# The network wave equation

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While his paper focuses on the heat equation on networks, given Neumann boundary conditions instead of Dirichlet, the eigenvalues of the heat equation are given to be constant $\mu$ such that

$\frac{{\partial }^{2}u}{\partial {x}^{2}}=\mu u$

Since $\mu$ has to be less than or equal to zero in order for a solution to the heat equation, the eigenvalues of the heat equation are the squares of the eigenvalues of the wave equation

$\frac{{\partial }^{2}u}{\partial {x}^{2}}={\lambda }^{2}u$

Therefore, taking the square roots of the eigenvalues $\mu$ , we can recover the eigenvalues of the wave equation on our network. For the rest of our stated examples, the eigenvalues mentioned are assumed to be eigenvalues of the wave equation.

Von Below studies the network heat equation with Neumann conditions at the endpoints

$\frac{\partial {u}_{j}}{\partial t}={a}_{j}\frac{{\partial }^{2}{u}_{j}}{\partial {x}_{j}^{2}}$

where ${u}_{j}$ is the temperature of the $j$ th edge, ${a}_{j}$ is the positive diffusion coefficient of the $j$ th edge, and ${x}_{j}\in \left[0,{\ell }_{j}\right]$ is position on the $j$ th edge, which has length ${\ell }_{j}$ . The notation is the same as the notation with which the network wave equation was described.

By defining simply a set of nodes and connections between them, Von Below calculates the eigenvalues and eigenmodes of a network with the help of spectral graph theory (a study of the connections between a graph and the eigenvalues of its adjacency matrix). The eigenvalues of the heat equation on networks of strings progress according to several different patterns, some of which are common to every network, some of which are network-specific and dependent on the eigenvalues of the network graph's row-normalized adjacency matrix.

For nonspecific parameter values, the best we can hope to do is to solve a transcendental equation for our eigenvalues. For specific parameter values, such as when ${a}_{j}={\ell }_{j}^{2}$ , we can solve analytically for both the eigenfunctions and eigenvalues of the heat equation, breaking both down into cases. There are, in general, 5 total classes into which eigenpairs of the heat equation on a network can be organized, with each network having at most 4 classes applicable to it.

For each one of these classes mentioned, there is an infinite number of eigenvalues associated with it. Organizing the progression by $k$ instead of by magnitude gives more insight into the nonlinear progression of our network eigenvalues - there is a clear bifurcating/splitting patten of the eigenvalues as $k$ increases. As the progression of eigenvalues splits into two separate patterns, each pattern is linear with different rates.

The fundamental difference here is that the eigenvalues for the one dimensional transverse network wave equation progress with two different rates. While there are two distinct sets of eigenvalues for our three dimensional string, to have eigenvalues progressing at different rates for one dimensional transverse string motion is atypical. This isn't a complete explanation for the irregular progression of the eigenvalues for the three dimensional wave equation; however, it does offer some clues as to the nonlinear ordering of those eigenvalues.

## Acknowledgments

Thanks to Jeremy Morell, whose work I built upon, as well as Robert Likamwa, who helped nail down a lot of details concerning visualization and graphics. Thanks to Dr. Cox and Dr. Embree for their guidance their trust that I could actually do something with the project they gave me.

This work was partially supported by NSF DMS Grant 0240058

#### Questions & Answers

more explanation on GDP
it is a country total out put of goods and services divided by the total population of the country.I think it can also be derived from the country labour force,,because it mostly depend on the labour force and the level of technology .
Tantoh
labour force and technological progress leads to greater production increases the GDP
Ahmed
What is economics?
by this time
Emmanuel
It is a social science that analyses production,distribution and consumption of goods and services
Emmanuel
A social science that study human behavior in relationship with decision making
Jessica
What are the typical patterns of GDP for a high-income economy like the United States in the long run and the short run?
What are the limitation and significant of macroeconomic
explain the significance of concerpt of opportunity cost in planning
what is meant by the price elasticity of demand?
when price of a commodity increase it's demand contracts , and whe the price of a commodity decreases it's demand expands so the degree of change in demand in response to change in own price of the commodity is called PED . Ed = percentage change in quantity demanded / percentage change in price
shaswat
What are the limitations of macroeconomic and their segnificant
Discuss the role of competition in stimulating economic growth?
competition stimulate economic growth because in such types of economy,they is no monopoly power every supplier will want to produce to meet customers choice which brings about quality production and attract invested and customers into such economy
Koka
competition creates Monopoly because of economy of scale. it's not antithesis but different side of same coin
toko
competition result in high economic growth since every firm will intend to provide quality services and products to meet customers needs and requirements unlike in Monopoly situation where a firm just provide what it want to resulting in large stock piles of unwanted products ,ie inefficiency, howev
Mark
microeconomics study part of the economy but macroeconomic study the whole economy
studying the whole economy, solving the problem of the economy and building up the economy
Olokun
micro means small while macro means large
Olokun
standard of living is the footsteps of an economy because it plays important role for country to have crucial view about their budget ,import and export
Olokun
it will be differ because economic agent will only take their views on some part of household
Olokun
can opportunity cost be zero
how many types of transportation do we have
Jacob
yes. when a customer's purchasing power is high, he may have d ability to purchase all he needs, dt makes opportunity cost zero
George
please can give more explanation on this question
OBED
what are the factors production
Labour capital entrepreneurs
Leta
Land,capital, labour,and the entrepreneur
Tantoh
I will like to know use of calculus in economics
do they use it in economics?
Pranav
I want to know if I should take calculus or statistics and probability my senior year of highschool
Yahir
yes for example in monopolistic competitive market..... TR=TC* & THIS CALCULATED BY CHANGING( DERIVATIVE LAW) MR =MC ** WILL BE THE FORMULA THAT USE.
Leta
please in which topic in economic is the question coming from.
Tantoh
from PCF in economics
Leta
why is unitary proportional to responsiveness
any tip for igcse economics exam?pls
well
The
What is a market
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
price of the related goods 2 price of the given commodity 3 income of the consumer 4 taste and preference 5 expectation in the future price
John
pls the taste and preference
Nas
explain briefly
Nas
a consumer taste and preference commodity changes for a time the man becomes
John
sorry sorry
John
is when the price of a commodity becomes high and can't afford example Samsung instead of iPhone
John
consumers who have high intense for goods will purchase the goods even if the price of that commodity increases because he or she preferred that commodity.people will be prefer iphone as its price increase
Yussif
as usual bad taste of preference is when a consumer regrets from one commodity to another in terms of the price
John
thanks alot
Nas
you're welcome
John
#Preference; #Income #Test
Dereje
#price Of Commodity #Income #Taste #Preference
Dereje
#Market is The Place Where Buyers And Sellers Are Exchanging Their Goods And service. #
Dereje
difference between macro and micro economics
Lawrence
Microeconomic Study about individual consumers market But Macroeconomis Study General economic Process Such As #Aggregate Demand #Aggregate Supples #GDp= #GNp
Dereje
nice so can u run down a brife discussion on GDP
Lawrence
good
Chinex
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
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