Related papers: A Combination Theorem for PD(n)-Pairs
We proved that any even number not less than 6 can be expressed as the sum of two old primes, $2n=p_i+p_j$
In this paper, we present a variant of the Balog-Szemer\'edi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon.
We prove an interpolation theorem for bounded free holomorphic functions.
We obtain a correct generalization of Shokurov's non-vanishing theorem for log canonical pairs. It implies the base point free theorem for log canonical pairs. We also prove the rationality theorem for log canonical pairs. As a corollary,…
The Integration Theory of Linear Ordinary Differential Equation.
We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the theory of complements for real coefficients.…
Using a new technique, we prove a rich family of special cases of the matroid intersection conjecture. Roughly, we prove the conjecture for pairs of tame matroids which have a common decomposition by 2-separations into finite parts.
In this article we will derive a combinatorial formula for the partition function p(n). In the second part of the paper we will establish connection between partitions and q-binomial coefficients and give new interpretation for q-binomial…
We show that there is a strong minimal pair in the computably enumerable Turing degrees.
In this short note, we will show that the metric of Deligne's pairing is continous.
A conjugate code pair is defined as a pair of linear codes either of which contains the dual of the other. A conjugate code pair represents the essential structure of the corresponding Calderbank-Shor-Steane (CSS) quantum code. It is known…
We prove a new theorem on additive Levy processes and show that this theorem implies several proved theorems and a hard conjectured theorem.
We show for a class of sequences $(a_n)_{n\geq 1}$ of distinct positive integers, that for no $\alpha$ the sequence $(\left\{a_n \alpha \right\})_{n \geq 1}$ does have Poissonian pair correlation. This class contains for example all…
We show the existence of $(\epsilon,n)$-complements for $(\epsilon,\mathbb{R})$-complementary surface pairs when the coefficients of boundaries belong to a DCC set.
We give a proof of the Howe duality conjecture in local theta correspondence for symplectic-orthogonal or unitary dual pairs in arbitrary residual characteristic.
We prove several congruences for trinomial coefficients.
A very simple but useful almost sure convergence theorem of probability is given.
The pair correlation statistic is an important concept in real uniform distribution theory. Therefore, sequences in the unit interval with (weak) Poissonian pair correlations have attracted a lot of attention in recent time. The aim of this…
Several results about the union-closed sets conjecture are presented.
We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.