Warmup

 

If a coconut tree begins producing 52 coconuts per year, 18 years after it being planted and it continues the average production till it is 80 years old, how many coconuts did it produce total?

Warm-up

 

If the average coconut gives 1.5 cups of coconut water, how many coconuts do you need for one gallon?

African American Mathematicians part 2

 

Today, I’ll be sharing more information on African American Mathematicians both male and female. Each and everyone of these mathematicians made an impact on their field and on history.  For instance, Martha Euphremia Lofton Haynes gained the title of the first african american woman to receive a Phd in mathematics from the Catholic University of America in 1943. Although christened Martha, she preferred using Euphemia.  Her father was a prominent dentist in Washington D.C and her mother although a stay at home mother, was actively involved in the Catholic Church.

After graduating from high school in 1909, she obtained her B.S. in mathematics four years later.  In 1917 she married Harold Haynes who eventually became superintendent of the african american school district in Washington, D.C.  Eventually she went to the University of Chicago to get her Master’s degree in Education with a significant number of classes in mathematics.  She went on to obtain her Phd in Mathematics in 1943.  She spent 47 years teaching in Washington D.C and taught everything from first grade to high school.  In addition to being the first African American woman to receive a Phd in mathematics, she was also the first woman to chair the District of Columbia School Board.  

During her lifetime, she also established the department of mathematics at Mines teacher college where she taught mathematics.  In addition, she taught mathematics at the District of Columbia Teachers College and occasionally taught at Howard University.  When she died, she left $700,000 to the Catholic University to support a Chair of Education and provide monies for a student loan program for the School of Education.

Then there was Walter Richard Talbet who was born in 1909 in Pittsburgh, Pennsylvania.  After graduating from high school, Walter attended the University of Pittsburgh in 1927 where he began studying physics.  Unfortunately, the depression hit so in order to afford college, Walter worked several part time jobs.  This meant he could not continue studying physics so he changed his major to mathematics.

He received his Bachelor of Science in mathematics in 1931 and continued studying at the University of Pittsburgh where he received his Phd in 1934.  Upon graduation, he accepted a staff position at Lincoln University in Jefferson City, Missouri.  He began as an assistant professor of mathematics but over time became a full professor, then Dean of Men and finally the head of the mathematics department in 1944.

He remained at Lincoln University until 1963 when he accepted the position as head of the mathematics department at Morgan State University till he retired and passed away the day after Christmas in 1977.  It is said that Dr. Talbot improved the ability of African Americans to study mathematics.  

As you can see both people went on to great careers in education.  Let me know what you think, I’d love to hear.  Have a great day.

Tropical Geometry

Monday, we looked at tropical arithmetic and algebra but that is only half of it.  The other half is tropical geometry which takes questions on Algebraic varieties and transforms these questions so they apply to polyhedral complexes.  Tropical geometry developed as an offshoot to Tropical Algebra due to questions that arose in computer science.  In addition, this topic is often referred to as tropical algebraic geometry.

There are three approaches to tropical geometry.  First is the synthetic approach, and is considered algebraic geometry applied to a tropical semifield.  It led to finding tropical theorems equivalent to those applied to algebraic curves and their associated geometry.  The second is the valuation theoretical approach which looks at tropical versions as shadows for the algebraic ones.  The final approach is the degeneration theoretical approach which is related to the degeneration theoretical approach for algebraic varieties.

Tropical geometry looks at things differently such as when polynomials which may be continuous in algebra are turned into piecewise functions. In fact, tropical geometry makes it easier to do polyhedral geometry because algebraic geometry doesn’t explain polyhedrals as well. It has been found that many of the invariant varieties are preserved when using tropical geometry.

Furthermore, tropical geometry works well with enumerative geometry, especially Mikhalken’s work.  Tropical geometry has developed its own formulas to determine the number of rational curves based on its degree. Furthermore, tropical geometry looks at solution spaces.  

If you look at this site, you can read four lectures by Dianne Maclagan given back in 2008/2009 to students taking a class in this topic.  The four lectures cover an introduction, fundamental theorems, more examples and explanations, finishing off with enumerative geometry.  Although it is only 67 pages long, it provides some fascinating information.

If you notice, tropical math encompasses both algebra and geometry to work together to form a new way of explaining things.  Let me know what you think, I’d love to hear.  Have a great day.

Tropical Math - Arithmetic/Algebra

 

I just heard this term the other day and the first thing my mind did was to picture numbers lying on the beach, sipping drinks, while sitting under umbrellas. Yes, I know it's not that but there are certain visuals we associate with certain words.  

Tropical math actually covers two different mathematical  sub groups - Algebra and Geometry.  It began developing around the beginning of this century and the adjective "tropical" came from several French mathematicians including Jean-Eric Pin.  

It is based on the "tropical semi-ring" which uses a set of real numbers with the additional element of infinity.  The tropical sum is defined as the sum of their minimums while the product is actually their sum. An example of a tropical sum for 4 and 8 is 4 because 4 is the minimum of both numbers while the product is 12 because 4 + 8 = 12. The notation used for tropical sum is a circle with a cross in the middle and tropical product is indicated by a circle with a dot inside it.  

What makes this even more interesting is that these two operations are actually commutative and can have the distributive property applied to them.  Furthermore, infinity is considered a natural element of addition and zero is a natural element for multiplication.  Unfortunately, subtraction within tropical arithmetic is a bit harder because there is no value for the phrase "10 - 3" thus it is best to stick with addition and multiplication.

As far as polynomials go, tropical monomials are actually linear functions with integer coefficients and a tropical polynomial is actually defined as "a finite linear functions with integer coefficients". Tropical polynomials are also continuous, composed of a piecewise of linear functions and it is concave. In addition, the Fundamental Theorem of Algebra apply to tropical linear functions.

Furthermore, curves in tropical algebra is shown in a hypersurface composed of all "roots" of the polynomials.  This can be extended to polynomials in two variables in which the curve is contained in the plane of real numbers squared with both bounded and unbounded edges.  In addition, the slopes are rational and if the sum of all vectors is taken, the result is zero.

When graphing such polynomials, one sees three half-rays heading off in three different directions. The degree of the polynomial determines the number of half-rays, vertices, bounded and unbounded edges but they half-rays tend to head off to the north, east, and southwestern directions.  This leads to the idea of linear spaces in which solving linear equations requires a person to determine the intersections of a certain number of hyperplanes.  

This is a brief look at tropical arithmetic and algebra but there is also the field on tropical geometry.  I'll be providing a short look at tropical geometry on Wednesday.  Let me know what you think, I'd love to hear. If you want to learn more about this check the internet for some really interesting reading.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If there about 540 peanuts are used to make 12 ounces of peanut butter, how many peanuts are needed for 32 ounces of peanut butter.

Warm-up.

 If one pound of pecans yields 4 cups of pecans, how many pounds of pecans will you need for 128 pies if one pie uses 1 3/4 cups of pecans?