A Novel Discriminant Approximation of Periodic Differential Equations

A new approximation of the discriminant of a second order periodic differential equation is presented as a recursive summation of the evaluation of its excitation function at different values of time. The new approximation is obtained, at first, by means of Walsh functions and then, by using some algebraic properties the dependence on the Walsh functions is eliminated. This new approximation is then used to calculate the boundaries of stability. We prove that by letting the summation elements number approach to infinite, the discriminant approximation can be rewritten as a summation of definite integrals. Finally we prove that the definite integrals summation is equivalent to the discriminant approximation made by Lyapunov which consists in an alternating series of coefficients defined by multiple definite integrals, that is, a series of the form $A=A_{0}-A_{1}+\ldots +\left( -1\right) ^{n}A_{n}$, where each coefficient $A_{n}$ is defined as an $n-$multiple definite integral

Efficient Quantum State Preparation with Walsh Series

A new approximate Quantum State Preparation (QSP) method is introduced, called the Walsh Series Loader (WSL). The WSL approximates quantum states defined by real-valued functions of single real variables with a depth independent of the number $n$ of qubits. Two approaches are presented: the first one approximates the target quantum state by a Walsh Series truncated at order $O(1/\sqrt{\epsilon})$, where $\epsilon$ is the precision of the approximation in terms of infidelity. The circuit depth is also $O(1/\sqrt{\epsilon})$, the size is $O(n+1/\sqrt{\epsilon})$ and only one ancilla qubit is needed. The second method represents accurately quantum states with sparse Walsh series. The WSL loads $s$-sparse Walsh Series into $n$-qubits with a depth doubly-sparse in $s$ and $k$, the maximum number of bits with value $1$ in the binary decomposition of the Walsh function indices. The associated quantum circuit approximates the sparse Walsh Series up to an error $\epsilon$ with a depth $O(sk)$, a size $O(n+sk)$ and one ancilla qubit. In both cases, the protocol is a Repeat-Until-Success (RUS) procedure with a probability of success $P=\Theta(\epsilon)$, giving an averaged total time of $O(1/\epsilon^{3/2})$ for the WSL (resp. $O(sk/\epsilon)$ for the sparse WSL). Amplitude amplification can be used to reduce by a factor $O(1/\sqrt{\epsilon})$ the total time dependency with $\epsilon$ but increases the size and depth of the associated quantum circuits, making them linearly dependent on $n$. These protocols give overall efficient algorithms with no exponential scaling in any parameter. They can be generalized to any complex-valued, multi-variate, almost-everywhere-differentiable function. The Repeat-Until-Success Walsh Series Loader is so far the only method which prepares a quantum state with a circuit depth and an averaged total time independent of the number of qubits

Almost Everywhere Strong Summability of two-dimensional Walsh-Fourier Series

It is proved a BMO-estimation for quadratic partial sums of two-dimensional Walsh-Fourier series from which it is derived an almost everywhere exponential summability of quadratic partial sums of double Walsh-Fourier series.Comment: arXiv admin note: substantial text overlap with arXiv:1310.821

On the partial sums of Walsh-Fourier series

In this paper we investigate some convergence and divergence of some specific subsequences of partial sums with respect to Walsh system on the martingale Hardy spaces. By using these results we obtain relationship of the ratio of convergence of the partial sums of the Walsh series with the modulus of continuity of martingale. These conditions are in a sense necessary and sufficient

Properties and examples of Faber--Walsh polynomials

The Faber--Walsh polynomials are a direct generalization of the (classical) Faber polynomials from simply connected sets to sets with several simply connected components. In this paper we derive new properties of the Faber--Walsh polynomials, where we focus on results of interest in numerical linear algebra, and on the relation between the Faber--Walsh polynomials and the classical Faber and Chebyshev polynomials. Moreover, we present examples of Faber--Walsh polynomials for two real intervals as well as some non-real sets consisting of several simply connected components.Comment: Minor rewording in Section 3, which now explicitly mentions the Bernstein-Walsh inequalit

Maximal operators of Vilenkin-N\"orlund means

In this paper we prove and discuss some new $\left(H_{p},weak-L_{p}\right)$ type inequalities of maximal operators of Vilenkin-N\"orlund means with monotone coefficients. We also apply these results to prove a.e. convergence of such Vilenkin-N\"orlund means. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out

On everywhere divergence of the strong Φ\Phi-means of Walsh-Fourier series

Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems established by Rodin in 1990. We prove, that if the growth of a function $\Phi(t):[0,\infty)\to[0,\infty)$ is bigger than the exponent, then the strong $\Phi$-summability of a Walsh-Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the author of this paper.Comment: 8 page

The Optimal Dyadic Derivative

We show that the best approximation to the difference operators on the cyclic groups of order $2^n$ by a dyadic convolution operator are the restrictions of a generalized dyadic derivative. This answers a question on the "intuitive" interpretation of the dyadic derivative posed by Butzer and Wagner more than 30 years ago

A note on the norm convergence by Vilenkin-Fej\'er means

The main aim of this paper is to find necessary and sufficient conditions for the convergence of Fej\'er means in terms of the modulus of continuity on the Hardy spaces $H_{p},$ when $0<p\leq 1/2.$Comment:

Time-resolved magnetic sensing with electronic spins in diamond

Quantum probes can measure time-varying fields with high sensitivity and spatial resolution, enabling the study of biological, material, and physical phenomena at the nanometer scale. In particular, nitrogen-vacancy centers in diamond have recently emerged as promising sensors of magnetic and electric fields. Although coherent control techniques have measured the amplitude of constant or oscillating fields, these techniques are not suitable for measuring time-varying fields with unknown dynamics. Here we introduce a coherent acquisition method to accurately reconstruct the temporal profile of time-varying fields using Walsh sequences. These decoupling sequences act as digital filters that efficiently extract spectral coefficients while suppressing decoherence, thus providing improved sensitivity over existing strategies. We experimentally reconstruct the magnetic field radiated by a physical model of a neuron using a single electronic spin in diamond and discuss practical applications. These results will be useful to implement time-resolved magnetic sensing with quantum probes at the nanometer scale.Comment: 8+12 page