Gradient configurations and quadratic functions

1963 ◽  
Vol 59 (2) ◽  
pp. 287-305
Author(s):  
S. N. Afriat
Keyword(s):  
Quadratic Function ◽  
Quadratic Functions ◽  
Second Law ◽  
Convex Region ◽  
Preference Functions ◽  
Preference System ◽  
Image Position ◽  
Infinite Variety ◽  
Increasing Function ◽  
General Method

A normal preference system for a combination of goods is represented by an increasing function φ with convex levels. From Gossen's law, that preference and price directions coincide in equilibrium (a special consequence of his Second Law), it follows that, on the data that xr is the vector of quantities purchased at prices given by a vector pr (r = 1,…, k), the gradient gr = g(xr) of the function φ at the point xr is given byfor some positive multiplier μr;. There may be considered the class of preference functions thus satisfying Gossen's law in respect to the data, and thus with gradients taking prescribed directions at k prescribed points. In particular, there may be considered the subclass of these which are quadratic in some convex region containing the points xr. By choosing any multipliers μr, there is obtained a set of gradients gr associated with the points xr. It is asked if there exists a quadratic function which is increasing and has convex levels in a convex neighbourhood of the points xr, and whose gradient at xr is gr; also it is required to characterize the class of such functions, which, if any exist, form an infinite variety. This is the background of the questions which are going to be investigated, and which are of importance in a general method of empirical preference analysis in economics.

Pengembangan Media Pembelajaran Interaktif Berdasarkan Teori Apos pada Materi Fungsi Kuadrat

2018 ◽  
Vol 6 (1) ◽  
pp. 1-10
Author(s):  
Arie Gusman ◽  
Kamid Kamid ◽  
Syamsurizal Syamsurizal
Keyword(s):  
Learning Theory ◽  
Teaching And Learning ◽  
Quadratic Function ◽  
Test Group ◽  
Quadratic Functions ◽  
High School Mathematics ◽  
Constructivist Learning Theory ◽  
Average Value ◽  
Long Time ◽  
Action Process

Learning quadratic functions that had been performed by the majority of vocational school and high school mathematics teacher in Kuala Tungkal is still using conventional learning media. The use of conventional learning media is experiencing a lot of obstacles, such as: a fairly long time in describing the graph function, especially when analyzing some quadratic function graphs with various characteristics. APOS is one of the constructivist learning theory which states that students learn through several stages, namely: action – process – object – schema. And to integrate into media APOS writer adapting ADDIE development model. The effectiveness of the use of media-based learning theory APOS seen from the student activity sheet can be concluded more increased activity of students in the learning process. Study of the test results, students were able to meet the completeness criteria stipulated minimum is 75. With an average value of learning outcomes, namely 87.14. It can be seen from the students' responses on a test group of small and large groups where it is concluded that researchers develop learning media can be categorized as good / interesting in the teaching and learning of mathematics.

scholarly journals Computability and the algebra of fields: Some affine constructions

1980 ◽  
Vol 45 (1) ◽  
pp. 103-120 ◽  
Author(s):  
J. V. Tucker
Keyword(s):  
Algebraic Structure ◽  
Algebraic System ◽  
Finiteness Condition ◽  
Coordinate Systems ◽  
Natural Numbers ◽  
Algebraic Foundation ◽  
Image Position ◽  
Technical Idea ◽  
Natural Way ◽  
General Method

A natural way of studying the computability of an algebraic structure or process is to apply some of the theory of the recursive functions to the algebra under consideration through the manufacture of appropriate coordinate systems from the natural numbers. An algebraic structure A = (A; σ1,…, σk) is computable if it possesses a recursive coordinate system in the following precise sense: associated to A there is a pair (α, Ω) consisting of a recursive set of natural numbers Ω and a surjection α: Ω → A so that (i) the relation defined on Ω by n ≡α m iff α(n) = α(m) in A is recursive, and (ii) each of the operations of A may be effectively followed in Ω, that is, for each (say) r-ary operation σ on A there is an r argument recursive function on Ω which commutes the diagramwherein αr is r-fold α × … × α.This concept of a computable algebraic system is the independent technical idea of M.O.Rabin [18] and A.I.Mal'cev [14]. From these first papers one may learn of the strength and elegance of the general method of coordinatising; note-worthy for us is the fact that computability is a finiteness condition of algebra—an isomorphism invariant possessed of all finite algebraic systems—and that it serves to set upon an algebraic foundation the combinatorial idea that a system can be combinatorially presented and have effectively decidable term or word problem.

A non-Markovian birth process with logarithmic growth

1975 ◽  
Vol 12 (04) ◽  
pp. 673-683
Author(s):  
G. R. Grimmett
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Birth Process ◽  
Image Position ◽  
Almost All ◽  
Increasing Function ◽  
Logarithmic Growth

I show that the sumof independent random variables converges in distribution when suitably normalised, so long as theXksatisfy the following two conditions:μ(n)= E |Xn|is comparable withE|Sn| for largen,andXk/μ(k) converges in distribution. Also I consider the associated birth processX(t) = max{n:Sn≦t} when eachXkis positive, and I show that there exists a continuous increasing functionv(t) such thatfor some variableYwith specified distribution, and for almost allu. The functionv, satisfiesv(t) =A(1 +o(t)) logt. The Markovian birth process with parameters λn= λn, where 0 < λ < 1, is an example of such a process.

Prediction of vernalisation in three Australian Vrn responsive wheats

2003 ◽  
Vol 54 (3) ◽  
pp. 283 ◽  
Author(s):  
L. D. J. Penrose ◽  
H. M. Rawson ◽  
M. Zajac
Keyword(s):  
Quadratic Function ◽  
Linear Interpolation ◽  
Development Stage ◽  
Linear Functions ◽  
Quadratic Functions ◽  
Published Data ◽  
Floral Initiation ◽  
Maximum Temperatures ◽  
Double Ridge ◽  
Interpolation Functions

This study sought to better estimate vernalisation in winter wheats, so that their early development and time of anthesis can be better predicted. For this, an accurate relationship between temperature and the effectiveness of vernalisation is required. Using previously published data, our study found that the relationship between temperature and effectiveness of vernalisation can be suitably described by a quadratic function. In contrast, most previous studies used linear interpolation functions to describe vernalising effectiveness. These consist of a series of linear functions of temperature over adjoining temperature ranges. An advantage of quadratic functions is that they allow effectiveness of vernalisation to be described in terms of underlying physiological processes, and require the estimation of fewer parameters to predict wheat development. Our study found the cardinal temperatures for vernalisation to be –3�C, 6.5�C, and 15.9�C, that is for the lower, optimum, and maximum temperatures respectively. To allow for different upper temperature limits for vernalisation, 2 quadratic temperature-vernalising effectiveness functions were used to predict accumulated daily vernalisation at 3 field sites. These predictions of daily vernalisation were compared with corresponding estimates produced with 3 previously proposed linear interpolation functions. Varying degrees of agreement were found between estimates produced by the 2 types of vernalising effectiveness functions. Equations that have been developed to predict floral initiation in winter wheats have not been previously evaluated in Australian field environments. These equations utilise the same underlying relationship between accumulated daily vernalisation and a measure of floral initiation, often the appearance of double ridges. Two of these equations were used to predict the appearance of double ridges for a field-grown Australian winter wheat, JF87%014. Neither equation could satisfactorily predict the timing of the double ridge development stage for this wheat, whatever vernalising effectiveness function was used to predict vernalisation in the field. Both equations had greatest difficulty in predicting the double ridge stage, in environments where vernalisation most delayed development. This finding suggests that equations currently predicting floral initiation in winter wheats do not utilise an accurate relationship between accumulated vernalisation and floral initiation. An alternative method of predicting anthesis in winter wheats is to predict final leaf number, but this approach has not been reliably applied in environments where vernalising temperatures vary.

scholarly journals Eggplant growth as affected by bovine manure and magnesium thermophosphate rates

2008 ◽  
Vol 65 (1) ◽  
pp. 77-86 ◽  
Author(s):  
Marinice Oliveira Cardoso ◽  
Walter Esfrain Pereira ◽  
Ademar Pereira de Oliveira ◽  
Adailson Pereira de Souza
Keyword(s):  
Leaf Area ◽  
Dry Matter ◽  
Quadratic Function ◽  
Potassium Sulfate ◽  
Linear Functions ◽  
Quadratic Functions ◽  
Positive Interaction ◽  
Relative Growth ◽  
High K ◽  
Cow Urine

Plant growth is influenced by nutrient availability. The objective of this research was to study, under greenhouse conditions, eggplant growth as affected by rates of bovine manure and magnesium thermophosphate (g kg-1 and mg kg-1, respectively), according to a "Box central composite" matrix: 4.15-259; 4.15-1509; 24.15-259; 24.15-1509; 0.0-884; 28.3-884; 14.15-0,0; 14.15-1768; 14.15-884. Potassium sulfate (170 mg kg-1) and 200 mL per pot of cow urine solution were applied four times, but the concentration of the last two applications (200 mL/H2O L) was twice of that of the first two. Additional treatments: magnesium thermophosphate without cow urine and triple superphosphate with urea, both with nutrient levels equivalent to the bovine manure, P2O5 and potassium sulfate to the combination 14.15-884. The experimental design consisted of randomized blocks with four replicates. Leaf area (LA) and LA ratio increased as quadratic functions with manure rates, with negative interaction for thermophosphate. Leaf dry matter mass (DMM) had an increasing quadratic function with rates for both fertilizers. The higher combined rates of both fertilizers resulted in the smallest specific leaf area, but also the highest values of shoot and root DMM, total DMM and, with positive interaction in relation to root shoot dry matter ratio. The relative growth rate in stem height, and also in diameter, increased with manure, according to quadratic and linear functions, respectively. The cow urine effect was, in general, lower than that of urea. The plant's overall growth was more influenced by manure. Root DMM and shoot DMM were greater with high K and P.

Additive functions of intervals and Hausdorff measure

1946 ◽  
Vol 42 (1) ◽  
pp. 15-23 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Hausdorff Measure ◽  
Convex Sets ◽  
Infinite Sequence ◽  
Small Positive Number ◽  
Additive Functions ◽  
Image Position ◽  
Bounded Sets ◽  
Increasing Function

Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a set E in Rq, relative to the function h(t), is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We define

A General Method of Four-Bar Linkage Mobility Analysis

1987 ◽  
Vol 109 (2) ◽  
pp. 197-203 ◽  
Author(s):  
J. Angeles ◽  
A. Bernier
Keyword(s):  
Optimization Algorithm ◽  
Quadratic Function ◽  
Mobility Analysis ◽  
Geometric Problem ◽  
Four Bar Linkage ◽  
General Method

The problem of mobility associated with four-bar linkages is addressed in this paper. The mobility analysis is reduced to finding the global extrema of a quadratic function on a cylinder, which then leads to the geometric problem of finding the intersections of a circle and a hyperbola. The method proposed here produces an efficient mobility analysis that can be readily integrated into any suitable optimization algorithm.

An irreducible non-convex region

1953 ◽  
Vol 49 (2) ◽  
pp. 194-200 ◽  
Author(s):  
Kathleen Ollerenshaw ◽  
L. J. Mordell
Keyword(s):  
Convex Region ◽  
Image Position

Some years ago I showed ((4), § 6, pp. 88–91) that the star domain K defined by the inequalitieshas the minimum determinant Δ(K) = 2 and has an infinity of singular critical lattices. In this note I show that there is a unique irreducible star domain . That ís to say, there is just one star domain H contained in but different from K for which Δ(H) = Δ(K) = 2, and such that Δ(H′) < 2 for every star domain H′ contained in but different from H.

Type-raising operations on cardinal and ordinal numbers in Quine's “New foundations”

1973 ◽  
Vol 38 (1) ◽  
pp. 59-68 ◽  
Author(s):  
C. Ward Henson
Keyword(s):  
Set Theory ◽  
Initial Segment ◽  
Initial Section ◽  
Ordinal Numbers ◽  
Cardinal Numbers ◽  
Basic Facts ◽  
Image Position ◽  
The Axiom Of Choice ◽  
Increasing Function ◽  
Standard Operations

In this paper we develop certain methods of proof in Quine's set theory NF which have no counterparts elsewhere. These ideas were first used by Specker [5] in his disproof of the Axiom of Choice in NF. They depend on the properties of two related operations, T(n) on cardinal numbers and U(α) on ordinal numbers, which are defined by the equationsfor each set x and well ordering R. (Here and below we use Rosser's notation [3].) The definitions insure that the formulas T(x) = y and U(x) = y are stratified when y is assigned a type one higher than x. The importance of T and U stems from the following facts: (i) each of T and U is a 1-1, order preserving operation from its domain onto a proper initial section of its domain; (ii) Tand U commute with most of the standard operations on cardinal and ordinal numbers.These basic facts are discussed in §1. In §2 we prove in NF that the exponential function 2n is not 1-1. Indeed, there exist cardinal numbers m and n which satisfyIn §3 we prove the following technical result, which has many important applications. Suppose f is an increasing function from an initial segment S of the set NO of ordinal numbers into NO and that f commutes with U.

On a problem of Littlewood concerning Riccati's equation

1969 ◽  
Vol 65 (3) ◽  
pp. 651-662 ◽  
Author(s):  
H. P. F. Swinnerton-Dyer
Keyword(s):  
Riccati Equation ◽  
Initial Condition ◽  
Large Parameter ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Image Position ◽  
Increasing Function ◽  
Parameter Theory

The study of the Riccati equationplays an essential part in the ‘large parameter’ theory of the inhomogeneous van der Pol equation; see for example Littlewood(1), (2). The crucial result is Lemma B of (1), restated and proved as Lemma 5 of (2); for the present paper the relevant parts of it are as follows:Lemma 1. Let z = z(x) be the solution of (1·1) which satisfies the initial condition z = 0 at x = 0, and assume α > 0. Then there is a unique β0 = β0(α) with the property that(i) if β > β0 then z → − ∞ as x → + ∞;(ii) if β < β0 then z → + ∞ at a vertical asymptote x = x0(α,β);(iii) if β = β0 then z ≥ 0 in 0 ≤ x < + ∞ and z = x + β0 + o(1) as x → + ∞.Moreover, β0(α) is a continuous monotone increasing function of α.

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