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Proofs
Proof means sufficient evidence to establish the truth of something. It is obtained from deductive reasoning, rather than from empirical arguments. Proof must show that a statement is true in all cases, without a single exception.
1,294 Questions
How does standard normal distribution differ from normal distribution?
The standard normal distribution has a mean of 0 and a standard deviation of 1.
Displacement= Volume x Density
for example to find the displacement of a ship you would do this formula:
underwater volume(m3) x density(t/m3)
so if you are a deck cadet like me this is the formula you would use to find the displacement of a ship in the first ship stability test!
THIS IS WRONG! THERE ARE MANY DIFFERENT TYPES OF DISPLACEMENT FOR-INSTANCE THOSE USED IN PHYSICS ALL HAVE DIFFERENT FORMULE
Take the square root of that number. If you get a whole number, the answer is yes; otherwise, the answer is no.
What is proof of Lami's theorem?
As you probably know, Lami's Theorem only applies to objects in equilibrium, with 3 coplanar (in the same plane) concurrent (intersecting at the same point) forces acting on it. It works because you add vectors together from tip to tail and also taking direction into account, and because the net force of an object in equilibrium is zero.
Let's look at an object for which Lami's Theorem works.
Now, let's add all these forces together, tip to tail.
The force vectors have to do this (form a closed shape) because the object is in equilibrium, and this makes the net force zero. When the net force is zero, the forces should cancel each other out entirely, meaning that adding the vectors will result in zero.
(If we added the force vectors of an object NOT in equilibrium, we would obtain a shape that:
· is not a proper closed shape, i.e. you add the vectors and they form 1. a wonky line, or 2. a weird triangle thingy where you haven't used the entirety of a vector for the shape.
1.
2.
· is some other shape.
This would indicate a net force being present.)
Let's take our added-up forces shape and add some details to it. (By the way, this shape is called a forces triangle.)
All I did was lengthen the force lines in the direction of the vector.
Now:
I just used the original diagram, and found out which angles are between which vectors, and inserted them here into the diagram.
Then, the inside angles must look like:
Now, what's the sine rule again?
For sides a,b,and c and included angles A,B and C:
Let's do it for our forces triangle!
but we know that sin (180-α/β/θ)=sin (α/β/θ), so
And that's Lami's Theorem!
Did Charles Forbin make the first computer?
No! Charles Forbin was a character in a 1970 sci-fi film called "The Forbin Project". Dr Forbin led a team which build a super-computer - called Colossus - which became a rogue sentient machine.
Line drawing simple dda algorithm in c?
#include<stdio.h>
#include<conio.h>
#include<graphics.h>
#include<ctype.h>
#include<math.h>
#include<stdlib.h>
void draw(int x1,int y1,int x2,int y2);
void main()
{
int x1,y1,x2,y2;
int gdriver=DETECT,gmode,gerror;
initgraph(&gdriver,&gmode,"c:\\tc\\bgi:");
printf("\n Enter the x and y value for starting point:\n");
scanf("%d%d",&x1,&y1);
printf("\n Enter the x and y value for ending point:\n");
scanf("%d%d",&x2,&y2);
printf("\n The Line is shown below: \n");
draw(x1,y1,x2,y2);
getch();
}
void draw(int x1,int y1,int x2,int y2)
{
float x,y,xinc,yinc,dx,dy;
int k;
int step;
dx=x2-x1;
dy=y2-y1;
if(abs(dx)>abs(dy))
step=abs(dx);
else
step=abs(dy);
xinc=dx/step;
yinc=dy/step;
x=x1;
y=y1;
putpixel(x,y,1);
for(k=1;k<=step;k++)
{
x=x+xinc;
y=y+yinc;
putpixel(x,y,2);
}
}
Yes. A graph is bipartite if it contains no odd cycles. Since a tree contains no cycles at all, it is bipartite.
What is the complement of Y if Y equals 8X-20 degrees?
Complementary angles add up to 90 degrees so if y = 8x-20 degrees then the complement of y is 90 - (8x - 20) = 110 - 8x degrees.
Show an analytical function of constant absolute value is constant?
This is based on exercise 10 on p 165.4 of "Introduction to Complex Analysis" by Nevanlinna and Paatero, Chelsea Publishing, NY.
With w(z) == u(x,y) + i v(x,y), ( |w(z)| = constant ) ==>
1) uu + vv = const. Using notation ux == du/dx, uy == du/dy, etc., 1) ==>
2) 0 = u ux + v vx and 0 = u uy + v vy.
Using the Cauchy-Riemann equations, ux = vy and uy = -vx to eliminate derivatives of v, 2) becomes
3) 0 = u ux - v uy and 0 = u uy + v ux. Exercise 10 asserts that from this one can show that
4) 0 = (uu + vv)(ux ux + uy uy). I have not obtained that formula, but using 3) one can form the combination
5) 0 = (u ux - v uy)(u ux + v uy) + (u uy + v ux)(u uy - v ux) which simlifies to
6) 0 = (uu - vv)(ux ux + uy uy).
If w(z) is not a constant, then from the continuity of derivatives of analytic functions there must exist some domain throughout which ux or uy is non-zero. 6) indicates that in that domain either w(z) must vanish or its argument must be constant (45 or 225 degrees) so that in that domain w(z) must vanish or its modulus and argument must both be constant making w(z) constant. But if w(z) is constant in a domain, then by the uniqueness theorem (Nevanlinna and Paatero page 139) it is constant altogether.
Why mathematical induction is a deductive process?
"Mathematical induction" is a misleading name.
Ordinarily, "induction" means observing that something is true in all known examples and concluding that it is always true. A famous example is "all swans are white", which was believed true for a long time. Eventually black swans were discovered in Australia.
Mathematical induction is quite different. The principle of mathematical induction says that:
* if some statement S(n) about a number is true for the number 1, and
* the conditional statement S(k) true implies S(k+1) true, for each k
then S(n) is true for all n. (You can start with 0 instead of 1 if appropriate.)
This principle is a theorem of set theory. It can be used in deduction like any other theorem. The principle of definition by mathematical induction (as in the definition of the factorial function) is also a theorem of set theory.
Although it is true that mathematical induction is a theorem of set theory, it is more true in spirit to say that it is built into the foundations of mathematics as a fundamental deductive principle. In set theory the Axiom of Infinity essentially contains the principle of mathematical induction.
My reference for set theory as a foundation for mathematics is the classic text "Naive Set Theory" by Paul Halmos. Warning: This is an advanced book, despite the title. Set theory at this level really only makes sense after several years of college/university mathematics study.
What are the dimensions of a 100 gallon propane tank?
approximately 2 ft diameter by 4 ft height, Its volume must be slightly more than 100 gallons to allow expansion in hot weather and room for vapor.
Given an undirected graph G and an integer k?
Given an undirected graph G=(V,E) and an integer k, find induced subgraph H=(U,F) of G of maximum size (maximum in terms of the number of vertices) such that all vertices of H have degree at least k
How do you solve patterns and squences?
The answer depends on what the precise question is and on the level of mathematical knowledge that you are expected to have.
What are the 4 fundamental laws in mathematics?
I don't know why there should be 4 laws (=axioms) specifically.
In mathematics you can choose whatever system of axioms and laws and work your way with those. Even "logic" (propositional calculus) can be redefined in meaningful ways.
the most commonly used system is Zermolo-Fraenkel+choice:
http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Axioms
It has 9 axioms though, not 4.
One might want to take into consideration the rules of "logic" as basic laws:
http://en.wikipedia.org/wiki/Propositional_calculus
Another common set of axioms that can be created inside the ZFC system is peano arithmetic:
http://en.wikipedia.org/wiki/Peano_arithmetic
I hope I understood your question.
The short answer is "there is no such thing".
I think the questioner may have meant the 5 fundmental laws in mathematics, also known as the axioms of arithmetic, these are as follows:
A1 - for any such real numbers a and b, a+b=b+a, the commutative law
A2 - for any such real numbers a,b and c, a+(b+c) = (a+b)+c, the associative law
A3 - for any real number a there exists an identity, 0, such that, a+0 = a, the identity law
A4 - for any real number a there exists a number -a such that a+(-a)=0, the inverse law
A5 - for any real numbers a and b, there exists a real number c, such that a+b=c, the closure property.
These 5 axioms, when combined with the axioms of multiplication and a bit of logic/analytical thinking, can build up every number field, and from there extend into differentiation, complex functions, statistics, finance, mechanics and virtually every area of mathematics.
A pendulum clock that keeps correct time on earth is taken to moonit will run at 6 times faster?
No. It will run 2.45 times as SLOW.
What is the probability that a family of 2 children has exactly 2 boys?
Probability equals the number of ways an event can occur divided by the total number of events. The total number of events is (b=boy, g=girl) is bb, bg, gb, gg. The probability is then 1/4.
What is the value of moment of inertia of ellipse about its centroidal axis?
If an ellipse has a radius A long the x-axis and B along the y-axis (A > B) then the moment of inertia about the x-axis is 0.25*pi*ab^3
Are pressure and vacuum inversely proportional or directly?
I am not sure if they are proportional, but they are inversely related. High pressure makes a low vacuum, and low pressure makes a high vacuum.
