Discontinuous Quantity Research Articles

Testing Neural Network (NN) algorithms implemented in MATLAB for the study of rainfall distribution patterns at different locations in Nigeria

In Nigeria, accurate information concerning rainfall distribution is generally difficult to obtain because it is a discontinuous quantity. Therefore, special tools and equipment are required for its recording and interpretation. This has made information dissemination regarding rainfall distribution across different locations in Nigeria difficult. The scenario has made everyone sorely dependent on the Nigeria Meteorological Agency (NiMet) as the only source of recording and interpreting rainfall distribution. Rainfall distributed at different locations in Nigeria at the same time and season for a particular year is not even. Therefore, it is uncertain for one to conclude that the rainfall distribution pattern at one location is or will be the same or different at another location at a particular time and season of the year. Therefore, the objective of this work is to test neural network (NN) algorithms implemented in MATLAB for the study of rainfall distribution patterns at different locations in Nigeria using satellite-based data obtained from The European Centre for Medium-Range Weather Forecasting (ECMWF) to have alternative methods of interpreting rainfall data in Nigeria. After data conversion, the designed neural network algorithms implemented in MATLAB were adopted to contour daily rainfall incidences in all the locations for the years of study. The result of the study showed that locations with latitude closer to 4o, received the highest amount of rainfall distributions and locations with latitude farther to 4o, received the lowest amount of rainfall distributions within the same years of the study. The results also showed that rainfall distribution patterns can be interpreted using NN implemented algorithms in MATLAB and they vary at different locations in Nigeria. Therefore, NN algorithms implemented in MATLAB may serve as another method for fast and reliable interpretation of rainfall distribution recorded data at any chosen location.

Ageing of an oscillator due to frequency switching

If an oscillator is driven by a force that switches between two frequencies, the dynamics it exhibits depends on the precise manner of switching. Here we take a one-dimensional oscillator and consider scenarios in which switching occurs either: (i) between two driving forces which have different frequencies, or (ii) as a single forcing whose frequency switches between two values. The difference is subtle, but its effect on the long term behaviour is severe, and occurs because the expressions of (i) and (ii) are linear and nonlinear, respectively, in terms of a discontinuous quantity (e.g. a sign or Heaviside step function) that represents the switch between frequencies. In scenario (i) the oscillator can be described as a Filippov system, and we will show it has a stable periodic orbit. In scenario (ii) the oscillator exhibits hidden dynamics, which lies outside the theory of Filippov’s systems, and causes the system to be increasingly (as time passes) dominated by sliding along the frequency-switching threshold, and in particular if periodic orbits do exist, they too exhibit sliding. We show that the behaviour persists, at least asymptotically, if the systems are regularized (i.e. if the switch is modelled in the manner of (i) or (ii) but with a smooth rather than discontinuous transition).

Implications of single or multiple pressure degrees of freedom at fractures in fluid-saturated porous media

The physical consequences of modelling the fluid pressure across a fracture using one, two or three degrees of freedom are elucidated. The implications are demonstrated for each model through numerical examples for different boundary value problems. When fracture propagation is mainly driven by mechanical loads a single pressure degree of freedom is normally sufficient. Modelling of the pressure as a discontinuous quantity can be done using a double degree of freedom, similar to the modelling of displacements. Historically, this has been proposed first, but it appears to be less well applicable, except for cases where there is no significant fluid transport along the fracture, as in shear failures. Modelling the pressure with a triple degree of freedom for the pressure at the fracture is the most versatile approach, and is physically the most reasonable and efficient approach to model the propagation of internally pressurised cracks (hydraulic fracturing).

A Hyperspherical Adaptive Sparse-Grid Method for High-Dimensional Discontinuity Detection

This work proposes and analyzes a hyperspherical adaptive hierarchical sparse-grid method for detecting jump discontinuities of functions in high-dimensional spaces. The method is motivated by the theoretical and computational inefficiencies of well-known adaptive sparse-grid methods for discontinuity detection. Our novel approach constructs a function representation of the discontinuity hypersurface of an $N$-dimensional discontinuous quantity of interest, by virtue of a hyperspherical transformation. Then, a sparse-grid approximation of the transformed function is built in the hyperspherical coordinate system, whose value at each point is estimated by solving a one-dimensional discontinuity detection problem. Due to the smoothness of the hypersurface, the new technique can identify jump discontinuities with significantly reduced computational cost, compared to existing methods. Moreover, hierarchical acceleration techniques are also incorporated to further reduce the overall complexity. Rigorous complexity analyses of the new method are provided as are several numerical examples that illustrate the effectiveness of the approach.

Proportional Reasoning in Young Children: Part-Part Comparisons about Continuous and Discontinuous Quantity

Previous studies showed that 6-year-old children can make proportional judgements when the first-order relations are accessible to them and that children use the half boundary in order to make such judgements. In these studies “half” proved to be an important boundary in children's initial understanding of proportion. These studies involved non-numerical tasks (continuous quantities) in which proportional judgements were established on the basis of part-part relations (ratio). This raises a question about ratio comparisons that are numerically represented (discontinuous quantities). Would children again use the half boundary in ratio comparisons between continuous and discontinuous quantities? In this experiment 6- to 8-year olds were given two tasks: In the sliced task, the standard and the choices were discontinuous quantities; in the non-sliced task, the standard was a continuous quantity and the choices were discontinuous quantities. Children's justifications reflected different levels of understanding of proportions and characteristics of the tasks. The results revealed that “dhalf” plays a crucial role in children's proportional reasoning even when discontinuous quantities are involved.

The development of understanding of quantity in children with down's syndrome

Thirty‐six children with Down's Syndrome were pretested on tasks involving conservation of discontinuous and continuous quantity. Their order of development of understanding was compared with that of twenty non‐retarded children who had been given exactly the same tasks. Similarity in sequence of development was clear and supported by scalogram analysis. From the sample of Down's children, two groups, each of eight children, were matched for understanding of conservation and for age. None of these children recognised conservation of continuous quantity. The experimental group was given individual teaching, experiences and explanations, building understanding of conservation from its components. Post tested immediately after teaching and again two weeks later, all the experimental group recognised conservation of continuous as well as discontinuous quantity. Their understanding was generalised and durable. The control group children did not show such understanding on either post test. The difference betwee...

Conservation Skills: A Replicated Study of Order of Acquisition across Tasks

Abstract Developmental patterns in conservation skills were examined in four samples of children (N = 390) 4 to 11 years of age. Judgments of conservation and explanations were elicited for seven conservation tasks assessing concepts from number to volume (adapted from Goldschmid & Bentler 1968a). Across all ages and samples, judgment-only and judgment-plus-explanation scores were highly correlated. Explanations provided little additional information on level of conservation skill. The greatest increment in conservation skill occurred between kindergarten and Grade 1, with some further increase between Grades 1 and 3 or 4. Order of acquisition of number, area, and volume were consistent in all samples, with number easiest and volume hardest. Conservation of substance, discontinuous quantity, continuous quantity and weight were always more difficult than area and easier than volume, but there was no evidence of consistent differences in difficulty among the matter and weight tasks. These findings confirm t...

«Why do you ask?» context and communication in the conservation task

Modifications of conservation tasks designed to increase their ‘social intelligibility’ have frequently been shown to improve young children’s performance. Several studies in this area have involved introducing an emphasis on fairness of distribution, but the significance of this factor has not been independently assessed. Another factor to which attention has recently been drawn concerns the child’s understanding of the experimenter’s intentions in asking the conservation question. Perner (1984) has reported that young children responded more accurately when the question was asked by an experimenter who had not witnessed the earlier stages of the procedure. This paper reports an experiment in which these two factors — the emphasis on fairness and the introduction of a naive experimenter—were manipulated independently. One hundred and twelve 4–6 year olds were divided between four conditions and tested in pairs on a task involving conservation of discontinuous quantity. The results offered clear evidence that emphasising fairness through the device of a competitive game did increase the frequency of correct responses. The introduction of a second experimenter to ask the conservation question also had a significant, if more limited, facilitative effect. Possible social and cognitive processes underlying these results are discussed.

Longitudinal development of conservation skills in learning disabled children.

A three-year longitudinal investigation was conducted on the development of conservation skills in learning disabled (LD) children. Six measures of conservation (space, number, substance, weight, continuous and discontinuous quantity) from the Concept Assessment Kit were administered to 31 newly identified LD students and 33 normally achieving children (NLD) during each of three years. Results indicated that the LD group demonstrated a developmental delay in attaining the stage of concrete operations compared to the NLD group. However, when this stage was achieved, the LD group appeared to acquire specific concepts at the same rate as normally achieving children. For the LD children, total conservation scores and age were more important predictors of academic achievement than was verbal intelligence during all three years of the study whereas, for the NLD group, this was true only during the first year. It appeared that delayed transition between preoperational and concrete operational thought may be an important factor in understanding the continued school failure of learning disabled children.

Knowledge of Number Conservation and the Acquisition of Quantity Conservation in First Graders

Summary First graders were tested on number conservation, conservation of continuous quantity (mass), and conservation of discontinuous quantity. The 24 girls and 19 boys who failed the two quantity conservation tasks were exposed to a classroom demonstration on conservation and retested on all conservation measures. Posttest quantity conservation performance was assessed in relationship to performance on the number conservation task. A chi-square test for the homogeneity of the distributions of number correct on quantity conservation showed a significant difference between number conservers and number nonconservers. Children who understood number conservation improved their quantity conservation performance more often than children who showed no understanding of number conservation.

THE CONSERVATION TASK AS AN INTERACTIONAL SETTING

Summary. Two experiments are reported, the first being a partial replication of McGarrigle and Donaldson (1975). Sixty children, average age 5 years 6 months, were tested on conservation of length both in a standard condition where the transformation was clearly a deliberate act of the tester and in an ‘accidental’ condition, where the transformation was occasioned by an errant teddy bear. Overall frequency of conserving judgments was lower than that obtained by McGarrigle and Donaldson, but their findings of a higher success rate in the accidental condition was confirmed. The second experiment represents a different approach to the same issue. Eighty children, average age 6 years 0 months, were tested on conservation of discontinuous quantity in pairs. Half of the children were given a standard presentation, while for the other half the transformation was made to seem incidental to a competitive game. Conservation rates were 5 and 70 per cent respectively. The implications of these results for our understanding of the transition to operational thinking are discussed.

Conservation Ability of Elderly Men Living in the Community and an Institution

The present study assessed possible differences in decline of ability for logical operations for 10 elderly community residents and two elderly institutionalized groups ( ns = 10). While 100% conservation was shown by the community group on two-dimensional space, number, substance, continuous quantity, weight, and discontinuous quantity, and 80% conservation of volume, the institutionalized groups showed much lower rates. The implication is that active elderly persons who reside in their own homes will not show the apparent decrement noted in earlier studies of institutionalized persons. The relationship between logical operations and fluid intelligence was supported by the diminished conservation performance of neurologically impaired institutionalized groups. Other sampling effects which might account for the differing results among studies were age and length of institutionalization.

The empirical violation of conservation laws and its relation to structural change

Concrete operational Ss were provided empirical evidence of nonconservation of discontinuous quantity and weight. The findings were that few Ss accepted this evidence. This was interpreted as supporting the organismic-developmental claim that lower forms of reasoning are transformed into structurally more advanced forms. It was also found that (1) probing provides a more accurate assessment of Ss' operational levels, (2) one can empirically distinguish between logicomathematical and physical knowledge, and (3) the former is understood by Ss to be nomically necessary and the latter, in the case of weight, may not be.

Conservation and Sex

Sex differences have often been compared incidentally in studies of Piagetian conservation. Most results showed no sex differences related to the acquisition of conservation (1, 2, 3, 6, 7, 9) and support Piaget's conclusions that sex differences are insignificant. Conversely, several studies have reported sex differences in favor of male Ss (4, 5. 8) . Goldschmid's (4) examination of the relation of sex to types of conservation indicated that boys performed on a higher level than girls on all tested tasks of conservation including conservation of substance, weight, quantity, discontinuous quantity, number, area, distance, length, twoand three-dimensional space. In the present study 46 boys, 35 girls in first grade in two public elementary schools in the Grand Forks, North Dakota area were given five conservation of number and substance tasks. Girls conserved more often on 4 of the 5 tasks and significantly ( p < .01) so on cwo conservation of substance tasks. The one task on which boys conserved more often (conservation of number) showed no significant differences. These results are in contradiction to earlier reported studies and indicate that sex differences are related to conservation of substance. These conservation tasks were administered by a young woman but in most studies E has been male. It seems sex of tester should be evaluated.

Note on the Concept Assessment Kit—Conservation

The Concept Assessment Kit-Conservation (Goldschmid & Bender, 1968) was devcloped to measure Piagetian conservation behavior in children and is composed of six tasks, tach measuring a different type of conservation: two-dimensional space, number, substance, continuous quantity, weight, and discontinuous quantity. Each task has two parts, one of which is termed behavioral, i.e., the child indicates his judgment of equaliry or inequality of the materials, and the other is called explanatory, i.e., the child gives a reason for his judgment. One point is given for each correct response, yielding a maximum possible score of 12 (6 for behavior and 6 for explanation). The materials used for the six tasks include wooden blocks (two-dimensional space), red and white poker chips (number), Play-Doh (substance and weight), water in beakers (continuous quantity), and corn seeds in beakers (discontinuous quantity).

A Pilot Investigation of the Effects of Training Techniques Designed to Accelerate Children's Acquisition of Conservation of Discontinuous Quantity

AUTHOR Feigenbaum, Kenneth D. TITLE A Pilot Investiqation of the Effects of Training Techniques Designed to Accelerate Childrens' Acquisition of Conservation of Discontinuous Quantity. Final Report. INSTITUTION Antioch Coll., Yellow Springs, Ohio. SPONS AGENCY Office of Education (DREW), Washington, D.C. Bureau of Research. BUREAU NO BR-6-8415 PUB DATE Aug 68 CONTRACT OEC-3-7-068415-0428 NOTE 41p.

SCALOGRAM ANALYSIS OF LOGICAL AND PERCEPTUAL COMPONENTS OF CONSERVATION OF DISCONTINUOUS QUANTITY

SCHWARTZ, MARILYN MILLER, and SCHOLNICK, ELLIN KOFSKY. Scalogram Analysis of Logical and Perceptual Components of Conservation of Discontinuous Quantity. CHID DEVELOPMENT, 1970, 41, 695-705. The role of stimulus factors and logical judgments in conservation of discontinuous quantity was explored using 2-choice, nonverbal discrimination tasks. 40 preschoolers estimated the amount of candy in 2 containers under conditions varying the relative diameters of the 2 containers (identical or different) and the type of judgment required (direct comparison, identity, and equivalence estimates). The tasks formed a Guttman scale. The easiest items involved identical containers. There was a significant interaction between stimulus setting and judgments. When identical containers were used, identity judgments were least accurate, but when the 2 containers differed, identity was easiest followed by equivalence and comparison judgments.

Scalogram Analysis of Logical and Perceptual Components of Conservation of Discontinuous Quantity

Scalogram Analysis of Logical and Perceptual Components of Conservation of Discontinuous Quantity

THE TRAINING OF CONSERVATION OF NUMBER IN YOUNG CHILDREN

An approach to the training of conservation of number through the presentation of a logical sequence of component concepts or steps was evaluated in a series of 3 experiments using 130 kindergarten children. The Ss were given individual instruction by Es. In addition, part of the instruction was presented by slightly older conserving peers who acted as assistant teachers. The experimental Ss showed significant growth in conservation while the control Ss showed no noticeable growth. The effects of the training were retained at 3 months and generalizability of these effects to conservation of discontinuous quantity was demonstrated. No sex differences were found and only slight differences were revealed in conservation learning between SES groups.

On propagation op shock waves in an elasto-plastic medium: PMM vol. 33, no. 1, 1969, pp. 143–147

Investigations of shock waves are well known in gases [1,2], in elastic media [3] and many others. These investigations pertain to the case where it is possible to obtain directly a closed system of equations at the discontinuities. For the majority of rheological media it is impossible to write directly the determining equations at the discontinuties. This situation arises becauses it is necessary on ths shock wave to know the character of change of some discontinuous quantity ƒ in its dependence on another discontinuous quantity φ. In this work it is proposed to find the dependence ƒ (φ) from the analysis of the shock transition.