We give an example of a compact set K ⊂ [0, 1] such that the space ℇ(K) of Whitney functions is isomorphic to the space s of rapidly decreasing sequences, and hence there exists a linear continuous extension operator . At the same time, Markov’s inequality is not satisfied for certain polynomials on K.
Let G be a split semisimple algebraic group over with trivial center. Let S be a compact oriented surface, with or without boundary. We define representations of the fundamental group of S to G(), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related...
Given a probability measure μ with non-polar compact support K, we define the n-th Widom factor W²ₙ(μ) as the ratio of the Hilbert norm of the monic n-th orthogonal polynomial and the n-th power of the logarithmic capacity of K. If μ is regular in the Stahl-Totik sense then the sequence has subexponential growth. For measures from the Szegő class on [-1,1] this sequence converges to some proper value. We calculate the corresponding limit for the measure that generates the Jacobi polynomials, analyze...
We consider a compact set K ⊂ ℝ in the form of the union of a sequence of segments. By means of nearly Chebyshev polynomials for K, the modulus of continuity of the Green functions is estimated. Markov’s constants of the corresponding set are evaluated.
A cluster ensemble is a pair of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group . The space is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism . The space is equipped with a closed -form, possibly degenerate, and the space has a Poisson structure. The map is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role...
We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type, which we call a. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs and areif the Newton polygons of the corresponding partition functions coincide up to translation. We define...
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