Geometric Definition and 790 Threads

Dear members, Do you maybe know some good reference for the application of differential geometric methods in biology, especially evolutionary theory, game theory, population ecology. I kinda feel that the application of mathematical methods in biology in general is a field of increasing...

Or is it something separate that acts on a geometric space? So we know that the Euclidean space is a vector space. But is it geometric? I ask this because in group theory, the group elements are the operators acting on another set, but clearly we see that this doesn't mean that the group...

What is the Geometric Proof for the product of two negative real numbers being a positive real number?

Prop 5.11 from John M. Lee's "Introduction to Topological Manifolds":If K is a simplicial complex whose geometric realization is a 1-manifold, each vertex of K lies one exactly two edges. This proposition confuses me. If we look at the geometric realization of a simplex with two vertices, then...

Hello! I am reading so very introductory stuff on geometric algebra and at a point the author says that, as a rule for calculation geometric products, we have that ##e_{12..n}=e_1\wedge e_2 \wedge ...\wedge e_n = e_1e_2...e_n##, with ##e_i## the orthonormal basis of an n-dimensional space, and I...

Hiya everyone, Alright ? I have a simple theoretical question. In a decreasing geometric series, is it true to say that the ratio q has to be 0<q<1, assuming that all members of the series are positive ? What if they weren't all positive ? Thank you in advance !

Hello all, Three consecutive elements of a geometric series are: m-3i, 8+i, n+17i where n and m are real numbers. I need to find n and m. I have tried using the conjugate in order to find (8+i)/(m-3i) and (n+17i)/(8+i), and was hopeful that at the end I will be able to compare the real and...

Given two positive numbers a and b, we define the geometric mean and the arithmetic mean as follows G. M. = sqrt{ab} A. M. = (a + b)/2 If a = 1 and b = 2, which is larger, G. M. or A. M. ? G. M. = sqrt{1•2} G. M. = sqrt{2} A. M. = (1 + 2)/2 A. M = 3/2 Conclusion: G. M. > A. M. Correct...

Homework Statement [/B] Given a general triangle ABC, find the geometric locus of points such that the three orthoprojection onto the sides of the triangle are aligned. Homework Equations Let's call A', B', and C' the orthoprojection of a given point M onto (AB) , (BC) , and (AC). M satisfies...

Hi All, I would like to know which are the journals of mathematics that publish papers on geometric constructions. Most of the journals on geometry I have found tends to an analytic approach. Best wishes, DaTario

My Statistics textbook does not prove this. The author think it is commons sense. I am not sure about this proof. Thank you.

Homework Statement $$z^2 + z|z| + |z|^2=0$$ The locus of ##z## represents- a) Circle b) Ellipse c) Pair of Straight Lines d) None of these Homework Equations ##z\bar{z} = |z|^2## The Attempt at a Solution Let ##z = r(cosx + isinx)## Using this in the given equation ##r^2(cos2x + isin2x) +...

find the sum of this infinite geometric series: 1 - √2 + 2 - 2√2 + ... a.) .414 b.) -2.414 c.) series diverges d.) 2 I found that the common difference is 2, so I calculated this: S∞= -.414/-1 s∞= .414 So i got that the answer is A, but will you check this?

Homework Statement Determine the order of the poles for the given function. f(z)=\frac{1}{1+e^z} Homework EquationsThe Attempt at a Solution I know if you set the denominator equal to zero you get z=ln(-1) But if you expand the function as a geometric series , 1-e^{z}+e^{2z}... I...

Homework Statement A farsighted boy has a near point at 2.3 m and requires eyeglasses to correct his vision. Corrective lenses are available in increments in power of 0.25 diopters. The eyeglasses should have lenses of the lowest power for which the near point is no further than 25 cm. The...

Need help with a homework question! The question gives: The first three terms of a geometric sequence are sin(x), sin(2x) and 4sin(x)cos^2(x) for -π/2 < x < π/2. First I had to find the common ratio which is 2cos(x) Then the question asks to find the values of x for which the geometric series...

Hello, quick question here I am studying mathematical astronomy / the history of , and I have noted that by raising the average sidereal period of any planet in our solar system, to the power .666666, that you are left with the average distance of that planet from the Sun, in AU I was told...

Homework Statement I have a range of numbers numbers n_i, each with a different weight w_i that sum up to 1. To keep things simple, let's take the case where we have three numbers with the following weights: n_i w_i ------------------------------...

Hey! :o We have two vectors $\vec{u}, \vec{v}\in \mathbb{R}^2$. I want to describe geometrically the set of vectors $\vec{z}$, for which it holds that $$\vec{z}=\lambda{u}+(1-\lambda)\vec{v}$$ with $0\leq \lambda \leq 1$. Does this set describe all the points that are on the line that...

I am investigating the mathematical properties of a vector-product. I am wondering if it might be old-hat in GA (which is new to me)? I am using the working-title "spin-product" for a vector-product that combines RANDOM rotation-only of a direction-vector [a unit 1-vector; say...

Hello! I would really appreciate it if somebody could recommend to me any books on geometric phases especially on Berry's phase. Thanks!

Here's an interesting article from Quanta magazine: https://www.quantamagazine.org/20170223-bootstrap-geometry-theory-space/ and some backstory: https://en.wikipedia.org/wiki/Bootstrap_model

Homework Statement Hi, I have the probabilty density: ##p_{n}=(1-p)^{n}p , n=0,1,2... ## and I am asked to find the characteristic function: ##p(k)= <e^{ikn}> ## and then use this to determine the mean and variance of the distribution. Homework Equations [/B] I have the general expression...

Hi all, I'm reading a paragraph from "Geometric Algebra for Physicists" - Chris Doran, Anthony Lasenby. I'm quite interested in applying GA to QM but I've got to a stage where I am not following part of the chapter and am wondering if someone can shed a little light for me. The part...

Homework Statement I want to show that ## \sum\limits_{n=1}^{\infty} log (1-q^n) = -\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty} \frac{q^{n.m}}{m} ##, where ##q^{n}=e^{2\pi i n t} ## , ##t## [1] a complex number in the upper plane.Homework Equations Only that ## e^{x} =...

Hi folks, I am a bit confused with the extreme condition used in the calculus of variations: δ = 0 I don't understand this rule to find extreme solutions (maximum or minimum) If in normal differential calculus we have a function y = y(x) and represent it graphically, you see that at the...

$\tiny{242.WS10.a}$ \begin{align*} &\textsf{use the techniques of geometric series} \\ &-\textsf {telescoping series, p-series, n-th term } \\ &-\textsf{divergence test, integral test, comparison test,} \\ &-\textsf{limit comparison test,ratio test, root test, } \\ &-\textsf {absolute...

$\tiny{206.10.3.17}$ $\textsf{Evaluate the following geometric sum.}$ $$\displaystyle S_n=\frac{1}{2}+ \frac{1}{8}+\frac{1}{32}+\frac{1}{128}+\cdots + \frac{1}{8192}$$ $\textsf{This becomes}$ $$\displaystyle S_n=\sum_{n=1}^{\infty}\frac{1}{2^{2n-1}}=\frac{2}{3}$$ $\textsf{How is this morphed...

$\tiny{242.10.3.27}$ evaluate $$S_j=\sum_{j=1}^{\infty}3^{-3j}=$$ rewrite $$S_j=\sum_{j=1}^{\infty} 27^{j-1}$$ using the geometric formula $$\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{1-r}, \left| r \right|<1$$ how do we get $a$ and $r$ to get the answer of $\frac{1}{26}$ ☕

$\tiny{206.10.3.54}$ $\text{Write the repeating decimal first as a geometric series} \\$ $\text{and then as fraction (a ratio of two intergers)} \\$ $\text{Write the repeating decimal as a geometric series} $ $6.94\overline{32}=6.94323232 \\$ $\displaystyle A.\ \ \...

$\displaystyle\text{if} \left| r \right|< 1 \text{ the geometric series } a+ar+ar^2+\cdots ar^{n-1}+\cdots \text{converges} $ $\displaystyle\text{to} \frac{a}{(1-r)}.$ $$\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{(1-r)}, \ \ \left| r \right|< 1$$ $\text{if} \left| r \right|\ge 1 \text{, the series...

a textbook I'm reading has pointed out geometric similarities between atomic orbitals and multipoles. do these similarities originate from a mutual dependence on the spherical harmonics? if so, how does something like a dipole or a quadrupole depend on the Ylm's? Note that my I did my...

Homework Statement I Have a differential equation y'' -xy'-y=0 and I must solve it by means of a power series and find the general term. I actually solved the most of it but I have problem to decide it in term of a ∑ notation! Homework Equations y'' -xy'-y=0 The Attempt at a Solution I know...

If the second term is 6 and the 5th term of a geometric progression is 48.Find the first term and the common difference of it The sum of certain number of terms of the above progression from first term is 381.Find the number of terms of it. Any ideas on how to begin ?

Hello, I was just wondering if there is a geometric interpretation of the trace in the same way that the determinant is the volume of the vectors that make up a parallelepiped. Thanks!

Calculate the sum for the infinite geometric series $4+2+1+\frac{1}{2}+...$ all I know is the ratio is $\frac{1}{2}$ $\displaystyle\sum_{n}^{\infty}a{r}^{n}$ assume this is used

This is a rough sketch, (Happy) Now apart from constructing the triangle can you help me to located the point D & Obtain the location of point E on side AB such that ACDE is a trapezium ,only using an straight edge and a compass. (Crying)

Hello, do you know of any books similar in style to Callahan's Advanced Calculus book(a book that explains the geometrical intuition behind the math)? This goes for any subject in mathematics(but especially for subjects like vector calculus, differential geometry, topology). Thanks in advance!

need a hand with a revision question, I don't quite understand how to go about solving it question is attached below

Looks like this question is going to make a long thread. :) This is what the problem states Using the pair of compass and a straight edge, Then, After That, Thereafter, Thereafter, The question states to mark the point where the tangent and AD produced as 'X'.Now can you help me...

I recently found that formulation of quantum mechanics as a hamiltonian flow in a Kahler manifold, where there is a classical hamiltonian, hamilton equations, poisson brackets and etc. And while the mathematics in terms of differential geometry is all fine and good, I'm having problem finding...

Homework Statement Homework Equations no equations required 3. The Attempt at a Solution a) so for part c) i came up with two formula's for the tortoise series: the first formula (for the toroise series) is Sn = 20n This formula makes sense and agrees with part a). for example, if the...

Homework Statement Let a,b and c be lengths of sides in a triangle, show that √(a+b-c)+√(a-b+c)+√(-a+b+c)≤√a+√b+√c The Attempt at a Solution With Ravi-transformation the expressions can be written as √(2x)+√(2y)+√(2z)≤√(x+y)+√(y+z)+√(x+z). Im stuck with this inequality. Can´t find a way to...

I am trying to learn Geometric Algebra from the textbook by Doran and Lasenby. They claim in chapter 4 that the geometric product ab between two vectors a and b is defined according to the axioms i) associativity: (ab)c = a(bc) = abc ii) distributive over addition: a(b+c) = ab+ac iii) The...

Homework Statement in this question , Lm / Lp = 1/ 6 ? or Lp / Lm = 1/6 ? Lp = length of prototype , Lm = Length of model Homework EquationsThe Attempt at a Solution i really have no idea... can someone help please?

Homework Statement Background from previous parts of the question: A simple isotropic dielectric occupies the region x>0, with vacuum in region x<0. I've found the wave equations for the electric field Incident, reflected and transmitted to prove Snell's law (Sinθ/Sinθ = c/c' = √εr) and the law...

Here is a question that I have a problem with, It doesn't seem to have a solution: An increasing sequence that is made of 4 positive numbers, The first three of it are arithmetic series. and the last three are geometric series. The last number minus the first number is equal to 30. Find the sum...

I'm a little confused on geometric series. My book says that a geometric series is a series of the type: n=1 to ∞, ∑arn-1 If r<1 the series converges to a/(1-r), otherwise the series diverges. So let's say we have a series: n=1 to ∞, ∑An, with An = 1/2n An can be re-written as (1/2)n, which...

My professor states that "A matrix is diagonalizable if and only if the sum of the geometric multiplicities of the eigen values equals the size of the matrix". I have to prove this and proofs are my biggest weakness; but, I understand that geometric multiplicites means the dimensions of the...

Is there a simple geometric interpretation of the Einstein tensor ? I know about its algebraic definitions ( i.e. via contraction of Riemann's double dual, as a combination of Ricci tensor and Ricci scalar etc etc ), but I am finding it hard to actually understand it geometrically...