By A. V. Akopyan

The publication is dedicated to the houses of conics (plane curves of moment measure) that may be formulated and proved utilizing purely undemanding geometry. beginning with the well known optical homes of conics, the authors stream to much less trivial effects, either classical and modern. particularly, the bankruptcy on projective homes of conics includes a targeted research of the polar correspondence, pencils of conics, and the Poncelet theorem. within the bankruptcy on metric homes of conics the authors speak about, particularly, inscribed conics, normals to conics, and the Poncelet theorem for confocal ellipses. The booklet demonstrates the benefit of merely geometric equipment of learning conics. It comprises over 50 workouts and difficulties aimed toward advancing geometric instinct of the reader. The publication additionally comprises greater than a hundred rigorously ready figures, with a view to support the reader to higher comprehend the fabric offered

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Sur le modele de turbulence de Benoit Mandelbrot. C. R. Acad. Sci. : Sur le chaos multiplicatif. Ann. Sci. Math. Que. : Positive martingales and random measures. Chi. Ann. Math. : Sur certaines martingales de B. Mandelbrot Adv. Math. : Multiplications aléatoires et dimensions de Hausdorff. Ann. Inst. Henri Poincaré Probab. Stat. : Produits de poids aléatoires indépendants et applications. In: Fractal Geometry and Analysis (Montreal, PQ, 1989), pp. 277–324. NATO Adv. Sci. Inst. Ser. C Math. Phys.

From that time on, various properties of singular spectral measures are studied extensively [Dai12, DHJ09, DHS09, DHSW11, LaW02, LaW06]. In particular, many exotic spectra were discovered and they do not appear in their absolutely continuous counterpart. Here, we list some of the interesting ones. -R. Dai et al. (1) There exists a spectrum of a singularly continuous measure μ such that k is also a spectrum of μ for some k = 1; (2) There exists a so that E( ) is a maximal orthogonal collection of exponentials for L 2 (μ), but not a basis; (3) There exists a spectrum of a singularly continuous measure so that its Beurling dimension is zero.

Then define the random measures μ B,n (dx) = 1 E B,n (w)e−HB (w) bn dx, if x ∈ Iw . They form a martingale which converges almost surely to a random measure μ B supported on E B = n≥1 w∈E B,n Iw . Moreover, it is possible to choose (Nk )k≥1 suitably so that all the measures μ B are simultaneously defined and non degenerate conditionally on E B = ∅. The sequence (Nk )k≥1 can also be fixed so that for all γ such that ϕ∗ (γ) ≥ 0, each time B = (qk )k≥1 is such that limk→∞ ϕ Ak (qk ) = γ and limk→∞ ϕ∗Ak (ϕ Ak (qk )) = ϕ∗ (γ), then, conditionally on {E B = ∅}, μ B is exact dimensional with dimension ϕ∗ (γ) and carried both by the set E H (γ) = {x : ν(In (x)) (x|n) = γ} and the set E ν (γ) = x ∈ [0, 1] : limn→∞ log limn→∞ nHlog(b) −n log(b) = ν(B(x,r ) γ ∩ x ∈ [0, 1] : limr →0+ log log(r = γ .