Inverse Exponent Research Articles

An infinite family of 0-APN monomials with two parameters

We consider an infinite family of exponents e(l, k) with two parameters, l and k, and derive sufficient conditions for e(l, k) to be 0-APN over F2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb F}_{2^n}$$\\end{document}. These conditions allow us to generate, for each choice of l and k, an infinite list of dimensions n where xe(l,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x^{e(l,k)}$$\\end{document} is 0-APN much more efficiently than in general. We observe that the Gold and Inverse exponents, as well as the inverses of the Gold exponents can be expressed in the form e(l, k) for suitable l and k. We characterize all cases in which e(l, k) can be cyclotomic equivalent to a representative from the Gold, Kasami, Welch, Niho, and Inverse families of exponents. We characterize when e(l, k) can lie in the same cyclotomic coset as the Dobbertin exponent (without considering inverses) and provide computational data showing that the Dobbertin inverse is never equivalent to e(l, k). We computationally test the APN-ness of e(l, k) for small values of l and k over F2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb F}_{2^n}$$\\end{document} for n≤100\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\le 100$$\\end{document}, and sketch the limits to which such tests can be performed using currently available technology. We conclude that there are no APN monomials among the tested functions, outside of the known classes.

The effect of the vacuum drying process on the appearance of cracks in the seeds

Annotation Purpose. Theoretically analyze the effect of vacuum drying parameters on the appearance of cracks in seeds. Methods. The theoretical positions of drying capillary-porous bodies and physicochemical foundations of the food industry are used. The analysis of mathematical dependencies was performed based on the methods of analytical algebra and mathematical analysis. Results. Based on the analysis of the movement of moisture during drying of a single seed, it was found that drying without cracks on the surface of the seed is possible when the rates of moisture diffusion inside the seed and evaporation of moisture from the surface layers are equal. The analytical dependence, which implements this condition and obtained using the differential equation of diffusion, made it possible to obtain the dependence of the vacuum in the drying chamber on the temperature of seed heating. This dependence shows that it is possible to realize the condition of drying without cracks only by changing the vacuum in the drying chamber in an inverse exponential manner. Heating seeds reduces the vacuum in the drying chamber, at which drying begins. Conclusions. An analysis of the operational parameters of vacuum drying of seeds revealed that for guaranteed drying of seeds without cracks, it is necessary that the rate of diffusion of moisture inside the seed be equal to the rate of evaporation of moisture from the surface layers. Technically, this is achieved by varying the vacuum in the drying chamber in time by the inverse exponent, using data from the sensor for heating the seeds. An increase in the heating temperature of the seeds requires a proportional decrease in the initial vacuum in the drying chamber. Keywords: seeds, vacuum in the drying chamber, heating temperature, evaporation rate, vacuum drying, diffusion, equivalent ball of seed.

Diversity time-period and diversity-time-area relationships exemplified by the human microbiome

We extend the ecological laws of species-time relationship (STR) and species-time-area relationship (STAR) to general diversity time-period relationship (DTR) and diversity-time-area relationship (DTAR), and test the extensions with the human vaginal microbiome datasets by building 1460 DTR/DTAR models. Our extensions were inspired by the observation that Hill numbers, well regarded as the most appropriate measure of alpha-diversity and also particularly suitable for multiplicative beta-diversity partitioning, are actually in the units of effective species, and therefore, should be able to substitute for species in the STR and STAR. We found that the traditional power law (PL) model is only applicable for DTR at diversity order zero (i.e., species richness); at higher diversity orders (q = 1–4), the power law with exponent cutoff (PLEC) and power law with inverse exponent cutoff (PLIEC) are more appropriate. In particular, PLEC has an advantage over PLIEC in predicting maximal accumulation diversity (MAD) over time. In fact, with the DTR extensions, we can construct DTR and MAD profiles. To the best of our knowledge, this is the first comprehensive investigation of the DTR/DTAR in human microbiome. Methodologically, our DTR/DTAR profiles can characterize general diversity scaling beyond species richness, covering both alpha- and beta-diversity regimes across different diversity orders.

Making Logarithms Accessible – Operational and Structural Basic Models for Logarithms

Logarithms have a reputation for being difficult and inaccessible. As an analysis of their historical, mathematical and educational background suggests, this problem might be due to the way in which logarithms are interpreted and explained in textbooks: as the inverse of exponents. If this conclusion is right, additional interpretations of logarithms are required. By combining the theoretical construct of ‘Grundvorstellungen’ (translated as ‘basic models’) and the distinction between operational and structural conceptions, I identify and elaborate four interpretations of logarithms: (i) the basic model of ‘multiplicative measuring’, (ii) the basic model of ‘counting the number of digits’, (iii) the basic model of ‘decreasing the hierarchy level’, and (iv) the basic model of ‘inverse exponent’. Three models (i–iii) reflect operational conceptions and interpret logarithms in contexts familiar to students. In combination with (iv), a structural basic model, this paper argues on a theoretical level that they could help to make logarithms accessible and understandable to students. Following the tradition of ‘Stoffdidaktik’ (‘subject-matter didactics’), the study thus aims to unpack some of the content knowledge required for the teaching of logarithms.

Exact Solution of the Dirac Equation for the Yukawa Potential with Scalar and Vector Potentials and Tensor Interaction

We present exact solutions of the Dirac equation with Yukawa potential in the presence of a Coulomb-like tensor potential. For this goal we expand the Yukawa form of the nuclear potential in its mesonic clouds by using Taylor extension to the power of seventh and bring out its simple form. In order to obtain the energy eigenvalue and the corresponding wave functions in closed forms for this potential (with great powers and inverse exponent), we use ansatz method. We also regard the effects of spin-spin, spin-isospin, and isospin-isospin interactions on the relativistic energy spectra of nucleon. By using the obtained results, we have calculated the deuteron mass. The results of our model show that the deuteron spectrum is very close to the ones obtained in experiments.

Boundary at infinity of symmetric rank one spaces

We show that canonical Carnot-Caratheodory spherical and horospherical metrics, which are defined on the boundary at infinity of every rank one symmetric space of non-compact type, are visual, i.e., they are bilipschitz equivalent with universal bilipschitz constants to the inverse exponent of Gromov products based in the space and on the boundary at infinity respectively.

Algebraic Immunity of S-Boxes Based on Power Mappings: Analysis and Construction

The algebraic immunity of an S-box depends on the number and type of linearly independent multivariate equations it satisfies. In this paper, techniques are developed to find the number of linearly independent, multivariate, bi-affine, and quadratic equations for S-boxes based on power mappings. These techniques can be used to prove the exact number of equations for any class of power mappings. Two algorithms to calculate the number of bi-affine and quadratic equations for any (n,n) S-box based on power mapping are also presented. The time complexity of both algorithms is only O(n 2) . To design algebraically immune S-boxes, four new classes of S-boxes that guarantee zero bi-affine equations and one class of S-boxes that guarantees zero quadratic equations are presented. The algebraic immunity of power mappings based on Kasami, Niho, Dobbertin, Gold, Welch, and inverse exponents are discussed along with other cryptographic properties and several cryptographically strong S-boxes are identified. It is conjectured that a known Kasami-like highly nonlinear power mapping is differentially 4 -uniform. Finally, an open problem to find an (n,n) bijective nonlinear S-box with more than 5n quadratic equations is solved.

Speciation-rate dependence in species–area relationships

The general tendency for species number ( S ) to increase with sampled area ( A ) constitutes one of the most robust empirical laws of ecology, quantified by species–area relationships (SAR). In many ecosystems, SAR curves display a power-law dependence, S ∝ A z . The exponent z is always less than one but shows significant variation in different ecosystems. We study the multitype voter model as one of the simplest models able to reproduce SAR similar to those observed in real ecosystems in terms of basic ecological processes such as birth, dispersal and speciation. Within the model, the species–area exponent z depends on the dimensionless speciation rate ν , even though the detailed dependence is still matter of controversy. We present extensive numerical simulations in a broad range of speciation rates from ν = 10 - 3 down to ν = 10 - 11 , where the model reproduces values of the exponent observed in nature. In particular, we show that the inverse of the species–area exponent linearly depends on the logarithm of ν . Further, we compare the model outcomes with field data collected from previous studies, for which we separate the effect of the speciation rate from that of the different species lifespans. We find a good linear relationship between inverse exponents and logarithm of species lifespans. However, the slope sets bounds on the speciation rates that can hardly be justified on evolutionary basis, suggesting that additional effects should be taken into account to consistently interpret the observed exponents.

Inversion formula of multifractal energy dissipation in three-dimensional fully developed turbulence

The concept of inverse statistics in turbulence has attracted much attention in recent years. It is argued that the scaling exponents of the direct structure functions and the inverse structure functions satisfy an inversion formula. This proposition has already been verified by numerical data using the shell model. However, no direct evidence was reported for experimental three-dimensional turbulence. We propose to test the inversion formula using experimental data of three-dimensional fully developed turbulence by considering the energy dissipation rates instead of the usual efforts on the structure functions. The moments of the exit distances are shown to exhibit nice multifractality. The inversion formula between the direct and inverse exponents is then verified.

Autoignition characteristics of aircraft-type fuels

An applied research program was undertaken to evaluate the autoignition characteristics of five liquid hydrocarbon fuels in air over ranges of air temperature, pressure, and equivalence ratio appropriate to advanced aircraft gas turbine engines. Ignition delay times were measured using a continuous flow test apparatus which permitted independent variation and evaluation of the effects of temperature, pressure, flow rate, and fuel/air ratio on ignition delay time. A multiple conical tube fuel injector, consisting of 19 parallel venturi elements with independent fuel control to each element, was used for the testing. Measurements of the spray distribution produced were made by isokinetically sampling the flow at reduced temperatures with water injection, and indicated that a nearly uniform fuel/air mixture distribution would be obtained. Parametric tests to map the ignition delay characteristics of Jet-A, JP-4, No. 2 diesel, cetane, and an experimental referee broad specification (ERBS) fuel were conducted at pressures of 10, 15, 20, 25 and 30 atm, inlet air temperatures up to 1000K, and fuel/air equivalence ratios of 0.3, 0.5, 0.7, and 1.0. Ignition delay times in the range of 1–50 msec at freestream flow velocities ranging from 20 to 100 m/sec were obtained. In accord with classical chemical kinetics, the ignition delay times for all fuels tested appeared to correlate with the inverse of pressure and the inverse exponent of temperature, viz., t= A pn exp E RT In general, the data were very repeatable. With the exception of pure cetane, which has the shortest ignition delay times, the difference between the fuels tested did not appear to be significant. The apparent global activation energies for the typical gas turbine fuels ranged from 38 to 43 kcal/mole, while the activation energy determined for cetane was 50 kcal/mole. In addition, the data indicate that, for lean mixtures, ignition delay times decrease with increasing equivalence ratio. It was also noted that physical (apparatus dependent) phenomena, such as mixing (i.e., length) and airstream cooling (due to fuel heating, vaporization, and convective heat loss) can have an important effect on the ignition delay.

Inverse cross-modality matching: A test of ratio judgment consistency for group and individual data

Direct and inverse cross-modality matches made by 20 subjects were assessed for ratio judgment consistency. Each subject matched apparent duration to loudness, and vice versa, in both a directly and an inversely proportional manner. All four tasks were repeated twice so that individual differences could be examined using interrepetition correlations. Group data exhibited the appropriate inverse relationship indicative of consistency, although the inverse matches were slightly curvilinear and resembled earlier studies with inverse attribute scales. Some individuals showed a high degree of consistency but many departed widely from inverse proportionality. Individual differences in exponents, which occurred for both types of tasks, were not removed by S. S. Stevens’ (1971) regression balance procedure. However, interrepetition correlations for the differences in the absolute values of the direct and inverse exponents of individuals were nonsignificant, suggesting that when subjects’ exponents differed for the two types of tasks they did so on a random basis. The latter finding implies that subjects would give inversely proportional matches were it not for random factors. The findings were discussed in relation to other types of ratio consistency.

Velocity Dependence of the Collisional Excitation Transfer Processes of a Resonant Type

The velocity dependence or the temperature dependence of the excitation transfer processes A*+A→A+A*,A*+B→A+B++e, were discussed within the limit of the straight-line trajectory and the impact-parameter method. It has been shown that the velocity dependence corresponds directly with the radial part of the interaction in some cases. In the discrete—discrete resonant case, the cross section σ is proportional to v−2/(n−1) and in the discrete—continuum case it is proportional to v−2/(2n−1), where v is the relative incident velocity of the atom and n is the inverse exponent of the radial part of the interaction.

Radiators for aircraft engines

An applied research program was undertaken to evaluate the autoignition characteristics of five liquid hydrocarbon fuels in air over ranges of air temperature, pressure, and equivalence ratio appropriate to advanced aircraft gas turbine engines. Ignition delay times were measured using a continuous flow test apparatus which permitted independent variation and evaluation of the effects of temperature, pressure, flow rate, and fuel/air ratio on ignition delay time. A multiple conical tube fuel injector, consisting of 19 parallel venturi elements with independent fuel control to each element, was used for the testing. Measurements of the spray distribution produced were made by isokinetically sampling the flow at reduced temperatures with water injection, and indicated that a nearly uniform fuel/air mixture distribution would be obtained.Parametric tests to map the ignition delay characteristics of Jet-A, JP-4, No. 2 diesel, cetane, and an experimental referee broad specification (ERBS) fuel were conducted at pressures of 10, 15, 20, 25 and 30 atm, inlet air temperatures up to 1000K, and fuel/air equivalence ratios of 0.3, 0.5, 0.7, and 1.0. Ignition delay times in the range of 1–50 msec at freestream flow velocities ranging from 20 to 100 m/sec were obtained. In accord with classical chemical kinetics, the ignition delay times for all fuels tested appeared to correlate with the inverse of pressure and the inverse exponent of temperature, viz., In general, the data were very repeatable. With the exception of pure cetane, which has the shortest ignition delay times, the difference between the fuels tested did not appear to be significant. The apparent global activation energies for the typical gas turbine fuels ranged from 38 to 43 kcal/mole, while the activation energy determined for cetane was 50 kcal/mole. In addition, the data indicate that, for lean mixtures, ignition delay times decrease with increasing equivalence ratio. It was also noted that physical (apparatus dependent) phenomena, such as mixing (i.e., length) and airstream cooling (due to fuel heating, vaporization, and convective heat loss) can have an important effect on the ignition delay.