Characterizations for functional dependency and Boyce-Codd normal form families

A functional dependency (fd) family was recently defined [20] as the set of all instances satisfying some set of functional dependencies. A Boyce-Codd normal form, abbreviated BCNF, family is defined here as an fd-family specified by some BCNF set of functional dependencies. The purpose of this paper is to present set-theoretic/algebraic characterizations relating to both types of families.Two characterizations of $\text{F}$($\text{I}$), the smallest fd-family containing the family $\text{I}$ of instances, are established. The first involves the notion of agreement, a concept related to that of a closed set of attributes. The second describes $\text{F}$($\text{I}$) as the smallest family of instances containing $\text{I}$ and closed under four specific operations on instances. Companion results are also given for BCNF- families.The remaining results concern characterizations involving the well-known operations of projection, join and union. Two characterizations for when the projection of an fd-family is again an fd-family are given. Several corollaries are obtained, including the effective decidability of whether a projection of an fd-family is an fd-family. The problem for BCNF-families disappears since it is shown that the projection of a BCNF-family is always a BCNF-family. Analogous to results for fd-families presented in [20], characterizations of when the join and union of BCNF-families are BCNF-families are given. Finally, the collections of all fd-families and all BCNF-families are characterized in terms of inverse projection operations and intersection.     

Estimating the quality of computational algorithms — Part 1

Computational algorithms for approximate calculation of the mapping $\text{A:X → Y}$ on some element $\text{f}$ of the compactum $\text{X}$ are considered, where $\text{A}$ is a one-to-one continuous mapping. Topology on $\text{X}$ and $\text{Y}$ is set by imbedding them into some Banach spaces. The relation between $\text{?(N; X)}$ and $\text{?(N);Y}$ plays the main role in the problems concerning the construction of such computational algorithms. Here $\text{?(N);X)}$ is a function reciprocal to the ?-entropy of the compactum $\text{X}$. The computational algorithm $\text{U}$ is characterized first of all by the volume of the input data $\text{N}$ and that accuracy ? with which elements $\text{g}$ on the output are defined. If $\text{N}$ is given (but under condition that a table of any $\text{f}$ is admissible), then inf $\text{? = ?}{}_1\text{(N)}$ exists.For the given concrete algorithm the quantity $\text{?/?}{}_1\text{(N)}$ characterizes its qualities and is one of the most important characteristics of the algorithm. In this paper some of the simplest current algorithms are analyzed, and their quality is determined as well. A number of recommendations relative to the questions of the construction of computational algorithms is proposed.     

Remote sensing data of SP Mountain and SP Lava flow in North-Central Arizona

Multifrequency airborne radar image data of SP Mountain [Official name of feature (U.S. Geological Survey, 1970)] and SP flow (and vicinity) in north-central Arizona were obtained in diverse viewing directions and direct and cross-polarization, then compared with surface and aerial photography, LANDSAT multispectral scanner data, airborne thermal infrared imagery, surface geology, and surface roughness statistics. The extremely blocky, basaltic andesite of SP flow is significantly brighter on direct-polarization $\text{K-}\text{band}$ (0.9-cm wavelength) images than on cross-polarized images taken simultaneously. Conversely, for the longer wavelength (25 cm) $\text{L-}\text{band}$ radar images, the cross-polarization image returns from SP flow are brighter than the direct-polarized image. This effect is explained by multiple scattering and the strong wavelength dependence of polarization effects caused by the rectilinear basaltic andesite scatters. Two distinct types of surface relief on SP flow, one extremely blocky, the other subdued, are found to be clearly discriminated on the visible and thermal wavelength images but are separable only on the longer wavelength $\text{L-}\text{band}$ radar image data. The inability of the $\text{K-}\text{and}\text{X- (3-}\text{cm wavelength}\text{)}$ band radars to portray the differences in roughness between the two SP flow surface units is attributed to the radar frequency dependence of the surface-relief scale, which, described as the Rayleigh criterion, represents the transition between quasispecular and primarily diffuse backscatter.     

Multi-search optimization techniques

Two multi-search optimization techniques are developed. The motive is to show that increasing the search directions results in a better rate of convergence. This idea was first investigated by Miele and Cantrell [1]. Their technique is computationally attractive, but exact line-search is needed in two directions. Therefore the problem of line-search has been more complicated since at the end of each iteration two stopping functions have to satisfied. Also, built-in safeguards were used to ensure the stability of the descent process. For nonquadratic functions Miele's method (as well as many other optimization techniques) needs restarting every $\text{n}$ + 1 iterations in order to improve the convergence rate. Extending this two-direction search of Miele to an $\text{n-}\text{direction}$ search in order to further improve the rate of convergence would highly complicate the search process, since exactly $\text{n}$ line-searches are needed together with $\text{n}$ simultaneous stopping functions to be satisfied besides the resulting increase in the built-in safeguards to impractical limits. In the techniques presented the optimum step-size in each direction is obtained by simply minimizing a quadratic function of a linearized gradient, and only sufficient decrease is needed along the search directions without the need of exact line-searches. Instead, simple successive halving is used till the function starts to increase; then a new iteration is started without the use of any simultaneous stopping functions at the end of each iteration. Also, the $\text{n-}\text{direction}$ case has been developed in exactly the same fashion as the two-direction case without quiring the use of $\text{n}$ simultaneous stopping functions a property which is not possible if the method of Miele is to be extended to the $\text{n-}\text{direction}$ case. The highly effective rate of convergence obtained by using the $\text{n-}\text{direction}$ search really justifies the would-be increase in the required computational effort. Furthermore, no restarting has been needed to improve the convergence rate. The results of two tried examples show that the multi-search optimization techniques that have been developed have excellent convergence rates without endangering the stability requirement.     

Circuit size is nonlinear in depth

Two fundamental complexity measures for a Boolean function f are its circuit depth d(f) and its circuit size c(f). It is shown that $\text{c?}\text{1}\text{4}\text{d·}\text{log}{}_2\text{d}$ for all f.     

A note on the accuracy and convergence of finite element approximations of the convection equation

This paper is concerned with the development of error estimates and a proof of convergence for finite element approximations of a class of first-order hyperbolic equations. Two techniques are described for determining energy and $\text{L}{}_2$ error estimates.     

New finite element results for the square cavity

New results for the recirculating flow inside a square cavity obtained by a finite element method are presented. The full Navier-Stokes equations in the form of a single, fourth order equation for stremfunction is recast into arestricted variational principles, which form finite element discretization. A triangular element with Hermitian interpolation is used, such that the velocities are continuous and the incompressibility is satisfied exactly. The resulting nonlinear system is solved by Newton-Raphson iteration. Calculations are carried out with several gridworks of progressive show: (1) the convergence of solutions with refinement for fixed Reynolds number $\text{R}$; and (2) the loss of accuracy with $\text{R}$ for fixed gridwork. The range of $\text{R}$ covered is from $\text{10}{}^{\mathrm{?4}}$ to 3450. The features illustrated include the enlargement of the inviscid core of rigid rotation, the intensification of primary and secondary vortices and the appearance of a third secondary vortex near the upper upstream corner at $\text{R =1500}$.     

A note on ‘geometric transforms’ of digital sets

We define a ‘geometric transform’ on the digital plane as a function ? that takes pairs (P, S), where S is a set and P a point of S, into nonnegative integers, and where $\text{?(S, P)}$ depends only on the positions of the points of S relative to P. Transforms of this type are useful for segmenting and describing S. Two examples are ‘distance transforms’, for which $\text{?(S, P)}$ is the distance from P to S, and ‘isovist transforms’, where $\text{?(S, P)}$, is the area of the part S visible from P. This note characterizes geometric transforms that have certain simple set-theoretic properties, e.g., such that $\text{?(S?T,P) = ?(S,P)∧?(T,P)}$ for all S, T, P. It is shown that a geometric transform has this intersection property if and only if it is defined in a special way in terms of a ‘neighborhood base’; the class of such ‘neighborhood transforms’ is a generalization of the class of distance transforms.     

A decomposition theorem for hyper-algebraic extensions of language families

In modern theories of rewriting structures, hyper-sentential and hyper-algebraic extensions of languages-families have abstracted the imminent features of iterated parallel substitution. After introducing the concept of a (depth-bounded) translation, we show that for each language L hyper-algebraic over a natural family $\text{F}$ there are $\text{F}$-translations Δ, ?D and languages L₁, …, L_m hyper-sentential over $\text{F}$ such that

, for some p, q ? 0.Two specializations of this result are given, when more assumptions are made about $\text{F}$. These are, firstly, a translation theorem and, secondly, an alphabetic homomorphism theorem for hyper-algebraic extensions (an alphabetic homomorphism is a letter-to-letter or letter-to-ε homomorphism).     

Effects of cadmium and copper treatments and water stress on the thermal emission from peas (Pisum sativum L.): Controlled environment experiments

Pea plants (Pisum sativum L., cv. Feltham First) were grown in hydroponic solution and in vermiculite in controlled conditions and were treated with various concentrations of Cd and Cu in the two types of rooting medium. Relative canopy temperatures ($\text{Tini}$) were estimated from thermal images obtained with an AGA “Thermovision 680” infrared camera. Stomatal resistances ($\text{R}$), leaf temperatures measured with thermocouples ($\text{Tini}$) and plant water potentials (ψ) were also recorded. Metal-treated plants in hydroponic solution appeared to show a slight decrease in $\text{Tini}$ (in the order of 0.2 °C) relative to control plants, at concentrations up to 100 ppm Cd or Cu. At 200 ppm Cd or Cu, $\text{Tini}$ of treated plants was about 1 °C higher than control, associated with visual toxicity symptoms. $\text{R}$ in these plants was typically 5–10 sec cm^?1, compared with under 5 sec cm^?1 in control plants, and probably limited evaporative cooling. Plants in vermiculite gave results consistent with those above, and were also used for a water stress experiment. Prewilting water stress caused an increase in $\text{Tini}$ of up to 3.5 °C relative to fully watered control plants and caused an increase in $\text{R}$ up to about 20 sec cm^?1. These effects obscured thermal manifestations of metal toxicity. It is therefore suggested that thermal effects of geochemical metal anomalies may be most readily observed when vegetation is not under water stress, although it is also pointed out that metal-stressed and normal plants may show different responses to soil water deficits.     

On some bounded semiAFLs and AFLs

Goldstine [8] has conjectured that not all context-free languages are contained in $\text{F}$_∩($\text{B}$)—the smallest AFL closed under intersection and containing all bounded languages. We prove this by showing that the linear context-free language pal is not in $\text{F}$_∩($\text{B}$). Inclusion relations among various subfamilies of $\text{F}$∩($\text{B}$) as well as their closure properties are studied. It is shown that all languages defined by polynomials whose coefficients are natural numbers are in $\text{M}$_∩(prod)—the smallest intersection-closed semiAFL containing the language $\text{prod}\text{=a}{}^{\mathrm{m}}\text{b}{}^{\mathrm{n}}\text{c}{}^{\mathrm{mn}}\text{s|m,n in N}$. This implies that the corresponding full semiAFL$\text{M}\text{?}$_∩ (prod) is equal to the smallest intersection-closed full semiAFL containing all recursively enumerable bounded languages. An analogous result for all bounded languages is also obtained. Since pal is not in$\text{M}\text{?}$_∩(prod), it follows that $\text{M}\text{?}$^lin_∩(prod) is a semiAFL closed under intersection and linear-erasing homomorphism but is not closed under Kleene⁺, homomorphism, or nonerasing homomorphic replication. This solves a problem considered by Book and Greibach [2]. Finally, nonprincipality of some semiAFLs and AFLs is established. As a consequence, we solve a problem of Ginsburg [6].     

Improved lower bounds on the number of multiplications/divisions which are necessary to evaluate polynomials

We improve some lower bounds which have been obtained by Strassen and Lipton. In particular there exist polynomials of degree n with 0–1 coefficients that cannot be evaluated with less than $\text{n}\text{/(4}\text{log}\text{n)}$ nonscalar multiplications/divisions. The evaluation of p(x) $\text{∑}\text{δ=0}\text{n}\text{e}{}^{\mathrm{<mtext>2\pi i</mtext><mtext>2\delta </mtext>}}\text{x}{}^{\mathrm{\delta }}$ requires at least $\text{n}\text{(12}\text{log}\text{n)}$ multiplications/divisions and at least $\text{n/(8}\text{log}\text{n)}$ nonscalar multiplications/divisions. We specify polynomials with algebraic coefficients that require $\text{1}\text{2}\text{n}$ multiplications/divisions.     

A method for the calculation of eigenfunction expansions

In this paper, a method is presented for the calculation of the coefficients of the series expansion of a function f(t), in the base orthonormal set made up by the eigenfunctions of the self-adjoint operator $\text{L: L(x(t)) = (}\text{d}\text{dt}\text{)( p(t)(}\text{dx(t)}\text{dt}\text{))?q(t)x(t)}$. We show that the values of the numbers $\text{t}\text{}$x_k> can be obtained by solving the differential equation L + λ) y(t) = Kf(t), in the interval of definition, for each of the eigenvalues λ of L and by using as initial conditions those which determine one of its associated orthonormal functions. This makes the method specially interesting for its implementation on a hybrid computer: One advantage of the proposed method is that the analysis of f(t) does not require the simultaneous presence of the functions of the base set and the problem signal, thus eliminating both the problems of the synchronized generation of signals and the need for storing it in memory.