Logarithmic Approximation Research Articles

Linear Rescaling to Accurately Interpret Logarithms

Abstract The standard approximation of a natural logarithm in statistical analysis interprets a linear change of p in ln(X) as a (1 + p) proportional change in X, which is only accurate for small values of p. I suggest base-(1 + p) logarithms, where p is chosen ahead of time. A one-unit change in log1 + p (X) is exactly equivalent to a (1 + p) proportional change in X. This avoids an approximation applied too broadly, makes exact interpretation easier and less error-prone, improves approximation quality when approximations are used, makes the change of interest a one-log-unit change like other regression variables, and reduces error from the use of log(1 + X).

Low-Cost Linearity Range Enhancement for Linear Variable Differential Transformer

This research proposes a low-cost and low-complexity technique to enhance the linearity range of linear variable differential transformer (LVDT) using logarithmic approximation and summing/subtracting operation. The proposed technique utilizes the feedforward approach and thus encounters no divide-by-zero behavior from inverse functions. The nonlinearity of LVDT can be counteracted by simply manipulating the parameters of the canceller circuit. Besides, the proposed technique can be applied to various LVDT nonlinearity types. The proposed linear enhancing circuit consists of operational amplifiers and a small number of passive elements. The nonlinearity cancellation could be further enhanced using a temperature-compensation circuit. Simulations were carried out using two different LVDT nonlinearity types: LVDT#1 and #2. The simulation results showed that the linearity enhancing circuit effectively enhances the LVDT linearity range, with the linearity errors of 0.51% and 1.83%, respectively. To validate, experiments were undertaken with the identical LVDT nonlinearity types using a circuit prototype, and the measured linearity errors are around 2.5% and 5.5%, respectively. The simulation and experimental results are agreeable with the theory, indicating that the proposed technique enhances the linearity range.

Approximating Minimum Representations of Key Horn Functions

Horn functions form a subclass of Boolean functions and appear in many different areas of computer science and mathematics as a general tool to describe implications and dependencies. Finding minimum sized representations for such functions with respect to most commonly used measures is a computationally hard problem that remains hard even for the important subclass of key Horn functions. In this paper we provide logarithmic factor approximation algorithms for key Horn functions with respect to all measures studied in the literature for which the problem is known to be hard.

The Proposition of Three Approaching Ways to Implement Tan-sigmoid Activation Function in FPGA

Hyperbolic tangents and Sigmoid are commonly used for Artificial Neural Networks as activation functions. The complex equation of the activation function is one of the most difficult to be implemented in hardware because containing division and exponential, which gives non-linear behavior. The challenge is building a tan-sigmoid function in hardware with efficient performance. Therefore, this work will focus on implementing the activation function in FPGA. To overcome this challenge, a different approach was proposed in this paper, efficient hardware-implemented for tan-sigmoid in terms of the number of slices occupied and the resources utilization are designed. In this work, three approaches are proposed: tan-sigmoid using the log approximation method, tan-sigmoid using segmentation method, and tan-sigmoid using the polynomial method. These approaches are efficiently implemented in the Xilinx Spartan-3A xc3s700a-4fg484 platform. Hardware synthesis and FPGA implementations illustrate that the proposed tan-sigmoid only takes up to 1% of logic resources in the first and second proposed approaches. While, 4% showed in the third proposed approach, with the best efficiency and significantly confirmed the lowest implementation costs than the traditional approach.

Pollard's Ro-Method for Finding a Discrete Logarithm in the Case of its Low Weight

n this paper, we present a modification of Pollard's ro-method for searching for a discrete logarithm in the case when it is expressed by a binary vector of relatively small weight. In this case, it is necessary to change the procedure for iterating through random elements of the group under consideration in order to remain in the set of discrete logarithms expressed by binary vectors of low weight. We assume that this weight does not exceed half the length of the input word in the algorithm under consideration. To the other hand the case, when some close approximation of discrete logarithm is known considered. The paper uses estimates for binomial coefficients, the birthday paradox theorem, and the Berry-Esseen theorem for the Bernoulli scheme.

Entanglement entropy production in deep inelastic scattering

Deep inelastic scattering (DIS) samples a part of the wave function of a hadron in the vicinity of the light cone. Lipatov constructed a spin chain which describes the amplitude of DIS in leading logarithmic approximation. Kharzeev and Levin proposed the entanglement entropy as an observable in DIS [Phys. Rev. D 95, 114008 (2017)], and suggested a relation between the entanglement entropy and parton distributions. Here we represent the DIS process as a local quench in the Lipatov's spin chain, and study the time evolution of the produced entanglement entropy. We show that the resulting entanglement entropy depends on time logarithmically, $\mathcal S(t)=1/3 \ln{(t/\tau)}$ with $\tau = 1/m$ for $1/m \le t\le (mx)^{-1}$, where $m$ is the proton mass and $x$ is the Bjorken $x$. The central charge $c$ of Lipatov's spin chain is determined here to be $c=1$; using the proposed relation between the entanglement entropy and parton distributions, this corresponds to the gluon structure function growing at small $x$ as $xG(x) \sim 1/x^{1/3}$.

Investigation of the characteristics of the InGaAs/InAlGaAs superlattice for 1300 nm range vertical-cavity surface-emitting lasers

X-ray structural analysis and photoluminescence spectroscopy techniques were used to study heterostructures based on InGaAs/InAlGaAs superlattice for active regions of 1300 nm range lasers grown by molecular beam epitaxy. It is shown that the grown heterostructures have a high crystal quality. The perpendicular lattice mismatch of the average crystal lattice constant of the InGaAs/InAlGaAs superlattice with respect to the crystal lattice constant of the InP substrate is estimated at ~+0.01%. An analysis of the photoluminescence spectra made it possible to conclude that the contribution of Auger recombination is insignificant in the studied range of excitation power density. Studies of vertical-cavity surface-emitting lasers with an active region based on the InGaAs/InAlGaAs superlattice made it possible to estimate the gain coefficient at a level of 650 cm-1 for the standard logarithmic approximation of the dependence of the gain on the current density. The transparency current density of the laser was 400-630 A/cm2, which is comparable to the record low values for the case of highly strained InGaAs-GaAs and InGaAsN-GaAs quantum wells in the spectral ranges of 1300 nm, respectively. Keywords: superlattice, vertical-cavity surface-emitting laser, optical gain.

Quaternion-based color image completion via logarithmic approximation

In color image processing, the objective of image completion is to restore missing entries from the incomplete observation image. Recent improvements have assisted in resolving the rank minimization issue, thereby promoting the realization of completion. This paper adopts the new quaternion matrix logarithmic norm to approximate rank in accordance with the quaternion matrix framework. Unlike the traditional matrix completion method, which handles RGB channels separately, the quaternion-based method avoids the destruction of the structure of images by placing the color image in a pure quaternion matrix. Furthermore, the logarithmic norm induces a more accurate rank surrogate. Based on the logarithmic norm, it is possible to exploit not only the factorization strategy but also the truncated technique, thus achieving successful image restoration. The alternating minimization framework renders it possible to optimize the two strategies, and mathematically detailed validation of the convergence analysis is provided. The experimental results demonstrate that the use of logarithmic surrogates in the quaternion domain is a superior strategy for solving the problem of color image completion.

Cone-Size Dependence of Jet Suppression in Heavy-Ion Collisions.

The strong suppression of high-p_{T} jets in heavy-ion collisions is a result of elastic and inelastic energy loss suffered by the jet multiprong collection of color charges that are resolved by medium interactions. Hence, quenching effects depend on the fluctuations of the jet substructure that are probed by the cone-size dependence of the spectrum. In this Letter, we present the first complete, analytic calculation of the inclusive R-dependent jet spectrum in PbPb collisions at LHC energies, including resummation of energy loss effects from hard, vacuumlike emissions occurring in the medium and modeling of soft energy flow and recovery at the jet cone. Both the geometry of the collision and the local medium properties, such as the temperature and fluid velocity, are given by a hydrodynamic evolution of the medium, leaving only the coupling constant in the medium as a free parameter. The calculation yields a good description of the centrality and p_{T} dependence of jet suppression for R=0.4 together with a mild cone-size dependence, which is in agreement with recent experimental results. Gauging the theoretical uncertainties, we find that the largest sensitivity resides in the leading logarithmic approximation of the phase space of resolved splittings, which can be improved systematically, while nonperturbative modeling of the soft-gluon sector is of relatively minor importance up to large cone sizes.

The looming health hazard: A wave of HPV-related cancers in Japan is becoming a reality due to the continued suspension of the governmental recommendation of HPV vaccine

The looming health hazard: A wave of HPV-related cancers in Japan is becoming a reality due to the continued suspension of the governmental recommendation of HPV vaccine

Two-level hub Steiner trees

We study a fundamental class of two-layer network design problems. A hub layer is configured by establishing hubs at selected nodes at considerable cost so that the routes between hubs can be operated cheaply. The remaining edges in the network are operated at regular cost. The resulting problem is to determine the set of nodes to open hubs and the set of edges to establish in order to find a network of minimum total cost.We consider the case where the network is required to form a Steiner tree spanning a given set of terminal vertices. When edge costs are non-metric, we show logarithmic approximation hardness even for the special case of spanning trees. On the other hand, we show a polynomial-time reduction for Steiner trees to its corresponding node-weighted version thus proving a logarithmic approximation factor. When edge costs are metric, we show the problem is only a constant factor harder to approximate than its original version (with no hub installation) using a similar reduction.

Новые подходы к оценке и прогнозу заболеваемости энтеровирусной (неполио) инфекцией в российской федерации с использованием математических моделей

At present it is impossible to develop epidemiologic surveillance and control over any infection regarding studies on dynamics of morbidity, seasonality and periodicity without using mathematical modeling techniques. Our research goal was to study regularities in manifestations of epidemic process for enterovirus (non-polio) infection (EVnI) in the Russian Federation over 14 years (2006–2019) using mathematical models (linear, logarithmic, power, and exponential approximation).An optimal mathematical model was selected using three statistical parameters, namely determination coefficient, Fischer’s exact test, and standard error. Periodicity of rises and falls in morbidity was calculated with Fourier one- dimensional spectral analysis. Intra-year dynamics of morbidity with EVnI was estimated basing on monthly spread of the disease cases on the RF territory. Classic seasonal decomposition, Census I technique, was applied to analyze time series of monthly morbidity. It was determined that EVnI epidemic process was unevenly spread over years in the RF in the examined period of time (2006–2019) and there were two opposite trends in it; the first one lasted from 2006 to 2010 when morbidity was declining and the second was from 2010 to 2019 when it was growing. Having analyzed manifestations of EVnI epidemi- ologic process in long-term dynamics given its uneven spread as per years, we established that it was advisable to use mathematical models approximated as per separate time periods. Average long-term morbidity with EVnI amounted to 8.09 0/0000 in the RF in 2010–2019 with growth rate being equal to 17.7 %. Maximum value was registered in 2017 (16.32 0/0000). An unfavorable prediction for further epidemic situation development was revealed for the examined pe- riod. The epidemic process was characterized with 4-year periodicity and summer-autumn seasonality with peaks usually occurring in August and September. Rates that characterized intensity of the trends in long-term morbidity dynamics and were calculated with mathematical models differed authentically from those obtained via conventional calculations of average values (χ=11.08; d.f.=1; p=0.0009).

NEW APPROACHES TO ASSESSING AND FORECASTING MORBIDITY WITH ENTEROVIRUS (NON-POLIO) INFECTION IN THE RUSSIAN FEDERATION USING MATHEMATICAL MODELS

At present it is impossible to develop epidemiologic surveillance and control over any infection regarding studies on dynamics of morbidity, seasonality and periodicity without using mathematical modeling techniques. Our research goal was to study regularities in manifestations of epidemic process for enterovirus (non-polio) infection (EVnI) in the Russian Federation over 14 years (2006–2019) using mathematical models (linear, logarithmic, power, and exponential approximation).An optimal mathematical model was selected using three statistical parameters, namely determination coefficient, Fischer’s exact test, and standard error. Periodicity of rises and falls in morbidity was calculated with Fourier one- dimensional spectral analysis. Intra-year dynamics of morbidity with EVnI was estimated basing on monthly spread of the disease cases on the RF territory. Classic seasonal decomposition, Census I technique, was applied to analyze time series of monthly morbidity. It was determined that EVnI epidemic process was unevenly spread over years in the RF in the examined period of time (2006–2019) and there were two opposite trends in it; the first one lasted from 2006 to 2010 when morbidity was declining and the second was from 2010 to 2019 when it was growing. Having analyzed manifestations of EVnI epidemi- ologic process in long-term dynamics given its uneven spread as per years, we established that it was advisable to use mathematical models approximated as per separate time periods. Average long-term morbidity with EVnI amounted to 8.09 0/0000 in the RF in 2010–2019 with growth rate being equal to 17.7 %. Maximum value was registered in 2017 (16.32 0/0000). An unfavorable prediction for further epidemic situation development was revealed for the examined pe- riod. The epidemic process was characterized with 4-year periodicity and summer-autumn seasonality with peaks usually occurring in August and September. Rates that characterized intensity of the trends in long-term morbidity dynamics and were calculated with mathematical models differed authentically from those obtained via conventional calculations of average values (χ=11.08; d.f.=1; p=0.0009).

An Improved Taylor Algorithm for Computing the Matrix Logarithm

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson–Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison.

Hunting BFKL in semi-hard reactions at the LHC

The agreement between calculations inspired by the resummation of energy logarithms, known as BFKL approach, and experimental data in the semi-hard sector of QCD has become manifest after a wealthy series of phenomenological analyses. However, the contingency that the same data could be concurrently portrayed at the hand of fixed-order, DGLAP-based calculations, has been pointed out recently, but not yet punctually addressed. Taking advantage of the richness of configurations gained by combining the acceptances of CMS and CASTOR detectors, we give results in the full next-to-leading logarithmic approximation of cross sections, azimuthal correlations and azimuthal distributions for three distinct semi-hard processes, each of them featuring a peculiar final-state exclusiveness. Then, making use of disjoint intervals for the transverse momenta of the emitted objects, i.e. kappa -windows, we clearly highlight how high-energy resummed and fixed-order driven predictions for semi-hard sensitive observables can be decisively discriminated in the kinematic ranges typical of current and forthcoming analyses at the LHC. The scale-optimization issue is also introduced and explored, while the uncertainty coming from the use of different PDF and FF set is deservedly handled. Finally, a brief overview of JETHAD, a numerical tool recently developed, suited for the computation of inclusive semi-hard reactions is provided.

Budget Feasible Mechanisms Over Graphs

This paper studies the budget-feasible mechanism design over graphs, where a buyer wishes to procure items from sellers, and all participants (the buyer and sellers) can only directly interact with their neighbors during the auction campaign. The problem for the buyer is to use the limited budget to incentivize sellers to propagate auction information to their neighbors, thereby more sellers will be informed of the auction and more item value will be procured. An impossibility result shows that the large-market assumption is necessary. We propose efficient budget-feasible diffusion mechanisms for large markets that simultaneously guarantee individual rationality, budget-feasibility, strong budget-balance, incentive-compatibility to report private costs and diffuse auction information. Moreover, the proposed mechanisms achieve logarithmic approximation that the total procured value is within a logarithmic factor of the optimal solution. Compared to most related budget-feasible mechanisms, which do not take the individual interactions among sellers into account, our mechanisms can incentivize sellers to further propagate auction information to other potential sellers. Meanwhile, existing related diffusion mechanisms only focus on seller-centric auctions and fail to satisfy the budget-feasibility of the buyer.

A Hybrid Radix-4 and Approximate Logarithmic Multiplier for Energy Efficient Image Processing

Multiplication is an essential image processing operation commonly implemented in hardware DSP cores. To improve DSP cores’ area, speed, or energy efficiency, we can approximate multiplication. We present an approximate multiplier that generates two partial products using hybrid radix-4 and logarithmic encoding of the input operands. It uses the exact radix-4 encoding to generate the partial product from the three most significant bits and the logarithmic approximation with mantissa trimming to approximate the partial product from the remaining least-significant bits. The proposed multiplier fills the gap between highly accurate approximate non-logarithmic multipliers with a complex design and less accurate approximate logarithmic multipliers with a more straightforward design. We evaluated the multiplier’s efficiency in terms of error, energy (power-delay-product) and area utilisation using NanGate 45 nm. The experimental results show that the proposed multiplier exhibits good area utilisation and energy consumption and behaves well in image processing applications.

Generalised online colouring problems in overlap graphs

In this paper we consider an online version of different colouring problems in overlap graphs, motivated by some stacking problems. The instance is a system of time intervals presented in non-decreasing order of the left endpoint. We consider the usual colouring problem as well as b-bounded colouring (colour class have a maximum capacity b) and the same problems in the complement graph. We also consider the case where at most b intervals of the same colour can intersect. For all these versions we obtain a logarithmic competitive ratio w.r.t. the maximum ratio of interval lengths while the best known ratio for the usual colouring was linear. To our knowledge it is the first time the other variants are considered in online overlap graphs. Moreover, in the offline case, a pre-processing allows us to deduce a logarithmic approximation ratio w.r.t. the maximum number of pairwise disjoint intervals in the system. Our method is based on a partition of the overlap graph into permutation graphs, leading to a competitive-preserving reduction of the problem in overlap graphs to the same problem in permutation graphs. We think that this new partition problem by itself is of interest for future work.

Exact summation of leading logs around Toverline{T} deformation of O(N + 1)-symmetric 2D QFTs

We consider a general (beyond Toverline{T} ) deformation of the 2D O(N + 1) σ-model by the irrelevant dimension-four operators. The theory deformed in this most general way is not integrable, and the S-matrix loses its factorization properties. We perform the all-order summation of the leading infrared logs for the 2 → 2 scattering amplitude and provide the exact result for the 2 → 2 S-matrix in the leading logarithmic approximation. These results can provide us with new insights into the properties of the theories deformed by irrelevant operators more general than the Toverline{T} deformation.

Joint Resource Allocation and Trajectory Optimization With QoS in UAV-Based NOMA Wireless Networks

Replacing base stations with unmanned aerial vehicles (UAVs) to serve the communication of ground users has attracted a lot of attention recently. In this paper, we study the joint resource allocation and UAV trajectory optimization for maximizing the total energy efficiency in UAV-based non-orthogonal multiple access (NOMA) downlink wireless networks with the quality of service (QoS) requirements. To handle the user scheduling problem, a heuristic algorithm based on matching and swapping theory is proposed first to allocate users that access UAV in each subperiod, then the transmit power allocation problem which considers the maximum transmit power and minimum user date rate is transformed to a convex optimization problem using logarithmic approximation. Meanwhile, the successive convex optimization is used in UAV trajectory optimization problem and a joint optimization algorithm is presented with the algorithm’s convergence and computational complexity. Finally, numerical results are provided to support the rationality of the proposed algorithm.