inextricably linked

The hypothesis of linguistic relativity, also known as the Sapir–Whorf hypothesis /səˌpɪər ˈwɔːrf/, the Whorf hypothesis, or Whorfianism, is a principle suggesting that the structure of a language influences its speakers' worldview or cognition, and thus people's perceptions are relative to their spoken language.

Different people have different characteristics when it comes to research. Some like to throw out ideas, others to criticize them, others to work out details, others to re-explain ideas in a different language, others to formulate different but related problems, others to step back from a big muddle of ideas and fashion some more coherent picture out of them, and so on. A hugely collaborative project would make it possible for people to specialize

we don’t have to confine ourselves to a purely probabilistic argument: different people know different things, so the knowledge that a large group can bring to bear on a problem is significantly greater than the knowledge that one or two individuals will have. This is not just knowledge of different areas of mathematics, but also the rather harder to describe knowledge of particular little tricks that work well for certain types of subproblem, or the kind of expertise that might enable someone to say, “That idea that you thought was a bit speculative is rather similar to a technique used to solve such-and-such a problem, so it might well have a chance of working,” or “The lemma you suggested trying to prove is known to be false,” and so on—the type of thing that one can take weeks or months to discover if one is working on one’s own

d[V3(K−Λ3−3|E|2)]= 0.

One can check that (56) is the flow ofΣin(M\∂M,V−2g)by equidistantsurface

eachΦt(Σ0) := Σt,t∈[0,δ), is a compact almost properly embedded surface

here exists a se-quence of initial data that satisfy all the hypothesis of item (i) and suchthat in the limit the equality in (3) is achieved. In this limit, the radius,the charge and the total mass of this sequence tend to zero.

A standard computation using the Gauss equation shows that∂f∂t(0,t′) =ddt|Σ0|g(t)(t′) =−∫Σ0(R−Ric(ν,ν))dμ=−4πχ(Σ0)−∫Σ0(Ric(ν,ν) +|A|2)dμ,where all geometric quantities are computed with respect tog(t′).

A computation in coordinates shows that the Ricci tensor ofhis given byRich(X,X) =−(1V∆gV)h(X,X),Rich(X,Z) = 0,Rich(Y,Z) =Ricg(Y,Z)−1V(HessgV)(Y,Z)

The structure of the metrichnear the singular set clearly implies thatgeodesics realizing the distance between a point inNand a component of∂Mmeets∂Morthogonally. The proof of this fact is essentially the sameas the proof of the Gauss’ Lemma.

the maximum principle above, yieldsSmin(t)≥Smin(0)1−2tmSmin(0)(5.3)for allt≥0 as long as the flow exists

Theorem 4.4Let(g(t),φ(t))solve(RH)αwithα(t)≡α >0. ThenSandSdefined as above satisfy thefollowing evolution equations∂∂tS=△S+ 2|Sij|2+ 2α|τgφ|2,∂∂tSij=△LSij+ 2ατgφ∇i∇jφ.(4.14)Proof.This follows directly by combining the evolution equations from Proposition4.2withthose from Proposition4.3.Remark.Note that in contrast to the evolution of Rc,R,∇φ⊗∇φand|∇φ|2the evolutionequations in Theorem4.4for the combinations Rc−α∇φ⊗∇φandR−α|∇φ|2donotdepend on the intrinsic curvature ofN.

What we read, how we read, and why we read change how we think.

usingA−1=1det(A)adj(A)

the scalar curvatureRofds2is given byR= (1−u−2)Rρ+u−2n−1∑i,jR0ijij+ 2n−1∑i=1Rnini= (1−u−2)Rρ+u−2R0−2u−1∆ρu+ 2u−3∂u∂ρH0whereR0is the scalar curvature ofNwith respect tods20andRρis the scalar curvatureof Σρwith the induced metric.

Let Σ0be a compact strictly convex hypersurface inRn,Xbe the position vector ofa point on Σ0, and letNbe the unit outward normal of Σ0atX. Let Σrbe the convexhypersurface described byY=X+rN, withr≥0. The Euclidean space outside Σ0canbe represented by(Σ0×(0,∞),dr2+gr)wheregris the induced metric on Σr. Consider the following initial value problem(2.1)2H0∂u∂r= 2u2∆ru+ (u−u3)Rron Σ0×[0,∞)u(x,0) =u0(x)whereu0(x)>0 is a smooth function on Σ0,H0andRrare the mean curvature and scalarcurvature of Σrrespectively, and ∆ris the Laplacian operator on Σr.

Given a functionRonN, we want to find the equation forusuch that(1.2)ds2=u2dρ2+gρhas scalar curvatureR.

Let Σ be a smooth compact manifold without boundary with dimensionn−1 and letN= [a,∞)×Σ equipped with a Riemannian metric of the form(1.1)ds20=dρ2+gρfor a point (ρ,x)∈N. Heregρis the induced metric on Σρwhich is the level surfaceρ=constant

However, this re-scaling of the space and time coordinates is in fact the result of Einstein ignoring the principle of the constancy of the speed of light by applying the usual vectorial velocity addition and then subsequently trying to compensate for this error by making a further error and changing the given length and time definitions (see my page regarding the Speed of Light).

if the corresponding distances are identical in both systems, then the delay times will also be identical and there won't be any difference in the clock readings afterwards

An all-star international team of astrophysicists used an exquisitely sensitive, $1.1 billion set of twin instruments known as the Laser Interferometer Gravitational-wave Observatory, or LIGO, to detect a gravitational wave generated by the collision of two black holes 1.3 billion light-years from Earth.

Lemma 2.3.(2.1) has a unique solutionufor allrwhich satisfies the estimates in Lemma2.2.

Let Σ0be a smooth compact strictly convex hypersurface inRn. Letrbe the distance function from Σ0. Then the metric on the exteriorNof Σ0is given bydr2+gr, wheregris the induced metric on Σr, which is the hypersurface with distancerfrom Σ0. The functionuwith prescribed scalar curvatureR= 0 is given by2H0∂u∂r= 2u2∆ru+ (u−u3)RrwhereH0is the mean curvature of Σr,Rris the scalar curvature of ΣrandR0is the scalarcurvature of Σrwith the induced metric fromRnand ∆ris the Laplacian on Σr.

If the space doctor’s ideas were wrong, your phone wouldn’t be able to tell where it was.

θ dμ ≥ p 16 π | Σ |

GIBBONS-PENROSE INEQUALITY

"the correct perception"-which would mean "the adequate expression of an object in the subject"-is a contradictory impossibility. For between two absolutely different spheres, as between subject and object, there is no causality, no correctness, and no expression; there is, at most, an aesthetic relation:

For the truth is, Socrates, that you, who pretend to be engaged in the pursuit of truth, are appealing now to the popular and vulgar notions of right, which are not natural, but only conventional

appears to be seeml