Fractional Part Research Articles

Stability and Synchronization of Fractional-Order Complex-Valued Inertial Neural Networks: A Direct Approach

This paper is dedicated to the asymptotic stability and synchronization for a type of fractional complex-valued inertial neural network by developing a direct analysis method. First, a new fractional differential inequality is presented for nonnegative functions, which provides an effective tool for the convergence analysis of fractional-order systems. Moreover, instead of the previous separation analysis for complex-valued neural networks, a class of Lyapunov functions composed of the complex-valued states and their fractional derivatives is constructed, and some compact stability criteria are derived. In synchronization analysis, unlike the existing control schemes for reduced-order subsystems, some feedback and adaptive control schemes, formed by the linear part and the fractional derivative part, are directly designed for the response fractional inertial neural networks, and some synchronization conditions are derived using the established fractional inequality. Finally, the theoretical analysis is supported via two numerical examples.

Robust Watermarking Scheme for Vector Geographic Data Based on the Ratio Invariance of DWT–CSVD Coefficients

Traditional frequency-domain watermarking algorithms for vector geographic data suffer from disadvantages such as the random watermark embedding position, unpredictable embedding strength, and difficulty in resisting multiple attacks at the same time. To address these problems, we propose a novel watermarking algorithm based on the geometric invariance of the ratios of discrete wavelet transform (DWT) and complex singular value decomposition (CSVD) coefficients, which embeds the watermark information in a new embedding domain. The proposed scheme first extracts feature points from the original vector geographic data using the Douglas–Peucker algorithm, and then constructs a complex sequence based on the feature points set. The two-level DWT is then performed on the complex sequence to obtain the low frequency coefficients (L2) and high frequency coefficients (H2). On this premise, the CSVD algorithm is utilized to calculate the singular values of L2 and H2, and the ratio of the singular values of L2 and H2 is acquired as the watermark embedding domain. During the watermark embedding process, a new watermark sequence is created by the fusion of the original watermark index value and bits value to improve the recognition of the watermark information, and the decimal part at different positions of the ratio is altered by the new watermark sequence to control the watermark embedding strength. The experimental results show that the proposed watermarking algorithm is not only robust to common attacks such as geometric, cropping, simplification, and coordinate point editing, but also can extract watermark images with a high probability under random multiple attacks.

Heatmap Regression via Randomized Rounding.

Heatmap regression has become the mainstream methodology for deep learning-based semantic landmark localization, including in facial landmark localization and human pose estimation. Though heatmap regression is robust to large variations in pose, illumination, and occlusion in unconstrained settings, it usually suffers from a sub-pixel localization problem. Specifically, considering that the activation point indices in heatmaps are always integers, quantization error thus appears when using heatmaps as the representation of numerical coordinates. Previous methods to overcome the sub-pixel localization problem usually rely on high-resolution heatmaps. As a result, there is always a trade-off between achieving localization accuracy and computational cost, where the computational complexity of heatmap regression depends on the heatmap resolution in a quadratic manner. In this paper, we formally analyze the quantization error of vanilla heatmap regression and propose a simple yet effective quantization system to address the sub-pixel localization problem. The proposed quantization system induced by the randomized rounding operation 1) encodes the fractional part of numerical coordinates into the ground truth heatmap using a probabilistic approach during training; and 2) decodes the predicted numerical coordinates from a set of activation points during testing. We prove that the proposed quantization system for heatmap regression is unbiased and lossless. Experimental results on popular facial landmark localization datasets (WFLW, 300W, COFW, and AFLW) and human pose estimation datasets (MPII and COCO) demonstrate the effectiveness of the proposed method for efficient and accurate semantic landmark localization. Code is available at http://github.com/baoshengyu/H3R.

A Remarkable Summation Formula, Lattice Tilings, and Fluctuations

We derive and prove an explicit formula for the sum of the fractional parts of certain geometric series. Although the proof is straightforward, we have been unable to locate any reference to this result. This summation formula allows us to efficiently analyze the average behavior of certain common nonlinear dynamical systems, such as the angle-doubling map, modulo 1. In particular, one can use this information to analyze how the behavior of individual orbits deviates from the global average (called fluctuations). More generally, the formula is valid in , where expanding maps give rise to so-called number systems. To illustrate the usefulness in this setting, we compute the fluctuations of a certain map on the plane.

Occluded object detection and exposure in cluttered environments with automated hyperspectral anomaly detection.

Cluttered environments with partial object occlusions pose significant challenges to robot manipulation. In settings composed of one dominant object type and various undesirable contaminants, occlusions make it difficult to both recognize and isolate undesirable objects. Spatial features alone are not always sufficiently distinct to reliably identify anomalies under multiple layers of clutter, with only a fractional part of the object exposed. We create a multi-modal data representation of cluttered object scenes pairing depth data with a registered hyperspectral data cube. Hyperspectral imaging provides pixel-wise Visible Near-Infrared (VNIR) reflectance spectral curves which are invariant in similar material types. Spectral reflectance data is grounded in the chemical-physical properties of an object, making spectral curves an excellent modality to differentiate inter-class material types. Our approach proposes a new automated method to perform hyperspectral anomaly detection in cluttered workspaces with the goal of improving robot manipulation. We first assume the dominance of a single material class, and coarsely identify the dominant, non-anomalous class. Next these labels are used to train an unsupervised autoencoder to identify anomalous pixels through reconstruction error. To tie our anomaly detection to robot actions, we then apply a set of heuristically-evaluated motion primitives to perturb and further expose local areas containing anomalies. The utility of this approach is demonstrated in numerous cluttered environments including organic and inorganic materials. In each of our four constructed scenarios, our proposed anomaly detection method is able to consistently increase the exposed surface area of anomalies. Our work advances robot perception for cluttered environments by incorporating multi-modal anomaly detection aided by hyperspectral sensing into detecting fractional object presence without need for laboriously curated labels.

On the correlations of $n^\alpha$ mod 1

A well known result in the theory of uniform distribution modulo 1 (which goes back to Fejér and Csillag) states that the fractional parts \{n^{\alpha}\} of the sequence (n^{\alpha})_{n\ge1} are uniformly distributed in the unit interval whenever \alpha>0 is not an integer. For sharpening this knowledge to local statistics, the k -level correlation functions of the sequence (\{n^{\alpha}\})_{n\geq1} are of fundamental importance. We prove that for each k\ge2, the k -level correlation function R_{k} is Poissonian for almost every \alpha>4k^{2}-4k-1 .

1D Sine-Map-Coupling-Logistic-Map for 3D model encryption

With the rise of technologies of VR technology, AR technology, and 3D printing, the application of 3D models has become more and more extensive. The data of the 3D model is the floating point and has a unique storage format, and the traditional 2D image encryption algorithms are unsuitable for 3D models. Therefore, based on 1D Sine-Map-Coupling-Logistic-Map (1D-SMCLM), a 3D model encryption algorithm is designed in this paper. The 1D-SMCLM is a new chaotic system with large parameter space and good chaotic characteristics. The keystream generated by the 1D-SMCLM has good randomness and is very suitable for cryptographic systems. In the new encryption algorithm (SMCLM-3ME), the vertices of the 3D models are divided into integer and decimal vertices. The integer part is encrypted by the strategy of simultaneous scrambling and diffusion. The 3D ciphertext model is obtained by combining the integer and fractional parts. Experimental results show that the SMCLM-IE exhibits excellent performance.

A fast algorithm for computing the Smith normal form with multipliers for a nonsingular integer matrix

A Las Vegas randomized algorithm is given to compute the Smith multipliers for a nonsingular integer matrix A, that is, unimodular matrices U and V such that AV=US, with S the Smith normal form of A. The expected running time of the algorithm is about the same as required to multiply together two matrices of the same dimension and size of entries as A. Explicit bounds are given for the size of the entries in both unimodular multipliers. The main tool used by the algorithm is the Smith massager, a relaxed version of V, the unimodular matrix specifying the column operations of the Smith computation. From the perspective of efficiency, the main tools used are fast linear system solving and partial linearization of integer matrices. As an application of the Smith with multipliers algorithm, a fast algorithm is given to find the fractional part of the inverse of the input matrix.

Rational points on nonlinear horocycles and pigeonhole statistics for the fractional parts of

AbstractIn this paper, we investigate pigeonhole statistics for the fractional parts of the sequence $\sqrt {n}$ . Namely, we partition the unit circle $ \mathbb {T} = \mathbb {R}/\mathbb {Z}$ into N intervals and show that the proportion of intervals containing exactly j points of the sequence $(\sqrt {n} + \mathbb {Z})_{n=1}^N$ converges in the limit as $N \to \infty $ . More generally, we investigate how the limiting distribution of the first $sN$ points of the sequence varies with the parameter $s \geq 0$ . A natural way to examine this is via point processes—random measures on $[0,\infty )$ which represent the arrival times of the points of our sequence to a random interval from our partition. We show that the sequence of point processes we obtain converges in distribution and give an explicit description of the limiting process in terms of random affine unimodular lattices. Our work uses ergodic theory in the space of affine unimodular lattices, building upon work of Elkies and McMullen [Gaps in $\sqrt {n}$ mod 1 and ergodic theory. Duke Math. J.123 (2004), 95–139]. We prove a generalisation of equidistribution of rational points on expanding horocycles in the modular surface, working instead on nonlinear horocycle sections.

SOLVING LINEAR SYSTEM OF VOLTERRA INTEGRO-FRACTIONAL DELAY DIFFERENTIAL EQUATIONS USING LAPLACE ADOMIAN DECOMPOSITION WITH MODIFICATION

This study uses for the first time a Laplace Adomian decomposition with a modified technique for solving linear system of Volterra integro-fractional delay differential equations. The fractional part is a Caputo type, and the delay part is a retarded delay time lag. This strategy is essentially based on an elegant combination of the Laplace transform strategy, the series expansion strategy, and the modified Adomian polynomial. This technique computes analytical work using multiple sorts of kernels, which have qualities such as difference kernels and simple degenerate kernels. This method is good for solving the problem like that and getting a good result. These are illustrated by assessing the efficiency and accuracy of the cause technique and solving specific examples.

A 3D model encryption scheme based on a cascaded chaotic system

With the birth of the metaverse, 3D models have received extensive attention, and the security of information transmission continues to be an important issue. In this paper, we propose a 3D model encryption method based on a 2D chaotic system constructed via the coupling of the logistic map and infinite collapse (2D-LAIC) and on semi-tensor product (STP) theory. In terms of Lyapunov exponents, NIST test results, bifurcation diagrams, etc., 2D-LAIC exhibits better dynamical behavior than classical chaotic systems. 2D-LAIC can generate an unpredictable keystream, which is highly suitable for cryptography. Therefore, we propose a new 3D model encryption algorithm based on 2D-LAIC, named 3DME-SC. For a 3D model of the floating-point data type, XOR and STP processing are applied to the integer part and fractional part, respectively, of the model to obtain a 3D ciphertext model. The keystream required for XOR and STP processing is generated by 2D-LAIC. The results of a detailed security analysis and a comparative experimental analysis show that 3DME-SC exhibits good performance and effectiveness.(Code: https://github.com/Gao5211996/3D-model-encryption)

A game-theoretic approach for power pricing in a resilient supply chain considering a dual channel biorefining structure and the hybrid power plant

This study investigates power pricing decisions in a resilient supply chain, including the main biomass supplier, biorefinery, the hybrid power plant, and the backup supplier. The power plant's main fuel is considered biofuel derived from Jatropha Curcas L. (JCL). However, due to unavoidable uncertainties in biofuel supply links, the shortage in meeting demand is inevitable. Therefore, for increasing the supply chain's resiliency, the fossil fuel backup supplier is taken into account to make up the shortage with anticipated fossil fuel. Due to environmental concerns, the penalty cost is considered for excessive use of fossil fuel. The model is studied under two cases by considering a dual-channel biorefining. Different game-theoretic models are established for varied power configurations and interactions of supply chain members. It is observed from the numerical study that, depending on the fractional part of the power plant's requirements of biofuel and JCL supplied by the biorefinery (Case 1) and the main supplier (Case 2), respectively the performance of the supply chain increases in the presence of the backup supplier. Additionally, the sensitivity analysis shows that Case 1 is more profitable than Case 2 for chain members. Moreover, in contrast to the fossil fuel replacement coefficient, the greater the convertibility degree of the JCL, the larger the profits for the members.

Mechanism and Suppression of Frequency-Decrease Effect Related to the Low Modulation Frequency Ratio in a High-Ratio MMC-Based DC Transformer

The high-ratio MMC-based DC transformer is the crucial device for the interconnection between different DC distribution networks. Compared to the typical MMC, the transformer has a similar structure and low modulation frequency ratios (FR). Thus, it also has the severe frequency-decrease effect and similar operating performance deterioration. However, because of the extremely lower FRs that are close to 1, the feature of the frequency-decrease effect in the transformer has different manifestations. And because of the two-stage structure which combines the MMC and rectifiers, more factors emerged in the transformer to determine the frequency-decrease effect. Therefore, this study further develops the theory and elaborates the frequency effect's mechanism that is originally from the MMC to adapt to the more complex structure of the transformer. A quantitative analysis of the frequency-decrease effect is then provided, demonstrating that because of the transformer's two-stage structures, two types of FRs and concomitant factors exist in the transformer, thus having a combined effect on the harmonic distributions of capacitor voltages and causing more proportional voltages' harmonics distorting into the slope harmonics. Based on the analysis, the study entirely redesigns the FRs and regulates the modulation strategy of the transformer to fundamentally eliminate the frequency-decrease effect. The study also confirms that the FRs in the two sides cannot be set to the same, and their decimal part cannot cause extra low-frequency harmonics. The frequency-decrease effect, the deteriorations, and the efficiency and comparisons of the proposed methods are finally demonstrated via simulations and experiments.

Astigmatic transformation of a fractional-order edge dislocation

It is shown theoretically that an astigmatic transformation of an edge dislocation (straight line of zero intensity) of the ν-th order (ν=n+α is a real positive number, n is integer, 0<α<1 is the fractional part of the number) forms at twice the focal length from a cylindrical lens n optical elliptical vortices (screw dislocations) with a topological charge of –1, located on a straight line perpendicular to the edge dislocation. Coordinates of these points are zeros of the Tricomi function. At some distance from these vortices and on the same straight line, another additional vortex with a topological charge of –1 is also generated, which moves to the periphery if α decreases to zero, or approaches n vortices if α tends to 1. In addition, at the periphery in the beam cross-section, a countable number of optical vortices (intensity zeros) are formed, all with a topological charge of –1, which are located on diverging curved lines (such as hyperbolas) equidistant from a straight line on which the main n intensity zeros are located. These "accompanying" vortices approach the center of the beam, following the additional "passenger" vortex, if 0<α<0.5, or move to the periphery, leaving the "passenger" next to the main vortices, if 0.5<α<1. At α=0 and α=1, the "accompanying" vortices are situated at infinity. The topological charge of the entire beam at fractional ν is infinite. The numerical simulation confirms theoretical predictions.

On expansions involving the Riemann zeta function and its derivatives

By studying the spectral aspects of the fractional part function in a well-known separable Hilbert space, we show, among other things, a rational approximation of the Riemann zeta function and its derivatives valid on every vertical line in the right half-planes ℜs>1/2 and ℜs>0. Moreover, we provide some discussions and explicit computations related to the fractional part function.

A Novel Technique for Secure Data Cryptosystem Based on Chaotic Key Image Generation

The advancements in Information and Communication Technology (ICT), within the previous decades, has significantly changed people’s transmit or store their information over the Internet or networks. So, one of the main challenges is to keep these information safe against attacks. Many researchers and institutions realized the importance and benefits of cryptography in achieving the efficiency and effectiveness of various aspects of secure communication.This work adopts a novel technique for secure data cryptosystem based on chaos theory. The proposed algorithm generate 2-Dimensional key matrix having the same dimensions of the original image that includes random numbers obtained from the 1-Dimensional logistic chaotic map for given control parameters, which is then processed by converting the fractional parts of them through a function into a set of non-repeating numbers that leads to a vast number of unpredicted probabilities (the factorial of rows times columns). Double layers of rows and columns permutation are made to the values of numbers for a specified number of stages. Then, XOR is performed between the key matrix and the original image, which represent an active resolve for data encryption for any type of files (text, image, audio, video, … etc). The results proved that the proposed encryption technique is very promising when tested on more than 500 image samples according to security measurements where the histograms of cipher images are very flatten compared with that for original images, while the averages of Mean Square Error is very high (10115.4) and Peak Signal to Noise Ratio is very low (8.17), besides Correlation near zero and Entropy close to 8 (7.9975).

On arithmetic series involving the fractional part function

We present new relationships between the work of H. Davenport and A. I. Popov. A new general formula involving the Von Mangoldt function is presented, as well as a criteria for the Riemann Hypothesis.

Influence of titanium on structure formation, liquation processes and microhardness of structural components of Al – Ni – Ti alloys synthesized from the oxide phases by SHS metallurgy

The paper is devoted to revealing the regularities of the influence of titanium (0.91, 2.42, 3.19, 3.39, 4.32 and 8.81 wt.%) on structure formation, nature of element distribution and microhardness of structural components in Al – Ni – Ti alloys by aluminothermy during SHS metallurgy. As the initial composition of the charge were selected the following materials in fractional parts: Al:NiO2:CaF2:NaNO3:TiO2 = 10:10:12:6:X, where X = 1.5, 4.5, 5.0, 7.0, 10.0. The structural components in Al – Ni – Ti alloys have been identified by electron microscopy and X-ray spectral analysis of elements. In the alloys with 0.91–4.32 wt.% Ti the following phases crystallize: β'-phase (solid solution of Ni in the nickel aluminide AlNi) Al3Ni2, Al3Ti, Al3Ni and α-solid solution of Ni and Ti in aluminum. In an alloy with 8.81 wt.% Ti the β'-phase turns into a titanium-doped nickel aluminide Al(NiTi) (composition in at.%: 50.53 Al; 1.47 Ti; 48.0 Ni). The increase of titanium content in Al – Ni – Ti alloys increases the solubility of Ni in the β'-phase and at titanium concentration in the alloy 8.81 wt.% in the aluminide Al(NiTi) up to 48 at.% Ni is dissolved compared to the solubility of nickel (38 at.%) in the alloy with 0.91 wt.% Ti. Increasing the nickel content in the above phases contributes to their microhardness from 13 GPa to 14.8 GPa at 8.81 wt.% Ti. Increasing the titanium content in Al – Ni – Ti alloys to 4.32 wt.% increases the solubility of nickel in the nickel aluminide Al3Ni, with a higher concentration of titanium (8.81 wt.%) in the nickel aluminide with titanium Al(NiTi) dissolves up to 48.53 at.% Ni, while in the alloy with 0.91 wt.% Ti – only about 1.0 at.% Ni. At the same time, the Al and Ti content in titanium aluminide Al3Ti decreases and its microhardness increases. It was not possible to determine the microhardness of Al(NiTi) aluminide because of the formation of a porous structure. In nickel aluminide Al3Ni, an increase in titanium content leads to an increase in nickel concentration to 4.32 wt.% Ti followed by a slight increase to 8.81 wt.% Ti. Despite increasing the nickel content and decreasing the aluminum concentration, the microhardness of the nickel aluminide decreases. Apparently, this circumstance is caused by the formation of a porous structure in this phase. The research was carried out at the Center for Collective Use “Applied Materials Science” of the Federal State Budgetary Educational Institution of Higher Education “TOGU” with the financial support of the Ministry of Science and Education of the Russian Federation within the framework of research work No state reg. of the state task AAAA-A20-120021490002-1.

A Low-Power ADPLL with Calibration-Free RO-Based Injection-Locking TDC for BLE Applications

This paper proposes a low-power all-digital phase-locked loop (ADPLL) with calibration-free ring oscillator (RO)-based injection-locking time to digital converter (TDC) for BLE applications. The RO is reused as the delay cell of TDC, and the quantization step of TDC is always tracked with the RO period; hence no calibration is needed in this architecture. We adopt RO tuning to lower the injection-locking bandwidth so as to decrease the power consumption of the injection current. Moreover, the fractional part of phase error detection is turned down in the coarse tuning of ADPLL to save power. An LC-based digital-controlled oscillator (LCDCO) with a 6.4 nH inductor and a resistive bias is used to have a low power and better phase noise performance. The ADPLL is fabricated in 40 nm CMOS with a 1 V supply and consumes 1.4 mW when it is locked. The measured phase noise is −114 dBc/Hz at 1 MHz offset. The test results show significant power saving. Thus, it can be a promising candidate for BLE applications.

Computational ghost imaging encryption with a pattern compression from 3D to 0D

The principle of computational ghost imaging (GI) offers a potential application in optical encryption. Nevertheless, large numbers of keys composed of random or specific patterns set an obstacle to its application. Here, we propose a series of pattern compression methods based on computational GI, in which thousands of patterns are replaced by a single standard image (i.e., two-dimensional data), a sequence of numbers (i.e., one-dimensional data) or the fractional part of an irrational number (i.e., zero-dimensional data). Different pattern compression methods are tested in both simulations and experiments, and their error tolerances in encryption are further discussed. Our proposed methods can greatly reduce the pattern amount and enhance encryption security, which pushes forward the application of computational GI, especially in optical encryption.