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This article discusses the class of Periodic Generalized Poisson Integer-Valued Generalized Autoregressive Conditional Heteroscedastic (PGPINGARCH) models. The model, in addition to properly capture the periodic feature in the autocovariance structure, encompasses different types of dispersions, with this conditional marginal distribution. The main theoretical properties of this model are developed, in particular, the first two moment periodically stationary conditions, while the closed form of these moments are derived. Moreover, the existence of the higher order moment and their closed forms are established. The periodic autocovariance structure is studied. The estimation is done by the Yule Walker and the Conditional Maximum Likelihood methods and their performance is shown via an simulation study. Moreover, an application on Campylobacteriosis time series is provided, which indicates that the proposed models performs better than other models in the literature. [ABSTRACT FROM AUTHOR]

We present a modified quadrature method to solve the logarithmic-kernel integral equations, because of its importance in mathematical physics. At first, the quadrature method is constructed by the trapezoidal rule, which is used to discrete the integral equation. In addition, when the integral curve can be represented by a periodic function, the error asymptotic expansion with single parameter will be proved, which shows that the quadrature method has high accuracy order O (h 3) , and can improve the accuracy of numerical solutions to O (h 5) order accuracy by extrapolation algorithm once. Three numerical examples show our theoretical analysis. [ABSTRACT FROM AUTHOR]

This correction notice addresses errors in the original publication of a narrative review on low-density lipoprotein cholesterol-lowering drugs. The correction clarifies that no dose adjustment is necessary for patients with mild hepatic impairment, but the use of these drugs is not recommended for patients with moderate or severe hepatic impairment. Additionally, the correction updates the references in the publication and adjusts the order of some references. The authors assure that these corrections do not affect the scientific conclusions of the original publication. [Extracted from the article]

We revisit the classic problem of the effective diffusion constant of a Brownian particle in a square lattice of reflecting impenetrable hard disks. This diffusion constant is also related to the effective conductivity of non-conducting and infinitely conductive disks in the same geometry. We show how a recently derived Green's function for the periodic lattice can be exploited to derive a series expansion of the diffusion constant in terms of the disk's volume fraction φ. Second, we propose a variant of the Fick–Jacobs approximation to study the large volume fraction limit. This combination of analytical results is shown to describe the behavior of the diffusion constant for all volume fractions. [ABSTRACT FROM AUTHOR]

In this paper, we study a free boundary problem modeling spherically symmetric tumor growth with angiogenesis and a periodic supply of nutrients. The mathematical model is a free boundary problem since the external radius of the tumor denoted by R (t) changes with time. The characteristic of this model is the consideration of both angiogenesis and periodic external nutrient supply. The cells inside the tumor absorb nutrient u (r , t) through blood vessels and attracts blood vessels at a rate proportional to α. Thus on the boundary, we have u r (r , t) + α (u (r , t) − ψ (t)) = 0 , r = R (t) , t > 0 , where ψ (t) is the nutrient concentration provided externally. Considering that the nutrient provided externally to the tumor are generally provided periodically, in this paper, we assume that ψ (t) is a periodic function. Sufficient conditions for a tumor to disappear are given. We investigate the existence, uniqueness, and stability of solutions. The results show that when the nutrient concentration exceeds a certain value and c is sufficiently small, the solutions of the model can be arbitrarily close to the unique periodic function as t → ∞ . [ABSTRACT FROM AUTHOR]

We consider the ordinary differential equations defined by a trigonometric polynomial field and we prove that any solution x admits a rotation vector ρ ∈ R n . More precisely, the function t ↦ x (t) − ρ t is bounded on time and it is a weak almost periodic function of slopeρ. [ABSTRACT FROM AUTHOR]

A second-order ordinary differential equation with an essential nonlinearity and an external perturbation in the form of a continuous periodic function is studied. Nonlinearity is given by a symmetrical relay characteristic with hysteresis, dead zone, and saturation zones. The traversal of the characteristic without entering the saturation zones is considered for some finite time and time commensurate with the period of the perturbation function. Conditions for the existence of an oscillating solution with a closed phase trajectory and four switching points during one traversal of the characteristic are obtained. Existence theorems for periodic solutions, including solutions with a symmetric trajectory, are proved. A numerical example is given. [ABSTRACT FROM AUTHOR]

We investigate a novel geometric Iwasawa theory for ${\mathbf Z}_p$ -extensions of function fields over a perfect field k of characteristic $p>0$ by replacing the usual study of p -torsion in class groups with the study of p -torsion class group schemes. That is, if $\cdots \to X_2 \to X_1 \to X_0$ is the tower of curves over k associated with a ${\mathbf Z}_p$ -extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p -torsion group scheme in the Jacobian of $X_n$ as $n\rightarrow \infty $. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of $X_n$ equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the $k[V]$ -module structure of the space $M_n:=H^0(X_n, \Omega ^1_{X_n/k})$ of global regular differential forms as $n\rightarrow \infty.$ For example, for each tower in a basic class of ${\mathbf Z}_p$ -towers, we conjecture that the dimension of the kernel of $V^r$ on $M_n$ is given by $a_r p^{2n} + \lambda _r n + c_r(n)$ for all n sufficiently large, where $a_r, \lambda _r$ are rational constants and $c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$ is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on ${\mathbf Z}_p$ -towers of curves, and we prove our conjectures in the case $p=2$ and $r=1$. [ABSTRACT FROM AUTHOR]

We discuss the possibility of localizing an electron in a highly excited Rydberg state. The second-order correlation of emitted photons is the tool for the determination of electron position. This second-order correlation of emitted radiation and, therefore, the correlation of operators describing the acceleration of the electron allows for a partial localization of the electron in its orbit. The correlation function is found by approximating the transition matrix elements by their values in the classical limit. It is shown that the second-order correlation, depending on two times, is a function of the time difference and is a periodic function of this argument with the period equal to the period of the corresponding classical motion. The function has sharp maxima corresponding to large electron acceleration in the vicinity of the “perihelion.” This allows the localization of the electron in its consecutive approach to the perihelion point. [ABSTRACT FROM AUTHOR]

It is known that the functional reproducing kernel Hilbert space (FRKHS) theory lays the functional analysis foundation for learning non-point-evaluation functional data with kernel-regularized learning. In the present paper, we investigate the convergence of regression learning associated with Radon reproducing kernel Hilbert spaces (RRKHSs) and a parameterized loss. We provide a kind of online learning algorithm and establish an upper bound for the learning rate, and it shows that the learning rate may be improved by adjusting the parameter in the loss function. As incidental conclusions, we provide a method of constructing FRKHSs with Fourier analysis, some RRKHSs are defined for the classes of periodic functions defined on $ [-\pi, \;\pi]^s $; the functions defined on the interval $ [-1, 1] $; the functions defined on $ R_+ = [0, +\infty) $ and the functions defined on the unit sphere. [ABSTRACT FROM AUTHOR]

We obtain some results regarding the problem of the periodicity of entire functions f(z) when differential polynomials P(z, f) with constant coefficients generated by f(z) are periodic. We provide some sufficient conditions that ensure the periodicity of f(z) , and we discuss some properties of periodic functions. Our results generalize and improve some earlier ones and have an importance concerning entire solutions of differential equations of the form P (z , f) = h (z) , where h(z) is a periodic function. [ABSTRACT FROM AUTHOR]

In bistable perception, observers experience alternations between two interpretations of an unchanging stimulus. Neurophysiological studies of bistable perception typically partition neural measurements into stimulus-based epochs and assess neuronal differences between epochs based on subjects' perceptual reports. Computational studies replicate statistical properties of percept durations with modeling principles like competitive attractors or Bayesian inference. However, bridging neuro-behavioral findings with modeling theory requires the analysis of single-trial dynamic data. Here, we propose an algorithm for extracting nonstationary time series features from single-trial electrocorticography (ECoG) data. We applied the proposed algorithm to 5-min ECoG recordings from human primary auditory cortex obtained during perceptual alternations in an auditory triplet streaming task (six subjects: four male, two female). We report two ensembles of emergent neuronal features in all trial blocks. One ensemble consists of periodic functions that encode a stereotypical response to the stimulus. The other comprises more transient features and encodes dynamics associated with bistable perception at multiple time scales: minutes (within-trial alternations), seconds (duration of individual percepts), and milliseconds (switches between percepts). Within the second ensemble, we identified a slowly drifting rhythm that correlates with the perceptual states and several oscillators with phase shifts near perceptual switches. Projections of single-trial ECoG data onto these features establish low-dimensional attractor-like geometric structures invariant across subjects and stimulus types. These findings provide supporting neural evidence for computational models with oscillatory-driven attractor-based principles. The feature extraction techniques described here generalize across recording modality and are appropriate when hypothesized low-dimensional dynamics characterize an underlying neural system. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the Bohr ρ -almost periodic type sequences and the generalized ρ -almost periodic type sequences of the form F : I × X → Y , where ∅ ≠ I ⊆ Z n , X and Y are complex Banach spaces and ρ is a general binary relation on Y. We provide many structural results, observations and open problems about the introduced classes of ρ -almost periodic sequences. Certain applications of the established theoretical results to the abstract Volterra integro-difference equations are also given. [ABSTRACT FROM AUTHOR]

With the development of quantum technology, quantum computing has an increasingly significant impact on cryptanalysis. Several quantum algorithms, such as Simon's algorithm, Grover's algorithm, the Bernstein–Vazirani algorithm, Shor's algorithm, and the Grover-meets-Simon algorithm, have been proposed successively. However, almost all cryptanalysis is based on the quantum chosen-plaintext attack (qCPA) model. This paper focuses on a powerful cryptanalytic model, quantum related-key attack (qRKA), and proposes a strategy of qRKAs against symmetric ciphers using Simon's algorithm. We construct a periodic function to efficiently recover the secret key of symmetric ciphers if the attacked symmetric ciphers satisfy Simon's promise, and present the complexity analysis on specific symmetric ciphers. Then, we apply qRKA to the Even–Mansour cipher and SoEM construction, recover their secret keys, and show their complexity comparison in the distinct attack models. This work is of great significance for the qRKA cryptanalysis of existing provably secure cryptographic schemes and the design of future quantum secure cryptographic schemes. [ABSTRACT FROM AUTHOR]

In this paper we completely solve the open problem of finding the fundamental solution of the semidiscrete fractional-spatial damped wave equation. We combine operator theory and Laplace transform methods with properties of Bessel functions to show an explicit representation of the solution when initial conditions are given. Our findings extend known results from the literature and also provide new insights into the qualitative behavior of the solutions for the studied model. As an example, we show the existence of almost periodic solutions as well as their profile in the homogeneous case. [ABSTRACT FROM AUTHOR]

The OTR scheme is a well-known AE system cipher proposed at the EUROCRYPT 2014. Due to its characteristics of fast and compact operation, OTR has wide applications. At CRYPTO 2016, Kaplan et al. proposed a method of constructing Simon function based on rotation value to implement quantum periodic attacks on cryptographic schemes such as CBC-MAC, PMAC, GMAC, GCM and OCB. This paper modifies the assumptions of random numbers in this method, proposes a Simon construction method based on different functions, and constructs a Simon periodic function on the OTR scheme. In addition, a forgery attack on the association information of the OTR scheme is proposed based on the quantum truncation technique. [ABSTRACT FROM AUTHOR]

In this study, the exact solutions of the Biswas-Arshed equation with the beta time derivative, which has an important role and physically means that it represents the pulse propagation in an optical fiber, nuclear, and particle physics, are obtained using the modified exponential function method. Exact solutions consisting of hyperbolic, trigonometric, rational trigonometric, and rational function solutions demonstrate the competence and relevance of the proposed method. In addition, the physical properties of the obtained exact solutions are shown by making graphical representations according to different parameter values. It is seen that the method used is an effective technique, since these solution functions obtained with all these cases have periodic function properties. [ABSTRACT FROM AUTHOR]

Let C ~ be the space of continuous 2 π -periodic functions f, endowed with the uniform norm ‖ f ‖ : = max x ∈ R | f (x) | , and denote by ω k (f , t) , the k-th modulus of smoothness of f. Denote by C ~ r , the subspace of r times continuously differentiable functions f ∈ C ~ , and let T n , be the set of trigonometric polynomials T n of degree ≤ n (that is, of order ≤ 2 n + 1 ). Given a set Y s : = { y i } i = 1 2 s , of 2s points, s ≥ 1 , such that - π ≤ y 1 < y 2 < ⋯ < y 2 s < π , and a function f ∈ C ~ r , r ≥ 3 , that changes convexity exactly at the points Y s , namely, the points Y s are all the inflection points of f. We wish to approximate f by trigonometric polynomials which are coconvex with it, that is, satisfy f ′ ′ (x) T n ′ ′ (x) ≥ 0 , x ∈ R. We prove, in particular, that if r ≥ 3 , then for every k , s ≥ 1 , there exists a sequence { T n } n = N ∞ , N = N (r , k , Y s) , of trigonometric polynomials T n ∈ T n , coconvex with f, such that ‖ f - T n ‖ ≤ c (r , k , s) n r ω k (f (r) , 1 / n). It is known that one may not take N independent of Y s . [ABSTRACT FROM AUTHOR]

An estimate for the convergence rate of the Fourier series of a continuous periodic function of bounded variation is refined. [ABSTRACT FROM AUTHOR]

We study transport properties of an active Brownian particle with an Rayleigh–Helmholtz friction function in a biased periodic potential. In the absence of noise and depending on the parameters of the friction function and on the bias force, the motion of the particle can be in a locked state or in different running states. According to the type of solutions, the parameter plane of friction and bias force can be divided into four regions. In these different regimes, there is either only a locked state, only a running state, a bistability between locked and running states, or a bistability of two different running states (corresponding to a systematic motion to the left or right, respectively). In the presence of noise, the mean velocity depends in different ways on the noise intensity for the various parameter regimes. These dependences are explored by means of numerical simulations and simple analytical estimates for limiting cases. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with some new asymptotic properties on mild solutions to a nonlocal Cauchy problem of integrodifferential equation in Banach spaces. Under some well-imposed conditions on the nonlocal Cauchy, the neutral and forced terms, respectively, we establish some existence results for weighted pseudo S-asymptotically (ω, k)-Bloch periodic mild solutions to the referenced equation on R + by suitable superposition theorems. The results show that the strict contraction of the nonlocal Cauchy and the neutral terms with the state variable has an appreciable effect on the existence and uniqueness of such a solution compared with the forced term. As an auxiliary result, the existence of weighted pseudo S-asymptotically (ω, k)-Bloch periodic mild solutions is deduced under the sublinear growth condition on the force term with its state variable. The existence of weighted pseudo S-asymptotically ω-antiperiodic mild solution is also obtained as a special example. [ABSTRACT FROM AUTHOR]

In this work, we studied the behavior of the solitonic waves and localized waves in nonlinear optical fibers. We use the generalized non-autonomous NLSE with space- and time-dependent coefficients. To construct exact soliton solutions, we use the transformation hypothesis and the JEF method. For constant values of the space-and time-dependent coefficients, we have pointed out W-shaped bright soliton, dark soliton, and Jacobi elliptic function solutions. When we consider the dispersion relation as a periodic function, we obtain new characteristics of the solitonic waves. We have used numerical simulation to seek the modulation's unstable and stable modes. It results in modulated wave patterns emerging for strong enough values of the cubic and quintic nonlinearities, and for specific times of simulation, rogue waves and breather solitons emerge in the structure. We have also demonstrated that when the quintic nonlinearity is absent, the breathing behavior of the solitonic waves generates localized energy in the system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the transport properties of the class of limit-periodic continuum Schrödinger operators whose potentials are approximated exponentially quickly by a sequence of periodic functions. For such an operator H, and XH(t) the Heisenberg evolution of the position operator, we show the limit of 1/t XH(t) Ψ as t → ∞ exists and is nonzero for Ψ ≠ 0 belonging to a dense subspace of initial states which are sufficiently regular and of suitably rapid decay. This is viewed as a particularly strong form of ballistic transport, and this is the first time it has been proven in a continuum almost periodic non-periodic setting. In particular, this statement implies that for the initial states considered, the second moment grows quadratically in time. [ABSTRACT FROM AUTHOR]

In this paper, we study the concept of weakly almost periodic functions on Fréchet algebras. For a Fréchet algebra , we show that WAP () = w a p (). We also show that * * is Arens regular if and only if both and WAP () * are Arens regular. Finally, for a sequence of Fréchet algebras ( n) , we prove that the Fréchet algebra ℓ 1 - ∏ n ∈ ℕ n is Arens regualr if and only if each n is Arens regular. [ABSTRACT FROM AUTHOR]

This paper proposes a novel method for constructing optical zero-correlation zone (OZCZ) codes based on the Mutually Orthogonal Complementary Sets (MOCS) matrix and the concatenation operation. The most important property of the obtained codes is that the periodic correlation function of the proposed codes is zero for the phase shifts within the zero-correlation zone. The procedure of construction is simple, flexible and offers increasing number of users M = 2 n + 1 . In the other hand, we have presented a fully reconfigurable hardware implementation based on very high description language of the proposed OZCZ codes for optical code-division multiple-access encoder system. The implementation is performed on an Altera field-programmable gate arrays device using ROM memory and an up counter. The simulation results have been found satisfactory and are in conformity with the theoretical observations. [ABSTRACT FROM AUTHOR]

In the current research, we investigate and establish Korovkin-type approximation theorems for linear operators defined on the space of all $% 2\pi $-periodic and real valued continuous functions on $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}$ by means of $\mathcal{A}$-summation process via statistical convergence with respect to power series method. We demonstrate with an example how our theory is more strong than previously studied. Additionally, we research the rate of convergence of positive linear operators defined on this space.

In this article, we found an undesirable feature in the theory of summability; that is, the Fourier series of 2π$$ 2\pi $$‐periodic functions is uniformly convergent to the functions via the Ceàsro mean. However, it does not preserve the uniform convergence for the arbitrary periodic functions. To overcome this limitation, the objective of the paper is to introduce and study the notion of deferred Ceàsro mean based on the q$$ q $$ integers for the Fourier series of arbitrary periodic functions. We further study the behavior of our proposed method under both ordinary and statistical versions of convergence and accordingly establish two Korovkin‐type approximation theorems for trigonometric test functions. Finally, we also provide some examples with geometrical illustrations in support of the effectiveness of our results established in this work. [ABSTRACT FROM AUTHOR]

In a former work, we introduced a new class of functions, which we call measure pseudo S$$ \mathcal{S} $$‐asymptotically omega periodic functions and give some properties of those functions as the topological properties of the space of such functions, the invariance by translation of this space, and the convolution product in this space. In this paper, we consider the same class of functions and give contributions as the stability by the integral operator; we present some examples and composition results. Then we apply the obtained results to investigate the existence of measure pseudo S$$ \mathcal{S} $$‐asymptotically omega periodic mild solutions for some nonlinear infinite delay integral equations arising in epidemiology; exactly, we consider the nonlinear infinite delay integral equation: x(t)=α(t)x(t−β)+∫−∞ta(t−s)f(s,x(s))ds+g(t,x(t))fort∈ℝ.$$ x(t)=\alpha (t)x\left(t-\beta \right)+{\int}_{-\infty}^ta\left(t-s\right)f\left(s,x(s)\right) ds+g\left(t,x(t)\right)\kern0.5em \mathrm{for}\kern0.4em t\in \mathbb{R}. $$Then we apply these results to a finite delay integral equation when the delay is time dependent x(t)=α(t)x(t−β)+∫t−γ(t)tf(s,x(s))ds+g(t,x(t))fort∈ℝ.$$ x(t)=\alpha (t)x\left(t-\beta \right)+{\int}_{t-\gamma (t)}^tf\left(s,x(s)\right) ds+g\left(t,x(t)\right)\kern0.5em \mathrm{for}\kern0.4em t\in \mathbb{R}. $$ [ABSTRACT FROM AUTHOR]

The average deformed fiber is a continuous and smooth function of the fourth order. The deflection and rotation of beams can be determined by various methods available in the literature. Thus, in this paper, the expression of the average deformed fiber is defined in advance, then it is considered an infinite sum of sinusoidal functions that are successively evaluated using sinusoidal trigonometric series if it is considered a periodic function, or using Fourier series if it is considered a non-periodic function. The method is examined by solving several beam problems. The results indicate that the method can be used with confidence for solving any bending beam problem. [ABSTRACT FROM AUTHOR]

Shunting inhibitory cellular neural networks with compartmental periodic unpredictable coefficients and inputs is the focus of this research. A new algorithm is suggested, to enlarge the set of known unpredictable functions by applying diagonalization in arguments of functions of several variables. Sufficient conditions for the existence and uniqueness of exponentially stable unpredictable and Poisson stable outputs are obtained. To attain theoretical results, the included intervals method and the contraction mapping principle are used. Appropriate examples with numerical simulations that support the theoretical results are provided. It is shown how dynamics of the neural network depend on a new numerical characteristic, the degree of periodicity. [ABSTRACT FROM AUTHOR]

Within the realms of human thoughts on nature, Fourier analysis is considered as one of the greatest ideas currently put forwarded. The Fourier transform shows that any periodic function can be rewritten as the sum of sinusoidal functions. Having a Fourier transform view on real-world problems like the DNA sequence of genes, would make things intuitively simple to understand in comparison with their initial formal domain view. In this study we used discrete Fourier transform (DFT) on DNA sequences of a set of genes in the bovine genome known to govern milk production, in order to develop a new gene clustering algorithm. The implementation of this algorithm is very user-friendly and requires only simple routine mathematical operations. By transforming the configuration of gene sequences into frequency domain, we sought to elucidate important features and reveal hidden gene properties. This is biologically appealing since no information is lost via this transformation and we are therefore not reducing the number of degrees of freedom. The results from different clustering methods were integrated using evidence accumulation algorithms to provide in insilico validation of our results. We propose using candidate gene sequences accompanied by other genes of biologically unknown function. These will then be assigned some degree of relevant annotation by using our proposed algorithm. Current knowledge in biological gene clustering investigation is also lacking, and so DFT-based methods will help shine a light on use of these algorithms for biological insight. [ABSTRACT FROM AUTHOR]

Lanthanide luminescent materials play key roles in modern society, but their first‐principles treatment remains a great challenge due to complex manifold of electronic excited states and the difficulty in performing excited state structural relaxations that is necessary to model luminescent properties. Herein, we propose a practical approach that combines embedded cluster model (ECM) based multi‐configurational wave function theory (WFT) and occupancy constrained density‐functional theory plus the Hubbard U correction (OC‐DFT + U) to treat lanthanide doped luminescent materials, using LaF3:Ce3+, a typical scintillator with low symmetry, as a case study. We show that the combined approach yields accurate absorption energies with an error on the order of 200 cm−1, but the emission energies are significantly underestimated, the origin of which is further clarified by vibrationally resolved absorption and emission spectra calculation. This work demonstrates the possibility of combining ECM‐based wave function theory and periodic DFT into a comprehensive computational scheme for lanthanide luminescent materials and highlights the limitations of the current implementation of OC‐DFT + U for excited state structural optimization. [ABSTRACT FROM AUTHOR]

A signature of synaptic potentiation conductance has been observed in an α-Fe2O3/p-Si device fabricated using spin coating. The conductance of the device in dark conditions and illumination with a white light source was characterized as a function of the application of a periodic bias (voltage) with a triangular profile. The conductance of the device increases with the number of voltage cycles applied and plateaus to its maximum value of 0.70 μS under dark conditions and 12.00 μS under illumination, and this mimics the analog synaptic weight change with the action potential of a neuron. In the range of applied voltage from 0 V to 0.7 V, the conduction mechanism corresponds to trap-assisted tunneling (TAT) and in the range of 0.7–5 V it corresponds to the Poole–Frenkel emission (PFE). The conductance as a function of electrical pulses was fitted with a Hill function, which is a measure of cooperation in biological systems. In this case, it allows one to determine the turn-on threshold (K) of the device in terms of the number of voltage pulses, which are found to be 3 and 166 under dark and illumination conditions, respectively. The gradual conductance change and activation after a certain number of pulses perfectly mimics the synaptic potentiation of neurons. In addition, the threshold parameter extracted from the Hill equation fit, acting as the number of pulses for synaptic activation, is found to have programmability with the intensity of the light illumination. [ABSTRACT FROM AUTHOR]

In this paper, we analyze Levitan and Bebutov metrical approximations of functions F : Λ x X → Y by trigonometric polynomials and ρ-periodic type functions, where ∅= Λ ⊆ Rn, X and Y are complex Banach spaces, and ρ is a general binary relation on Y . We also analyze various classes of multidimensional Levitan almost periodic functions in general metric and multidimensional Bebutov uniformly recurrent functions in general metric. We provide several applications of our theoretical results to the abstract Volterra integro-differential equations and the partial differential equations. [ABSTRACT FROM AUTHOR]

Based on the triply periodic minimal surface (TPMS), 3D auxetic structures are successfully implemented using a dual-period function. A series of shape-controllable, dual-period deformation functions are obtained by summarising the characteristics of periodic deformation functions and applying Bezier curve fitting methods. Then, with the geometry originating from the Schwarz primitive (P) of TPMS, the periodic shape transformation of TPMS is achieved using the dual-period deformation functions. The property (negative Poisson's ratio) of the auxetic structure is investigated based on the control parameters (the TPMS c value, periodic function η, and deformation index γ). The auxetic structures can exhibit excellent 3D negative Poisson's ratio properties, and the Poisson's ratio can be effectively adjusted. Moreover, a heterostructure with positive and negative Poisson's ratio structures is obtained and applied to a stem in the hip joint. The simulation proves that the heterostructure can effectively prevent the failure of the bone implant. [ABSTRACT FROM AUTHOR]

Spectral decomposition is a robust attribute in the characterisation of post-stack seismic data. Reflected wavelets from specific geological lithologies show distinct patterns in the frequency spectrum, which is related to the wavelet frequency content and the interference pattern of a structure within a layer. Generally, the frequency spectrum of a wavelet has a smooth pattern through the entire bandwidth. However, the structure of a layer, such as its thickness and lateral distribution, will appear as a periodic function in the frequency domain. This research investigates the distinct patterns of reflective layers with different thicknesses using instantaneous amplitude and fast sparse S-transform (ST). To verify the efficiency of both methods, we have examined these approaches on channel-shaped and wedge-shaped synthetic models. The results showed that the fast sparse ST could predict features of both synthetic models better than the instantaneous amplitude. Next, the fast sparse ST is performed on experimental field data selected from a time interval containing a hydrocarbon channel reservoir. We used a sparse deconvolution algorithm to suppress the wavelet effect and balance all the frequencies. The results of this research showed the efficiency of the fast sparse ST in predicting a hydrocarbon channel reservoir's thickness. [ABSTRACT FROM AUTHOR]

BACKGROUND: Patient data in current computed tomography scanner machines are transferred through several communication channels, such as WiFi, to the mobile channel platform. Therefore, patient information is an important security concern. Medical imaging must be protected using various methods. OBJECTIVE: The current hardware-dependent method for generating random bits exhibits predictable or inconvenient physical characteristics. Therefore, a more flexible random-bit generation technique is to be devised. METHODS: We propose a deterministic random bit generation algorithm that uses a mathematical periodic function. RESULTS: After randomizing the image using the proposed random bit, the performance is analyzed and compared with that of the processed image. CONCLUSION: The random bit generation method using a mathematical algorithm shows higher entropy than the random bit generated by hardware. [ABSTRACT FROM AUTHOR]

We present a general method for establishing quantitative Voronovskaya-type estimates of convolution operators on homogeneous Banach spaces of periodic functions of one real variable or of functions on the real line. The method is based on properties of the Fourier transform of the kernel of the operator. We illustrate the elegance and the efficiency of this approach on two convolution operators--the Riesz typical means, and, in particular, the Fejér operator, and the generalized singular integral of Picard. A noteworthy feature of the former is the fact that, though the operator itself is saturated, the convergence in its Voronovskaya-type estimate can be of an arbitrary fast power-type provided that the function is smooth enough in a certain sense. [ABSTRACT FROM AUTHOR]

Lanthanide luminescent materials play key roles in modern society, but their first‐principles treatment remains a great challenge due to complex manifold of electronic excited states and the difficulty in performing excited state structural relaxations that is necessary to model luminescent properties. Herein, we propose a practical approach that combines embedded cluster model (ECM) based multi‐configurational wave function theory (WFT) and occupancy constrained density‐functional theory plus the Hubbard U correction (OC‐DFT + U) to treat lanthanide doped luminescent materials, using LaF3:Ce3+, a typical scintillator with low symmetry, as a case study. We show that the combined approach yields accurate absorption energies with an error on the order of 200 cm−1, but the emission energies are significantly underestimated, the origin of which is further clarified by vibrationally resolved absorption and emission spectra calculation. This work demonstrates the possibility of combining ECM‐based wave function theory and periodic DFT into a comprehensive computational scheme for lanthanide luminescent materials and highlights the limitations of the current implementation of OC‐DFT + U for excited state structural optimization. [ABSTRACT FROM AUTHOR]

We analyze the existence and types of unbounded Fatou components for elliptic functions and other meromorphic functions with doubly periodic Julia sets. We show that apart from Herman rings and Siegel disks, all types of dynamics can occur in these domains, which are called toral bands. We show that toral bands are not necessarily periodic, and we give results about the number of distinct residue classes of critical points in each toral band. [ABSTRACT FROM AUTHOR]

The dynamic response transfer rate function (TRF) is increasingly used in the field of structural modal parameter identification because it does not depend on the white noise assumption of the excitation. In this paper, the interference of periodic excitation on structural modal parameter identification using TRF is analyzed theoretically for a class of civil engineering structures with obvious periodic components in excitation, and then an identification method of structural real modal parameters is proposed. First, a delayed TRF is constructed, and the pseudo-frequency response function is further obtained to identify the periodic spurious poles of the whole system. Then, the effective identification of the real modal parameters of the structure is achieved by comparing the system poles identified via conventional TRF. Finally, the feasibility and robustness of the proposed method were verified using a calculation example with four-degrees-of-freedom system. In addition, the modal parameters of a structure under periodic excitation were effectively identified by taking a pumping station as an example, and the results show that the method accurately identified the structural modal parameters when the excitation contained periodic components, which has wider prospects for technical applications. [ABSTRACT FROM AUTHOR]

In this article, we first introduce and study the basic concepts of deferred Euler and deferred Nörlund product summability means of Fourier series of arbitrary periodic functions. We then estimate the degree of approximation of Fourier series of an arbitrary periodic function in the generalized Zygmund class based upon our proposed product deferred summability means. Moreover, we discuss some important concluding remarks in connection with our findings. Finally, we suggest a direction for future studies on this subject, which are based upon the basic notion of statistical product deferred summability means of Fourier series of arbitrary periodic functions in the generalized Zygmund class. [ABSTRACT FROM AUTHOR]

The goal of this paper is to define local weighted variable Sobolev spaces of fractional and negative order and their characterization by wavelets. We first consider local weighted variable Sobolev spaces by means of weak derivatives and obtain a wavelet characterization for these spaces. Using the Bessel potentials, we next define local weighted variable Sobolev spaces of fractional order. We show that Sobolev spaces obtained by weak derivatives and those by the Bessel potentials coincide. Finally, using duality, we define local weighted variable Sobolev spaces with negative order. We also show that local weighted variable Sobolev spaces are closed under complex interpolation. Some examples are given including the applications to weighted uniformly local Lebesgue spaces with variable exponents and periodic function spaces as a by-product, although the exponent is constant. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new properties of (ω , c) -periodic and asymptotically (ω , c) -periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations: (1) { c D t α u (t) = A u (t) + c D t α − 1 f (t , u (t)) , 1 < α < 2 , t ∈ R , u (0) = 0 and (2) { c D t α u (t) = A u (t) + c D t α − 1 f (t , u (t − h)) , 1 < α < 2 , t , h ∈ R + , u (0) = 0 where c D t α (⋅) (1 < α < 2) stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space X and the function f (t , x) is (ω , c) -periodic or asymptotically (ω , c) -periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations. [ABSTRACT FROM AUTHOR]

We study the second moduli of continuity for functions from the space of continuous and periodic real functions and the space of functions uniformly continuous on the entire axis. In both spaces, we give negative answer to the question about the necessity of a condition formulated by V. E. Geit, which is sufficient for a function to be the second module of continuity. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and systematically analyze the classes of (pre-) (B , ρ , (t k)) -piecewise continuous almost periodic functions and (pre-) (B , ρ , (t k)) -piecewise continuous uniformly recurrent functions with values in complex Banach spaces. We weaken substantially, or remove completely, the assumption that the sequence (t k) of possible first kind discontinuities of the function under consideration is a Wexler sequence (in order to achieve these aims, we use certain results about Stepanov almost periodic type functions). We provide many applications in the analysis of the existence and uniqueness of almost periodic type solutions for various classes of the abstract impulsive Volterra integro-differential inclusions. [ABSTRACT FROM AUTHOR]

A second-order ordinary differential equation with a three-position hysteresis relay characteristic and a periodic perturbation function is considered. The existence theorem is proved for an oscillatory solution with a complete traversal of the characteristic with a possible exit into its saturation zones in some finite time and with a closed phase trajectory of an arbitrary shape. Sufficient conditions for the existence of periodic solutions with arbitrary and symmetric phase trajectories are established, as well as conditions for the nonexistence of a periodic solution with a symmetric phase trajectory. Numerical examples are given. [ABSTRACT FROM AUTHOR]

A result of Monsky states that the Hilbert–Kunz function of a one-dimensional local ring of prime characteristic has a term \phi that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field, \phi is the zero function and is therefore immediately periodic with period 1. In additional examples produced by Kunz [Amer. J. Math. 91 (1969), pp. 772–784] and Monsky [Math. Ann. 263 (1983), pp. 43–49], \phi is immediately periodic with period 2. We show that, for every positive integer \pi, there exists a ring for which \phi is immediately periodic with period \pi. [ABSTRACT FROM AUTHOR]