Fractional Part Research Articles

A QUATERNIONIC FRACTIONAL BOREL–POMPEIU-TYPE FORMULA

In theoretical setting, associated with a fractional [Formula: see text]-Fueter operator that depends on an additional vector of complex parameters with fractional real parts, this paper establishes a fractional analog of Borel–Pompeiu formula as a first step to develop a fractional [Formula: see text]-hyperholomorphic function theory and the related operator calculus.

Emergent anomalies and generalized Luttinger theorems in metals and semimetals

Whether a given material is a metal or an insulator is determined by a single parameter, the number of electrons per unit cell per spin, that is, by the filling number. When the latter is integer, the material is an insulator or a compensated semimetal. When it is fractional, the material is a metal with a Fermi surface, enclosing a volume in momentum space, determined by the fractional part of the filling -- a statement known as Luttinger's theorem. Topological semimetals provide a realization of the metallic phase, distinct from ordinary metals, as they exist at integer fillings, which normally correspond to insulators. Here, the authors develop a framework that unifies Luttinger's theorem for ordinary metals and analogous exact statements for topological semimetals. They use the language of quantum anomalies of the emergent low-energy charge conservation symmetry at each point on the Fermi surface. This framework allows one to connect topological responses in all nodal semimetals to their low-energy properties, such as the distance between the point nodes in momentum space or the area, enclosed by the nodal line.

Middle school students’ construction of reciprocal reasoning with unknowns

Three iterative, after school design experiments with small groups of middle school students were conducted to investigate how students represent fractional relationships between two unknowns and whether they construct reciprocal reasoning with unknowns. Of the 22 students who participated, 9 had constructed fractions as multiples of unit fractions. Seven of these 9 students constructed reciprocal reasoning with unknowns. They did so by constructing a multiplicative relationship between a unit fraction of one unknown and the other unknown. The other 2 students did not make this construction. Instead, they represented relationships between unknowns with whole number multiplication and division, and by adding unknowns and fractional parts of unknowns. The students who constructed reciprocal reasoning demonstrated a link between fractions as measures and fractions as operators, one benefit of working on rational number knowledge and algebraic reasoning together. Implications for teaching include recommendations for supporting students to construct reciprocal reasoning with unknowns.

Mahler’s Conjecture on ξ(3/2) n mod 1

Abstract K. Mahler’s conjecture: There exists no ξ ∈ ℝ+ such that the fractional parts {ξ(3/2) n } satisfy 0 ≤ {ξ(3/2) n } < 1/2 for all n = 0, 1, 2,... Such a ξ, if exists, is called a Mahler’s Z-number. In this paper we prove that if ξ is a Z-number, then the sequence xn = {ξ(3/2) n }, n =1, 2,... has asymptotic distribution function c 0(x), where c 0(x)=1 for x ∈ (0, 1].

A DCT Fractional Bit Replacement Based Dual Watermarking Algorithm for Image Authentication

Abstract: Watermarking is a process of embedding a message inside a digital signal like an image, video or text. It is used for several key reasons such as authenticity verification, ownership recognition and hidden communication. In this paper, we discuss about image watermarking, where secret messages are stored in images. Introduction: We propose a dual watermarking approach, which is based on Discrete Cosine Transform, Discrete Wavelet Transform and Singular Value Decomposition methods. This paper considers one watermark as robust and other water mark as fragile. Method: The robust watermark is embedded in Discrete Wavelet Transform- Singular Value Decomposition - domain and is used to transmit hidden messages. The fragile watermark is embedded in Discrete Cosine Transform domain and is used for verification of secret message of the robust watermark. The proposed algorithm is tested in the experimental results section and shows promising results against denoising, rotation, translation and cropping attacks. Result: The results show that the performance of the proposed algorithm in terms of mean squared error, structural similarity and peak signal to noise ratio is S4considerable as compared with the existing methods. Discussion: We present the comparison results with Himanshu et. al. in table 10, from which we can see that our method performs better with gaussian noise and rotational attack only lacking with Salt and Pepper noise. Fig. 7 and Fig. 8, in terms of resulting PSNR shows the variation of noise variance and degree of rotation. From the graphs it is evident that out method performs better against Gaussian and rotational attack. Conclusion: In this paper a dual watermarking method is proposed in which one watermark is fragile which is called as authentication watermark whereas the other watermark is robust and is called as the information watermark. The authentication watermark is embedded in the fractional part of DCT domain in the cover image and the information watermark is embedded in the diagonal vector of the LL sub-band.

High resolution time of arrival estimation algorithm for B5G indoor positioning

In beyond 5G (B5G) indoor positioning time of arrival (TOA) positioning method is an important one. Base stations measure reference signals sent by terminals to calculate the TOA. The accuracy of position calculation is highly dependent on the TOA estimation resolution. This paper proposes a high-resolution TOA estimation algorithm based on Sounding Reference Signal (SRS) of 5G communication system. The algorithm achieves high resolution estimation by two steps. The Least Mean Square (LMS) adaptive filtering method is used as the first step to obtain the integer part of TOA, which is a coarse estimation of TOA. In the second step, we use weighted Generalized Cross-Correlation (GCC) algorithm to obtain the fractional part of TOA in frequency domain to get a more precise TOA estimation. Moreover, the stability of the estimated results is improved by the threshold value setting. The proposed algorithm is simulated in both Additive White Gaussian Noise (AWGN) and Rayleigh fading channels. The simulation results show that the TOA error achieves 0.56 m in AWGN and 0.96 m in Rayleigh fading channels for 80% of test samples respectively.

An Extension of the Three Gap Theorem to Interval Exchange Transformations

Abstract The three gap theorem asserts that for any real $\alpha $ and any positive integer $N$, the fractional parts of the sequence $0, \alpha , 2\alpha , \cdots , (N-1)\alpha $ have at most three distinct gap lengths. In this note, we extend the three gap theorem to arbitrary orbits of interval exchange transformations, and in the process, we generalize the three gap theorem and a result by Boshernitzan.

On the use of Egyptian fractions for stream ciphers

Within the scope of the mutual interference between modern number theory and chaos-based cryptography, and as it is well-known and already explored for the decimal or the continued fraction expansion of irrational numbers, interesting random-like behaviors seem to be hidden in Egyptian fraction expansions, thus suggesting new chaos-based encryption systems. In fact, at a practical level, and as will be shown through the present study, concatenation of involved denominators and binary expansion generally lead to pseudo-random streams which satisfactorily pass the NIST statistical test suite for randomness. Then, to a certain extent, and for cryptographic purposes, this can be considered as a new tool to complete the conventional class of already-in-use pseudo-random number generators. Some mathematical issues, however, have to be clarified, as for example the chaoticity of the process obtained after converting these denominators to fractional parts of a wandering sequence of real numbers in the unit interval. To the best of our knowledge, and from a cryptographic point of view, previous works on the subject are limited to statistical tests (and subsequent cryptographic analysis) of the resulting one-time pads or stream ciphers, but no rigorous study has been conducted to justify these methods within chaos theory. Considering Egyptian expansions as a working example and using the term “chaotic” in the sense of Devaney, this is what our proposal is aimed at in the present work.

Neural net implementation of steam properties on FPGA

Real time applications like model predictive control, monitoring and data reconciliation of power plants and industrial processes employ nonlinear mathematical models and require thermodynamic properties and their derivatives of working fluids. Applications like super heater temperature control based on energy balance and real time data reconciliation, require an efficient and a compact method for simultaneous estimation of thermodynamic properties, and their partial derivatives suitable for implementation in field-programmable gate array (FPGA). However, the complex mathematical formulations of these properties prohibit direct implementations in FPGAs. Single artificial neural network (ANN) architecture is used to replace the entire code in higher level languages, running into a few thousand lines. FPGA implementation of a compact neural network for the entire range of thermodynamic properties is presented. Large arguments in sigmoid function are factored into a product of integer and a fractional part which is represented using series approximation with five terms only and the integers are represented in look up table (LUT). This ensures optimum storage and computational burden for the above applications. The ANN is implemented in IEEE 754 floating point with synthesis in Xilinx ISE design suite using Verilog HDL. The results are presented for a typical pressure versus saturation temperature.

Fundamentals of metric theory of real numbers in their $\overline{Q_3}$-representation

In the paper we were studied encoding of fractional part of a real number with an infinite alphabet (set of digits) coinciding with the set of non-negative integers. The geometry of this encoding is generated by $Q_3$-representation of real numbers, which is a generalization of the classical ternary representation. The new representation has infinite alphabet, zero surfeit and can be efficiently used for specifying mathematical objects with fractal properties.
 We have been studied the functions that store the "tails" of $\overline{Q_3}$-representation of numbers and the set of such functions,some metric problems and some problems of probability theory are connected with $\overline{Q_3}$-representation.

On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function

The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on (0,∞), which is equivalent to the Riemann hypothesis. This involves dilations of the fractional part function by factors θ k ∈(0,1), k≥1. We develop probabilistic extensions of the Nyman-Beurling criterion by considering these θ k as random: this yields new structures and criteria, one of them having a significant overlap with the general strong Báez-Duarte criterion.

Intensity interference characteristics of fractional Laguerre–Gaussian optical vortex

The paper reports the intensity characteristics of fractional vortex beam, and controls the coupling between photon orbital angular momentum and spin angular momentum by coherently manipulating Laguerre–Gaussian vortex beam with linearly polarized plane light and its conjugate beam. By comparing theory with experiment, fractional vortex beams with different topological charges are generated by using pure phase liquid crystal spatial light modulator, and their intensity characteristics and interference characteristics are observed. The results show that the radial notch size of the spot is related to the fractional part of the topological charge value, and the closer the fractional part of the topological charge value is to the half integer, the more obvious the radial notch is. With the increase of integer part of topological charge, the spot radius also increases. In addition, interference with linearly polarized plane light reveals that the number of branches is equal to the number of topological charges. Interfere with its conjugate beam, and judge the topological charge number of vortex beam by the relationship between petal number and topological charge number. The petal number of integer vortex beam is twice the topological charge number, and fractional order vortex beam destroys the ring structure of Laguerre–Gaussian beam.

Quantification of DOM effects on tetracyclines transport during struvite recovery from swine wastewater

Struvite (MgNH4PO4·6H2O) recovered from livestock wastewater may impose a pharmacological threat to the environment, due to the extensive existence of antibiotics in the wastewater. In this study, tetracyclines (TCs) were selected as the typical antibiotics, and the individual processes of dissolved organic matters (DOM) evolution and their effects on TCs migration in struvite recovery from swine wastewater were discriminated and quantified. Results revealed that TCs transport was contributed by the adsorption of pure struvite crystals, struvite adsorbing DOM-TCs complex and DOM aggregation, which occupied 2.29–6.53%, 23.53–34.66%, and 59.09–74.19% of the total TCs migration amounts, respectively. A tangential flow filtration system was employed to divide DOM into five fractional parts on the basis of molecular weight cut-offs. Experimental results indicated that under alkaline conditions of struvite crystallization, DOMs with larger molecular weights, hydrolyzed to DOMs with smaller molecular weights, which consequently promoted TCs re-distribution in DOMs from higher molecular weights to those with lower molecular weights. Furthermore, a distribution model was developed to characterize TCs transport in struvite recovery by describing TCs distribution among various phases, including struvite adsorption, DOM-TCs complexing, DOM aggregation, and free state in the solution, respectively. These outcomes provided new understanding on DOM evolution and effects on antibiotics transport in phosphate recovery from wastewater.

A novel secure biomedical data aggregation using fully homomorphic encryption in WSN

A new method of secure data aggregation for decimal data having integer as well as fractional part using homomorphic encryption is described. The proposed homomorphic encryption provides addition, subtraction, multiplication, division and averaging operations in the cipher domain for both positive and negative numbers. The scheme uses integer matrices in finite field Zp as encryption and decryption keys. An embedded Digital signature along with data provides data integrity and authentication by signature verification at the receiving end. The proposed scheme is immune to chosen plaintext and chosen ciphertext attacks. In the case of homomorphic multiplication, the ciphertext expansion ratio grows linearly with the data size. The computational complexity of the proposed method for multiplication and division is relatively less by 22.87% compared to Brakerski and Vaikantanathan method when the size of the plaintext data is ten decimal digits.

A generalization of the Gauss–Kuzmin–Wirsing constant

The Gauss–Kuzmin–Wirsing constant is generalized to the transformation \(T_p(x)=\left\{ \dfrac{p}{x}\right\} \), where p is any given positive integer and \(\left\{ \dfrac{p}{x}\right\} \) denotes the fractional part of \(\dfrac{p}{x}\). A near-optimal estimate for the generalized constant is obtained.

Conservation of the half-integer topological charge on propagation of a superposition of two Bessel-Gaussian beams

We theoretically show that by continuously varying the amplitude of a constituent beam in the superposition of two Bessel-Gaussian (BG) beams with topological charges (TCs) $m$ and $n$ the TC of the entire superposition can be controlled. Between the constituent beams, there is a topological competition. For some parameters of the BG beams, TC of the superposition equals $n$, while for other parameters, it is equal to $m$. In the center of such a superposition (on the optical axis), there is an $m$-fold degenerate intensity null (optical vortex with the TC of $m<n$), whereas $n--m$ optical vortices with the unitary TC are in the beam periphery. We demonstrate that when the amplitudes of the constituent beams tend to be equal, peripheral vortices ``move'' to infinity. In this case, superposition of two BG beams has a half-integer TC, and this TC is conserved on propagation in space. The fractional part of the TC is, however, ``hidden'' in the infinity (``hidden'' phase). Conservation of the half-integer TC of an optical vortex on propagation is shown.

Proving the 6d Cardy formula and matching global gravitational anomalies

A Cardy formula for 6d superconformal field theories (SCFTs) conjectured by Di Pietro and Komargodski in [1] governs the universal behavior of the supersymmetric partition function on S^1_\beta \times S^5Sβ1×S5 in the limit of small \betaβ and fixed squashing of the S^5S5. For a general 6d SCFT, we study its 5d effective action, which is dominated by the supersymmetric completions of perturbatively gauge-invariant Chern-Simons terms in the small \betaβ limit. Explicitly evaluating these supersymmetric completions gives the precise squashing dependence in the Cardy formula. For SCFTs with a pure Higgs branch (also known as very Higgsable SCFTs), we determine the Chern-Simons levels by explicitly going onto the Higgs branch and integrating out the Kaluza-Klein modes of the 6d fields on S^1_\betaSβ1. We then discuss tensor branch flows, where an apparent mismatch between the formula in [1] and the free field answer requires an additional contribution from BPS strings. This ``missing contribution’’ is further sharpened by the relation between the fractional part of the Chern-Simons levels and the (mixed) global gravitational anomalies of the 6d SCFT. We also comment on the Cardy formula for 4d \mathcal{N}=2𝒩=2 SCFTs in relation to Higgs branch and Coulomb branch flows.

Real-Time Estimation of GPS-BDS Inter-System Biases: An Improved Particle Swarm Optimization Algorithm

The restart of the receiver will lead to the change in the non-overlapping frequency inter-system biases (ISB), which will make it difficult to apply the tightly combined RTK method of pre-calibrating ISB to the actual scene. Particle swarm optimization (PSO) algorithm can be used to estimate the fractional part of the inter-system phase bias (F-ISPB) in real time, which is not affected by the receiver restart. However, the standard PSO can easily fall into local optimum and cannot accurately estimate the value of F-ISPB. In this contribution, based on the characteristics of F-ISPB, we propose an improved PSO with adaptive search space and elite reservation strategy to estimate the F-ISPB in real time. When the value of F-ISPB is close to the boundary of the search space, the improved PSO will transform the search space so that F-ISPB will be located near the central region of the new search space, which will greatly reduce the situation of the standard PSO easily falling into local optimum. Since F-ISPB is very stable, an elite retention strategy will help us to estimate F-ISPB faster and more accurately. Three sets of short baseline static data were selected for testing. The results show that the inter-system differenced model based on the improved PSO has a higher ambiguity fixed rate and positioning accuracy than the inter-system differenced model based on the standard PSO and the classical intra-system differenced model, and the fewer the number of satellites, the more obvious the effect.

A generalized Davenport expansion

AbstractWe prove a new generalization of Davenport's Fourier expansion of the infinite series involving the fractional part function over arithmetic functions. A new Mellin transform related to the Riemann zeta function is also established.

Long-range correlations of sequences modulo 1

In this paper we consider the fractional parts of a general sequence, for example the sequence αn or αn2. We give a general method, which allows one to show that long-range correlations (correlations where the support of the test function grows as we consider more points) are Poissonian. We show that these statements about convergence can be reduced to bounds on associated Weyl sums. In particular we apply this methodology to the aforementioned examples. In so doing, we recover a recent result of Technau-Walker (2020) for the triple correlation of αn2 and generalize the result to higher moments. For both of the aforementioned sequences this is one of the only results which indicates the pseudo-random nature of the higher level (m≥3) correlations.