The rabbit (or Fibonacci) numbers form a sequence (a(1) = 1), 1, 2, 3, 5, 8,13,21,34,...,in which a(n)+2 =a(n+1) +a(n) for an y n=1,2,... . Find the greatest common divisor of the numbers a(100) and a(99).

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$De façon générale, cherchons une solution de : \[ dX_t = [a_1(t)X_t + a_0(t)]dt + [b_1(t)X_t + b_0(t)]dW_t, \quad X_0 \text{ donné} \] On commence par chercher une solution $F_t$ du système homogène: $dF_t = a_1(t)F_t dt + b_1(t)F_t dW_t$ avec $F_0 = 1$. Trivialement : \[ F_t = \exp\left[\int_0^t \left(a_1(s) - \frac{1}{2}b_1(s)^2\right)ds + \int_0^t b_1(s)dW_s\right] \] On va ensuite cherche une solution autour de $F_t$ en posant On pose $Y_t = X_t/F_t$ où par Itô : \[ d\left({1\over F$

The value of a text extends beyond its content to include the origin of the information it contains. This origin can be rooted in significant events or the accomplishments of individuals. A text may serve as a record of an event, providing valuable documentation for future reference. It can also convey the achievements of notable individuals, whether they made groundbreaking discoveries or created influential works. It's important to note that the author of a text may not always be the person re

How many ways are there to break up the number 64 into 10 natural summands (integers ≥ 1), whose maximum is 12? [Ways that differ only by the order of their summands do not count as different.]

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$explain: $$c = \sqrt{\frac{g \lambda}{2\pi}}$$$$c = \sqrt{g h}$$$$c = \sqrt{g d}$$$$\gamma_b = \frac{H_b}{h_b} \approx 0.78$$$

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You are a proof theorist. Read the paper below and extract two objects. DEFINITIONS - Derivation-Level Intension (DLI): the derivation of the paper's shared results (main theorem + companions), taken up to bureaucratic reordering of independent steps, together with the vocabulary of its intermediate statements. Definitions stay FOLDED (never unfold a defined symbol). Do not perform cut elimination: the intermediate lemmas and concepts the author introduces are exactly what the DLI mus