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We study the class of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. No model-theoretic characterization for arithmetic circuit classes is known so far. Inspired by Immerman's characterization of the Boolean circuit class , we remedy this situation and develop such a characterization of . Our characterization can be interpreted as follows: Functions in are exactly those functions counting winning strategies in first-order model checking games. A consequence of our results is a new model-theoretic characterization of , the class of languages accepted by constant-depth polynomial-size majority circuits. 相似文献
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In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension in , and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in , , with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of and the Hausdorff measure. 相似文献
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In this paper, we investigate the existence of multiple radial sign-changing solutions with the nodal characterization for a class of Kirchhoff type problems where , are radial and bounded away from below by positive numbers. Under some weak assumptions on , by taking advantage of the Gersgorin disc's theorem and Miranda theorem, we develop some new analytic techniques and prove that this problem admits infinitely many nodal solutions having a prescribed number of nodes k, whose energy is strictly increasing in k. Moreover, the asymptotic behaviors of as are established. These results improve and generalize the previous results in the literature. 相似文献
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A revised Yau's Curvature Difference Flow is considered to deform one convex curve to another one . It is proved that this flow exists globally on time interval and the evolving curve, preserving its convexity and bounded area A, converges to a fixed limiting curve (congruent to ) as time tends to infinity, where is the area bounded by the target curve . 相似文献
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The traveling wave solutions connecting two equilibria for a delayed Logistic equation in a cylinder are obtained for any delay . We attain our goal by using the approach based on the combination of Schauder fixed point theory and the weak coupled upper–lower solutions method. Moreover, we prove that there is a constant that serves as the minimal wave speed of such traveling wave solutions. 相似文献
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Barbara Niethammer Alessia Nota Sebastian Throm Juan J.L. Velázquez 《Journal of Differential Equations》2019,266(1):653-715
In this paper we consider the long-time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving in a random distribution of fixed particles. The volumes v of these particles are independently distributed according to a probability distribution which decays asymptotically as a power law . The validity of the equation has been rigorously proved in [22] taking as a starting point a particle model and for values of the exponent , but the model can be expected to be valid, on heuristic grounds, for . The resulting equation is a non-local linear degenerate parabolic equation. The solutions of this equation display a rich structure of different asymptotic behaviors according to the different values of the exponent σ. Here we show that for the linear Smoluchowski equation is well-posed and that there exists a unique self-similar profile which is asymptotically stable. 相似文献
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Jonathan Jaquette 《Journal of Differential Equations》2019,266(6):3818-3859
In this paper, we prove that Wright's equation has a unique slowly oscillating periodic solution for parameter values , up to time translation. This result proves Jones' Conjecture formulated in 1962, that there is a unique slowly oscillating periodic orbit for all . Furthermore, there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations. 相似文献
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The -dimensional nonlocal nonlinear Schrödinger–Maxwell–Bloch system with -symmetric potentials will be investigated in this paper. The -dimensional Darboux transformation will be constructed, some explicit solutions including periodic waves solutions, soliton solutions and complexiton solutions will be obtained under different backgrounds, and the interaction behaviors of those solutions will be discussed through some figures. 相似文献
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A finite support product of clones of Jensen's minimal singleton forcing is used to define a model in which any non-empty analytically definable set of reals contains an analytically definable real (the full basis theorem), but there is no analytically definable wellordering of the reals. 相似文献
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We compare Brouwer's bar theorem and Spector's bar recursion for the lowest type in the context of constructive reverse mathematics. To this end, we reformulate bar recursion as a logical principle stating the existence of a bar recursor for every function which serves as the stopping condition of bar recursion. We then show that the decidable bar induction is equivalent to the existence of a bar recursor for every continuous function from to with a continuous modulus. We also introduce fan recursion, the bar recursion for binary trees, and show that the decidable fan theorem is equivalent to the existence of a fan recursor for every continuous function from to with a continuous modulus. The equivalence for bar induction holds over the extensional version of intuitionistic arithmetic in all finite types augmented with the characteristic principles of Gödel's Dialectica interpretation. On the other hand, we show the equivalence for fan theorem without using such extra principles. 相似文献
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In this paper, we establish a stronger version of Artemov's arithmetical completeness theorem of the Logic of Proofs . Moreover, we prove a version of the uniform arithmetical completeness theorem of . 相似文献
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《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2021,38(5):1583-1603
We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation ϵ from a radial stationary solution . We use a modified energy method to prove the existence time of classical solutions from to a time scale of . Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time and space. 相似文献
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Sergio Conti Matteo Focardi Flaviana Iurlano 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2019,36(2):455-474
We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization to the vectorial situation of the decay estimate by De Giorgi, Carriero, and Leaci. This is based on replacing the coarea formula by a method to approximate functions with small jump set by Sobolev functions, and is restricted to two dimensions. The other two ingredients will appear in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy functions by ones. 相似文献
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S. Whitehouse's lifts of the Eulerian representations of to are reinterpreted, topologically and ring-theoretically, building on the first author's work on A. Ocneanu's theory of permutohedral blades. 相似文献
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