Images, posts & videos related to "Existence Theorem"

A Series of Figures Used in a Proof of the Erdรถs-Faber-Lovasz Conjecture for Hypergraphs Satisfying Certain Critรฆria & an Algorithm for Actually Yielding the Colouring Of Which the Theorem Guarantees the Existence

Understanding the theorem/proof of the existence of nash equilibrium(s).

Hi all,

I have a decent math background (master of data science), definitely not bachelor of mathematics level (maybe the equiv. of a first/second year math student), but I don't even know where to start with understanding the proof and theorems that prove nash equilibria in games.

Could anyone shed some light?

The Four-Color theorem follows trivially from the non-existence of a complete planar graph with 5 vertices. reddit.com/r/math/commentโฆ

TIL the 'mountain pass theorem' is an existence theorem from the calculus of variations. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. en.wikipedia.org/wiki/Mouโฆ

The existence and uniqueness theorem

I'm confused as to how it works and why it's so important for differential equations

Why/When does Kakutani's fixed point theorem imply the existence of a nash equilibrium?

For example, in the game of "choose a number, whoever chooses a higher number wins", there is no nash equilibrium, even though it is zero-sum. Kakutani's fixed-point theorem requires that the domain is compact.

I'm reading the Wikipedia article and it seems that they're only applying Kakutani's fixed point theorem to the case where the number of choices is finite, so the set of strategies is "play choice i with probability p_i". Can we get the existence of nash equilibriums in more general situations?

[Differential Equations] Question about applying existence and uniqueness theorem.

**Consider the differential equation:**

dy/dt = y/(t^2)

**a. show that the constant function y(t) = 0 is a solution.**

I know that the existence theorem states that if f(t, y) is continuous near (t, y) then the solution exists.

I reasoned that f(t, y) = y/(t^2) is continuous near any (t, y) where t โ 0 so the solution y(t) = 0 exists where t โ 0.

Is this a valid reasoning?

**b. Show that there are infinitely many other functions that satisfy the differential equation, that agree with this solution when t โค 0, but that are nonzero when t > 0. [Hint: You need to define these functions using language like "y(t) = ... when t โค 0 and y(t) = ... when t > 0."]**

This is the part that I am having the most trouble with. I believe part of the answer needs to touch upon why y(t) = 0 is not unique, however, I'm not sure exactly how I go about explaining this. I know that the uniqueness theorem states that if โf/โy is continuous near (t, y) then the solution is unique. Ho

... keep reading on reddit โกTIL that Bayes' theorem was first used to try to mathematically prove the existence of God scientificamerican.com/arโฆ

Theorems where existence is known, but there is no known example.

Just going through my notes on Ergodic Theory as I revise for my final, found the statement that

> One can show that `[; x_n = \alpha^n ;]`

is uniformly distributed mod 1 for almost all `[; \alpha > 1 ;]`

, however not a single example of such an `[; \alpha ;]`

is known!

I love facts like this, where something has been proven to be true almost everywhere (ie. in this case picking a random `[; \alpha \in \mathbb{R} ;]`

gives probability 1 of the statement being true), yet noone can find an example for which the statement is known to hold.

I know I've seen more of these before, and would love to see some more cool examples.

unfortunately existence theorems are often nonconstructive

Does the completeness theorem of FOL imply the existence of non-standard models of arithmetic in which a Gรถdelsentence codifying its own unprovability is false?

If yes: what consequences does this have? Informally, the negation of the Gรถdelsentence implies the existences of a proof of it. Does this mean that we have to accept 'inconsistent models', in which a falsity is provable? Is this where dialetheism and paraconsistent logic comes in?

If we deny that there can be 'inconsistent models', do we have to deny that the negation of a Gรถdelsentences actually implies the existence of a proof?

Are there any theorems that affirm existence of a kind of object, but for which we absolutely cannot find an example?

I was reading about Tarski's circle-squaring problem and I started wondering, what if we cant ever find an example? I imagine there are theorem that prove sets exists but for which you can never find its elements.

I'm not sure what the definition of "finding" should be. Some could consider an "implicit definition" of the object to not be count as finding the object. But the definition of implicit/explicit seems a bit arbitrary.

Can we use Bayes' theorem to prove the existence of god? reddit.com/r/math/commentโฆ

Identity, Haecceity, & the Godzilla Problem [abstract + link to PDF]: "In standard first order predicate logic with identity it is usually taken that a=a is a theorem for any term a. It is easily shown that this enables the apparent proof of a theorem stating the existence of any entity whatsoever." arxiv.org/abs/1709.04607

So I recently learned that there is a polynomial such that the existence of integer roots is an undecidable statement in ZFC (Matiyasevich's Theorem). I have questions.

> One can write down a concrete polynomial PโZ[x1,...x9] such that the statement "there are integers m1,...,m9 with P(m1,...,m9)=0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent).[16] This follows from Yuri Matiyasevich's resolution of Hilbert's tenth problem; **the polynomial is constructed so that it has an integer root if and only if ZFC is inconsistent**.

https://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC#Number_theory

My first question:

Does the undecidability of the problem imply that P has in fact no integer roots and we just can't prove it? Because if P had an integer root we could just plug it in, check that it's zero and call it a proof of the existence of an integer root of P.

My problem with that though is that I always thought that Gรถdel's completeness theorem guarantees that every (semantically) true statement is provable. So does that mean that there is a model of ZFC where P has integer roots and a model of ZFC where P do

... keep reading on reddit โกA shape whose impossibility might have been an elegant theorem, but whose existence may be much more elegant." gomboc.eu/gomboc_english.โฆ

TIL Pythagoras was described as a cult leader and may have not existed all, a mythical figure, described as having a golden thigh and being a son of Apollo. His famous theorem was discovered prior to his "existence" by the Egyptians history.com/news/history-โฆ

"Carrier uses a scientific method of examining historical claims (rare among historians) by using Bayes' theorem as a method to establish the limits of the probability of Jesus's existence."

Remember when Churches were supposedly the ones that pumped millions out of the ignorant masses by promises of revelation? Well Richard Carrier is so sure that Jesus didn't real he had to write another book about it.

German Computer Scientists 'Prove' God Exists "Two scientists have formalized a theorem regarding the existence of God penned by mathematician Gรถdel. But the God angle is somewhat of a red herring -- the real step forward is the example it sets of how computers can make scientific progress simpler" spiegel.de/international/โฆ

Calling on critical atheists: What are your views on the applicability of Gรถdel's incompleteness theorem on the unprovability of the existence of a God?

If you don't know the theorem, you can read up on it here or read a discussion of it here. It effectively states that in any system of logic there exists true statements that are absolutely unprovable from within the system.

Many theists claim that the existence of god is not provable, yet true. I accept their claim of non-provability and say that if his existence is not provable, it is also of no consequence to me and therefore I won't accept that it should be true. My father (an atheist and studied philosopher) always counters my argument using Gรถdel's incompleteness theorem and says that though

... keep reading on reddit โกThe Complexity of Theorem-Proving Procedures: The first paper to establish the existence of an NP-complete problem. 4mhz.de/cook.html

Born today : April 21st - Michel Rolle, Mathematician, "best known for Rolle's theorem ... needed to prove both the mean value theorem and the existence of Taylor series", "the first published description... of the Gaussian elimination algorithm" en.wikipedia.org/wiki/Micโฆ

[Differential Equations] Existence and Uniqueness Theorem

I can't figure out how to completely answer this question:

Initial condition for the differential equation, dy/dt = y(y-1)(y-3), is given. What does the Existence and Uniqueness Theorem say about the corresponding solution?

y(0) = 4, y(0) = 0, y(0) = 2, y(0) = -1

All I can think of saying is that since the equation satisfies the hypotheses for the Existence and Uniqueness Theorem, a unique solution exists around (0, 4), (0, 0), (0, 2), and (0, -1) respectively. Is there anything else I should be noting?

Thanks.

"Existence Theorem of the /r/badeconomics Subreddit"

Existence Theorem of the /r/badeconomics Subreddit: Everyone in /r/badeconomics is a good economist.

Proof:

Let economics be a set (denoted as E) and let โgood economicsโ and โbad economicsโ be subsets of economics (denoted by GE and BE, respectively). Let GEโฉBE=ร. Let โgood economistsโ know that GEโฉBE=ร. Let โbad economistsโ not know that GEโฉBE=ร and let bad economists think that BE=GE. Furthermore, let bad economists think that there exists another set โtrue bad economicsโ TBE such that TBEโE and TBEโฉBE= ร. Let good economists know that there does not exist a set TBE.

Assume that we in /r/badeconomics were not good economists. Thus, we would not post anything to /r/badeconomics, because we would think that bad economics is actually good economics (since BE=GE). Instead, we would post elements of TBE. However, we in /r/badeconomics post elements of BE. But, this contradicts our assumption that we in /r/badeconomcis are not good economists. Thus, we in /r/badeconomics a

... keep reading on reddit โกTheorem On Christian Existence And Reputation

Hello. Fellow Atheists:

I am an aspiring physics student, so I thought I would throw my take on physics in relation to my school peers as they are denouncing my atheist lifestyle.

X=vt velocity is the strength of an idea and t is the amount of traction the idea has so even though my ideas are better than my peers since the other people believes different things they still laugh at me.

V=d/t

again my idea reputation is the strength minus traction

EXCELeration= v/t

My ideas excel if the idea reputation increase over time

What ideas need to work is that if others let my ideas excel then they will have a huge idea reputation over a long period.

Since I am alone I am not getting idea reputation so how do I get people to help?

Two scientists have formalized a theorem regarding the existence of God penned by mathematician Kurt Gรถdel. But the God angle is somewhat of a red herring -- the real step forward is the example it sets of how computers can make scientific progress simpler. spiegel.de/international/โฆ

My friends and family keep getting displaced from existence and/or impaled by spontaneously replicated hyperspheres, but the Banach-Tarski Theorem won't stop being valid.

Existence and Uniqueness Theorem for Discrete Ordinary Difference Equations?

Does anyone know, or can anyone point me to a statement of when a nonlinear discrete ordinary difference equation will have a unique solution for all n a non-negative integer (or all integers).

For example the sequences {-2, 3, 8, 63, 3968,.... } and {2, 3, 8, 63, 3968,....} both satisfy x[n+1] = x[n]^2 - 1 for all non-negative integers 'n' with the 'initial condition' x[1] = 3. However - they aren't (obviously) the same.

Something along the lines of the Picard-Lindelhof thoerem would be ideal, but I am unsure of how to formulate either continuity or Lipschitz continuity in the discrete case.

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