Search Results (18)

Viscoelastic relaxation time and frequency spectra are useful for describing, analyzing, comparing, and improving the mechanical properties of materials. The spectra are typically obtained using the stress or oscillatory shear measurements. Over the last 80 years, dozens of mathematical models and algorithms were proposed to identify relaxation spectra models using different analytical and numerical tools. Some models and identification algorithms are intended for specific materials, while others are general and can be applied for an arbitrary rheological material. The identified relaxation spectrum model always depends on the identification method applied and on the specific measurements used in the identification process. The stress relaxation experiment data consist of the sampling times used in the experiment and the noise-corrupted relaxation modulus measurements. The aim of this paper is to build a model of the spectrum that asymptotically does not depend on the sampling times used in the experiment as the number of measurements tends to infinity. Broad model classes, determined by a finite series of various basis functions, are assumed for the relaxation spectra approximation. Both orthogonal series expansions based on the Legendre, Laguerre, and Chebyshev functions and non-orthogonal basis functions, like power exponential and modified Bessel functions of the second kind, are considered. It is proved that, even when the true spectrum description is entirely unfamiliar, the approximate sampling-times-independent spectra optimal models can be determined using modulus measurements for appropriately randomly selected sampling times. The recovered spectra models are strongly consistent estimates of the desirable models corresponding to the relaxation modulus models, being optimal for the deterministic integral weighted square error. A complete identification algorithm leading to the relaxation spectra models is presented that requires solving a sequence of weighted least-squares relaxation modulus approximation problems and a random selection of the sampling times. The problems of relaxation spectra identification are ill-posed; solution stability is ensured by applying Tikhonov regularization. Stochastic convergence analysis is conducted and the convergence with an exponential rate is demonstrated. Simulation studies are presented for the Kohlrausch–Williams–Watts spectrum with short relaxation times, the uni- and double-mode Gauss-like spectra with intermediate relaxation times, and the Baumgaertel–Schausberger–Winter spectrum with long relaxation times. Models using spectrum expansions on different basis series are applied. These studies have shown that sampling times randomization provides the sequence of the optimal spectra models that asymptotically converge to sampling-times-independent models. The noise robustness of the identified model was shown both by analytical analysis and numerical studies. Full article

The arithmetic average of the first n primes, ${\overline{p}}_n={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{p}_i$, exhibits very many interesting and subtle properties. Since the transformation from $p_n\to {\overline{p}}_n$ is extremely easy to invert, $p_n=n{\overline{p}}_n-(n-1){\overline{p}}_{n-1}$, it is clear that these two sequences $p_n⟷{\overline{p}}_n$ must ultimately carry exactly the same information. But the averaged sequence ${\overline{p}}_n$, while very closely correlated with the primes, (${\overline{p}}_n\sim {\displaystyle \frac{1}{2}}{p}_n$), is much “smoother” and much better behaved. Using extensions of various standard results, I shall demonstrate that the prime-averaged sequence ${\overline{p}}_n$ satisfies prime-averaged analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Fourges, Firoozbakht, Nicholson, and Farhadian conjectures. (So these prime-averaged analogues are not conjectures; they are theorems). The crucial key to enabling this pleasant behaviour is the “smoothing” process inherent in averaging. While the asymptotic behaviour of the two sequences is very closely correlated, the local fluctuations are quite different. Full article

The variance function (VF) is central to natural exponential family (NEF) theory. Prompted by an online query about whether, beyond the classical normal NEF, other real-line NEFs with bounded VFs exist, we establish three complementary sets of sufficient conditions that yield many such families. One set imposes a polynomial-growth bound on the NEF’s generating measure, ensuring rapid tail decay and a uniformly bounded VF. A second set relies on the Legendre duality, requiring a uniform positive lower bound on the second derivative of the generating function, which likewise ensures a bounded VF. The third set starts from the standard normal distribution and constructs an explicit sequence of NEFs whose Laplace transforms and VFs remain bounded. Collectively, these results reveal a remarkably broad class of NEFs whose Laplace transforms are not expressible in elementary form (apart from those stemming from the standard normal case), yet can be handled easily using modern symbolic and numerical software. Worked examples show that NEFs with bounded VFs are far more varied than previously recognized, offering practical alternatives to the normal and other classical models for real-data analysis across many fields. Full article

In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials ${\left\{K_n\right\}}_{n=0}^{\infty }$—which they named Krein–Sobolev polynomials—that are orthogonal in the classical Sobolev space $H^1[-1,1]$ with respect to the (positive-definite) inner product ${(f,g)}_{1,c}:=-{\displaystyle \frac{\left(f\left(1\right)-f(-1)\right)\left(\overline{g}\left(1\right)-\overline{g}(-1)\right)}{2}}+{\int }_{-1}^1(f^{\prime }\left(x\right){\overline{g}}^{\prime }\left(x\right)+cf\left(x\right)\overline{g}\left(x\right))dx,$ where c is a fixed, positive constant. These polynomials generalize the Althammer (or Legendre–Sobolev) polynomials first studied by Althammer and Schäfke. The Krein–Sobolev polynomials were found as a result of a left-definite spectral study of the self-adjoint Krein Laplacian operator ${\mathcal{K}}_c$$(c>0)$ in $L^2(-1,1).$ Other than $K_0$ and $K_1,$ these polynomials are not eigenfunctions of ${\mathcal{K}}_c.$ As shown by Littlejohn and Quintero, the sequence ${\left\{K_n\right\}}_{n=0}^{\infty }$ forms a complete orthogonal set in the first left-definite space $(H^1[-1,1],{(·,·)}_{1,c})$ associated with $({\mathcal{K}}_c,$$L^2(-1,1))$. Furthermore, they show that, for $n\ge 1,$$K_n\left(x\right)$ has n distinct zeros in $(-1,1).$ In this note, we find an explicit formula for Krein–Sobolev polynomials ${\left\{K_n\right\}}_{n=0}^{\infty }$. Full article

Compressed sensing (CS) is an innovative signal acquisition and reconstruction technology that has broken through the limit of the Nyquist sampling theory. It is widely employed to optimize the measurement processes in various applications. One of the core challenges of CS is the construction of a measurement matrix. However, traditional random measurement matrices are often impractical. Additionally, many existing deterministic binary measurement matrices fail to provide the required flexibility for practical applications. In this study, inspired by the observation that pseudo-random sequences share similar properties with random sequences, we constructed a deterministic sparse measurement matrix with a flexible measurement number based on an pseudo-random sequence—the Legendre sequence. Empirical analysis of the phase transition and an assessment of the practical features of the proposed measurement matrix were conducted. We validated the effectiveness of the proposed measurement matrix on randomly synthesized signals and images. The results of our simulations reveal that our proposed measurement matrix performs better than several other measurement matrices, particularly in terms of accuracy and efficiency. Full article

This research aims to introduce and examine a new type of polynomial called the ${\Delta }_h$ Legendre–Appell polynomials. We use the monomiality principle and operational rules to define the ${\Delta }_h$ Legendre–Appell polynomials and explore their properties. We derive the generating function and recurrence relations for these polynomials and their explicit formulas, recurrence relations, and summation formulas. We also verify the monomiality principle for these polynomials and express them in determinant form. Additionally, we establish similar results for the ${\Delta }_h$ Legendre–Bernoulli, Euler, and Genocchi polynomials. Full article

A multi-constellation, multi-frequency Global Navigation Satellite System (GNSS) receiver is capable of simultaneously receiving signals from multiple satellite constellations across various frequency bands. This allows for increased observations, thereby enhancing navigation accuracy, continuity, effectiveness, and reliability. The spread spectrum code structures used in satellite navigation signals differ among systems. Compatible code generators are employed in multi-constellation, multi-frequency GNSS receivers to support tasks such as signal acquisition and tracking. There are three main types of spread spectrum code structures: Linear Feedback Shift Register (LFSR), Legendre sequences, and Memory codes. The Indian Regional Navigation Satellite System (IRNSS) released the L1-SPS (Standard Positioning Service) signal format in August 2023, which utilizes the Interleaved Z4-linear ranging code (IZ4 code) as its spread spectrum code. Currently, there is no universal code generator design compatible with the IZ4 code. In this paper, a proposed universal code generator is based on the hardware structure of the IRNSS IZ4 code generator. It achieves compatibility with all LFSR-based spread spectrum codes and enables parallel generation of multiple sets of GNSS signal spread spectrum codes, thereby improving hardware utilization efficiency. The proposed structure is implemented and validated using FPGA design, and resource consumption is provided as part of the validation results. Full article

Recently, multi-frequency multi-constellation receivers have been actively studied, which are single receivers that process multiple global navigation satellite system (GNSS) signals for high accuracy and reliability. However, in order for a single receiver to process multiple GNSS signals, it requires as many code generators as the number of supported GNSS signals, and this is one of the problems that must be solved in implementing an efficient multi-frequency multi-constellation receiver. This paper proposes an area-efficient universal code generator that can support both GPS L1C signals and BDS B1C signals. The proposed architecture alleviates the area problem by sharing common hardware in a time-multiplex mode without degrading the overall system performance. According to the result of the synthesis using the CMOS 65 nm process, the proposed universal code generator has an area reduced by 98%, 93%, and 60% compared to the previous memory-based universal code generator (MB UCG), the Legendre-generation universal code generator (LG UCG), and the Weil-generation universal code generator (WG UCG), respectively. Furthermore, the proposed generator is applicable to all Legendre sequence-based codes. Full article

The goal of this paper is to derive Hermite–Hadamard–Fejér-type inequalities for higher-order convex functions and a general three-point integral formula involving harmonic sequences of polynomials and w-harmonic sequences of functions. In special cases, Hermite–Hadamard–Fejér-type estimates are derived for various classical quadrature formulae such as the Gauss–Legendre three-point quadrature formula and the Gauss–Chebyshev three-point quadrature formula of the first and of the second kind. Full article

In this paper, we consider the inverse scattering problem and, in particular, the problem of reconstructing the spectral density associated with the Yukawian potentials from the sequence of the partial-waves $f_{\ell }$ of the Fourier–Legendre expansion of the scattering amplitude. We prove that if the partial-waves $f_{\ell }$ satisfy a suitable Hausdorff-type condition, then they can be uniquely interpolated by a function $\tilde{f}\left(\lambda \right)\in \mathbb{C}$, analytic in a half-plane. Assuming also the Martin condition to hold, we can prove that the Fourier–Legendre expansion of the scattering amplitude converges uniformly to a function $f\left(\theta \right)\in \mathbb{C}$ ($\theta$ being the complexified scattering angle), which is analytic in a strip contained in the $\theta$-plane. This result is obtained mainly through geometrical methods by replacing the analysis on the complex $cos\theta$-plane with the analysis on a suitable complex hyperboloid. The double analytic symmetry of the scattering amplitude is therefore made manifest by its analyticity properties in the $\lambda$- and $\theta$-planes. The function $f\left(\theta \right)$ is shown to have a holomorphic extension to a cut-domain, and from the discontinuity across the cuts we can iteratively reconstruct the spectral density $\sigma \left(\mu \right)$ associated with the class of Yukawian potentials. A reconstruction algorithm which makes use of Pollaczeck and Laguerre polynomials is finally given. Full article

We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some recent arithmetic properties of the generalized Lucas, and generalized Pell–Lucas sequences, to define some new types of pseudoprimes of levels $k^{+}$ and $k^{-}$ and parameter a. For these novel pseudoprime sequences we investigate some basic properties and calculate numerous associated integer sequences which we have added to the Online Encyclopedia of Integer Sequences. Full article

A remarkably large of number of polynomials have been presented and studied. Among several important polynomials, Legendre polynomials, Gould-Hopper polynomials, and Sheffer polynomials have been intensively investigated. In this paper, we aim to incorporate the above-referred three polynomials to introduce the Legendre-Gould Hopper-based Sheffer polynomials by modifying the classical generating function of the Sheffer polynomials. In addition, we investigate diverse properties and formulas for these newly introduced polynomials. Full article

The paper discusses a measurement approach for the room impulse response (RIR), which is insensitive to the nonlinearities that affect the measurement instruments. The approach employs as measurement signals the perfect periodic sequences for Wiener nonlinear (WN) filters. Perfect periodic sequences (PPSs) are periodic sequences that guarantee the perfect orthogonality of a filter basis functions over a period. The PPSs for WN filters are appealing for RIR measurement, since their sample distribution is almost Gaussian and provides a low excitation to the highest amplitudes. RIR measurement using PPSs for WN filters is studied and its advantages and limitations are discussed. The derivation of PPSs for WN filters suitable for RIR measurement is detailed. Limitations in the identification given by the underestimation of RIR memory, order of nonlinearity, and effect of measurement noise are analysed and estimated. Finally, experimental results, which involve both simulations using signals affected by real nonlinear devices and real RIR measurements in the presence of nonlinearities, compare the proposed approach with the ones that are based on PPSs for Legendre nonlinear filter, maximal length sequences, and exponential sweeps. Full article

Recently Edemskiy proposed a method for computing the linear complexity of generalized cyclotomic binary sequences of period $p^{n+1}$ , where $p=dR+1$ is an odd prime, $d,R$ are two non-negative integers, and $n>0$ is a positive integer. In this paper we determine the exact values of autocorrelation of these sequences of period $p^{n+1}$ $(n\ge 0)$ with special subsets. The method is based on certain identities involving character sums. Our results on the autocorrelation values include those of Legendre sequences, prime-square sequences, and prime cube sequences. Full article

The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is $f\left(t\right)=\frac{1}{1+at+b{t}^2}$ , we can give the exact coefficient expression of the power series expansion of $f^x\left(t\right)$ for $x\in \mathbf{R}$ with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials. Full article