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On Polynomially Bounded Harmonic Functions on the $Z^d$ Lattice

100%
Nayar P.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2009
|
tom 57
|
nr 3
231-242
EN
We prove that if $f: Z^d → R$ is harmonic and there exists a polynomial $W: Z^d → R$ such that f + W is nonnegative, then f is a polynomial.
2
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FKN Theorem on the biased cube

100%
Nayar P.
Colloquium Mathematicum
|
2014
|
tom 137
|
nr 2
253-261
EN
We consider Boolean functions defined on the discrete cube ${-γ, γ^{-1}}ⁿ$ equipped with a product probability measure $μ^{⊗ n}$, where $μ = βδ_{-γ} + αδ_{γ^{-1}}$ and γ = √(α/β). This normalization ensures that the coordinate functions $(x_i)_{i=1,...,n}$ are orthonormal in $L₂({-γ,γ^{-1}}ⁿ,μ^{⊗ n})$. We prove that if the spectrum of a Boolean function is concentrated on the first two Fourier levels, then the function is close to a certain function of one variable. Our theorem strengthens the non-symmetric FKN Theorem due to Jendrej, Oleszkiewicz and Wojtaszczyk. Moreover, in the symmetric case α = β = 1/2 we prove that if a [-1,1]-valued function defined on the discrete cube is close to a certain affine function, then it is also close to a [-1,1]-valued affine function.
3
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A Note on the Rational Cuspidal Curves

63%
Nayar P., Pilat B.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2014
|
tom 62
|
nr 2
117-123
EN
In this short note we give an elementary combinatorial argument, showing that the conjecture of J. Fernández de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández and A. Némethi [Proc. London Math. Soc. 92 (2006), 99-138, Conjecture 1] follows from Theorem 5.4 of Brodzik and Livingston [arXiv:1304.1062] in the case of rational cuspidal curves with two critical points.
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