We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval (−1, 1) are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit translation invariant correlations. Then we calculate the correlations between the straightened zeros of the SO(2) random p...

We prove that there is an absolute constants c1 > 0 such that for every {a0, a1, . . . , an} ⊂ [1, M ] , 1 ≤ M ≤ exp((1/64)n) , there are b0, b1, . . . bn ∈ {−1, 0, 1}

Fredholm determinant asymptotics of convolution operators on large finite intervals with rational symbols having real zeros are studied. The explicit asymptotic formulae obtained can be considered as a direct extension of the Ahiezer-Kac formula to symbols with real zeros.

An important feature of the frontal method for the solution of large sparse systems of linear equations is that the frontal matrix is held as a dense matrix. This allows efficient dense linear algebra kernels, in particular, the Level 3 Basic Linear Algebra Subprograms (BLAS) to be used during the numerical factorization. However, the frontal matrix may contain a significant number of zeros and...

In this article we suggest a new odd entire function of order one. We discover that it is the solution heat equation. propose conjecture has only real zeros. The obtained result gives connection with problem and number theory.