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The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These…
This article considers the error term of the primes counting function. It applies some recent results on the densities of prime numbers in short intervals to derive an improvement of the error term from subexponential size to fractional…
To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…
Alpoge and Granville (separately) gave novel proofs that the primes are infinite that use Ramsey Theory. In particular, they use Van der Waerden's Theorem and some number theory. We prove the primes are infinite using an easier theorem from…
One of the greatest difficulties encountered by all in their first proof intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results…
This paper is part of series on self-contained papers in which a large part, if not the full extent, of the asymptotic limit theory of summands of independent random variables is exposed. Each paper of the series may be taken as review…
This paper has been withdrawn by the authors due to a mistake in the proof of Theorem 1.
On [3, p. 199] one says "We mention parenthetically that the proof of [99, Lemma 41.3] is incorrect, and we do not know whether it, [99, Theorem 41.5] and [99, Theorem 41.6] are true". The previously cited reference [99] is our reference…
We consider first order expressible properties of random perfect graphs. That is, we pick a graph $G_n$ uniformly at random from all (labelled) perfect graphs on $n$ vertices and consider the probability that it satisfies some graph…
We argue that the field theory that descibes randomly branched polymers is not generally conformally invariant in two dimensions at its critical point. In particular, we show (i) that the most natural formulation of conformal invariance for…
A mathematics student's first introduction to the fundamental theorem of finite fields (FTFF) often occurs in an advanced abstract algebra course and invokes the power of Galois theory to prove it. Yet the combinatorial and algebraic coding…
We prove limit theorems for the number of fixed points occurring in a random pattern-avoiding permutation distributed according to a one-parameter family of biased distributions. The bias parameter exponentially tilts the distribution…
A well-known discovery of Feige's is the following: Let $X_1, \ldots, X_n$ be nonnegative independent random variables, with $\mathbb{E}[X_i] \leq 1 \;\forall i$, and let $X = \sum_{i=1}^n X_i$. Then for any $n$, \[\Pr[X < \mathbb{E}[X] +…
This paper has been withdrawn by the author due to a crucial argument error at p.10.
In this article we prove a general theorem which establishes the existence of limiting distributions for a wide class of error terms from prime number theory. As a corollary to our main theorem, we deduce previous results of Wintner (1935),…
Bell's theorem is reformulated and proved in the pure mathematical terms of automata theory, avoiding any physical or ontological notions. It is stated that no pair of finite probabilistic sequential machines can reproduce in its output the…
This paper has been withdrawn by the author due to a crucial sign error in equation 1.
This version of the paper corrects an inaccuracy in the proof of Theorem 2.9 in the published version. The main results remain unchanged.
This paper have serious error in the proof of main theorem 1.1.Result is not proved.
This paper has been withdrawn by the author due to a mistake in the proof of the main theorem.