Related papers: Kolmogorov complexity and the asymptotic bound for…
In this paper, we present explicit and computable error bounds for the asymptotic expansions of the Hermite polynomials with Plancherel--Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory…
The asymptotic rate vs. distance problem is a long-standing fundamental problem in coding theory. The best upper bound to date was given in 1977 and has received since then numerous proofs and interpretations. Here we provide a new,…
New non-asymptotic random coding theorems (with error probability $\epsilon$ and finite block length $n$) based on Gallager parity check ensemble and Shannon random code ensemble with a fixed codeword type are established for discrete input…
This paper presents a saddlepoint approximation of the random-coding union bound of Polyanskiy et al. for i.i.d. random coding over discrete memoryless channels. The approximation is single-letter, and can thus be computed efficiently.…
The purpose of this thesis is to give a formal definition of quantum Kolmogorov complexity (QC), and rigorous mathematical proofs of its basic properties. The definition used here is similar to that by Berthiaume, van Dam, and Laplante. It…
We provide a tight asymptotic characterization of the error exponent for classical-quantum channel coding assisted by activated non-signaling correlations. Namely, we find that the optimal exponent--also called reliability function--is…
Asymptotically tight lower bounds are derived for the Input/Output (I/O) complexity of a class of dynamic programming algorithms including matrix chain multiplication, optimal polygon triangulation, and the construction of optimal binary…
In this paper, we derive non-asymptotic achievability and converse bounds on the random number generation with/without side-information. Our bounds are efficiently computable in the sense that the computational complexity does not depend on…
The classical coding theorem in Kolmogorov complexity states that if an $n$-bit string $x$ is sampled with probability $\delta$ by an algorithm with prefix-free domain then K$(x) \leq \log(1/\delta) + O(1)$. In a recent work, Lu and…
In 1991, Vladimir Maz'ya, Serguei Nazarov and Boris Plamenevskij developed the theory of compound asymptotics for elliptic boundary value problems in singularly perturbed domains. They considered a harmonic function whose domain contains a…
We construct a probability model seemingly unrelated to the considered stochastic process of coagulation and fragmentation. By proving for this model the local limit theorem, we establish the asymptotic formula for the partition function of…
By Kolmogorov Complexity,two number-theoretic problems are solved in different way than before,one problem is Maxim Kontsevich and Don Bernard Zagier's Problem 3 \emph{Exhibit at least one number which does not belong to} $ \mathcal{P}$…
We introduce a notion of Kolmogorov complexity of unitary transformation, which can (roughly) be understood as the least possible amount of information required to fully describe and reconstruct a given finite unitary transformation. In the…
The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets…
We give a general statement of the convolution method so that one can provide explicit asymptotic estimations for all averages of square-free supported arithmetic functions that have a sufficiently regular order on the prime numbers and…
Kolmogorov complexity of a finite binary word reflects both algorithmic structure and the empirical distribution of symbols appearing in the word. Words with symbol frequencies far from one half have smaller combinatorial richness and…
Quantum computers hold the promise of solving computational problems which are intractable using conventional methods. For fault-tolerant operation quantum computers must correct errors occurring due to unavoidable decoherence and limited…
A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not…
We introduce a new quantum decoder based on a variant of the pretty good measurement, but defined via an alternative matrix quotient. We use this decoder to show new lower bounds on the error exponent both in the one-shot and asymptotic…
We construct an asymptotic approximation to the solution of a transmission problem for a body containing a region occupied by many small inclusions. The cluster of inclusions is characterised by two small parameters that determine the…