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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

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Collineations and Translations

What are the fixed points of a translation T? A fixed point is a point that is not moved by a given collineation, i.e. P = TP

How many of the 500 students are enrolled in all three classes?

500 students enrolled in at least 2 of 3 classes: English, Math, History. 150 students enrolled in both history and English, 300 enrolled in math & history, 170 enrolled in both math and English. How many of the 500 students are enrolled in all three classes?

Path connected subsets

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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.

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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

Path components

(See attached file for full problem description with proper symbols and equations) --- Let X be a topological space. Mapping a point to the path component which contains x establishes a map . Show that for any continuous map between topological spaces, there exists a map such that the following holds: ? ? for two co

Homomorphisms, Bijection Map and Continuous Map

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

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Infinite Series : Geometric Progression

Summation of series from n=2 to infinity of (pi/4)^(n/2) I don't know if I should set the whole thing up as a power of e and then calculate it, or calculate it as geometric series.

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

The sum of an 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?


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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?