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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

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.

FKN Theorem on the biased cube

100%

Nayar P.

Colloquium Mathematicum

2014

tom 137

nr 2

253-261

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.

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

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.