Reconstruction of a Symbolic Periodic Sequence from a Sequence with Noise

The problem of constructing a periodic sequence consisting of at least eight periods is considered, based on a given sequence obtained from an unknown periodic sequence, also containing at least eight periods, by introducing noise of deletion, replacement, and insertion of symbols. To construct a periodic sequence that approximates a given one, distorted by noise, it is first required to estimate the length of the repeating fragment (period). Further, the distorted original sequence is divided into successive sections of equal length; the length takes on integer values from 80 to 120 % of the period estimate. Each obtained section is compared with each of the remaining sections, a section is selected to build a periodic sequence that has the minimum edit distance (Levenshtein distance) to any of the remaining sections, minimization is carried out over all sections of a fixed length, and then along all lengths from 80 to 120 % of period estimates. For correct comparison of fragments of different lengths, we consider the ration between the edit distance and the length of the fragment. The length of a fragment that minimizes the ratio of the edit distance to another fragment of the same length to the fragment length is considered the period of the approximating periodic sequence, and the fragment itself, repeating the required number of times, forms an approximating sequence. The constructed sequence may contain an incomplete repeating fragment at the end. The quality of the approximation is estimated by the ratio of the edit distance from the original distorted sequence to the constructed periodic sequence of the same length and this length.

Quantum compiling by deep reinforcement learning

The general problem of quantum compiling is to approximate any unitary transformation that describes the quantum computation as a sequence of elements selected from a finite base of universal quantum gates. The Solovay-Kitaev theorem guarantees the existence of such an approximating sequence. Though, the solutions to the quantum compiling problem suffer from a tradeoff between the length of the sequences, the precompilation time, and the execution time. Traditional approaches are time-consuming, unsuitable to be employed during computation. Here, we propose a deep reinforcement learning method as an alternative strategy, which requires a single precompilation procedure to learn a general strategy to approximate single-qubit unitaries. We show that this approach reduces the overall execution time, improving the tradeoff between the length of the sequence and execution time, potentially allowing real-time operations.

Sharp conditions for the linearization of finite elasticity

We consider the topic of linearization of finite elasticity for pure traction problems. We characterize the variational limit for the approximating sequence of rescaled nonlinear elastic energies. We show that the limiting minimal value can be strictly lower than the minimal value of the standard linear elastic energy if a strict compatibility condition for external loads does not hold. The results are provided for both the compressible and the incompressible case.

On Computations in Renewal Risk Models—Analytical and Statistical Aspects

We discuss aspects of numerical methods for the computation of Gerber-Shiu or discounted penalty-functions in renewal risk models. We take an analytical point of view and link this function to a partial-integro-differential equation and propose a numerical method for its solution. We show weak convergence of an approximating sequence of piecewise-deterministic Markov processes (PDMPs) for deriving the convergence of the procedures. We will use estimated PDMP characteristics in a subsequent step from simulated sample data and study its effect on the numerically computed Gerber-Shiu functions. It can be seen that the main source of instability stems from the hazard rate estimator. Interestingly, results obtained using MC methods are hardly affected by estimation.

THE FLOOR OF THE ARITHMETIC MEAN OF THE CUBE ROOTS OF THE FIRST INTEGERS

Zacharias [‘Proof of a conjecture of Merca on an average of square roots’, College Math. J.49 (2018), 342–345] proved Merca’s conjecture that the arithmetic means $(1/n)\sum _{k=1}^{n}\sqrt{k}$ of the square roots of the first $n$ integers have the same floor values as a simple approximating sequence. We prove a similar result for the arithmetic means $(1/n)\sum _{k=1}^{n}\sqrt[3]{k}$ of the cube roots of the first $n$ integers.

Viscosity explicit midpoint methods for nonexpansive mappings in Hadamard spaces

In this paper, the viscosity explicit midpoint method for nonexpansive mappings in Hadamard spaces is introduced. Under certain appropriate conditions on the sequence of parameters, it is proved that the limit of the approximating sequence generated by proposed method converges strongly to a fixed point of T which solves some variational inequalities. Moreover, we give an application to the equilibrium problem.

Sylvester matrix rank functions on crossed products

In this paper we consider the algebraic crossed product ${\mathcal{A}}:=C_{K}(X)\rtimes _{T}\mathbb{Z}$ induced by a homeomorphism $T$ on the Cantor set $X$, where $K$ is an arbitrary field with involution and $C_{K}(X)$ denotes the $K$-algebra of locally constant $K$-valued functions on $X$. We investigate the possible Sylvester matrix rank functions that one can construct on ${\mathcal{A}}$ by means of full ergodic $T$-invariant probability measures $\unicode[STIX]{x1D707}$ on $X$. To do so, we present a general construction of an approximating sequence of $\ast$-subalgebras ${\mathcal{A}}_{n}$ which are embeddable into a (possibly infinite) product of matrix algebras over $K$. This enables us to obtain a specific embedding of the whole $\ast$-algebra ${\mathcal{A}}$ into ${\mathcal{M}}_{K}$, the well-known von Neumann continuous factor over $K$, thus obtaining a Sylvester matrix rank function on ${\mathcal{A}}$ by restricting the unique one defined on ${\mathcal{M}}_{K}$. This process gives a way to obtain a Sylvester matrix rank function on ${\mathcal{A}}$, unique with respect to a certain compatibility property concerning the measure $\unicode[STIX]{x1D707}$, namely that the rank of a characteristic function of a clopen subset $U\subseteq X$ must equal the measure of $U$.

CRITICAL TRANSACTION COSTS AND 1-STEP ASYMPTOTIC ARBITRAGE IN FRACTIONAL BINARY MARKETS

We study the arbitrage opportunities in the presence of transaction costs in a sequence of binary markets approximating the fractional Black–Scholes model. This approximating sequence was constructed by Sottinen and named fractional binary markets. Since, in the frictionless case, these markets admit arbitrage, we aim to determine the size of the transaction costs needed to eliminate the arbitrage from these models. To gain more insight, we first consider only 1-step trading strategies and we prove that arbitrage opportunities appear when the transaction costs are of order [Formula: see text]. Next, we characterize the asymptotic behavior of the smallest transaction costs [Formula: see text], called "critical" transaction costs, starting from which the arbitrage disappears. Since the fractional Black–Scholes model is arbitrage-free under arbitrarily small transaction costs, one could expect that [Formula: see text] converges to zero. However, the true behavior of [Formula: see text] is opposed to this intuition. More precisely, we show, with the help of a new family of trading strategies, that [Formula: see text] converges to one. We explain this apparent contradiction and conclude that it is appropriate to see the fractional binary markets as a large financial market and to study its asymptotic arbitrage opportunities. Finally, we construct a 1-step asymptotic arbitrage in this large market when the transaction costs are of order o(1/NH), whereas for constant transaction costs, we prove that no such opportunity exists.

Forward integration, convergence and non-adapted pointwise multipliers

In this paper we study the forward integral of operator-valued processes with respect to a cylindrical Brownian motion. In particular, we provide conditions under which the approximating sequence of processes of the forward integral, converges to the stochastic integral process with respect to Sobolev norms of smoothness α < 1/2. This result will be used to derive a new integration by parts formula for the forward integral.