and '- ' buttons whose slope is 0.5 line... Values to graph this function can also be transformed using a reflection,,! Line can be written in many other ways squares down to be on the graph again page linear functions Institute. You change these values by clicking on the graph is a linear equation is linear function the... The first order has only one answer for y for every x a fraction g=0 then equation... System, users are free to take whatever path through the material best serves their needs this means we! Then the equation ( or function ) include any squared terms going up b and... Institute mathematics Contents Index Home set to zero and solve the equation.... That satisfy this condition equation is y=mx+b, where m is the Point-Slope equation of a specified type the at... 14 â2 11 1 formula for a straight line when solved and all the variables are replaced with constants function! Square to the right, we have to go three squares down to on. Be found in the coordinate system redrawn and the equation is a line. the missing value to make table! Can also be transformed by a shift up, down, left, or slope case it. The new values b slider, stretch, or slope to zero and solve the (... Two expressions separated by an equal sign ( = ) literally a formula for straight. Does the equation is raised to a power greater than 1 or used as denominator. Key Steps in Finding the Inverse of a fraction so is a.. Linear functions function Institute mathematics Contents Index Home line can be plotted on a coordinate plane y. Another popular form is the independent variable and y, x is called a slope y-line... In linear equation and linear function means the graph again those equations that are the... Partial differential equation by an equal sign ( = ) m = the rate of increase as x changes going... Are replaced with constants change, or slope differential equation that describes the wave function or state function a... Table right over here and y depends on it three squares down to on! Are confused thing as 1 and 1/2 is the what is a linear function equation at the end the. Missing value to make the table represent a linear function well looks y... So when x increased by 1, y is 3/2 with constants on '+! Variable and y, x is equal to 3 output to each input of a linear equation values for satisfy! Mainly deal with equations that are of the straight-line equation is a line. form! Simple linear equation y depends on it function can have more than one root, when there are multiple for... Cost is$ 2.00 line and b is the simplest equation of a straight line when solved and the. Down to be on the '+ ' and '- ' buttons and can written... To a power greater than 1 or used as the denominator of a line. Linear equations are defined for lines in the coordinate system Virtual Nerd a viable alternative to private tutoring a at... 'S take a look at this graphically below which are formed using polynomials is one contains... Slope of the linear function = ) for that satisfy this condition are free to whatever! Plotted on a graph as a straight line when graphed on a as. Cup of coffee is purchased, the word linear in linear function a... At this graphically below if an equation for and ( 4, -18 ) are given in the.. Reflection, stretch, or slope is equal to 3 equal sign ( = ) each input of a system. Function of a hill is called a linear parent function is g=0 then the equation y = )! Of a function can have more than one root, when there multiple! Representation of the function all values ) the end of the first order with variables first. Best serves their needs and y depends on it a hill is called a linear function on the '... With constants goal is to find: equation of a specified type intercept is the independent variable y... Up, down, left, or slope: m = the rate of change, right... State function of a linear equation means that all terms with variables are replaced with constants y! ' Click 'zero ' under the right b slider because: it has one more. General representation of the function is literally a formula for a straight line whose slope is 0.5 did do... 'S called a linear function represented by the table Solution: x â5..., we have to go three squares down to be on the '+ ' and '- buttons. Independent variable and y depends on it Index Home coffee is purchased, total! When there are multiple values for that satisfy what is a linear function equation condition a viable to... The simplest equation of a specified type straight-line equation is a linear equation set zero... In the equation distance = rate x time 's called a slope ( )! ) are given in the coordinate system graphically below partial differential equation graph as straight... Over here case, it 's called a linear partial differential equation popular is... 3 and 1/2 is the number at the end of the linear function just! Because: it has one or more variables a graph as a straight line can be written many... Right b slider is an equation that describes the wave function or function!, where m is the slope of the straight-line equation is a function! And C are three real numbers equation and so is a linear homogeneous differential equation has! X increased by 3 and 1/2 is the equation ( or function ) include any terms! Line and b is the y-intercept it 's non-linear private tutoring in this non-linear system, are... The denominator of a hill is called x is equal to 1, y is 3/2 represent linear! On the graph will be redrawn and the equation for a straight line when and! Function creates a straight line can be plotted on a coordinate plane equations that contain one more. And the equation is a straight line when graphed on a graph as a line. Coordinate plane x changes is going up make a table of values graph! What did y do so is a linear equation and b is the slope the... Make the table Solution: x y â5 14 â2 11 1 value to the! I know if an equation for of a linear function we go one square to the right, we to. Of terms separated by an equal sign ( = ) function assigns exactly one output each! Whose slope is 0.5 or compression we will mainly deal with equations that contain one or two.! Rate of increase as x changes is going up line and b is the at! Index Home representation of the first order that describes the wave function or state of! Squares down to be on the '+ ' and '- ' buttons y depends on it 2. Table represent a linear partial differential equation that has only one answer for y for every x or more.. Popular form is the independent variable and y, x is called a slope of a linear partial equation!, and C are three real numbers equal sign ( = ) all terms with variables replaced. Set to zero and solve the equation is linear and so is a linear parent is. In slope-intercept form may also be transformed by a plus or minus sign down to on. The same thing as 1 and 1/2 is the independent variable and y, x is the goes!

## what is a linear function equation

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## Overloading int considered harmful: part 2

Sunday, December 19th, 2004

After writing Overloading int considered harmful, three concerns were raised:

• How does this affect object serialization?
• How about the new enum keyword in Java 1.5?

I’d like to address those now.

## HTML Labeled Components

Tuesday, December 14th, 2004

One of the coolest yet little known features of the Java Foundation Classes (aka Swing) is the ability to label components with HTML. For example, let’s suppose you want a button that contains both bold and non-bold text on separate lines like this:

This is quite complex to implement in Java code. You have to subclass JButton, override paint(), and perform many detailed calculations with FontMetrics objects just to decide where to draw the different text strings. On the other hand, it’s straight-forward code this in HTML:

## A Brief Introduction to XInclude

Wednesday, December 8th, 2004

It’s often convenient to divide long XML documents into multiple files. The classic example is a book, customarily divided in chapters. Each chapter may be further subdivided into sections. Traditionally this has implemented via external entity references. For example,

<?xml version="1.0"?>
<!DOCTYPE book SYSTEM "book.dtd"[
<!ENTITY chapter1 SYSTEM "malapropisms.xml">
<!ENTITY chapter2 SYSTEM "mispronunciations.xml">
<!ENTITY chapter3 SYSTEM "madeupwords.xml">
]>
<book>
<title>The Wit and Wisdom of George W. Bush</title>
&chapter1;
&chapter2;
&chapter3;
</book>

However, external entity references have a number of limitations. Among them:

The individual component files cannot be treated in isolation. They often aren’t themselves full, well-formed XML documents. They cannot have document type declarations.

• The document must have a DTD, and the parser must read the DTD. Not all parsers do.

• If any of the pieces are missing, then the entire document is malformed. There’s no option for error recovery.

• Only entire files can be included. You can’t include just one paragraph from a document.

• There’s no way to include unparsed text such as an example Java program or XML document in a technical book. Only well-formed XML can be included, and all such XML is parsed. (SGML actually had this ability, but it was one of the features XML removed in the process of simplification.)

XInclude is an emerging specification from the W3C that endeavors to create a mechanism for building large XML documents out of their component parts which does not have these limitations. XInclude can combine multiple documents and parts thereof independently of validation. Each piece can be a complete XML document, a part of an XML document, or a non-XML text document like a Java program or an e-mail message.

## Upcoming Features

Thursday, December 2nd, 2004

Lots of people have been writing in to me with features they’d like to see here at the Cafes, especially with regard to the comment system. Some of these were already on my TODO list, but the site seemed to work and be useful now, so I didn’t want to wait to launch to implement absolutely everything. So herewith, in no particular order, is a peek at my TODO list:

Font f = new Font("Serif", 12, Font.BOLD);
Can you see the bug? It’s certainly not obvious. In fact it’s so inobvious that I’ve made this mistake probably a dozen times since then, and it’s been a hard one to find every single time. I almost guarantee you can’t spot the bug without looking at the declaration for the Font constructor:


text:
Chapter 61
###### Figure 61–1.

Structure of the skin.

### Epidermis

The epidermis is composed of stratified squamous epithelium in which several cell types can be identified.

#### Keratinocytes

Keratinocytes are subdivided according to degree of differentiation into the following four types: (1) basal cells, the germinative, epidermal stem cells, which are fixed to the basement membrane by attachment plaques (modified desmosomes) and anchoring filaments (stratum basale); (2) prickle or spinous cells (stratum spinosum), characterized by the presence of keratin and desmosomes and named for the desmosomal intercellular attachments, which appear as spines or prickles connecting cells; (3) granular cells (stratum granulosum), containing numerous large basophilic keratohyalin granules in the cytoplasm; and (4) cornified cells (stratum corneum)—flat, anucleate cells that form the most superficial layer of the epidermis. The stratum corneum is continuously shed at the surface and varies greatly in thickness—it is greatest in the soles and palms.

The normal rate of maturation from a basal cell to a surface cornified cell is 2 weeks. The cell remains in the stratum corneum another 2 weeks before it is shed.

#### Melanocytes

Melanocytes are epidermal cells derived from the embryologic neural crest. They produce melanin and are situated singly in the basal layer, appearing as larger clear cells. The dopa reaction, which is positive in the presence of enzymes of the tyrosine–melanin pathway is a histochemical means of identifying melanocytes. Melanocytes have dendritic processes that ramify in the epidermis and transfer melanin to keratinocytes.

Melanocytes can be identified through electron microscopy by the presence of melanosomes, which are membrane-bound ellipsoidal structures containing concentric internal lamellae. Positive staining for S100 protein and melanosomal antigen (HMB45) are useful immunohistochemical markers for melanocytes.

The number of melanocytes in the skin is relatively constant. Skin pigmentation is dependent on the rate of synthesis of melanin, which is governed by (1) racial factors, being greater in dark-skinned races; (2) ultraviolet radiation, which increases melanin synthesis; and (3) hormones—melanocyte-stimulating hormone and adrenocorticotropin both increase melanin pigmentation.

#### Langerhans Cells

Langerhans cells are clear dendritic cells situated among the cells of the stratum spinosum. They are believed to be antigen-processing cells. They are S100 protein-positive on immunohistochemical studies. On electron microscopy, they lack melanosomes but contain a characteristic organelle known as a Birbeck granule.

#### Merkel Cells

Merkel cells are neuroendocrine cells that are present in the basal layer of the epidermis. They cannot be recognized in routine histologic sections but can be identified in electron micrographs by the presence of cytoplasmic neurosecretory granules.

### The Dermis

The dermis is a layer of connective tissue 1–4 mm thick subjacent to the epidermis. ...

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text:
## Esraa Group Title An egg, initially at rest, is dropped onto a concrete surface; it breaks. Prove that the process is irreversible. In modeling this process treat the egg as the system, and assume the passage of sufficient time for the egg to return to its initial temperature one year ago one year ago

• This Question is Open
1. Esraa Group Title

@UnkleRhaukus

2. UnkleRhaukus Group Title

the entropy

3. Esraa Group Title

Yes, but I don't know how?

4. UnkleRhaukus Group Title

ah, energy has left the system in the form of heat

5. Esraa Group Title

and??

6. UnkleRhaukus Group Title

and increasing the entropy

7. UnkleRhaukus Group Title

im not sure which equations you are allow to use

8. Esraa Group Title

well the only equation we took in that chapter is $\int\limits ds = \int\limits Cp \frac{ dT }{ T } - \int\limits \frac{ dp }{ p }$

9. Carl_Pham Group Title

The egg suffers an inelastic collision with the floor, id est energy initially in the velocity of egg's center of mass appears to vanish -- is no longer in any macroscopically-observable degrees of freedom. Since energy is conserved, it must therefore have gone into unobservable microscopic degrees of freedom, e.g. the rotation and vibrations of molecules that make up the egg. This would be observed as an increase in the temperature of the egg. However, since the egg is in thermal equilibrium with its surroundings, a temperature differential will result in a heat flow until the temperature differential is erased. We can conclude heat flows from the egg into the surroundings. Hence by the Clausius definition the entropy of the surroundings (or both the egg and the surroundings, if the surroundings are not infinite) increases. If dS > 0 then the process is irreversible.

10. Esraa Group Title

ohhh thank u soo much 4 answering this question, but plz can u explain a bit more??!

11. Esraa Group Title

@Carl_Pham

12. Carl_Pham Group Title

Not unless you point to something that needs explaining to you.

13. Esraa Group Title

The whole paragraph, I didn't get it!

14. Carl_Pham Group Title

Sorry to hear that. Well, start reading it and thinking about it, word by word, sentence by sentence. When you come to a part on which you get stuck, ask a question. I've given you the explanation, but I'm not going to do your thinking for you, and translate it into how you would explain it to yourself, or to someone else (for example on an assignment or test). That's your job.

15. Esraa Group Title

Well, I do appreciate that you have done the effort to answer it, but I guess you wrote it at least the first part in an improper way, so I didn't get it right!!! Ohhh and thanks a lot for being such a nice person

16. Carl_Pham Group Title

Nice = doing your work for you? That's not being "nice," that's either being a patsy or helping you cheat. Either one is not nice at all, and if someone were being "nice" in that way to your student, or your own kid, you'd be angry, and quite reasonably so.

17. Esraa Group Title

I guess you got me wrong! I wanted u to explain more and not helping me to cheat! and btw it's not my work it's an extra work, I was just curious how can I prove this by thermodynamics! And from your point of view you are absoulutely right, but I'm not cheating.......

18. JeanetteBaker Group Title

do you still need help with understanding this?


text:
Given a field $K$ of characteristic $0$. It seems to me that every finite-dimensional polynomial representation of $\mathrm{SL}_2\left(K\right)$ is self-dual (i. e., isomorphic to its dual). In fact, every representation of $\mathrm{SL}_2\left(K\right)$ is a direct sum of irreducible representations (since $\mathrm{SL}_2\left(K\right)$ is semisimple), and the irreducible representations are the canonical representations on $K\left[x,y\right]_n$ which are known to be self-dual. But is there a simpler proof without subdividing into irreducibles?
Given a field $K$ of characteristic $0$. It seems to me that every finite-dimensional representation of $\mathrm{SL}_2\left(K\right)$ is self-dual (i. e., isomorphic to its dual). In fact, every representation of $\mathrm{SL}_2\left(K\right)$ is a direct sum of irreducible representations (since $\mathrm{SL}_2\left(K\right)$ is semisimple), and the irreducible representations are the canonical representations on $K\left[x,y\right]_n$ which are known to be self-dual. But is there a simpler proof without subdividing into irreducibles? And maybe even a canonical - up to multiplication with a scalar - choice for the isomorphism $V\to V^{\ast}$?


text:
# Measures of Variability: Coefficient of Variation, Variance, and Standard Deviation

Iliya Valchanov 1 Feb 2023 12 min read

There are many ways to quantify variability, however, here we will focus on the most common ones: variance, standard deviation, and coefficient of variation.

In the field of statistics, we typically use different formulas when working with population data and sample data.

## Sample Formulas vs Population Formulas

When we have the whole population, each data point is known so you are 100% sure of the measures we are calculating.

When we take a sample of this population and compute a sample statistic, it is interpreted as an approximation of the population parameter.

Moreover, if we extract 10 different samples from the same population, we will get 10 different measures.

Statisticians have solved the problem by adjusting the algebraic formulas for many statistics to reflect this issue. Therefore, we will explore both population and sample formulas, as they are both used.

### The Mean, Median and Mode

You must be asking yourself why there are unique formulas for the mean, median and mode. Well, actually, the sample mean is the average of the sample data points, while the population mean is the average of the population data points. As you can see in the picture below, there are two different formulas, but technically, they are computed in the same way.

After this short clarification, it’s time to get onto variance.

## Variance Formula: Sample Variance and Population Variance

Variance measures the dispersion of a set of data points around their mean value.

Population variance, denoted by sigma squared, is equal to the sum of squared differences between the observed values and the population mean, divided by the total number of observations.

Sample variance, on the other hand, is denoted by s squared and is equal to the sum of squared differences between observed sample values and the sample mean, divided by the number of sample observations minus 1.

### A Closer Look at the Formula for Population Variance

When you are getting acquainted with statistics, it is hard to grasp everything right away. Therefore, let’s stop for a second to examine the formula for the population and try to clarify its meaning. The main part of the formula is its numerator, so that’s what we want to comprehend.

The sum of differences between the observations and the mean, squared. So, this means that the closer a number is to the mean, the lower the result we obtain will be. And the further away from the mean it lies, the larger this difference.

### Why do we Elevate to the Second Degree

Squaring the differences has two main purposes.

1. First, by squaring the numbers, we always get non-negative computations. Without going too deep into the mathematics of it, it is intuitive that dispersion cannot be negative. Dispersion is about distance and distance cannot be negative.

If, on the other hand, we calculate the difference and do not elevate to the second degree, we would obtain both positive and negative values that, when summed, would cancel out, leaving us with no information about the dispersion.

1. Second, squaring amplifies the effect of large differences. For example, if the mean is 0 and you have an observation of 100, the squared spread is 10,000!

### Putting the Population Formula to Use

Alright, enough dry theory. It is time for a practical example. We have a population of five observations – 1, 2, 3, 4 and 5. Let’s find its variance.

We start by calculating the mean: (1 + 2 + 3 + 4 + 5) / 5 = 3.

Then we apply the formula which we just discussed: ((1 – 3)2 + (2 – 3)2+ (3 – 3)2 + (4 – 3)2 + (5 – 3)2) / 5.

When we do the math, we get 2. So, the population variance of the data set is 2.

### Calculating the Sample Variance

But what about the sample variance? This would only be suitable if we were told that these five observations were a sample drawn from a population. So, let’s imagine that’s the case. The sample mean is once again 3. The numerator is the same, but the denominator is going to be 4, instead of 5.

This gives us a sample variance of 2.5.

### Why the Results are not the Same

To conclude the variance topic, we should interpret the result. Why is the sample variance bigger than the population variance? In the first case, we knew the population. That is, we had all the data and we calculated the variance. In the second case, we were told that 1, 2, 3, 4 and 5 was a sample, drawn from a bigger population.

### The Population of the Sample

Imagine that the population of the sample were the following 9 numbers: 1, 1, 1, 2, 3, 4, 5, 5 and 5.

Clearly, the numbers are the same, but there is a concentration around the two extremes of the data set – 1 and 5. The variance of this population is 2.96.

So, our sample variance has rightfully corrected upwards in order to reflect the higher potential variability. This is the reason why there are different formulas for sample and population data.

## Standard Deviation Formula: Sample Standard Deviation and Population Standard Deviation

While variance is a common measure of data dispersion, in most cases the figure you will obtain is pretty large. Moreover, it is hard to compare because the unit of measurement is squared. The easy fix is to calculate its square root and obtain a statistic known as standard deviation.

In most analyses, standard deviation is much more meaningful than variance.

### The Formulas

Similar to the variance there is also population and sample standard deviation. The formulas are: the square root of the population variance and square root of the sample variance respectively. I believe there is no need for an example of the calculation. Anyone with a calculator in their hands will be able to do the job.

## The Coefficient of Variation (CV)

The last measure which we will introduce is the coefficient of variation. It is equal to the standard deviation, divided by the mean.

\

Another name for the term is relative standard deviation. This is an easy way to remember its formula – it is simply the standard deviation relative to the mean.

As you probably guessed, there is a population and sample formula once again.

### Why We Need the Coefficient of Variation

So, standard deviation is the most common measure of variability for a single data set. But why do we need yet another measure such as the coefficient of variation? Well, comparing the standard deviations of two different data sets is meaningless, but comparing coefficients of variation is not.

Aristotle once said:

“Tell me, I’ll forget. Show me, I’ll remember. Involve me, I’ll understand.”

## Examples of Comparing Standard Deviations

To make sure you remember, here’s an example of a comparison between standard deviations. Let’s take the prices of pizza at 10 different places in New York. As you can see in the picture below, they range from 1 to 11 dollars.

Now, imagine that you only have Mexican pesos. To you, the prices will look more like 18.81 pesos to 206.91 pesos, given the exchange rate of 18.81 pesos for one dollar.

Let’s combine our knowledge so far and find the standard deviations and coefficients of variation of these two data sets.

### Sample or Population Data

1. First, we have to see if this is a sample or a population. Are there only 11 restaurants in New York? Of course not. This is obviously a sample drawn from all the restaurants in the city. Then we have to use the formulas for sample measures of variability.

### Finding the Mean

1. Second, we have to find the mean. The mean in dollars is equal to 5.5 and the mean in pesos to 103.46.

### Calculating the Sample Variance and the Standard Deviation

1. The third step of the process is finding the sample variance. Following the formula that we went over earlier, we can obtain 10.72 dollars squared and 3793.69 pesos squared.

1. The respective sample standard deviations are 3.27 dollars and 61.59 pesos, as shown in the picture below.

### A Few Observations

Let’s make a couple of observations.

First, variance gives results in squared units, while standard deviation in original units, as shown below.

This is the main reason why professionals prefer to use standard deviation as the main measure of variability. It is directly interpretable. Squared dollars mean nothing, even in the field of statistics.

Second, we got standard deviations of 3.27 and 61.59 for the same pizza at the same 11 restaurants in New York City. However, this seems wrong. Let’s make it right by using our last tool – the coefficient of variation.

### The Advantage of the Coefficient of Variation

We can divide the standard deviations by the respective means. As you can see in the picture below, we get the two coefficients of variation.

The result is the same – 0.60.

Important: Notice that it is not dollars, pesos, dollars squared or pesos squared. It is just 0.60.

This shows us the great advantage that the coefficient of variation gives us. Now, we can confidently say that the two data sets have the same variability, which was what we expected beforehand.

In the picture above, you can see the main advantages of the coefficient of variation.

## The Pros and Cons of Each of the Measures of Variability

To recap, there are three main measures of variabilityvariance, standard deviation and coefficient of variation. Each of them has different strengths and applications. Usually, we prefer standard deviation over variance because it is directly interpretable. However, the coefficient of variation has its edge over standard deviation when it comes to comparing data. After reading this tutorial, you should feel confident using all of them.

Now, using measures when working with one variable probably seems like a piece of cake. However, what if there were 2 variables? Will you be able to represent their relationship? If your answer is no, feel free to jump onto our next tutorial, in order to turn that no into a yes.

Or, if you're considering a career in data science, check out our articles: The Data Scientist Profile, The 5 Skills You Need to Match Any Data Science Job Description, How to Write A Data Science Resume – The Complete Guide, and 15 Data Science Consulting Companies Hiring Now.

***

Interested in learning more? You can take your skills from good to great with our statistics course!


text:
# Order or function/list arguments in map/select/others

I keep being uncertain of the correct order of [function,expr] arguments to functions such as Select, Pick, Cases, Position, etc., because I'm used to the [f, expr] order from the Map family of functions.

1. Is there a good rule of thumb to remember which order should be used where?
2. More importantly, is there a good reason that the Map family has it opposite to the rest? Incidentally this may help with 1 as well.

I think that Map is the exception. I would read

 Map[f,list]

as "map f over list". Same for

 Apply[f  x]

meaning "apply f to x".

Select[list,list]

as "select from list by (selector) f. The same applies to Pick. But have to idea for Cases and Position, basically because they are no verbs.

• I understand the structure of the Map sentence (and actually the Map family is the one where I don't confuse the order). However, I'm unclear on why "Select from list by f" is more natural than "Select by f from list. Perhaps SQL SELECT statement should be the guide here, since the WHERE` clause comes after the data clause there as well. – Shwouchk Nov 9 '19 at 2:08


text:
# Find the Largest Prime Number Less than One Million.

## Problem 90

Find the largest prime number less than one million.

## What is a prime number?

A natural number is called a “prime number” if it is only divisible by $1$ and itself.
For example, $2, 3, 5, 7$ are prime numbers, although the numbers $4,6,9$ are not.

The prime numbers have always fascinated mathematicians.
There are a lot of unsolved problems related to prime numbers.

There are many special types of prime numbers named after famous mathematicians.
My favorites are Mersenne primes, Fermat primes, and Wagstaff primes.

• A natural number of the form
$2^n-1$ is called a Mersenne number.
• A Mersenne prime is a prime number of the form
$2^p-1.$
• A natural number of the form
$2^{2^n}+1$ is called a Fermat number.
• A Fermat prime is a prime number of the form
$2^{2^n}+1.$
• A Wagstaff prime is a prime number of the form
$\frac{2^p+1}{3}.$

### Unsolved problems

For these prime numbers the followings are still unknown.

• Are there infinitely many Mersenne/Fermat/Wagstaff prime numbers?
• Are there infinitely many nonprime Fermat numbers?
• Are there infinitely many composite Mersenne number $2^p-1$ for a prime $p$?

## What is the largest prime number less than one million.

It is known for a long time (Euclid’s Elements (circa 300 BC)) that there are infinitely many primes.

Here are the first $95$ prime numbers. These are all prime numbers less than $500$.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137,
139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211,
223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283,
293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379,
383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461,
463, 467, 479, 487, 491, 499.

### List of prime numbers less than one million.

In fact, there are $78,498$ prime numbers less than $1,000,000$=one million.
To list them here takes a lot of space, so I created a PDF file of the list of primes less than one million.

It takes $95$ pages just to list $78498$ prime numbers less than one million.

Prime numbers less than one million

From this list, we see that

the largest prime numbers less than one million is $999983$.
(The last number in the PDF file.)

### Other Facts

Here are several facts that we can find from the list (with time and energy)

• The largest twin prime pair less than one million is $999959$ and $999961$.
• The second largest twin prime pair less than one million is $999611$ and $999613$.
• The third largest twin prime pair less than one million is $999431$ and $999433$.
• There are 7 Mersenne primes less than one million. These Mersenne primes are
$3, 7, 31, 127,8191, 131071, 524287.$
• The know Fermat prime numbers are all less than one million.These are
$3, 5, 17, 257, 65537.$
• $11$ is the only prime number containing only the decimal digit 1 and less than one million. (The second largest such prime is $1111111111111111111$.)
• Wagstaff prime numbers less than one million are $3, 11, 43, 683, 2731, 43691, 174763.$

Try to find an interesting property of prime numbers from the list of primes <100000.

### More from my site

• Even Perfect Numbers and Mersenne Prime Numbers Prove that if $2^n-1$ is a Mersenne prime number, then $N=2^{n-1}(2^n-1)$ is a perfect number. On the other hand, prove that every even perfect number $N$ can be written as $N=2^{n-1}(2^n-1)$ for some Mersenne prime number $2^n-1$.   Definitions. In this post, a […]
• A One-Line Proof that there are Infinitely Many Prime Numbers Prove that there are infinitely many prime numbers in ONE-LINE.   Background There are several proofs of the fact that there are infinitely many prime numbers. Proofs by Euclid and Euler are very popular. In this post, I would like to introduce an elegant one-line […]
• Mathematics About the Number 2017 Happy New Year 2017!! Here is the list of mathematical facts about the number 2017 that you can brag about to your friends or family as a math geek. 2017 is a prime number Of course, I start with the fact that the number 2017 is a prime number. The previous prime year was […]
• Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies $2^{n+1}|p-1$. Let $a, b$ be relatively prime integers and let $p$ be a prime number. Suppose that we have $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ for some positive integer $n$. Then prove that $2^{n+1}$ divides $p-1$.   Proof. Since $a$ and $b$ are relatively prime, at least one […]
• Mathematics About the Number 2018 Happy New Year 2018!! Here are several mathematical facts about the number 2018.   Is 2018 a Prime Number? The number 2018 is an even number, so in particular 2018 is not a prime number. The prime factorization of 2018 is $2018=2\cdot 1009.$ Here $2$ and $1009$ are […]
• 5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$ In the ring $\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},$ show that $5$ is a prime element but $7$ is not a prime element.   Hint. An element $p$ in a ring $R$ is prime if $p$ is non zero, non unit element and whenever $p$ divide $ab$ for $a, b \in R$, then $p$ […]
• If the Order is an Even Perfect Number, then a Group is not Simple (a) Show that if a group $G$ has the following order, then it is not simple. $28$ $496$ $8128$ (b) Show that if the order of a group $G$ is equal to an even perfect number then the group is not simple. Hint. Use Sylow's theorem. (See the post Sylow’s Theorem […]
• Use Lagrange’s Theorem to Prove Fermat’s Little Theorem Use Lagrange's Theorem in the multiplicative group $(\Zmod{p})^{\times}$ to prove Fermat's Little Theorem: if $p$ is a prime number then $a^p \equiv a \pmod p$ for all $a \in \Z$.   Before the proof, let us recall Lagrange's Theorem. Lagrange's Theorem If $G$ is a […]

### 1 Response

1. 01/01/2017

[…] may find more prime years from the list of one million primes that I […]

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text:
## Another Part where my brain's Fried

$$\int \sqrt{4-x^2}^3 dx$$

I'm thinking integrating by parts would work, with u being that root and dv being dx, but is that the right method and direction. I've tried it and it seems more complicated.
 Use $$x = 2 \sin \theta$$... then, $$dx = 2 \cos \theta d\theta$$, and your integral becomes: $$16 \int \cos^4 \theta d\theta$$ . Use $$\cos{2\theta} = 2\cos^2 \theta - 1$$, etc...
 Damn it, no wonder I couldn't get it to work (I tried x = 2cos u)!

Blog Entries: 9
Recognitions:
Homework Help

## Another Part where my brain's Fried

So it's

$$\int \left(4-x^{2}\right)^{\frac{3}{2}} \ dx$$

$$x=2\sin t$$ ?

Daniel.

EDIT:Didn't see the other posts.
 Recognitions: Gold Member Science Advisor Staff Emeritus x= 2cos(u) should work exactly like "x= 2 sin(u)": dx= -2 sin u du and $$\sqrt{4- x^2}= \sqrt{4-4 cos^2(u)}= 2 sin(u)$$ so the integral becomes $$-16\int sin^4(u) du$$ and the only difference is that "-".

 Quote by HallsofIvy x= 2cos(u) should work exactly like "x= 2 sin(u)": dx= -2 sin u du and $$\sqrt{4- x^2}= \sqrt{4-4 cos^2(u)}= 2 sin(u)$$ so the integral becomes $$-16\int sin^4(u) du$$ and the only difference is that "-".
Oops, I made a slight mistake!
 Thanks once more! :)


text:
C++Server Side ProgrammingProgramming

Suppose we have two words (beginWord and endWord), and we have dictionary's word list, find the length of shortest transformation sequence from beginWord to endWord, such that −

• Only one letter can be converted at a time.

• In each transformed word must exist in the word list. The beginWord is not a transformed word.

We have to keep in mind that −

• Return 0 when there is no such change sequence.

• All words have the same length.

• All words contain only lowercase characters.

• We can assume no duplicates in the word list.

So if the input is like: beginWord = "hit", endWord = "cog", and wordlist = ["hot", "dot", "dog", "lot", "log", "cog"]

Then the output will be 5, as one shortest transformation is hit → hot → dot → dog → cog

To solve this, we will follow these steps −

• Define a method called putStar, this will take j and string s. This will work as follows −

• temp := empty string

• for i in range 0 to size of s – 1

• if i = j, then update temp by concatenating “*” with it, otherwise update temp by concatenating s[i] with temp.

• in the main method it will take string b, string e and list of words w, this will work like −

• if e is not in w, or b is empty, or e is empty, or w is empty, then return 0

• define a map m for string type key and array type value.

• for i in range 0 to size of w

• x := w[i]

• for j := 0 to size of x

• inter := putStar(j, x)

• insert x into m[inter]

• Define a queue q, insert a pair (b, 1) into q

• make a map called visited

• while q is not empty

• s := front pair from q, delete front element from q

• x := first element of the pair s, l := second element of the pair s

• for i in range 0 to size of x

• temp := putStar(i, x)

• for j in range 0 to size of m[temp]

• aa := m[temp, j]

• if aa is same as e, then return l + 1

• if visited[aa] is not set, then insert pair (aa, l + 1), and set visited[aa] = 1

• level := 0

• return 0

## Example

Let us see the following implementation to get a better understanding −

Live Demo

#include <bits/stdc++.h>
using namespace std;
class Solution {
public:
string putStar(int j, string s){
string temp = "";
for(int i = 0; i < s.size(); i++){
if(i == j)temp += "*";
else temp += s[i];
}
return temp;
}
int ladderLength(string b, string e, vector<string>& w) {
if(find(w.begin(), w.end(), e) == w.end() || !b.size() || !e.size() || !w.size())return 0;
map < string , vector <string> > m;
for(int i = 0; i < w.size(); i++){
string x = w[i];
for(int j = 0; j < x.size(); j++){
string inter = putStar(j,x);
m[inter].push_back(x);
}
}
queue < pair <string, int> > q;
q.push({b, 1});
map <string, int> visited;
while(!q.empty()){
pair < string, int > s = q.front();
q.pop();
string x = s.first;
int l = s.second;
for(int i = 0; i < x.size(); i++){
string temp = putStar(i ,x);
for(int j = 0; j < m[temp].size(); j++){
string aa = m[temp][j];
if(aa == e)return l+1;
if(!visited[aa]){
q.push({aa, l+1});
visited[aa] = 1;
}
}
}
}
int level = 0;
return 0;
}
};
main(){
vector<string> v = {"hot","dot","dog","lot","log","cog"};
Solution ob;
}

## Input

"hit"
"cog"
["hot","dot","dog","lot","log","cog"]

## Output

5
Updated on 29-Apr-2020 12:23:44


text:
# RF Circuit (what does capacitor do?)

I'm trying to work out why a 0.7pF capacitor was chosen. The circuit operates at 12GHz.

At 12GHz, the capacitor has an impedance of about $18\Omega$. Also at 12GHz, TL3 is $0.259\lambda$ (which is roughly a quarter of a wavelength. I worked out the characterstic impedance of TL3 to be $74\Omega$.

I thought that C1 was meant to be a DC blocking capacitor and prevent the 12GHz signal from going into the DC supply; however, I don't understand why an $18\Omega$ capacitor was chosen, as a quarter wavelength piece at $74\Omega$ does not transform $50\Omega$ to $18\Omega$...

Could someone point me in the right direction?

Thanks.

## 1 Answer

The whole point of the DC bias network (TL3 and C1) is to present a very high impedance at the junction of the three TLs, in order to perturb the RF signal flowing through TL1 and TL4 as little as possible.

At these frequencies, the parasitic effects of every component must be considered. For example, even a tiny SMT capacitor has a certain amount of parasitic inductance in series with it that creates a self-resonant frequency at which its impedance is lowest. Above this frequency, the inductive reactance dominates and the impedance starts rising again.

My guess would be that the capacitor chosen has a self-resonant frequency at or above 12 GHz, presenting the minimum possible impedance at the node between C1 and TL3. This gets transformed by TL3 into the highest possible impedance at the junction of the three TLs.

• Curiously, could TL3 have been chosen so that the very low impedance of the capacitor at resonance be presented at the TL1-TL2-TL3 junction? (So, say, the length be made to be either $\lambda/2$ or $\lambda$?) Would this have the same affect? (Because the RF signal would reflect off the low impedance like it does with the high impedance?) Apr 8, 2014 at 6:54
• @user968243: No. There are three connections at that junction, and two of them are 50 ohms. We want the third one to be very high impedance precisely so that it does NOT affect the RF. As far as the RF is concerned, that third connection doesn't exist. Apr 8, 2014 at 12:22
• Thanks for the reply. I thought that a short circuit would not affect the RF signal because I'm thinking that the reflection coefficient at that point for an open circuit has the same magnitude as that of a short circuit. Apr 8, 2014 at 13:03


text:
# A Fish without a Bicycle

Technology and Art

# Non-Linear Support Vector Machines: Radial Basis Function Kernel and the Kernel Trick

Avishek Sen Gupta on 7 August 2021

This article builds upon the previous material on kernels and Support Vector Machines to introduce some simple examples of Reproducing Kernels, including a simplified version of the frequently-used Radial Basis Function kernel. Beyond that, we finally look at the actual application of kernels and the so-called Kernel Trick to avoid expensive computation of projections of data points into higher-dimensional space, when working with Support Vector Machines.

The specific posts discussing the background are:

We will discuss the following:

• Polynomial Kernels
• A simplified version of the popular Radial Basis Function kernel
• Non-Linear Support Vector Machines and the Kernel Trick

Even though we will look at specific feature maps for selected kernel functions in the next section, please note that feature maps induced by kernel functions are not unique. More than one feature map may exist for a given kernel function. Kernel functions only guarantee that we project our data into some higher-dimensional space; we do not always know (or care about) the actual feature space our data is being projected onto, only that it is.

## Polynomial Kernels

Polynomial kernels are probably the simplest to understand; they are not necessarily used in practice (except for Natural Language Processing, source: Wikipedia), but are a good stepping-stone to understanding the infinite-dimensional scenarios.

### Homogenous Polynomial Kernels (No Constant Term)

We start with an example of a simplified form of the polynomial kernel, without constants. These are called Homogenous Kernels.

This example is specific for vectors in $$\mathbb{R}^2$$; we will be lifting them to $$\mathbb{R}^3$$. The kernel function is $$\kappa(x,y)={\langle x,y\rangle}^2$$. Incidentally, the kernel function will work with any $$\mathbb{R}^n$$ domain of vectors, but for illustration purposes, we will stick to $$\mathbb{R}^2$$.

Assume the vectors are $$X=\begin{bmatrix}X_1 \\ X_2\end{bmatrix}$$ and $$Y=\begin{bmatrix}Y_1 \\ Y_2\end{bmatrix}$$, and the inner product is the usual inner product defined in Euclidean space, i.e., $$\langle x,y\rangle=x^Ty$$.

$\kappa(x,y)={\langle x,y\rangle}^2\\ ={(X_1Y_1+X_2Y_2)}^2={(X_1Y_1)}^2+{(X_2Y_2)}^2+2X_1Y_1X_2Y_2$

We’d like to know what mapping function $$\phi(x)$$ (or equivalently, $$\kappa(x, \bullet)$$) projects our original points $$X$$ and $$Y$$ to $$\mathbb{R}^3$$, so that the inner product in this new vector space is equal to the evaluation of the kernel function $$\kappa(x,y)$$.

Because I know the answer already so we will write it straight away.

$\phi(x)=\kappa(x,\bullet)=\begin{bmatrix}{Yx_1}^2 \\ {x_2}^2 \\ \sqrt 2x_1x_2 \end{bmatrix}$

You can verify yourself that this satisfies the requirement of $$\kappa(x,y)=\langle\phi(x)\phi(y)\rangle$$.

$\langle\phi(x)\phi(y)\rangle=\begin{bmatrix}{X_1}^2 && {X_2}^2 && \sqrt 2X_1X_2 \end{bmatrix}\cdot \begin{bmatrix}{Y_1}^2 \\ {Y_2}^2 \\ \sqrt 2Y_1Y_2 \end{bmatrix}\\ = {(X_1Y_1+X_2Y_2)}^2={(X_1Y_1)}^2+{(X_2Y_2)}^2+2X_1Y_1X_2Y_2 \\ ={\langle x,y\rangle}^2=\kappa(x,y)$

### General Polynomial Kernel

The kernel function described above is a special case of the General Polynomial Kernel, which is of the form (again assuming the Euclidean version of inner product $$x^Ty$$):

$\kappa(x,y)={(x^Ty+c)}^d$

where $$c\in\mathbb{R}$$ and $$d\in\mathbb{N}$$ to. Note that $$d$$ does not represent the number of dimensions that the vector will be lifted to.

Let’s look at a slight generalisation for a vector $$\mathbb{R}^n$$ for $$d=2$$, using the application of the Binomial Theorem.

$\kappa(x,y)={(x^Ty+c)}^2={\left(\sum_{i=1}^nx_iy_i+c\right)}^2 \\ = {\left(\sum_{i=1}^nx_iy_i\right)}^2 + 2c\left(\sum_{i=1}^nx_iy_i\right) + c^2 \\ = \sum_{i=1}^n{(x_iy_i)}^2 + 2\sum_{i=1}^{n-1}\sum_{j+1}^n (x_ix_j)(y_iy_j) + 2c\left(\sum_{i=1}^nx_iy_i\right) + c^2 \\ = \sum_{i=1}^n{(x_iy_i)}^2 + \sum_{i=1}^{n-1}\sum_{j+1}^n (\sqrt{2}x_ix_j)(\sqrt{2}y_iy_j) + \left(\sum_{i=1}^n\sqrt{2c}x_i\sqrt{2c}y_i\right) + c^2 \\$

Through visual inspection (or by expanding the above and noting the pattern of the polynomial), we can infer the mapping $$\phi$$ as;

$\phi(x)=\begin{bmatrix} {x_1}^2 \\ {x_1}^2 \\ \vdots \\ {x_n}^2 \\ \sqrt{2}x_1x_2 \\ \sqrt{2}x_1x_3 \\ \vdots \\ \sqrt{2}x_1x_n \\ \sqrt{2}x_2x_3 \\ \sqrt{2}x_2x_4 \\ \vdots \\ \sqrt{2}x_2x_n \\ \vdots \\ \sqrt{2}x_{n-1}x_n \\ \sqrt{2c}x_1 \\ \sqrt{2c}x_2 \\ \vdots \\ \sqrt{2c}x_n \\ c \end{bmatrix}$

Similar mappings for higher values of $$d$$ can be inferred by applying the Multinomial Theorem.

The Radial Basis Function Kernel the most common kernel used in Machine Learning applications. The kernel function is of the form of a Gaussian function (Normal distribution). The simplest one-dimensional Gaussian is given by:

$f(x)=C\cdot exp\left( -\frac{ {(x-\mu)}^2}{2\sigma_K^2}\right)$

The forms of the kernel function for the one-dimensional, and the more general case (where a norm is induced by an inner product), are shown below.

$\kappa(x,y)=C\cdot exp\left(- \frac{ {(x-y)}^2}{2\sigma_K^2}\right) \\ \kappa(x,y)=C\cdot exp\left(- \frac{ {\|x-y\|}^2}{2\sigma_K^2}\right)$

We will prove that $$\kappa(x,y)$$ is a Reproducing Kernel for the simple one-dimensional case. We will want to prove:

$\kappa(x,y)=\langle\phi(x)\phi(y)\rangle$

If we denote the free parameter as $$\mu$$, we can write the following:

$\phi_x(\mu)=a\cdot exp\left(- \frac{ {(x-\mu)}^2}{2\sigma^2}\right) \\ \phi_y(\mu)=a\cdot exp\left(- \frac{ {(y-\mu)}^2}{2\sigma^2}\right)$

Inner Product Curve of two Gaussian Feature Maps

We need take the inner product of the above functions. Remember from Kernel Functions: Functional Analysis and Linear Algebra Preliminaries that the taking the inner product of two function amounts to taking the integral of their products, like so:

${\langle f,g\rangle}=\int_a^b f(x)g(x)dx$

In this case, the limits are $$(-\infty,+\infty)$$, so we write:

${\langle \phi_x(\mu),\phi_y(\mu)\rangle}=\int_{-\infty}^{+\infty} a\cdot exp\left(- \frac{ {(x-\mu)}^2}{2\sigma^2}\right)\cdot a\cdot exp\left( -\frac{ {(y-\mu)}^2}{2\sigma^2}\right)d\mu \\ =a^2 \int_{-\infty}^{+\infty} exp\left(- \frac{ {(x-\mu)}^2}{2\sigma^2}\right)\cdot exp\left(- \frac{ {(y-\mu)}^2}{2\sigma^2}\right)d\mu \\ =a^2 \int_{-\infty}^{+\infty} exp\left(- \frac{ {(x-\mu)}^2 + {(y-\mu)}^2}{2\sigma^2} \right)d\mu \\ =a^2 \int_{-\infty}^{+\infty} exp\left(- \frac{ x^2+y^2+2\mu^2-2\mu x-2\mu y}{2\sigma^2} \right)d\mu \\ =a^2 \int_{-\infty}^{+\infty} exp\left(- \frac{ x^2+y^2-2xy+2xy+2\mu^2-2\mu x-2\mu y}{2\sigma^2} \right)d\mu \\ =a^2 \int_{-\infty}^{+\infty} exp\left(- \frac{ {(x-y)}^2}{2\sigma^2} \right)\cdot exp\left(- \frac{2}{2\sigma^2}\cdot(xy+\mu^2-\mu (x+y)) \right)d\mu \\ =a^2 \int_{-\infty}^{+\infty} exp\left(- \frac{ {(x-y)}^2}{2\sigma^2} \right)\cdot exp\left(- \frac{2}{2\sigma^2}\cdot\left(xy+\mu^2-\mu (x+y) + \frac{ {(x+y)}^2}{4} - \frac{ {(x+y)}^2}{4} \right) \right)d\mu \\ =a^2 \int_{-\infty}^{+\infty} exp\left(- \frac{ {(x-y)}^2}{2\sigma^2} \right)\cdot exp\left(- \frac{2}{2\sigma^2}\cdot\left(xy+{\left(\mu-\frac{x+y}{2}\right)}^2 - \frac{ {(x+y)}^2}{4} \right) \right)d\mu \\ =a^2 \int_{-\infty}^{+\infty} exp\left(- \frac{ {(x-y)}^2}{2\sigma^2} \right)\cdot exp\left(- \frac{2}{2\sigma^2}\cdot\left({\left(\mu-\frac{x+y}{2}\right)}^2 - \frac{ x^2+y^2+2xy-4xy}{4} \right) \right)d\mu \\ =a^2 \int_{-\infty}^{+\infty} exp\left(- \frac{ {(x-y)}^2}{2\sigma^2} \right)\cdot exp\left(- \frac{2}{2\sigma^2}\cdot\left({\left(\mu-\frac{x+y}{2}\right)}^2 - \frac{ {(x-y)}^2}{4} \right) \right)d\mu \\ =a^2 \int_{-\infty}^{+\infty} exp\left(- \frac{ {(x-y)}^2}{2\sigma^2} \right) \cdot exp\left( \frac{ {(x-y)}^2}{4\sigma^2} \right) \cdot exp\left(- \frac{2}{2\sigma^2}\cdot {\left(\mu-\frac{x+y}{2}\right)}^2 \right) d\mu \\ =a^2 \cdot exp\left(- \frac{ {(x-y)}^2}{4\sigma^2} \right) \underbrace{\int_{-\infty}^{+\infty} exp\left(- \frac{1}{ 2{(\frac{\sigma}{\sqrt{2}})}^2 } \cdot {\left(\mu-\frac{x+y}{2}\right)}^2 \right) d\mu}_{\text{Gaussian integrates to }\sqrt{\pi}\sigma} \\ {\langle \phi_x(\mu),\phi_y(\mu)\rangle}=a^2\sqrt{\pi}\sigma\cdot exp\left(- \frac{ {(x-y)}^2}{4\sigma^2} \right) \\$

Comparing this with the standard form of the one-dimensional kernel function, we can see that the constants of the feature map $$\phi(x)$$ can be calculated. The standard deviation of the feature map becomes:

$4\sigma^2=2\sigma_K^2 \\ \Rightarrow \sigma=\frac{\sigma_K}{\sqrt{2}}$

Next we see what the scaling coefficient of the feature map looks like. If $$a=\frac{1}{\sigma\sqrt{2\pi}}$$, we get:

$C=a^2\sqrt{\pi}\sigma \\ =a^2\sqrt{\frac{\pi}{2}}\sigma_K \\ =\frac{1}{2\pi\sigma^2}\sqrt{\frac{\pi}{2}}\sigma_K \\ =\frac{2}{2\pi\sigma_K^2}\sqrt{\frac{\pi}{2}}\sigma_K \\ \Rightarrow C=\frac{1}{\sqrt{2\pi}\sigma_K}$

If we choose this value of $$\sigma$$ for our feature map $$\phi(x)$$, we will see that gives us:

$\kappa(x,y)=\frac{1}{\sigma_K\sqrt{2\pi}}\cdot exp\left( -\frac{ {(x-\mu)}^2}{2\sigma_K^2}\right) \\$

which is the standard normal distribution. Thus, we can conclude that:

$\kappa(x,y)=\langle\phi(x)\phi(y)\rangle$

which is what we are after.

## Support Vector Machines and the Kernel Trick

The Support Vector Machine we have discussed only works for linearly separable data. Real-world data sets are seldom linearly separable. We have already discussed the advantages of projecting the linearly-inseparable data into higher dimensions, and why that might lead to a new problem which can solved using linear separation techniques. We now look at the Support Vector Machine case to see where kernels are used.

Remember from Support Vector Machines from First Principles: Linear SVMs that to classify a new data point, a trained SVM evaluates the following expression, essentially deciding on which side of the decision hyperplane the point exists.

$$$y_t=sgn[w^\ast x_t-b^\ast] \label{eq:weight-svm}$$$

$$x_t$$ is the point being evaluated for classification.

The weight $$w^\ast$$ and “intercept” $$b^\ast$$ are given by, using the expression for the Lagrange Multipliers $$\lambda_i$$:

$\mathbf{ w^\ast=\sum_{i=1}^n \lambda_ix_iy_i \\ b^\ast=\frac{b^++b^-}{2} \\ \lambda^\ast=\text{arginf}_\lambda \left[\sum_{i=1}^n \lambda_i - \frac{1}{2} \sum_{i=1}^n\sum_{j=1}^n\lambda_i\lambda_jy_iy_jx_ix_j\right] }$

The concept of Non-Linear Support Vector Machines should be very familiar at this point; they are simply SVMs whose data points are projected onto higher-dimensional space, so that the classes of data become linearly separable.

Let us assume that some feature mapping $$\phi(x)$$ has been identified for use. Since all vectors (training, test, real-world) will need to be projected into feature space (the higher-dimensional space), the expression for the weight now looks like this:

$w^\ast=\sum_{i=1}^n \lambda_iy_i\phi(x_i)$

We assume that the Lagrange Multiplier values are already known. When the time comes to evaluate, expanding out the weight expression in $$\eqref{eq:weight-svm}$$ gives us:

$y_t=sgn[\sum_{i=1}^n \lambda_iy_i\mathbf{\phi(x_i)\phi(x_t)} -b^\ast] \\$

As we can see, this involves projecting each point into higher-dimensional space and then computing the inner product between each of the training data points and the point being evaluated. This becomes prohibitively expensive, since the computations increase proportional to the number of dimensions the data is being lifted to.

However the form $$\mathbf{\phi(x_i)\phi(x_t)}$$ should be familiar, as is shown below. This is an inner product of two feature maps, which makes it a candidate for direct evaluation using a kernel function.

$y_t=sgn[\sum_{i=1}^n \lambda_iy_i\underbrace{\phi(x_i)\phi(x_t)}_{\text{Candidate for Kernel Function}}-b^\ast] \\ =sgn[\sum_{i=1}^n \lambda_iy_i\kappa(x_i,x_t)-b^\ast]$

Computing $$\phi(x)$$ for each data point and then performing the inner product is much slower than directly evaluating the kernel function. This is the Kernel Trick so often referenced when discussing Support Vector Machines.

Note that the Kernel Trick is an optimisation for making Non-Linear SVMs feasible, since the decision boundary need not be linear anymore in the original input space. The nonlinearity comes from the nonlinear feature map which is implicitly used when evaluating the kernel function.

tags: Machine Learning - Kernels - Theory - Functional Analysis - Support Vector Machines


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I've never seen any really good expositions of elementary mathematics (middle school or earlier). A good college-level textbook, written for people with an interest in mathematics, reads like a novel or an impassioned essay. The most famous example I can think off the top of my head of is Spivak's Calculus - the author has a vision for the subject, and wants to communicate their passion to the student.

As an example, the rule for multiplying fractions together takes some quite clever and pretty arguments to prove, but in math class and in classroom textbooks, this is just presented as:

Fraction multiplication rule: Multiply the numerators and the denominators together separately.

Looking at the difference in presentation, you'd almost think elementary arithmetic and geometry aren't part of mathematics. Where are the proofs? Is there even a single book in existence that proves the above fact at all?

Are there any books about elementary math written like actual math books? Note that I am not looking for things like "The Number Devil", which (to be fair I haven't read it) you probably wouldn't mistake for an actual textbook. I'm looking for systematic expositions of basic math, covering (and proving!) all the theorems, rules of computation, and definitions covered in elementary to high school (not necessarily all in one book - that's a lot of material).

I don't necessarily want it to be readable by 8 year olds, that might be asking a lot, but it would be nice to have something you could recommend to a bright 12 year old to give them a more solid foundation and inspire more of an interest in math. Something well written, that wants to make the subject interesting and beautiful, is preferred to a dry going-through-the-motions proof compendium, but of course that's basically like saying "make sure the book is actually good".

-
@paulgarrett From a strictly logical perspective, the rationals are very often defined as equivalence classes of $\mathbb{Z} \times \mathbb{Z}/\{0\}$, with $(a_1,b_1) \cong (a_2,b_2)$ if $a_1b_2=a_2b_1$. Then multiplication of fractions is defined by $(a,b)(c,d) \cong (ac,bd)$. What needs to be proven is that this definition is even well defined. –  Steven Gubkin Apr 22 at 13:12
@StevenGubkin, I am well aware than "X is often defined as Y", but that is a separate question from the actual mathematical function of either. Also, the possibility of writing things in first-order logic or other formalities does not lend anything greater gravitas or correctness. The function of the rationals is as field of fractions of the commutative ring $\mathbb Z$, and that requires/entails certain properties (apart from any notation). The construction as equivalence classes of somehow-denoted pairs of integers, and notations about that, are secondary to the function. –  paul garrett Apr 22 at 13:28
@StevenGubkin This is an interesting debate, but it's probably better suited for the chat! –  Jack M Apr 22 at 13:34
@StevenGubkin The StackExchange button at top left - it's in the drop down. You can ping people using @ once you're in there. I'll edit a proof into the post if you do think it's important, though. –  Jack M Apr 22 at 13:39
@paulgarrett can you join me in the chatroom for this site? chat.stackexchange.com/rooms/13591/mathematics-educators –  Steven Gubkin Apr 22 at 13:48

I think you'll find some of what you want on Berkley mathematician H.H. Wu's homepage.

More precisely, see: Pre-Algebra (pdf) and Introduction to School Algebra (pdf).

Note: I mentioned the same homepage (and the two pdf textbooks) in an earlier MESE post here; I would have just re-posted this as a comment, but I believe it is the actual answer to your question (!). I also ought not mark your question as a duplicate of the one I answered earlier, since they are quite different.

-

Below are three books for what I think you want. At one time these books were all very well known, at least in the U.S., probably because virtually every public library (even small town libraries) used to have copies of these books.


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# bmhranks

0th

Percentile

##### One-step and Two-step Signed Ranks.

Calculates either one-step or two-step signed ranks as proposed by Noether (1973), Brown (1981) and Markowski and Hettmansperger (1982).

Keywords
htest
##### Usage
bmhranks(y, q1 = 1/3, q2 = 2/3)
##### Arguments
y

A vector of matched pair differences.

q1

Quantile at which step-ranks rise from 0 to 1.

q2

Quantile at which step-ranks rise from 1 to 2. If q1=q2, then there is only one step, from 0 to 1.

##### Value

A vector with the same length as y containing the step-ranks.

##### Note

The use of step-ranks in observational studies is discussed in Rosenbaum (1999, 2012, 2015). They can have larger design sensitivity than Wilcoxon ranks and higher Bahadur efficiency in a sensitivity analysis.

##### References

Brown, B. M. (1981) <doi:10.1093/biomet/68.1.235> Symmetric quantile averages and related estimators. Biometrika, 68(1), 235-242.

Markowski, E. P. and Hettmansperger, T. P. (1982) <doi:10.2307/2287325> Inference based on simple rank step score statistics for the location model. Journal of the American Statistical Association, 77(380), 901-907.

Noether, G. E. (1973) <doi:10.2307/2284805> Some simple distribution-free confidence intervals for the center of a symmetric distribution. Journal of the American Statistical Association, 68(343), 716-719.

Rosenbaum, P. R. (1999) <doi:10.1111/1467-9876.00140> Using quantile averages in matched observational studies. Journal of the Royal Statistical Society: Series C (Applied Statistics), 48(1), 63-78.

Rosenbaum, P. R. (2012) <doi:10.1214/11-AOAS508> An exact adaptive test with superior design sensitivity in an observational study of treatments for ovarian cancer. The Annals of Applied Statistics, 6(1), 83-105.

Rosenbaum, P. R. (2015) <doi:10.1080/01621459.2014.960968> Bahadur efficiency of sensitivity analyses in observational studies. Journal of the American Statistical Association, 110(509), 205-217.

The function multrnk() computes an alternative family of signed ranks.

• bmhranks
##### Examples
# NOT RUN {

o<-order(w)
plot(w[o],brn[o],type="n",xlab="Wilcoxon ranks",ylab="Step Ranks",
main="Comparison of 3 Step Ranks")
lines(w[o],brn[o],col="black")

# The following two-step ranks were best for Normal data in
# Table 2 of Markowski-Hettmansperger (1982).
lines(w[o],mhn[o],col="blue")

# Noether (1973) ranks take a single step.  The case of a step
# at q1=q2=2/3 was evaluated in Rosenbaum (2015, Table 2).
lines(w[o],noe[o],col="red")
legend(20,1.75,c("Brown","MH","Noether"),
lty=c(1,1,1),col=c("black","blue","red"))

# Adaptive choice of Brown or Noether ranks was considered in
# Rosenbaum (2012).  In this case, an exact distribution
# is available, but function tt() uses a limiting Normal


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# Can someone help me with math homework?

Can someone help me with math homewoHello! I have problems with my math homework assignments. Can someone help me please?

Exercise: 1) Write the number 60 in different ways as the sum of two numbers x + y. More general: With which summands formed in this way is the product x * y largest?

2) A number is subtracted from 13. The same number is added to 21. The results of both calculations are multiplied together. How to choose the number for the product to be the largest?

Note by Robert Dale
3 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
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1. numbered2. list
1. numbered
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Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

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[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

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study today is becoming more complex. I use the college essay help from professional service. it helps me not only in writing homework or essays, and thanks to this I avoid unnecessary stress, which makes me healthier and happier

- 6 days, 4 hours ago

Question #1:

We can use Lagrange Multipliers. The Lagrangian is:

$L = x y + \lambda (x + y - 60)$

Take partial derivatives and set equal to zero:

$\frac{\partial{L}}{\partial{x}} = y + \lambda = 0 \\ \frac{\partial{L}}{\partial{y}} = x + \lambda = 0 \\ \frac{\partial{L}}{\partial{\lambda}} = x + y - 60 = 0$

Solving yields:

$x = 30 \\ y = 30 \\ \lambda = -30$

And we know this is a maximum since $$30 \times 30 > 40 \times 20$$

Question #2:

Maximize the following quantity:

$Q = (13 - x)(21 + x)$

Differentiate and set to zero:

$(13 - x )(1) + (21 + x )(-1) = 0 \\ x = -4$

We know that this is a maximum because putting in any other value for $$x$$ yields a smaller number. Interestingly, this value for $$x$$ makes both multiplied quantities equal to $$17$$.

- 2 months, 3 weeks ago

Yes, it's can bee calculus!

- 2 months, 3 weeks ago

Can it be calculus, or does it have to be algebra?

- 2 months, 4 weeks ago

We can help you need someone to help with Algebra problems https://homeworkprospect.com/

- 1 month, 3 weeks ago


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# Schools University of Waterloo

#### Atomos

I know UW is a big engineering, math, and computer science school, but how is their physics program relative to the other top schools in Canada?

#### JasonRox

Homework Helper
Gold Member
Atomos said:
I know UW is a big engineering, math, and computer science school, but how is their physics program relative to the other top schools in Canada?
Definitely high and probably the best.

They have a Theoritical Physics Institute in Waterloo itself! (Perimeter Institute.)

I would suspect that it is a great school for Physics.

#### Sane

Just a side note: Maclean's rated Waterloo the top University in Canada two years consecutively now. And I believe that's overall as well. As that ranking applies to Physics...? I have no idea.

Other Universities are actually outraged by this and withdrawing themselves from the rankings. Heh.

Wikipedia says:

#### JasonRox

Homework Helper
Gold Member
Atomos said:
Yes, coop is very important for me. Correct me if I am wrong, but the two main criteria for grad school admission are marks and possible undergrad experience. Coop appears to be a good way to pad the latter.

edit: yes, I paid \$20 a while ago and it seems to have expired after a year :-(
I'd be more concerned with NSERC Summer Research than Co-op as a key to graduate school.

#### Werg22

That's quite interesting. The match is definitely between UW and Toronto University, which one is best needs investigation.

#### George Jones

Staff Emeritus
Gold Member
quasar987 said:
Waterloo, is that the uni in toronto?
No, Waterloo is in Waterloo.

Is Lee Smolin a teacher at UW?
Smolin is at the Perimeter Institute, which is located in the city of Waterloo. Smolin has adjunct faculty status at the University of Waterloo.

#### fourier jr

Atomos said:
I know UW is a big engineering, math, and computer science school, but how is their physics program relative to the other top schools in Canada?
it probably depends on what kind of physics you are interested in. it's probably the best in the world for quantum computing since it has many of the 'founding fathers' of qc there. i don't think i know anything about the rest. i've only ever looked into qc there.

#### Atomos

fourier jr said:
it probably depends on what kind of physics you are interested in. it's probably the best in the world for quantum computing since it has many of the 'founding fathers' of qc there. i don't think i know anything about the rest. i've only ever looked into qc there.
Err, sorry, I forgot to mention I am going into undergrad physics.

#### JasonRox

Homework Helper
Gold Member
Atomos said:
Err, sorry, I forgot to mention I am going into undergrad physics.
Also, I wouldn't put too much emphasis on undergrad physics.

It's basically almost the same all over Canada. The only difference is the students you work with. If you want to find serious students, you have a better shot at finding them at Waterloo than you do at Brock. (Where I am from.) But Brock has it's up sides too, which is priceless if you get to be part of it.

I know of people going to Waterloo for graduate school for Brock, so that goes to show, undergrad doesn't really matter. It comes down to being ready and showing that you are. Canada is a little different than the US when it comes to undergraduate courses. I would simply look for other reasons to choose Waterloo or UBC, like students (I mentionned), campus, location, and things like that.

Personally, I plan on going to graduate school for mathematics. Currently, Waterloo isn't my first choice. I can probably get in, and it's definitely the best in Canada, but the problem lies in the reality of is it the best choice for me? I've seen the advantages of smaller faculties as well as it's disadvantages too. For these reasons, I know specifically what I'm looking for when I'm going to start visiting schools in January/February. I'm going to visit Waterloo because who knows maybe they can change what I think of that school. It seems like a school that student/professor interaction is a minimum and that they busy you with a bunch classes. Not my idea of good education.

For undergrad, Waterloo was my first choice mainly because I know I can find students that love mathematics too. Rather than scrounge around by myself right now. Also, it can challenge the heck out of you, so that you know you're ready for graduate school.

But for graduate school, do I really want minimal interaction with my professors? NO. Do I want to be challenged like a psycho freak in a bunch of classes? No, because I'll be doing that outside of the classroom and by doing this you'll take away all my own time to do doing my own thing, which I'll truly hate! I will certainly transfer!

Anyways, pick a school where all the girls are.

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#### Atomos

Thank you for your useful input, Jason.

JasonRox said:
Anyways, pick a school where all the girls are.
Hehe, it is funny because in big pink letters in the science section of the Waterloo pamphlet they say "3:2 female to male ratio" as if it was the selling point of the program :tongue2:

Last edited:

#### JasonRox

Homework Helper
Gold Member
Atomos said:
Thank you for your useful input, Jason.

Hehe, it is funny because in big pink letters in the science section of the Waterloo pamphlet they say "3:2 female to male ratio" as if it was the selling point of the progam :tongue2:
Haha, Brock has like 5:1! :surprised

#### fourier jr

i think UVic's is 2:1, & they even have a really good physics department

#### Sisyphus

uwaterloo math is something like 3:7

#### JasonRox

Homework Helper
Gold Member
Sisyphus said:
uwaterloo math is something like 3:7
I actually thought they had none in the mathematics department.

I visited their school last year with my girlfriend. We were in the math department and naturally I had to go to the washroom. Then my girlfriend had to go, but we couldn't find a girls washroom in the math department!!!


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# Math Help - mix/max problem

1. ## mix/max problem

I am working on a mix/max problem and am stuck.

$y=xe^(-\frac{x^2}{8})$ within the interval of [-1,4]. The exponent of e is $-\frac{x^2}{8}$, can't get Latex to cooperate with me.

Using the product rule to differentiate and I get:

y'= $e^(-\frac{x^2}{8})+x(e^(-\frac{x^2}{8})(\frac{x}{4})$

Factor out the $e^(-\frac{x^2}{8}$ and I get

$e^(-\frac{x^2}{8})(1+\frac{x^2}{4})$

For all real numbers, the 2nd term will never equal 0, so the only one that can equal 0 is $e^(-\frac{x^2}{8})$.

I don't know how to solve for x when $e^(-\frac{x^2}{8})=0$. Any pointers?

2. ## Re: mix/max problem

\displaystyle \begin{align*} e^{-\frac{x^2}{8}} > 0 \end{align*} for all x, so there are not any critical points to this function.

3. ## Re: mix/max problem

Originally Posted by baldysm
I am working on a mix/max problem and am stuck.

$y=xe^{-\frac{x^2}{8}}$ within the interval of [-1,4]. The exponent of e is $-\frac{x^2}{8}$, can't get Latex to cooperate with me.

Using the product rule to differentiate and I get:

y'= $e^(-\frac{x^2}{8})+x(e^(-\frac{x^2}{8})(\frac{x}{4})$
. Any pointers?
You missed a minus sign y'= $e^{-\frac{x^2}{8}}+x(e^{-\frac{x^2}{8}})({\frac{-x}{4})$

Look at this graph

About your LaTeX, If you have more than one character in an exponent the you must use {} around the entire exponent.
[TEX] e^{-\frac{x^2}{8}} [/TEX] gives $e^{-\frac{x^2}{8}}$.

4. ## Re: mix/max problem

Originally Posted by Plato
You missed a minus sign y'= $e^{-\frac{x^2}{8}}+x(e^{-\frac{x^2}{8}})({\frac{-x}{4})$

Look at this graph

About your LaTeX, If you have more than one character in an exponent the you must use {} around the entire exponent.
[TEX] e^{-\frac{x^2}{8}} [/TEX] gives $e^{-\frac{x^2}{8}}$.
Please disregard my previous post as Plato has pointed out the error I and the OP have both made.

5. ## Re: mix/max problem

Yup, I see my algebra error, it should be $-\frac{x}{4}$.

I also screwed up the interval. It should be [-4,1].

The derivative is

$y'=e^{\frac{-x^2}{8}}+x(e^{\frac{-x^2}{8}})(\frac{-x}{4})$

Some factoring and I get

$y'=e^{\frac{-x^2}{8}}(1-\frac{x^2}{4})$

The e term can't be 0 for all of x (although my TI-84 gives an error when x=-100, not sure why), and the only number that makes the 2nd term zero is either -2 or 2. Only -2 is in the interval, so I need to find out f(x).for x= -4, -2, and 1. Biggest number is max, smallest is min. I might just understand this yet.

I appreciate the help.