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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. 


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)

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.

How to cut the rug into two pieces.

If there is a rectangular rug with a hole in the middle, how do you cut the rug into two pieces, so that when you put the pieces back together, you have a square rug with no hole? See attached file for full problem description (including diagram).

Cylindrical can holding tennis ball

A cylindrical can is just big enough to hold three tennis balls. The radius of a tennis ball is 5 cm. What is the volume of air that surrounds the tennis balls?

Geometric sequence

Please show how you arrived at the following solution Use the geometric sequence of numbers 1, 2, 4, 8,..... to find the following : a. What is r, the ratio between 2 consecutive terms b. Using the formula for the nth term of a geometric sequence, what is the 24th term.

Sequences and Series

I need assistance in discerning sequences and series including the classical grains and checkboard problem

Geometric Series and Sums : Checkerboard Story Problem - Get your checkerboard and place one penny on the first square. Then place two pennies on the next square. Then place four pennies on the ....

CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulled a young lady out of the flaming truck. Farmer Brown came out and gratefully thanked the traveling salesman for saving his daughter's life. Mr. Brown

Homomorphisms, Bijection Map and Continuous Map

1) Prove that the map GIVEN BY is a homomorphism between the real line and open interval (-1,1). 2) Let be the map given by a) show that f is a bijection map b) show that f is a continuous map c) If f a homomorphism? Justify your answer. Please see the attached file for the fully formatted problems.

Multipliers and Geometric Series: Application to U.S. Economy

Suppose that, throughout the U.S. economy, individuals spend 90% of every additional dollar that they earn. Economists would say that an individual's marginal propensity to consume is 0.90. For example, if Jane earns an additional dollar, she will spend 0.9(1)=$0.90 of it. The individual that earns $0.90(from Jane) will spend

Dimensions of the building

Three buildings abut as shown in the diagram below. What are the dimensions of the courtyard and what is the perimeter of the building? See attachment for diagram

Sum of infinite geometric series

A regular hexagon is inscribed in a circle of radius 100. The area of that hexagon outside the square is shaded. Another circle is inscribed in the hexagon, and hexagon is inscribed in the second circle. The area of the second circle which lies outside the hexagon is shaded. This process continues to infinity. What is the

Volume and water displaced.

When a chunk of iron is completely submerged into a cylindrical tank of water, the water rises 6 inches. The diameter of the tank is 18 inches. What is the volume of the chunk of iron?

Find the location of a pole in a rectangle.

Johnny green has a rectangular piece of ground. He places a pole in the gound five feet from the upper right corner. The same pole is nine feet from the lower right corner. It is also thirteen feet from the lower left corner. How far is the pole from the upper left corner?


Note: C = containment int = interior ext = exterior cl = closure Could you please prove if S C T, then a)int(S) C int(T) b)ext(T) C ext(S) c)cl(S) C cl(T)

Geometric and arithmetic series, pulleys in parallell

Resistances in series can be reduced to a unique resistance R such that eq(1) R= r1 + r2 +...+ rn in Parallel , we have eq (2) 1/ (1/r1 +1/r2 +...+ 1/rn) For the pulleys , to reduce the effort to keep a block and tackle (which has a mass M at he end) in equilibrium , the necessary force F to

Urysohn's lemma

A Hausdorff space is said to be completely regular if for each pt. x in X and closed set C with x not in C, there exists a continuous function f: X --> {0,1} s.t. f(x)=0 and f(C)={1}. Show that if a space is normal, it is completely regular. How do I use Urysohn's lemma along with Hausdorffiness to show this. Thank

Scale model

I have a model of Mercury that is 5 inches in diameter the actual diameter is 4880 km. What is my scale?


Find equation for hyperbola with center at (4,9); a=5; c=13; traverse axis parallel to the x axis


Find equation for parabola: focus at (11,7) and vertex at (6,7)


Phil and Fran are photographers who develop their own pictures and also restore old photographs. They have an enlargement and reducing machine that can change the size of photographs. A customer asks Fran to enlarge a 3 inch by 5 inch photograph to 8 inch by 10 inch. Can this be done without cutting or distorting the picture?