Rolle's theorem states that if a function,f, is continuous on a closed interval a,b, differentiable on the open interval (a,b) such that f(a)=f(b), then there exists a point c in the interval (a,b) such that f'(c)=0. That is, if these conditions are met, the function f...
We establish a relationship between stride generators and h-bases, and show that the result follows immediately if the stride generator underlying a basis has a particular property - here called "canonicality". The proof is lengthy (with a number of special cases to consider), but the ...
Let F be a function defined by \displaystyle{ F = \dfrac{ x^2 1}{ x 1 }. } Determine if \displaystyle{ \lim\limits_{x\to 1^ } F(x) } exists, and if it does, find its value. Show that the product of a complex number ''a'' + '...
A geodesic γ: R → M is called invariant under the isometry A if there exists a nonzero constant b such that A γ( t )=γ( t + b ) for any real t . We prove that if A is an isometry of compact type on a connected Riemannian manifold, and if A has a nonclosed invariant ...
Since ψ has exactly n vanishing moments, (6.15) proves that |Wg(u,s)|∼Ksn+1/2g(n)(u). We suppose that g(n)(u)≠ 0. For τ(q)≤ (n + 1/2)q, since |Wg(u,s)|q∼sq(n+1/2) has a faster asymptotic decay than sτ(q) when s goes to zero, one can verify that...
RISC Zero protocol Rewards up to$150,000 Reports9 Members5 The RISC Zero zkVM is a verifiable computer that works like a real embedded RISC-V micro-processor, enabling programmers to write ZK proofs like they write any other code.
The Sequel of Ingenious Proof of the Riemann Hypothesis that All Non-trivial Zeros are on a Fixed Line and Also Letting Old Methods Produce New Vitality: T... For the well-known Riemann hypothesis(abbreviated as RH), based on the relevant principle of complex functions and the idea of simpl...
However, there exists no stringent criterion for the absolute tumor uptake of radiopharmaceuticals. On the other hand, high uptake and prolonged retention of radiopharmaceuticals at the tumor site are necessary for tumor therapy to ensure treatment efficacy [9,10,11]. Although traditional dual-purpose...
Consider a function f that represents the speed of a car. The speed of the car is continuous since there are no points at which it would not exists (note that when the car is not moving its speed is zero). If the speed of a car is considered on a certain time interval, and the ...
Beyond a reasonable doubt establishes that no other reasonable explanation exists other than the evidence presented to the court. Commonly used in criminal prosecutions, such as murder trials, this is the highest legal standard of proof. Civil Case vs. Criminal Case The standard of burden of proo...