Algebraic Geometry

Algebraic geometry is a field of mathematics which combines two different branches of study, specifically algebra and linear algebra. Algebra and linear algebra can be defined as (1):

Algebra: A field of mathematics which analyzes polynomial equations using letters and symbols to represent unknown values.

Linear Algebra: The investigation of a system of linear equations using several variables.

Thus, algebraic geometry uses several variables for systems of polynomial equations. Algebraic geometry does not necessarily aim to solve these equations, but rather formulate statements about the geometric structure for the set of solutions belonging to these polynomial equations (algebraic varieties). When studying algebraic geometry, polynomial equations can be of various degrees (denoted as d). Polynomial equations of several variables are investigated to discover what their geometric structure might be.

Algebraic geometry is a rather abstract field of mathematics. It is a field which involves a lot of both algebra and geometry, although for a beginner, it may initially seem to be more “algebra heavy”. Algebraic geometry allows for two-dimensional shapes to be defined in three dimensions. For example, if a set of points defined as (x,y,z) satisfy a particular equation, such as x²y² + z² – 3 = 0, this can be translated into a 3-D shape.

In mathematics today, algebraic geometry is often utilized when studying higher complexity math problems related to geometry in other fields. Thus, developing an understanding of algebraic geometry is necessary for multiple fields of mathematics.

Reference:

http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf

BrainMass Solutions Available for Instant Download

Basic postulates, and theorems

Undefined terms, basic definitions of geometry, postulates, theorems, corollaries and problems based on the theorems.

Steps on Calculating the Angle of a Triangle

In a triangle ABC, the measure of angle B is 5 times the measure of angle A, and the measure of angle c is 3 degrees less than 4 times the measure of angle A. A is x degrees, B is 5x degrees and C is 4x-3 degrees.

Unit Conversions: Inches to Feet

Maria is using a cylindrical oxygen tank while scubadiving. It holds 80 cubic inches of air. How many cubic feet of air is she using? Round to the nearest hundredth if necessary.

Calculate the area of a playground

A playground is 37 ft by 52 ft and surrounded by chain-link fencing. Calculate the area of given playground. (Round to the nearest tenth)

Find the length or width of similar triangles or rectangles.

Please provide step-by-step solutions. 1 A drawing on a transparency is6.5 inches wide and 3 inches long. The width of the image of the drawing projected onto the screen is 13 feet. How long, in feet, is the drawing on the screen? 2. Triangles ABC and DEF are similar. The length of the sides of ABC are 56, 64, and 72

Proof that three lines with only one intersection are dependent

I want a solution that explains why three lines that intersect at only one point is not independent. And is there a way of proving that only one of them is dependent? How do we differentiate between dependent and independent linear systems? Please see the attached diagram for the full problem outline.

Geometry: Incidence geometry

1. Construct a model of incidence geometry that has neither the elliptic hyperbolic nor Euclidean parallel properties. 2. Consider a finite geometry where the points are interpreted to be the six vertices of a regular octedron and the lines are sets of exactly two points. see attached

Question on central force

A particle of unit mass is projected with speed (see attached) at right angles to the radius vector at a distance a from a fixed point O and is subjected to a force F(r) = μ/r^2, with μ a positive constant. Show that the particle moves on a path which passes the centre at a distance of 3a at its maximum. See attached

What is a fair price for the competition?

A community 5K run will award $50 to the winner. 55 people enter the race, and they each pay an entry fee of $20. Assuming they are all equally likely to win, what is a fair price for the competition? Round to the nearest cent. (Points : 6)

Illustrating a Convergent Geometric Series

Please work the attached problem. This problem is adapted from the famous Greek paradox "Achilles and the Tortoise": suppose a rabbit and a turtle are in a race. The rabbit is ten times as fast as the turtle, and the turtle is given a 100-foot head start. The paradox claims that the turtle will always be ahead of the rabbit si

A geometric series with a geometric rate

** Please see the attachment for the complete problem description ** iv) Series C and S are defined as follows: C=2cos x + 4 cos 2x + 8 cos 3x + ... + 2^n cos nx S=2 sin x + 4 sin 2x + 8 sin 3x + ... + 2^n sin nx Show that C+iS is a geometric series. Hence show that: S= 2sin x - 2^n+1 sin ((n+1)x) + 2^n+2 sin nx /

Closure of a Set in a Topoogical Subspace

Let Y be a subspace of X and let A be a subset of Y. Denote by Cl(A_X) the closure of A in the topological space X and by Cl(A_Y) the closure of A in the topological space Y. Prove that Cl(A_Y) is a subset of Cl(A_X) . Show that in general Cl(A_Y) not equal Cl(A_X). See the attached file.

Cauchy sequence

Show directly from the definition that a_n = 1 + ½! + 1/3! + ... + 1/n! is a Cauchy sequence.

Length of the shortest path

A spider is sitting at A, the midpoint of the edge of the ceiling in the room shown in Figure 11-53. It spies a fly on the floor at C, the midpoint of the edge of the floor. If the spider must walk along the wall, ceiling, or floor, what is the length of the shortest path the spider can travel to reach the fly?

Monthly paymnets

A real estate partnership predicts it will pay $300 at the end of each month to its partners over the next 6 months. Assuming the partners desire an 8% return compounded monthly on their investment how much should they pay?

Geometric Series and Sums

Please see the attached Microsoft Word document. Thanks for your expertise. Part A Find a specific geometric that sums up to 30. Part B Can you find a specific geometric series that sums to -30 ? Part C Can you find a specific geometric series summing up to - ? Part D Determine precisely which real numbers can be sums of

Convergence and Uniform Convergence

See attached. Let f_n(x):R-->Rbe the function... Show that f_n(x)-->f(x) for each x in R Show that f_n does not converge uniformly to f.

Economic Order Quantity: How Many Parts Should Be Ordered?

Tanaka Suzuki inventory control manager for Itex receives wheel bearings from wheel right a small producer of metal parts. Unfortunatley wheel right can only produce 500 wheel bearings per day Itex recieves 10000 wheel bearings from wheel-right per year. Since Itex operates 200 working days each year its average daily demand for

Nonempty Set Bounded Range

See attachment 1) Let > 0 and >0, and . Show that and are - neighborhoods of for appropriate values of . 2)Let and be nonempty sets and let : have bounded range in . Let : and be defined by , Prove that We sometimes express this by writing .

Suppose A, B, C, and D are sets.

College level proof before Real Analysis. (see attached)

Large shipment of TV sets is accepted - Acceptance sampling

A large shipment of TV sets is accepted upon delivery if an inspection of ten randomly selected TV sets yields no more than one defective TV. a) Find the probability that this shipment is accepted if 5% of the total shipment is defective. b) Find the probability that this shipment is not accepted if 10% of this shipment i

Sum of a Series (Geometric Progression)

Find the sum of the series: From k =0 to infinity for SIGMA 1/(4^k).

Magic Square

Problem: Create a 3 x 3 magic square using the whole numbers from 9 to 17 so the sum of the numbers in each row, column, and diagonal is 39.

Infinite Taylor Series

(b) Find an infinite series for ln 2 by setting x = 1/2. Now, find another infinite series for ln 2 by setting x = 2. (c) Find an infinite series for pi by setting x = 1 + i. Infinite Taylor Series. I only need help with parts (b) and (c). See attached file for full problem description.

Closed Form for an Infinite Series

Find a closed form for the infinite series 1+ x + x^2 + x^3 +.... show that the closed form is only valid if /x/< 1, where x may be a complex number (i.e. x=x1 +ix2) In the complex plane, draw a diagram that shows successive values of 1, 1+x, 1+x+x^2 , 1+x+x^2+x^3, etc. for x=i. Does the series converge for x=i?

Explore BrainMass