1,721,052 research outputs found
On some universal Morse–Sard type theorems
Full text linkThe classical Morse-Sard theorem claims that for a mapping v : R-n -> Rm+1 of class C-k the measure of critical values v(Z(v,m)) is zero under condition k >= n - m. Here the critical set, or m-critical set is defined as Z(v,m) = {x is an element of R-n : rank del v(x) <= m}. Further Dubovitskii in 1957 and independently Federer and Dubovitskii in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C-k category.Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R-n -> R-d belongs to the Holder class C-k,C-alpha, 0 <= alpha < 1, then for every q> m the identityH-mu (Z(v,m) boolean AND v(-1) (y)) = 0holds for H-q-almost all y is an element of R-d, where mu = n - m - (k + alpha) (q - m).Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa.The result is new even for the classical C-k-case (when alpha = 0); similar result is established for the Sobolev classes of mappings W-p(k)(R-n,R-d) with minimal integrability assumptions p = max(1, n/k), i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces.The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools). (C) 2020 Elsevier Masson SAS. All rights reserved
The Morse–Sard Theorem and Luzin N-Property: A New Synthesis for Smooth and Sobolev Mappings
Full text linkConsidering regular mappings of Euclidean spaces, we study the distortion of the Hausdorff dimension of a given set under restrictions on the rank of the gradient on the set. This problem was solved for the classical cases of k-smooth and Holder mappings by Dubovitskii, Bates, and Moreira. We solve the problem for Sobolev and fractional Sobolev classes as well. Here we study the Sobolev case under minimal integrability assumptions that guarantee in general only the continuity of a mapping (rather than differentiability everywhere). Some new facts are found out in the classical smooth case. The proofs are mostly based on our previous joint papers with Bourgain and Kristensen (2013, 2015)
Psychological Nash equilibria under ambiguity
Full text linkPsychological games aim to represent situations in which players may have belief-dependent moti-vations. In this setting, utility functions are directly dependent on the entire hierarchy of beliefs of each player. On the other hand, the literature on strategic ambiguity in classical games highlights that players may have ambiguous (or imprecise) beliefs about opponents' strategy choices. In this paper, we look at the issue of ambiguity in the framework of simultaneous psychological games by taking into account ambiguous hierarchies of beliefs and study a natural generalization of the psychological Nash equilibrium concept to this framework. We give an existence result for this new concept of equilibrium and provide examples that show that even an infinitesimal amount of ambiguity may alter significantly the equilibria of the game or can work as an equilibrium selection device. Finally, we look at the problem of stability of psychological equilibria with respect to ambiguous trembles on the entire hierarchy of correct beliefs and we provide a limit result that gives conditions so that sequences of psychological equilibria under ambiguous perturbation converge to psychological equilibria of the unperturbed game.(c) 2022 Elsevier B.V. All rights reserved
On the luzin N-property and the uncertainty principle for Sobolev mappings
Full text linkWe say that a mapping v from Rn to Rd satisfies the (tau, sigma) -N-property if H^sigma( v (E)) = 0 whenever H^tau (E) = 0, where H^tau means the Hausdorff measure. We prove that every mapping v of Sobolev class W_p^k (Rn,Rd ) with kp > n satisfies the (tau, sigma)-N-property for every 0 1 and for the critical value tau= tau* the corresponding (tau, sigma)-N-property fails in general. Nevertheless, this (tau, sigma)-N-property holds for tau = tau* if we assume in addition that the highest derivatives (Nabla^k)v belong to the Lorentz space L_p,1(Rn ) instead of L_p. We extend these results to the case of fractional Sobolev spaces as well. Also, we establish some Fubini-type theorems for N-properties and discuss their applications to the Morse-Sard theorem and its recent extensions
The liquid-crystalline properties of bis [n-[[4-[4-(alkoxy)benzoyloxy]2-hydroxypheyl]methylene]alkanamino]complexes of Cu (II), Pd (II), and Ni (II). A general view
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