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What is a Weber Fraction?
From Panamath
In mathematics and physics, Weber's law states that, as the ratio between the magnitudes of two stimuli increases, the more easily the difference between the two stimuli will be perceived.
Because the probability of distinguishing whether there are more blue or yellow dots increases as the disparity between the actual number of blue versus yellow dots increases, Weber's law also applies to the Panamath task. Models of the ANS utilize Gaussian curves that represent different numerosities, or mental representations of quantity. The spread of these curves is determined by the Weber fraction, hence allowing analysis and comparison of subject performance.
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Model Representations of the ANS
In modeling performance on tasks that engage the ANS, it is necessary first to specify a model for the underlying approximate number representations. It is generally agreed that each numerosity is mentally represented by a distribution of activation on an internal "number line." These distributions are inherently "noisy" and do not represent number exactly or discretely. This means that there is some error each time they represent a number; and this error can be thought of as a spread of activation around the number being represented.
The Mental Number Line
The mental number line is often modeled as having linearly increasing means and linearly increasing standard deviation. In such a format, the representation for e.g., cardinality seven is a probability density function that has its mean at 7 on the mental number line and a smooth degradation to either side of 7 such that 6 and 8 on the mental number line are also highly activated by instances of seven in the world. In Figure 1a I have drawn idealized curves which represent the ANS representations for numerosities 4-10 for an individual with Weber fraction = .125. You can think of these curves as representing the amount of activity generated in the mind by a particular array of items in the world with a different bump for each numerosity you might experience (e.g., 4 balls, 5 houses, 6 blue dots, etc). Rather than activating a single discrete value (e.g., 6) the curves are meant to indicate that a range of activity is present each time an array of (e.g., 6) items is presented. That is, an array of e.g., six items will greatly activate the ANS numerosity representation of 6, but because these representations are noisy this array will also activate representations of 5 and 7 etc with the amount of activation centered on 6 and gradually decreasing to either side of 6.
Neuronal Associations of the Mental Number Line
The bell-shaped representations of number depicted in Figure 1a are more than just a theoretical construct; "bumps" like these have been observed in neuronal recordings of the cortex of awake behaving monkeys as they engage in numerical discrimination tasks (e.g., shown an array of six dots, neurons that are preferentially tuned to representing 6 are most highly activated, while neurons tuned to 5 and 7 are also fairly active, and those tuned to 4 and 8 are active above their resting state but less active than those for 5, 6, and 7. These neurons are found in the monkey brain in the same region of cortex that has been found to support approximate number representations in human subjects. This type of spreading, "noisy" activation is common throughout the cortex and is not specific to representing approximate number. Rather, approximate number representations obey principles that operate quite generally throughout the mind/brain.
Interpreting the Gaussian Curves
The bell-shaped representations of number depicted in Figure 1a are more than just a theoretical construct; "bumps" like these have been observed in neuronal recordings of the cortex of awake behaving monkeys as they engage in numerical discrimination tasks (e.g., shown an array of six dots, neurons that are preferentially tuned to representing 6 are most highly activated, while neurons tuned to 5 and 7 are also fairly active, and those tuned to 4 and 8 are active above their resting state but less active than those for 5, 6, and 7. These neurons are found in the monkey brain in the same region of cortex that has been found to support approximate number representations in human subjects. This type of spreading, "noisy", activation is common throughout the cortex and is not specific to representing approximate number. Rather, approximate number representations obey principles that operate quite generally throughout the mind/brain.
When trying to discriminate one numerosity from another using the Gaussian representations in Figure 1a, the more overlap there is between the two Gaussians being compared the less accurately they can be discriminated.
Ratios that are closer to 1, where the two numbers being compared are close (e.g., 9 versus 10), give rise to Gaussians with greater overlap resulting in poorer discrimination (i.e., "ratio-dependent performance"). The ratio is calculated with this formula, where is the smaller number and is the larger number, respectively:
Visually, the curve for 5 in Figure 1a looks clearly different in shape from the curve for 4 (e.g., curve 4 is higher and skinnier than curve 5); and discriminating 4 from 5 is fairly easy. As you increase in number (i.e., move to the right in Figure 1a), the curves become more and more similar looking (e.g., is curve 9 higher and skinnier than curve 10?); and discrimination becomes harder.
But, it is not simply that larger numbers are harder to discriminate across the board. For example, performance at discriminating 16 from 20 (not shown) will be identical to performance discriminating 4 from 5 as these pairs differ by the same ratio (i.e., 5 / 4 = 1.125 = 20 / 16); and the curves representing these numbers overlap in the ANS such that the representation of 4 and 5 overlap in area to the same extent that 16 overlaps with 20 (i.e., although 16 and 20 each activate very wide curves with large standard deviations, these curves are far enough apart on the mental number line that their overlap is the same amount of area as the overlap between 5 and 4, i.e., they have the same discriminability). This is ratio-dependent performance.
Tasks That Give Rise to the Gaussian Curves
The Gaussian curves in Figure 1a are depictions of the mental representations of 4-10 in the ANS. Similar looking curves can be generated by asking subjects to make rapid responses that engage the ANS, such as asking subjects to press a button 9 times as quickly as possible while saying the word "the" repeatedly to disrupt explicit counting. In such tasks, the resulting curves, generated over many trials, represent the number of times the subject pressed the button when asked to press it e.g., 9 times. Because the subject can't count verbally and exactly while saying "the", they tend to rely on their ANS to tell them when they have reached the requested number of taps.
When this is the case, the variance in the number of taps across trials is the result of the noisiness of the underlying ANS representations and so can be thought of as another method for determining what the underlying Gaussian representations are. That is, if starting to tap and ending tapping etc did not contribute additional noise to the number of taps (i.e., if the ANS sense of how many taps had been made were the only source of over and under tapping) then the standard deviation of the number of taps for e.g., 9 across trials would be identical to the standard deviation of the underlying ANS number representation of e.g., 9.
When attempting to visualize what the noisy representations of the ANS are like, one can think of the Gaussian activations depicted in Figure 1a, and these representations affect performance in a variety of tasks including discriminating one number for another (e.g., 5 versus 4) and generating number-relevant behaviors (e.g., tapping 9 times).
Numerical Discrimination in the ANS
To understand how numerical discrimination is possible in the ANS, consider the task of briefly presenting a subject with two arrays, e.g., 5 yellow dots and 6 blue dots, and asking the subject to determine which array is greater in number (Figure 2a). The 5 yellow dots will activate the ANS curve representation of 5 and the 6 blue dots will activate the ANS curve representation of 6 (assume that the subject uses attention to select which dots to send to the ANS for enumerating and then stores and compares those numerosity representations bound to their respective colors) (Figure 2a-b).
ANS Modeling and Subtraction
An intuitive way to think about ordinal comparison within the ANS is to liken it to a subtraction; this will be mathematically equivalent to other ways of making an ordinal judgment within the ANS and my use of subtraction here should be thought of as one illustration among several mathematically equivalent illustrations.
Imagine that an operation within the ANS subtracts the smaller (i.e., five-yellow) representation from the larger (i.e., six-blue) representation (Figure 2b). Because the 5 and 6 representations are Gaussian curves, this subtraction results in a new Gaussian representation of the difference which is a Gaussian curve on the mental number line that has a mean of 1 (6 − 5 = 1) and a standard deviation of:
In Figure 2c (i.e., when subtracting one Gaussian random variable from another (i.e., ), the result is a new Gaussian random variable with the mean at the difference (6 − 5 = 1) and a variance that adds the variances of the original variables: . This results in a Gaussian curve that is centered on 1, but that extends to both the left and right of 0 (Figure 2c).
One can think of 0 as the demarcation line separating evidence "for" and "against" in that the area under the curve to the right of 0 is the portion of the resulting representation that correctly indicates that six is greater than five while the area under the curve to the left of 0 is the portion of the resulting representation that incorrectly indicates that five is greater than six. This area to the left of 0 results from the overlap between the original Gaussian representations, five and six, that were being discriminated in which some of the area of five-yellow is to the right (i.e., greater than) some of the area of six-blue (Figure 2b).
Interpreting Gaussian Overlap: Weber's Law
One can think of 0 as the demarcation line separating evidence "for" and "against" in that the area under the curve to the right of 0 is the portion of the resulting representation that correctly indicates that six is greater than five while the area under the curve to the left of 0 is the portion of the resulting representation that incorrectly indicates that five is greater than six. This area to the left of 0 results from the overlap between the original Gaussian representations, five and six, that were being discriminated in which some of the area of five-yellow is to the right (i.e., greater than) some of the area of six-blue (Figure 2b).
Another method would rely on assessing the total evidence for blue and the total evidence for yellow. Either of these ways of making a decision will have the result that, on a particular trial, the probability of the subject getting the trial correct will depend on the relative area under the curve to the left and right of 0 which is itself determined by the amount of overlap between the original Gaussian representations for the numerosities being compared (i.e., five and six).
The more overlap there is between the two Gaussian representations being compared, the less accurately they can be discriminated. Consider comparing a subject's performance on a 5 dots versus 6 dots trial to a trial involving 9 versus 10 dots. Using the curves in Figure 1a as a guide, we see that the overlapping area for the curves representing 5 and 6 is less than the overlapping area for the curves representing 9 and 10, because the curves flatten and spread as numerosity increases. This means that it will be easier for the subject to tell the difference between 5 and 6 than between 9 and 10, i.e., the resulting Gaussian for the subtraction will have more area to the right of 0 for the subtraction of 5 from 6 than for the subtraction of 9 from 10.
Across multiple trials the subject would give more correct responses on the 5 versus 6 trials than the 9 versus 10 trials owing to the greater evidence for 6 being larger than 5 (i.e., more area to the right of 0 in the subtraction, Figure 2c). The linear increase in the standard deviation of the curves representing the numerosites along the mental number line results in ratio-dependent performance whereby the discriminability of two numerosities increases as the ratio between them (e.g., biggernumber / smallernumber) increases. This is Weber's law.
How rapidly performance rises from chance (50%) to near-asymptotic performance (100%) in this kind of dot numerosity discrimination task is controlled by the subject's Weber fraction (w).
The Weber Fraction: Overview
The Weber fraction indexes the amount of spread in the subject's ANS number representations and therefore the overlap between any two numbers as a function of ratio (described in a succeeding section). The precision of the ANS varies across individuals with some people having a smaller Weber fraction (i.e., better performance and sharper Gaussian curves) and others having a larger Weber fraction (i.e., poorer performance owing to wider noisier Gaussian curves).
Numerical discrimination (e.g., determining which color, blue or yellow, has more dots in an array of dots flashed to quickly for explicit counting) is possible in the ANS through a process that attempts to determine which of the two resulting curves (e.g., five-yellow or six-blue) is further to the right on the mental number line. The fullness of these noisy curves is used to make this decision (and not just the mode, or mean or some other metric) and successful discrimination thereby depends on the amount of overlap between the two activated curves (i.e., ratio-dependent performance).
The amount of overlap is indexed by a subject's Weber fraction (w) with a larger Weber fraction indicating more noise, more overlap, and thereby worse discrimination performance. This model has been found to provide an accurate fit to data from rats, pigeons, and humans of all ages.
Relationship to the ANS
What does a Weber fraction (w) tell us about a subject's Approximate Number System (ANS) representations? A relationship exists between the Weber fraction (w), the standard deviations of the underlying Gaussian numerosity representations, and the discriminability of any two numbers; and a Weber fraction can be thought of in at least three ways, which sometimes leads to confusions.
These are: 1) the fraction by which a stimulus with numerosity n would need to be increased in order for a subject to detect this change and its direction on 50% of the trials (aka the difference threshold, aka the Just Noticeable Difference, J.N.D.), 2) the midpoint between subjective equality of two numbers and asymptotic performance in discrimination, and 3) the constant that describes the standard deviation of all of the numerosity representations on the mental number line. Option 3 is perhaps the least discussed in the psychophysics literature - it is rarely taught in psychophysics courses - but I will suggest that option 3 is the most productive way to understand the Weber fraction.
I first describe what a Weber fraction is and how it relates to the underlying number representations and then provide some suggestions for what I take to be a productive way of understanding the Weber fraction.
How to Think of the Weber Fraction
First, consider Figure 1a to represent the ANS number representations for a particular individual who has a Weber fraction = .125. If one presents the hypothetical subject in Figure 1a with the task of determining which of two colors has the greater number of dots on a trial where there are 16 blue dots and some number of yellow dots, this subject would require an increase of 2 dots from this standard (16 * .125 = 2, N2 = 16 + 2 = 18) in order to respond that yellow (18) is more numerous than blue (16) on 75% of the trials that present these two numerosities. That is, a subject's Weber fraction can be used to determine the amount by which one would need to change a particular number in order for that subject to correctly determine which number was larger on 75% of the trials (chance = 50%).
Conceived in this way, the Weber fraction describes a relationship between any numerosity and the numerosity that will be consistently discriminated from this standard. This gives one way of understanding why one might choose 75% correct performance as an interesting indicator of performance.
In order to specify what "consistently discriminated from" might mean one might also choose some other standard (e.g., 66.7% correct, or any other % above chance). From this point of view, often the view taught in courses on Psychophysics, the point is to estimate the steepness of the linear portion of the psychometric function (depicted in Figure 1b) and 66.7% would work for such purposes just as well as 75% or 80%; the choice of 75% correct is presented as more or less an arbitrary one.
However, as we will see below, 75% correct is special and the seemingly arbitrary reasons for choosing 75% correct as an index of performance find their non-arbitrariness in a mathematical relationship between correct discrimination, the Weber fraction, and the standard deviations of the underlying Gaussian representations.
Some readers, more familiar with research on the acuity of the ANS in infants 6-9 months of age and less familiar with the literature on adult psychophysics, may have come to believe that the Weber fraction describes the ratio below which a subject will fail to discriminate two numerosities (e.g., 6-month-olds succeed with a 1:2 ratio and fail with a 2:3 ratio). This suggests a categorical interpretation of the Weber fraction (e.g., a threshold where you will succeed if a numerical difference is "above threshold" and fail if it is "below threshold"). That is, some may have come to believe that performance should be near perfect with ratios easier than a subject's Weber fraction and at chance for ratios harder than a subject's Weber fraction. This is not what is seen in typical performance where a large number of trials test a subject's discrimination abilities across a wide variety of ratios.
Modeling Trends
Consider again the simple task of being briefly shown a display that includes some blue and yellow dots and being asked to determine on each flash if there had been more blue or more yellow dots. Percent correct on this numerical discrimination task is not a step function with poor performance "below threshold" and good performance "above threshold", but rather is a smoothly increasing function from near chance performance to success.
This performance and the range of individual differences, gathered from over 10,000 subjects between the ages of 8 and 85 years of age participating in this type of blue-yellow dots task, can be seen in Figure 3a-b. Every one of the more than 10,000 participants obeyed this kind of gradual increase in percent correct from a ratio of 1 (where the number of blue and yellow dots are equal) to easier ratios like 2 (where there might be 10 blue dots and only 5 yellow dots; 10 / 5 = 2). What changes from participant to participant is how steep the left side of the curve is, and these individual differences are shown in the figure by indicating the performance of the 10th and 90th percentile ranking of the more than 10,000 participants.
Figure 3b shows how the average Weber fraction improves over development. A steeper, quicker rise in the psychometric function (Figure 3a) indicates better sensitivity, better discrimination abilities and is indexed by the subject having a smaller Weber fraction (Figure 3b) (i.e., a smaller Weber fraction indicates less noise in the underlying ANS representations).
If one wished to localize the Weber fraction at some point along the smoothly increasing curve in Figure 3a it would be at the midpoint between subjective equality of the two numerosities being compared (typically occurring at a ratio = 1, where N1 = N2) and asymptotic performance (typically 100% correct, though asymptotic performance may be lower in unskilled subjects resulting in the midpoint falling at a percent correct lower than 75%).
The "Difference Threshold"
When subjects behave optimally, the Weber fraction is related to the ratio that results in 75% correct performance and not to the first ratio that results in chance, or above chance, performance. Indeed, the actual behavioral data from subjects seen in Figure 3a, and the modeled ideal behavior seen in Figure 1b, suggest that the subjects will always be above chance no matter how small the difference between N1 and N2. What changes is not whether a participant will succeed or fail to make a discrimination but rather the number of trials an experimenter will have to run in order to find statistically significant performance.
However, within the practical limits of testing real subjects, the infant literature's method of looking for a change from above chance performance to below chance performance is a reasonable approach for attempting to roughly locate the Weber fraction of subjects, like infants, who cannot participate in the large number of trials it takes to achieve the smooth data seen in Figure 3a.
The 75% correct point has been used as an indicator of the "difference threshold" because this point can be translated into a whole-number ratio through a mathematical relationship that holds between the Weber fraction and the ratio at which performance attains 75% correct.
Researchers occasionally suggest that "human adults can discriminate numerosities that differ by a ratio of 7:8," where 8 / 7 = 1.14 would be the whole number ratio nearest the Weber Ratio that results in 75% correct discrimination. Even this understanding of the Weber fraction as the midpoint between subjective equality and asymptotic performance, while it goes some way towards avoiding the mistaken belief that a difference threshold is a step function, misses the deeper continuous nature of discrimination within the ANS.
Understanding Gaussian Spreads
Let us consider a third way of understanding the Weber fraction (w): as a scaling factor that indexes the amount of noise for every numerical representation of the ANS. Understood in this way, described below, the Weber fraction is not specific to the task of numerical discrimination; indeed, it is wholly independent and prior to discrimination.
An animal that (bizarrely) could only represent a single numerical value in their ANS (e.g., could only represent approximately-twelve and no other numbers), and who could therefore never discriminate 12 from any other number (i.e., could not even perform a numerical discrimination task), would nonetheless have a Weber fraction and we could measure it.
Consider the Gaussian curves depicted in Figure 1a. The spread of each successive numerosity from 4 to 10 is steadily wider than the numerosity before it. This means that the discriminability of any two numerosities is a smoothly varying function, dependent on the ratio between the two numerosities to be discriminated.
In theory, such discrimination is never perfect because any two numerosities no matter how distant from one another will always share some overlap. At the same time, discrimination will never be entirely impossible, so long as the two numerosities are not identical, because any two numerosities, no matter how close (e.g., 67 and 68) will always have some non-overlapping area where the larger numerosity is detectably larger. Correct discrimination may occur on only a small percentage of trials if the two sets are very close in number, but it will never be impossible. This again motivates the intuition that percent correct in a discrimination task should be a smoothly increasing function from the point of subjective equality to asymptotic performance.
In Figure 1b I have drawn the expected percent correct for the ideal subject in Figure 1a whose w = .125 as derived by a model from classical psychophysics. Those who wish to translate this subject's Weber fraction into a whole number ratio can determine this ratio from the function in Figure 1b as the nearest whole number ratio to the Weber Ratio that falls at the midpoint between subjective equality (i.e., Weber Ratio = 1, Percent Correct = 50%) and asymptotic performance (i.e., Percent Correct = 100%) and would equal 8:9 (i.e., WR = 1.125) for the subject in Figure 1b.
Notice, the Weber fraction is equal to this Weber Ratio minus 1 (i.e., 1.125 − 1 = .125). It is the smoothly increasing spread in the underlying Gaussian representations depicted in Figure 1a that is the source of the smoothly increasing Percent Correct ideal performance in Figure 1b.
Noting the smoothly increasing spread of the Gaussian representations in Figure 1a motivates one to ask what is the parameter that determines the amount of spread in any each Gaussian representation on the mental number line.
Calculating Standard Deviation
In fact, it is the Weber fraction that determines the spread of every single representation on the mental number line by the following formula:
The standard deviation (SD) of the Gaussian representing any numerosity is that numerosity multiplied by the Weber fraction. Why is this the case? Intuitively, it is the standard deviations of the underlying Gaussian representations that determines the amount of overlap between the curves that represent any two numerosities, and it is the amount of overlap that determines how well any two numerosities can be discriminated.
The categorical views of the Weber fraction as a kind of threshold between successful discrimination and failure, or as the midpoint between subjective equality and asymptotic performance, choose to focus on only one particular point of what is actually a continuous and smooth function of increasing success at discrimination, and this entire function is determined by the Weber fraction as this fraction describes the standard deviation of any numerosity representation on the mental number line and thereby the degree of overlap between any two numerosities on the mental number line.
Viewed in this light, the Weber fraction describes all of the numerosity representations in the ANS. It is not specific to the task of discriminating two numerosities, and not specific to numerosity comparisons near "threshold". In fact, though this may never be the case in practice, given this understanding of the Weber fraction it would be possible to assess the Weber fraction for an animal who could only represent a single number in the ANS (e.g., can only represent approximately-12). It would be the standard deviation of the Gaussian that represents approximately-12 divided by the mean of this Gaussian.
The Weber Fraction as a Scaling Ratio
The Weber fraction (w) is the constant that describes the amount of variability in the underlying ANS number representations. It is a scaling factor by which one could take any one of the curves in Figure 1a and turn it into any of the other curves in Figure 1a. In the linear model in Figure 1a, the analog representation for any numerosity (e.g., n = 7) is a Gaussian random variable with a mean at n (e.g., n = 7) and a standard deviation of (n * w). This means that for a subject who has a Weber fraction of .125, the ANS representation for n = 7 will be a normal curve with a mean of 7 on the mental number line and a standard deviation of .125 * 7 = .875.
By substituting any number you like for n you can easily determine the shape of the underlying ANS representation, without ever having the participant engage in a numerical discrimination task that compares two numbers. This illustrates the power of understanding the Weber fraction as an index of internal noise: rather than simply telling us something about how well a subject will do at discriminating two numbers near their threshold, the Weber fraction (w) tells us the shape and overlap of every single number representation inside a person's Approximate Number System (ANS). The Weber fraction is about all of the representations, not just the ones "near threshold."
The inductive power of understanding the Weber fraction (w) to be an internal scaling factor is also seen when we compare the Weber fractions of different individuals. Individuals differ in the precision of their ANS representations. Some people have more noise and some people have less.
In Figure 4a I illustrate some idealized curves which represent the underlying ANS representations for a subject whose w = .125 and in Figure 4b for a subject whose w = .20. Crucially, one can see that the subject in Figure 4b has a greater degree of overlap between the Gaussian curves that represent numerosity than the subject in Figure 4a (recall, a bigger Weber fraction means more noise and fatter curves). It is this overlap that leads to difficulty in discriminating two stimuli that are close in numerosity (i.e., as one nears Weber Ratio = 1). The hypothetical subject in Figure 4b would have poorer discrimination in a numerical discrimination task (e.g., blue & yellow dots task) than the subject in Figure 4a.
In Figure 4c I have drawn the ideal performance for these two subjects in a numerical discrimination task. The midpoint between subjective equality (Percent Correct = 50%) and asymptotic performance (Percent Correct = 100%) in Figure 4c will fall at WR = 1.125 for the subject in Figure 4a and at WR = 1.20 for the subject in Figure 4b because these are equal to the Weber fraction for each subject plus one. An inspection of Figure 4c reveals that this is the case. This is not an accident; it derives from the relationship described above between the Weber fraction (w), the standard deviations of the underlying ANS number representations, and the overlap between any two ANS number representations.
The values for the Weber fractions in Figure 4 have been chosen so as to illustrate another value of understanding the Weber fraction to be an internal scaling factor; it empowers comparisons across individuals and formal models of individual differences. Converting the Weber fraction for each of these subjects into the nearest whole number fraction reveals that the Weber fraction for the subject in Figure 4a is 8:9 and for the subject in Figure 4b is 5:6 (i.e., 9 / 8 = 1.125; 6 / 5 = 1.20). Investigating the Gaussian curves in Figure 4a and 4b, which were created by treating the Weber fraction as an internal scaling factor to determine the standard deviations of the ANS Gaussian curves, reveals that the Gaussians for the numerosities 8 and 9 for the subject in Figure 4a are identical to the Gaussians for the numerosities 5 and 6 for the subject in Figure 4b. This is no accident. The only parameter that has been changed in constructing Figures 4a and 4b is the Weber fraction for the subject and this single parameter determines the spread in the Gaussians that represent every possible numerosity in the ANS of each subject.
In this way, understanding the Weber fraction to be an internal scaling factor that determines the spread of every ANS number representation not only empowers us to compare one number representation to another within a particular subject, it also empowers us to compare across individuals and to create mathematically tractable predictions about how the ANS representations of one person (e.g., the subject in Figure 4a) relate to the ANS representations of another (e.g., the subject in Figure 4b).
Deviations From the Panamath Model
It is not uncommon for performance in a discrimination task to deviate from the ideal performance depicted in e.g., Figure 1b and seen in actual data in Figure 2a. Two basic deviations capture almost all cases: 1) a "give-up" function wherein subjects give-up on their numerical discrimination abilities on hard ratios and guess at chance, and 2) a "lapse parameter" which tracks participants' tendency to have a lapse in attention such that they do not attend the e.g., blue and yellow dots on a particular trial. These two forms of deviation from the model presented in this chapter are tractable in that additional parameters can be included in the model to capture these behaviors.
The model I have described does not take account of participants' confidence in their answers, nor does it adjust its predictions based on sub-optimal engagement of the Approximate Number System (ANS). But, in practice, sub-optimal performance can occur, especially on difficult trials.
For example, consider the task of being shown a brief display of blue and yellow dots and being asked to guess which color had more dots. One a fairly easy trial (e.g., 27 yellow versus 7 blue), you would most likely get the correct answer and be highly confident in your answer. This would be true even if the display time was incredibly brief (e.g., 50 msec).
Effect of Difficulty on Trials
But, what will happen on a difficult trial of e.g., 27 yellow and 28 blue dots? On such a trial, a subject can tell fairly rapidly that the trial will be incredibly difficult and they will have very low confidence that their guess will be correct. On such trials, it is not unreasonable that a subject will simply give up on trying to determine whether there are more blue or more yellow dots and guess randomly (e.g., flip a coin). This kind of behavior has been observed in numerical discrimination tasks that present trials that are very difficult for that subject (e.g., presenting 3-year-olds with a brief display of 9 versus 10 objects) and it has also been observed in the discrimination performance of many animal models making a variety of ratio-dependent discriminations (but has yet to be discussed in detail in any literature). When this happens, performance will deviate from the model shown in Figure 1b in that percent correct will drop to chance before the Weber Ratio = 1. Participants will "give-up" on difficult trials and, rather than relying on their ANS representations, will guess randomly. This kind of performance can be seen in the left side of the curves diagramming the performance of children and adults in a difficult numerical discrimination task (Figure 5).
It is important to note that this kind of drop to chance performance is not predicted by any existing psychophysical model and it appears to be a deviation from the model caused by the subject giving up on the task; it does not appear to be the case, as of yet, that this type of deviation indicates that the underlying model of the ANS itself is flawed. We are currently attempting to specify the exact relationship between difficulty, Weber fraction, and the probability of giving up on a trial.
Effect of Attention
Another deviation can occur on the right side of the psychometric curve and can also be seen in Figure 5. As has been observed in previous numerical discrimination tasks, performance in unskilled subjects can fail to reach 100% correct even on easy trials, perhaps because of a tendency to suffer a lapse of attention on a subset of trials. As can be seen in the example data in Figure 5, this was the case for the 3-, 4- and 5-year-olds.
When this type of lapse occurs, it should not be dependent on the ratio that was presented on that trial. Rather, it is an event that occurs because of e.g., fatigue, irrespective of the ratio being presented on that particular trial. As such, this deviation is typically detectable as a flat cut in percent correct across the entire range of ratios presented in an experiment. It is because percent correct is highest on the easiest ratios that the deleterious affects of this lapse in attention are most visible on easier trials. The simplest way to account for this tendency in the model is to include a lapse parameter that is a constant probability of guessing randomly on any particular trial.
This parameter primarily lowers the model's asymptotic performance while retaining an accurate estimate of the Weber fraction (w). When required, I included this parameter in the model as follows, where is the probability of guessing randomly on any particular trial irrespective of ratio, is the probability of being incorrect given the model above and the chance level is .5 multiplied by 100 to return a percentage:
This type of guessing has also been observed in children (Figure 5) and in animal and other adult human participants for both numerical discrimination and other ratio-dependent dimensions.
What Do These Deviations Mean?
Each of these deviations from the model presented in this chapter can be captured by including an additional parameter in the model (i.e., a threshold for "giving up" as a function of ratio, and a lapse parameter; the probability of guessing randomly on any particular trial irrespective of ratio).
At the moment, the existing datasets do not suggest that deviation from the model presented here is so drastic as to recommend the abandoning of the model. Rather, as with all complex cognition, behaviors that engage the Approximate Number System (ANS) are occurring in a rich context that drives many other considerations that can affect performance. Giving up on a hard trial and being bored such that you fail to attend on a trial are two very basic considerations that can affect performance.
Conclusions and Summary
If one does some reading in hopes of understanding what a Weber fraction (w) is, the most common descriptions turn on the idea of a difference threshold (aka Just Noticeable Difference, J.N.D.). For example, "the fraction by which a stimulus with numerosity n would need to be increased in order for a subject to detect this change and its direction on 50% of the trials" is a common phrase; or, "the midpoint that indicates roughly successful discrimination performance."
Thinking about a Weber fraction in these ways promotes confusions and limits our inferences (e.g., the confusion that performance should change from chance performance at difficult ratios to above chance performance at easy ratios; as we've seen in Figure 3, the actual performance of subjects does not look this way but instead is a smooth function). It also does not promote the kind of understanding of the Approximate Number System (ANS) that highlights the systematic nature of the noise across various ANS number representations within an individual (e.g., understanding that the noise for any one ANS representation can be easily translated into an understanding of the noise for every single ANS number representation), nor the systematic relationships that exist across individuals (e.g., the comparison of the two subjects in Figure 4).
In this chapter I have tried to promote a third way of understanding the Weber fraction: it is a scaling factor that enables any ANS number representation to be turned into any other; or, equivalently, it is an index of the amount of noise in every one of the Approximate Number System (ANS) representations. Understood in this way, a Weber fraction does not require the commonsense notion of a "threshold" (i.e., a change from failure to success) and does not generate the same kinds of confusions that this commonsense notion gives rise to.
The commonsense notion of a "threshold," as a change from poor to good performance, is a productive one for first coming to understand how discrimination behavior works across many dimensions (e.g., approximate number, loudness, brightness etc). And so, it should not be abandoned. But it does not foster our intuitions about the underlying representations that give rise to this discrimination behavior (i.e., it doesn't give rise to an accurate picture of the systematic relationships between noise, numerosity, and discrimination performance). An understanding of a Weber fraction (w) as an index of the internal noise in the underlying ANS representations promotes productive intuitions and better theorizing about the Approximate Number System (ANS) and other ratio-dependent cognitive dimensions (e.g., line length, weight, brightness, etc).
A Weber fraction (w) indexes the amount of noise in the underlying approximate number representations, it specifies the standard deviations of the underlying Gaussian number curves for every numerosity in a subject's Approximate Numer System (ANS) and thereby the degree of overlap between any of these representations. For discrimination, the Weber fraction describes a subject's performance at discriminating any possible combination of numerosities, not just discriminability near their "discrimination threshold". And, the Weber fraction (w), understood as an internal scaling factor, describes all of the representations in the Approximate Number System (ANS) without any reference to a numerical discrimination task.

subset: commoncrawl


text:
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Rely on coming after, since it often comes after yours
Updated: 29 August 2016
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Sunday, 1 September 2019
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Friday, 14 August 2020


text:
Poulter boosts Cup hopes with 67
Ian Poulter is hopeful he is beginning to put his "horrendous" year behind him and play himself onto Europe's Ryder Cup team after a strong opening round at the Deutsche Bank Championship.
The Englishman has toiled through the season with shoulder, back and wrist injuries and, after starting his campaign with three top-10s on the PGA Tour, has only been able to record one more since.
Poulter has also not recorded any top-10s on the European Tour this season and that has left him on the outside looking in and requiring one of Paul McGinley's captain's picks to make it onto the team for Gleneagles next month.
Ian Poulter, pictured, is trying to impress Paul McGinley
But an opening 67 in Boston will have certainly caught the Irishman's eye, with Poulter sitting just four shots behind first-round leader Ryan Palmer.
Luke Donald will also be hoping for a good week as he too is relying on a wildcard and the 36-year-old began his tournament with a steady two-under 69.
McGinley makes his choices on Tuesday with Stephen Gallacher still in with a chance of playing his way onto the team, and displacing Graeme McDowell in the process, with a top-two finish at the Italian Open.
McGinley's choices look to come down to three of Gallacher, McDowell, Poulter, Donald, Lee Westwood and Francesco Molinari.
But Poulter, who has won 12 of his 15 previous Ryder Cup clashes, was a talisman during Europe's win at Medinah in 2012 and finally feels his game, and his fitness, is coming together.
"I'm feeling pretty good I just shot four under," he told Sky Sports 4 following his round.
"I feel my body is holding up well, I feel the work I've done in the last six weeks has been really really good.
"I'm coming into, I would say, a bit of form at the right time to try and press on this week and then have a couple of good weeks to finish and then keep my fingers crossed and see what happens for the Ryder Cup.
"It's been a horrendous year for injuries. I've been forced to take time off because I can't practise as hard as I've wanted to.
"And that's not good because I like to get out there, I like to have fun on the golf course and I like to work hard and I haven't been able to do that this year until now.
"I'm putting the reps in now, I'm feeling good, the body is feeling a lot better than it was in the middle part of the season and hopefully we can finish the second part of this season real strong."
Away from the race to Gleneagles is the small matter of the second tournament in the FedEx Cup play-offs, with Palmer holding a two-shot lead over fellow American Keegan Bradley.
Bradley, who is a likely pick for the United States team in Scotland due to his successful partnership with Phil Mickelson, set the pace amongst the early starters but Palmer overtook him with a stunning 63.
Starting on the back nine, the 37-year-old bogeyed the first but then birdied six of his next eight, including a run of four from 15 to 18, to turn in 35.
Birdies on two, five and nine followed to send Palmer to the top of the leaderboard.
Bradley's 65 left him on his own in second, one shot clear of fellow Americans Webb Simpson and Chesson Hadley and Australian Jason Day.
Poulter was joined at four under by Swede Carl Petterson, Scot Russell Knox and Americans Bill Haas and Jordan Spieth.
World number one Rory McIlroy was one under after a 70 containing five birdies and four bogeys while FedEx Cup leader Hunter Mahan struggled to a two-over 73.
The top 70 in the FedEx Cup rankings after the Deutsche Bank Championship will advance to next week's BMW Championship, with the overall winner taking home a 10million US dollar bonus.
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Reviews written by jimmyw247
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Home > System/Utilities > ZylBattery 1.20
ZylBattery 1.20 is a great operating system but there are times when the users are annoyed and want to uninstall the ZylBattery 1.20 from their computers. There are two ways that he can uninstall it manually and automatically. Firstly, he should apply necessary caution to take the back up precautions as there are chances of important files being created in this format that he cannot afford to loose.
go to the Add/Remove Programs Control Panel applet and click Uninstall ZylBattery 1.20.
If you want, you can also uninstall ZylBattery 1.20 from the command prompt by performing the following steps:
Start the computer in Safe Mode with Command Prompt support (press F8 during start up)
Navigate to %systemroot%\system32.
Type osuninst.exe and press Enter.
Follow the onscreen instructions.
If ZylBattery 1.20 is installed as a dual-boot system along with other operation system then you can simply format the partition on which it is installed. That is one of the reasons why I always recommend to install separate operating systems on separate partitions.
To uninstall ZylBattery 1.20 manually, there are some steps the user should follow. The user should go to 'Start' and then to 'Control panel'. The next step is to navigate to the option of 'Add/ Remove programs' and find the program from the list. Clicking on the 'Remove' option will help to follow the uninstall wizard and to finish the removal.
(Completely remove the ZylBattery 1.20)
Uninstalling ZylBattery 1.20 automatically is a great blessing for a novice PC user since it does the work efficiently and with the least hassles. To force uninstall ZylBattery 1.20 using a powerful uninstaller, certain steps should be effectively followed. The automatic uninstaller should be installed on your computer. Then the user should locate where the program is stored on the computer. The next concrete step is to click the selected folder and click 'Uninstall' to begin the process of removal. This method ensures that this program is thoroughly uninstalled from the computer without any entries left down in the program files. The users vouch for the convenience and excellence of uninstalling ZylBattery 1.20 with the help of a third party uninstaller since it thoroughly scans the computer and removes the program from the hard drive and the Windows registry.


text:
Skip to content
UH Hilo Home > Student Life Center > Reservations
Call 911 (or 9-911 from a campus phone) for a police, fire, or medical emergency. See the Actions to be Taken During Specific Emergencies. For general emergency info, read up on the Emergency Preparedness Information.
Campus Security, (808) 974-7911 or 7911 from a campus phone.
We also encourage you to submit a Silent Witness report if you wish to remain anonymous and it's a non-emergency safety or security concern.
The safety of the community and its members is very important to us. If you become aware of a situation that you believe may pose a threat, please submit a Student of Concern form to the UH Hilo Care Team.
If you are having a medical emergency and require immediate assistance, please call 911.
If you need non-emergency health services or assistance, please contact the Student Medical Services at (808) 932-7369.
Contact the Hawaiʻi County Civil Defense at (808) 935-0031.
See the UH Hilo Emergency Operations Plan and visit the Hawaiʻi County Civil Defense website.
Counseling Services main office (808) 932-7465 or 7465 from a campus phone.
Office hours are 7:45am - 4:30pm, Mondays through Fridays (except holidays). Emergencies or crises are handled as expediently as possible by an available counselor.
If you need assistance on an emergency or crisis basis outside of office hours:


text:
MB Madaera
Lost 31.7 lbs fat
Built 11.7 lbs muscle
Chris Madaera
Built 9 lbs muscle
Keelan Parham
Lost 30 lbs fat
Built 4 lbs muscle
Bob Marchesello
Lost 23.55 lbs fat
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Lost 26 lbs fat
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Lost 41 lbs fat
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Determine the Length of Your Workouts
Evaluate Your Progress
Keep Warm-Up in Perspective
"Doing more exercise with less intensity,"
Arthur Jones believes, "has all but
destroyed the actual great value
of weight training. Something
must be done . . . and quickly."
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This is one of 93 photos of Andy McCutcheon that are used in The New High-Intensity Training to illustrate the recommended exercises.
To find out more about McCutcheon and his training, click here.
Mission Statement
H.I.T. Acceptable Use Policy
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HIT Tip 4: Determine the Length of Your Workouts.
Generally, there are nine parts of your body that are most often subject to training. These nine are listed as follows:
• Hips • Thighs • Calves • Shoulders • Back • Chest • Arms • Waist • Neck
• Hips • Thighs • Calves • Shoulders • Back • Chest • Arms • Waist • Neck
HIT dictates that you train your entire body one to three times per week (as opposed to splitting it up), and that you perform one to two exercises per body part. Usually the total number of exercises per workout varies from seven to twelve, with more advanced trainees requiring fewer exercises because of their higher levels of strength.
Each exercise should be performed slowly and smoothly for one set of approximately 8 to 12 repetitions. Each set should take from 40 to 90 seconds. There should be no longer than 60 seconds rest between exercises. Thus, a typical workout should take between 20 and 30 minutes.
The next time you?re in the gym, time your workout from the beginning of the first exercise to the end of the last exercise. How long did it take?
If you?re like most bodybuilders, it probably took you well over an hour, or perhaps two hours. And that two hours may have been for only half of your body.
If you are spending more than 30 minutes in the gym actually training, then you are not getting anything close to maximum growth stimulation. You are probably not working hard enough and you are probably resting too long between sets.
Focus your attention on the fact that your exercise is going to become briefer as a result of your increased intensity.
Ellington Darden
California, USA
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Iowa, USA
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Armed Forces - Europe
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text:
1 x 10ml Totally Wicked Vape JuiceTotally Wicked
The Totally Wicked Titan product line is available in 18mg (1.8% nicotine), 11mg (1.1% nicotine), and 6mg (0.6% nicotine).
1 x 10ml Totally Wicked Vape Juice
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All orders placed before 2pm MST Mon-Fri are dispatched the same business day. Excludes public holidays. All quoted delivery times are estimates. Please click on each shipping option at checkout to see time estimates and pricing.
Generally, USPS delivery takes 3–5 business days for standard (First-Class) shipping, 2–3 business days for Priority, and 1–2 business days for Priority Express.
UPS shipping is available for all states that we ship to except New York. Transit time varies from 1 to 4 days based on distance from Boulder, Colorado. Click here and enter ZIP code 80301 to see approximate transit times based on your location.
Please note that UPS does not ship to P.O. boxes or APO/FPO/DPO addresses.
Shipping Restrictions
We do not ship outside of the United States of America, and we cannot ship to Arkansas, Maine, Vermont or Utah due to state law.
USPS ships to PO boxes, but UPS does not.
Orders that weigh over 16 oz. cannot be shipped via USPS First-Class.
Adult Signature-required Deliveries
We ship adult signature required via USPS Priority Mail to Delaware, Iowa, Massachusetts, Nevada, North Dakota, South Carolina, Texas, Virginia, and Wyoming. You will need to sign for your package when it is delivered. The shipping cost for Priority Mail orders under $80 that require adult signature upon delivery is $9.99. Free standard shipping is not available to states that require an adult signature upon delivery.
Please note that all UPS shipments require an adult signature upon delivery.
Haven't gotten your order yet? Click here for order tracking information.
In order to limit reselling of bulk orders, Electric Tobacconist has implemented a purchase limit for the following categories:
This limit is over a 30-day period. Essentially, the limit resets 30 days after reaching a category limit.
So, for example, if you purchased 40 pods on 01/01/2020, you would need to wait until 02/01/2020 (30 days) to order again.
You could have also purchased 2 vape kits on 01/01/2020, 2 on 01/14/2020, and 1 on 01/21/2020 for example. You would have to wait until 02/01/2020 to order again.
In accordance with state law, we require adult signature upon delivery in Delaware, Iowa, Massachusetts, Nevada, North Dakota, South Carolina, Texas, Virginia, and Wyoming.
Please note that all orders shipped via UPS Ground require an adult signature upon delivery.
Unfortunately, free USPS standard shipping is not available to states that require an adult signature upon delivery. Additionally, shipping costs $9.99 for Priority Mail orders under $80 that require an adult signature upon delivery.
No. However, we do ship to APO addresses. Additionally, our sister UK vape website ships to most European countries.
You must be at least 21 years of age to order from Electric Tobacconist. We do not ship to Maine, Vermont, Arkansas or Utah.
If you order by 2pm MST, Monday through Friday, your order will dispatch the same day. Our standard USPS delivery option means your order should arrive in 2-5 business days. If you need it quicker, you can select a faster delivery method (eg USPS Priority) in the checkout under "Shipping Method".
UPS shipping is available for all states that we ship to except New York. Transit time varies from 1 to 4 days based on distance from Boulder, Colorado. Click here and enter ZIP code 80301 to see approximate transit times based on your location.
Totally Wicked USA shut down its online and brick & mortar operations in the United States in April of 2019, though they continue to manufacture their incredible lines of e-liquid and quality hardware. Electric Tobacconist jumped at the opportunity to partner with Totally Wicked USA and strive to give Totally Wicked's loyal customers the same amount of attention, care, and experience they are accustomed to. As Totally Wicked's exclusive US retailer, we will continue to carry a variety of Totally Wicked liquids from the Patriot, Titan, and Red Label ranges, as well as Totally Wicked hardware.
PG (propylene glycol) and VG (vegetable glycerin) are diluants used in e-liquid. VG is more viscous (thicker), slightly sweet and produces larger vapor clouds. PG carries flavor better.
Most first time users tend to lean towards higher PG content, but the more experience you gain with vaping, you might want to try higher VG content, especially when using box mods.
Note: we do not recommend using VG content of higher than 50% with plastic clearomizers as they may cause cracking.
This e-liquid in this product is 70% propylene glycol (PG) and 30% vegetable glycerin
Due to FDA guidelines, all e-liquid products must be labeled with, "Warning: This product contains nicotine". Rest assured, any 0mg/ml product does not contain nicotine, despite this warning.
Unfortunately, no. As of March 1st, 2020, New Jersey prohibits the use of any coupons, price reduction promotions, or price rebates for vapor products.
While you may see discounts and promotions on the website, these sadly do not apply to New Jersey customers.
No, unfortunately, UPS Ground shipping is currently unavailable for orders being shipped to New York.


text:
Please contact our office within 72 hours of receiving the product that you wish to return. All returns must be authorized by our office. Some of our manufacturers may charge a restocking fee. Most unused, unopened, stocked inventory may be returned in its original packaging without a restocking charge.
In the unlikely event a custom order needs to be returned, please contact our office for assistance. Custom orders are not subject to our general return policy and will be addressed on an individual basis.
All shipping/transportation/delivery charges are the responsibility of the
returning party.
No personal information collected about our customers or subscribers (including email
addresses) will be used for any purposes other than those for which the information was
We will never sell, distribute, or otherwise disclose any customer information to
third parties. Period.


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Blind Item #1
This actor is really close to A+ list and in his own country probably is. Here he is a solid A lister who only does movies and has been in this space lots of times before. Over the weekend, our foreign born actor with A++ list name recognition was in a bar in Edinburgh where he met two women. After a while he became bored with small talk and drinking and decided to take one of them back to his place to have sex. How did they leave the bar? He grabbed her by her breast and led her out of the bar. Yes, let me clarify. He grabbed her breast with his hand and got a firm hold and then dragged her out of the bar just like that and she went willingly. Oh, and this will be revealed for sure. Have to call this guy out.


text:
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Ram 1500 Car Loan Refinance at Anderson Chrysler Dodge Jeep Ram near Crystal Lake 60014
Here at Anderson Chrysler Dodge Jeep Ram near Crystal Lake 60014, we offer Ram 1500 refinance car loans. There are several reasons as to why many people opt to refinance, in addition to the many advantages and benefits it can bring. Contact us today to learn more about refinancing your car loan; our Finance Department is always happy to help!
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Poster - 100lb Gloss
Please call 800.959.8365 to place an order for this product.
11" x 17"
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Greensboro Library, NCHome Menu
English Literacy Class
English for Speakers of Other Languages
- Date: -
- Location: Glenwood Branch Library
1901 W. Florida Street
Greensboro, North Carolina 27403
- Cost
This class is for ESOL beginning adult learners. Students will learn to strengthen and expand knowledge of complex sentence structure, grammar, easy reading, writing, and speaking skills. Please call 336-297-5000 to register or for more information.Return to previous page >>


text:
Product name :Stomach cancer test tissue array, with normal control tissue, including TNM, clinical stage and pathology grade, 12 cases 24 cores
Catalog Number :T012
Quantity :
Price :88.00 €
Pay now with :
Supplier :US Biomax
|Microarray Panel:
|Stomach carcinoma test tissue microarray, containing 10 cases of tumor, 2 normal tissue, duplicate cores per case, divided into two identical 12 core arrays for testing different antibody dilutions
|8 cols × 3 rows
|Core Diameter:
|1.5 mm
|5 µm
|Quality Control:
|Anti-Actin confirmed
|Routine histology procedures including ImmunoHistochemistry (IHC) and In Situ Hybridization (ISH), protocols which can be found on our support page.
|Unless specified, all TMA slides are not coated with extra layer of paraffin (tissue cores can be easily seen on the glass), so there is no need to bake, can be directly put into xylene for de-paraffin procedure.
*Tissue Microarray Slide Types:
- Unstained: unstained paraffin tissue microarray slide.
- Trial: tissue microarray trial slide. 10% - 25% of cores missing. Good for titrating antibody dilution and experiment conditions. Limited numbers are available, to 2 per item per order. Test slides (tissue arrays with catalog numbers ending with 241 or 242) are recommended as a substitute.
- H & E: Hematoxylin and Eosin stained tissue microarray slide.


text:
|1 Year
|Assembly Time
|120 mins
|Footend Height
|94 cm
|Headboard Height
|122 cm
|Bed Width
|147 cm
|Bed Length
|206 cm
|Bed Base
|Wooden Slats
|Julian Bowen
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Julian Bowen offers the most luxurious bed frames with or without a mattress, as well as long lasting bedroom furniture. Julian Bowen prides itself on their years of experience which allows them to offer the most luxurious of beds and bedroom furniture, at surprisingly affordable prices.
We are an official online retailer of Julian Bowen beds and furniture, so all our Julian Bowen products come with a manufacturer's warranty.
All our Julian Bowen beds, mattresses and bedroom furniture come via a professional White Glove delivery service. This specialised white glove delivery service is carried out by dedicated courier staff who will call you to arrange a delivery date, they will also call you up to 1 hour before attempting your delivery on the scheduled day of delivery. The friendly uniformed driver will unload your goods carefully and will deliver them to a room of choice. Your Julian Bowen bedroom furniture, beds and mattresses will be delivered within 10-21 working days.
Orders received before 1pm will be processed on the same working day for delivery within 10-21 days. Due to the specialised and trusted courier offering a sophisticated delivery service, there is no need for us to supply you with tracking details. The courier will contact you directly and they will keep you informed of the delivery process. We offer free UK delivery (mainland only) on all our Julian Bowen beds, mattresses and bedroom furniture. Some post code areas, such as Scottish Highlands, will incur an additional delivery fee which will be added during checkout. Such postcode areas are exempt from 7-10 day delivery, as our courier do not provide this service. Such postcodes include AB, FK, KW, IV, KY, DD, PA and PH. Deliveries to these areas will take approximately 14-28 working days. We apologise for any inconvenience this may cause you.
Unfortunately, as our White Glove delivery service does not operate on weekends, Bed SOS does not offer a Saturday Delivery service on any Julian Bowen products. Due to the deliveries being arranged around the couriers route as to when they are in your postcode area, we cannot offer the option of Pre-scheduled delivery date. We apologise for any inconvenience this may cause you. Please be assured that the white glove delivery service costs Bed SOS a premium and is therefore more reliable than other delivery services.
You must inspect all goods for any damages prior to signing for the goods. Deliveries are usually between 8am-6pm. It is the customers responsibility to ensure that either the customer or a nominated adult is home during this timescale to take delivery of the goods.
BedSOS is open.
BED SOS is still trading. Goods are leaving us on a daily basis via our third-party couriers. Please be aware that there may be some disruption with the couriers, due to the couriers having minimal staff on hand due to the Government Lockdown and individuals self-isolating during these unprecedented times. Unfortunately this is beyond our control.
Our Sales Team and support services are fully set up to work from home wherever possible. We will be able to answer incoming emails to our email@example.com email address, and we endeavour to do our utmost to provide the same high level of service.
We will continue to monitor Government and NHS advice during this COVID-19 pandemic.
Bed SOS would like to take this time to thank all our customers for their support during these difficult times.
Thank you for your understanding and support in advance.
Under the Consumer Contracts (Information, Cancellation and Additional Charges) Regulations 2013 you have the right to return your order for a full refund within 14 days of purchase. As we offer FREE delivery, it is the customer's responsibility to return non suitable goods to Bed SOS at a cost to themselves. We recommend using trusted couriers such as TNT, XDP or Tuffnells, full contact details can be obtained from email@example.com The returned items must be sufficiently packaged to withstand handling during your chosen couriers service. Due to hygiene reasons, mattresses must be unused and still have all original packaging and tags attached.
Bed SOS has the right to deduct value off a customers returned goods should the item(s) reach us damaged or show signs to suggest that they have been used beyond what is acceptable to decide if you are happy with the goods. We recommend that all customers returning goods use a service which insures them to the full value of the goods being returned.
Bed SOS will arrange the return/ replacement of goods found to be faulty free of charge, if reported to ourselves as being damaged within 10 days of receipt. Photographic evidence of the fault must be supplied to firstname.lastname@example.com Bed SOS will not be held responsible for customers returning goods who have initially stated they want to cancel and then state they are faulty, this does not affect your Statutory Rights. This will result in replacements part(s) being sent only. To discuss the matter further please contact us on 0800 009 6107 Mon - Fri 9am - 5pm.
If you are not completely satisfied with your purchase, simply return the item or items to us in their original condition within 14 days of receipt. We will issue a full refund upon receipt of any returned items, or exchange the item(s) for a different size/ colour if preferred. Please note, we can not refund delivery fees incurred by the customer upon return of any items, as depending on where the consumer lives and the delivery option chosen during checkout, we offer FREE Delivery on all goods purchased via Bed SOS. Bed SOS has the right to offer a diminished value on goods which are received back to ourselves in a damaged state. We advise all customers to use some form of courier which offers insurance, as this should cover you against transit damage. Bed SOS also has the right to offer a diminished value on goods which have been deemed as used beyond what is acceptable to decide if you are happy with the goods. Due to the T&C's with our suppliers, Bed SOS cannot accept returns on mattresses, specifically memory foam/ foam mattresses, if they have been removed from their vacuum sealed packaging. This is due to health and hygiene reasons, and will result in diminished value being offered to you, our customer. It is important that all vacuum sealed goods are inspected for the correct size information, as Bed SOS will not be held responsible for the return of opened goods which the customer has ordered through their own error.
• The item is your responsibility until it reaches us. For your own protection, we recommend that you send the parcel using a delivery service that insures you for the value of the goods.
• The cost of returning the item to us is your responsibility.
• The parcel can take up to 7 working days to be returned to our Warehouse, unless sent via special courier.
Here at BEDSOS® we use the latest payment & security software allowing our customers to experience a happy and smooth shopping experience.
We accept all major cards, credit cards, Paypal, Amazon Checkout and Pay4Later. Simply select your relevant payment option at checkout.
Unfortunately, we are unable to take over the phone payments. This is due to the 3D secure checks which the bank put in place to protect ourselves and our customers from a potential act of online fraud. For further information regarding this please call us on 0800 009 6107 Monday – Friday 9am to 5pm and we would be happy to discuss this in depth over the telephone.
For further information on our various payment options please email us or call us.


text:
- Nibbi MSS
- Coleção
Small group of correspondence.
Nibbi, Alessandra
27 resultados com objetos digitais Mostrar resultados com objetos digitais
Small group of correspondence.
Nibbi, Alessandra
Rosalind Frances Emily Paget Collection
Coloured drawings of hieroglyphs.
Paget, Rosalind Frances Emily
John Phillips Slides Collection
2,500 slides (35 mm), taken in 1962 and 1963 at sites in Egypt and in the Egyptian Museum, Cairo. Folder containing typewritten and handwritten lists/catalogues.
Phillips, John
1) Original measured perspective drawing of the tomb of Tutankhamun in the Valley of the Kings (1978) [corrected version].
2) Drawings of Eighteenth Dynasty tombs (incomplete original set and complete sets of photocopies/prints) (1980s).
3) Prints of the measured perspective drawing of the tomb of Tutankhamun in the Valley of the Kings (1978) [version with error].
4) Drawings of reconstructions of Egyptian boats (1993-4).
5) Models, plaster and resin molds and casts for plaque depicting scene from lid of box from the tomb of Tutankhamun (c. 1979) - the plaque is being sold in the British Museum shop since late 1990s (copyright issued in the late 1990s for a while, then reissued c. 2010).
6) Model, plaster and resin molds and casts for the mask of Tutankhamun (1978).
7) Models, plaster and resin molds and casts for small figurines (2 seated Isis suckling Horus, Bes amulet and scarab, 1970s).
8) Album of newspaper cuttings entitled "Illustrations", mostly from the Illustrated London News, bound in illustrated binding by Harold Parkinson (c. 1930).
9) Drawings of reconstructions of the Theban tomb-chapel of Nebamun, now in the British Museum in London, including sketches, tracings and a measured perspective acrylic (1997) [by Richard B. Parkinson].
Parkinson, Harold
Colin Christopher Walters Collection
Complete papers, including notes, drafts, lectures, annotated photocopies and notebooks.
Walters, Colin Christopher
Arthur Edward Pearse Brome Weigall Collection
Negatives, mainly connected with the tomb of Tutankhamun.
Weigall, Arthur Edward Pearse Brome
Henry Solomon Wellcome Collection
Journals, excavation records (Abu Geili, Jebel Moya, Saqadi, Dar el-Mek, etc.), indexes, maps, plans and photographs.
Wellcome, (Sir) Henry Solomon
Dodgson, Aquila
Complete papers, including manuscripts, photographs and digital files.
DuQuesne, Terence
Notes, negatives, transcripts, photographs and photostats of Coptic documents.
Kuhn, Karl-Heinz
Robert Elcum Horsfall Negatives Collection
Negatives of Egyptian sites and monuments taken before 1914.
Horsfall, (Capt) Robert Elcum
Arpag Mekhitarian Slides Collection
158 colour slides (35mm) of Theban Tomb scenes, made in Thebes in 1953-4.
Mekhitarian, Arpag
Photographs, tracings and squeezes of Dodecaschoenus graffiti.
Roeder, (Ernst) Günther
Fifteen notebooks on Ethiopic.
Harden, (Rt. Revd) John Mason
Notebook, and lithographs of some monuments by Madeley.
Lee, John
Griffith Institute Lantern Slides Collection
Lantern slides from various sources.
Sem título
Griffith Institute Postcards Collection
Antique and modern postcards from various sources.
Griffith Institute Squeeze Collection
Paper squeezes from various sources.
Stobart, (Revd) Henry
Griffith Institute Film and Media Collection
Film and media from various sources.
Robert Walter Sloley Collection
Various notes.
Sloley, Robert Walter
Album of watercolours, drawings, and tracings of Egypt produced close to the end of Lloyd's life. It is a folio volume (33 x 44 cm, pages 32.7 x 43 cm), containing 135 watercolours and drawings (74 of which are full-page and 1 folding) and 10 tracings (8 folding). They are all mounted and most are captioned either on the image itself or on the mount. Captions on the mount seem to have been added when the album was put together after Lloyd's death. The album also contains a loose watercolour, a loose drawing and a loose lithographed portrait of George Lloyd by Prisse d'Avennes, who published it in his Oriental Album: Characters, Costumes and Modes of Life in the Valley of the Nile (1848). One of the drawings in the album is a pencil portrait of Lloyd sketched by Prince A. Soltykoff.
Lloyd, George
John Frederick Dewey Collection
C. 10,500 colour slides (35mm) mostly of Egypt (temples, tombs and other ancient monuments), but also of different countries in the Middle East (Syria, Persia, Turkey, etc.) as well as the Far East. They are cross-referenced with an extensive card index and cover a period from the early 1980s until the early 2000s. They consist of images from virtually every site in the Egyptian Delta region and Nile Valley.
Dewey, John Frederick
Amelia Ann Blanford Edwards Collection
-Album of drawings and watercolours made during Amelia Edwards's visit to the Dolomites.
-Album of drawings and watercolours made during her visit to Egypt in 1873-4.
-Album with 26 pencil sketches and one watercolour entitled 'Small Egyptian Scraps, 1874', made during same visit to Egypt.
-Album, notebook, packet containing an assortment of drawings, letters, and notes, another packet contains copies of deeds and accounts issued following her death.
-Two watercolours of the temples on Philae island, assumed to have been painted by A. Edwards.
-First edition of Amelia B. Edwards, Pharaohs, Fellahs and Explorers (1891); Kate Griffith's (née Bradbury) copy, with handwritten dedication to 'My Katie - Novr. 1891 - A.B.E'.
Edwards, Amelia Ann Blanford


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About our fabric
All our fabrics and interiors are painstakingly sourced with durability and comfort in mind, and pass twelve independent British Standard tests (BS 2543) before we authorise them for sale.
Our delivery service
We deliver direct from our South Wales factories to your home using our fleet of 7.5 tonne lorries.
We deliver to any address on the UK Mainland or to a port of your choosing for onward shipment to any of the islands or even abroad. All deliveries are undertaken by a two man delivery team so all you have to do is open the door and, if you're feeling really sociable, put the kettle on.
Our delivery team will position the furniture in the room of your choosing, take away all packaging if you want them to and allow you to inspect your furniture before signing for safe receipt. They are fully house trained and extremely skilled in getting sofas into the tightest of spots.
Moulded Foam Seat Cushions
Characteristics: firm, supportive, sit-on feel
Look: tailored, linear, structured


text:
My personal favourite at the moment is... Anub'Arak.
Why? Because he's a really sad story who is often just cast aside.
While he's extremely loyal and helpful to Arthas during The Frozen Throne, there's much to prove that he's just another sad slave under the Lich King.
In "Path to Damnation", he meets with Kel'Thuzad, the soon to be damned mage and councilmember of The Six in the Kirin tor. The following dialogue occurs:
Anub'arak: The nerubian race, yes. Then the master came. As his influence spread, we made war upon him, foolishly believing we stood a chance. Many of us were slain and raised into undeath. In life I was a king. Today I am a crypt lord.
Kel'Thuzad: In return for immortality, you agreed to serve him. Remarkable.
Anub'arak: 'Agreed' implies choice.
This, strengthened by his dying words in Azjol-Nerub: "Ahhh... RAAAAAGH! Never thought... I would be free of him...", to me, says he's just another slave under Arthas and Ner'Zhul. The sad part is how few realize this. People who are partially interested in the lore tend to think of him as "just another evil right hand of the Lich King", and even his own people hate him for the betrayal they think he was capable of resisiting. He is, after all, dubbed "The Traitor King".
To me, it feels like being killed in Azjol-Nerub was a great blessing upon Anub'Araks storyline. It's the only peace he has found at soul since the Spider War.
Your turn :3


text:
Harriet Compston wrote this irresistible line about the English dancer Georgiana Cavendish, a direct descendent of her namesake, Georgiana, Duchess of Devonshire, in the January 2009 issue of Tatler:
"Georgiana's escape is big trashy novels, which she cuts in half with a bread knife to make them easier to read."
Read it and weep, librarians and second-hand booksellers. You're not going to get those novels for your Friends sales and sidewalk tables.
© 2009 Janice Harayda. All rights reserved.


text:
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Master Bedroom Floor Plans (with Ensuite) On Pinterest …
Ensuite Bathroom Design Floor Plans,ensuite bathroom design floor plans,Master bedroom floor plans (with ensuite) on Pinterest …
ensuite bathroom design floor plans – Small bathrooms may assume like a difficult architecture assignment to yield on; however, these spaces may acquaint a able architecture claiming to add to your plate. Creating a anatomic and storage-friendly ablution may be just what your home needs.
image ensuite bathroom design floor plans description :
post title : Ensuite Bathroom Design Floor Plans
post date : April 16, 2018
author : Rita Schroeder
image source : www.pinterest.com/houseplanshelp/master-bedroom-floor-plans-with-ensuite/
For added ample attending you charge to use added ablaze colors. The absolute best is white color, it can accord you best results. Try to abstain aphotic colors they can "close" the space. Set bigger mirror because it can visually access the space. At the end accomplish abiding the ablution to be able-bodied lit because the poor lighting creates activity of attenuated space.
For a lot of of us (especially those of us who are not religious), the ablution is the abutting we will get to accepting an accustomed ritual space. We use it to absolve and adorn ourselves and to accomplish added health-related rituals. So doesn't it deserve the aforementioned array of absorption as a church, temple or shrine?
Instead of bathtub that would advance to bisected a room, congenital battery that you can ample out an angle. It can be just a battery or battery allowance with battery tub. In contempo years, baths with shower-room became a hit, because they absorb beneath amplitude than the approved ones. A bottle aperture of the battery berth can creates an apparition of added amplitude because it allows you to see both ends of the room. If it is possible, use mini WC-shell and that way you will save added space. For accumulator you charge to use shelves from the attic to the ceiling, area you can adjournment all all-important things.
Just click download link in many Resolutions at the end of this sentence and you will be redirected on direct image file, and then you must right click on image and select "Save image as". 236 × 120
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If you found any images copyrighted to yours, please contact us and we will remove it. We don't intend to display any copyright protected images.We hope you can find what you need here. We always effort to show a picture with HD resolution or at least with perfect images. Master Bedroom Floor Plans (with Ensuite) On Pinterest … can be beneficial inspiration for those who seek an image according specific categories; you can find it in this site. Finally all pictures we have been displayed in this site will inspire you all..


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Prices and availability are subject to change from the time and date that you print this page.
For Customer Service, please call(888) 780-5873
Rocklin Large Planter
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Return/Exchange Policy
What you need to know.
At Crate and Barrel, we stand behind the quality of our products and want every purchase to be a positive experience. We will do our best to make things right if you are not satisfied.
Non-Furniture Items
We will accept returns and exchanges of non-furniture items if they are returned within 90 days of customer receipt, a valid proof of purchase is provided, and items are returned in good condition (unused and unwashed). Live botanicals cannot be returned or exchanged.
Stock Furniture, Rugs and Rug Pads
If you are not satisfied with your Stock furniture, rug or rug pad purchase upon receipt, you must contact us within 7 days of your delivery or pickup to arrange a return. The item must be returned within 30 days of delivery or pickup. All items will be inspected upon return. A restocking fee will apply, starting at 25% of the purchase price, for any items not returned in good condition or outside the stated time frame. For orders placed online, please call us at 888.780.5873. For orders placed through a store, please contact the store where the sale originated.
Additional Colors and Frames, Custom Furniture, and Personalization
These special-order items are made to your individual specifications; therefore, we cannot accept returns or exchanges. We require a 50% deposit for all Additional Color, Additional Frame and Custom order merchandise. Once you place your order for Additional Color, Additional Frame or Custom merchandise, you have 48 hours to change or cancel. After 48 hours, your 50% deposit is nonrefundable. The 48-hour time frame begins as soon as you place your order and confirm your fabric selection in a store or online. For Personalization orders, the full amount is required at time of purchase; personalized or monogrammed items cannot be returned, exchanged or canceled at any time and are nonrefundable.


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In point of fact, Islamists, like many Hispanic political activists (think: La Raza), are statists... Islamists, like many Hispanic activists, are the vanguard of a different culture that they passionately believe is superior to the culture of individual liberty.
There is no single-issue quick-fix to the challenge of ushering them into the Republican coalition. Rather, there is a choice to be made: either convince them that they are wrong, meaning make the unapologetic case for liberty and limited government; or fundamentally change who you are, meaning accommodate their statism.Translation: Mooslims and Messicans can't get into this party till they prove their Americanism to Andrew C. McCarthy! I'm beginning to think John Judis and Ruy Teixeira were right.


text:
Nobody deals with the really low-level stuff any more; I learned it from UNIX Gurus in the 90s. I was really lucky to have met some real experts, and was stupid not to have better understood the opportunity to pick their brains.
Write to a file
$ echo foo > file
Append to a file
$ echo foo >> file
Read from a file (1)
$ cat < file
Read from a file (2)$ cat file
Read lines from a file$ while read f > do > echo LINE: $f > done < file $