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The set of possible input values may be infinitely large, and may possibly be continuous and therefore uncountable such as the set of all real numbers , or all real numbers within some limited range. The set of possible output values may be finite or countably infinite. For example, vector quantization is the application of quantization to multi-dimensional vector-valued input data.
An analog-to-digital converter ADC can be modeled as two processes: sampling and quantization. Sampling converts a time-varying voltage signal into a discrete-time signal , a sequence of real numbers.
Quantization replaces each real number with an approximation from a finite set of discrete values. Most commonly, these discrete values are represented as fixed-point words. Though any number of quantization levels is possible, common word-lengths are 8-bit levels , bit 65, levels and bit Quantizing a sequence of numbers produces a sequence of quantization errors which is sometimes modeled as an additive random signal called quantization noise because of its stochastic behavior.
The more levels a quantizer uses, the lower is its quantization noise power. Rate�distortion optimized quantization is encountered in source coding for lossy data compression algorithms, where the purpose is to manage distortion within the limits of the bit rate supported by a communication channel or storage medium.
The analysis of quantization in this context involves studying the amount of data typically measured in digits or bits or bit rate that is used to represent the output of the quantizer, and studying the loss of precision that is introduced by the quantization process which is referred to as the distortion.
Most uniform quantizers for signed input data can be classified as being of one of two types: mid-riser and mid-tread. The terminology is based on what happens in the region around the value 0, and uses the analogy of viewing the input-output function of the quantizer as a stairway. Mid-tread quantizers have a zero-valued reconstruction level corresponding to a tread of a stairway , while mid-riser quantizers have a zero-valued classification threshold corresponding to a riser of a stairway.
Mid-tread quantization involves rounding. The formulas for mid-tread uniform quantization are provided in the previous section. Mid-riser quantization involves truncation. The input-output formula for a mid-riser uniform quantizer is given by:. Note that mid-riser uniform quantizers do not have a zero output value � their minimum output magnitude is half the step size.
In contrast, mid-tread quantizers do have a zero output level. For some applications, having a zero output signal representation may be a necessity. In general, a mid-riser or mid-tread quantizer may not actually be a uniform quantizer � i. The distinguishing characteristic of a mid-riser quantizer is that it has a classification threshold value that is exactly zero, and the distinguishing characteristic of a mid-tread quantizer is that is it has a reconstruction value that is exactly zero.
A dead-zone quantizer is a type of mid-tread quantizer with symmetric behavior around 0. The region around the zero output value of such a quantizer is referred to as the dead zone or deadband.
The dead zone can sometimes serve the same purpose as a noise gate or squelch function. Especially for compression applications, the dead-zone may be given a different width than that for the other steps. The general reconstruction rule for such a dead-zone quantizer is given by. A very commonly used special case e. In this case, the dead-zone quantizer is also a uniform quantizer, since the central dead-zone of this quantizer has the same width as all of its other steps, and all of its reconstruction values are equally spaced as well.
A common assumption for the analysis of quantization error is that it affects a signal processing system in a similar manner to that of additive white noise � having negligible correlation with the signal and an approximately flat power spectral density.
Additive noise behavior is not always a valid assumption. Quantization error for quantizers defined as described here is deterministically related to the signal and not entirely independent of it. Thus, periodic signals can create periodic quantization noise.
And in some cases it can even cause limit cycles to appear in digital signal processing systems. One way to ensure effective independence of the quantization error from the source signal is to perform dithered quantization sometimes with noise shaping , which involves adding random or pseudo-random noise to the signal prior to quantization. In the typical case, the original signal is much larger than one least significant bit LSB.
When this is the case, the quantization error is not significantly correlated with the signal, and has an approximately uniform distribution. In either case, the standard deviation, as a percentage of the full signal range, changes by a factor of 2 for each 1-bit change in the number of quantization bits. At lower amplitudes the quantization error becomes dependent on the input signal, resulting in distortion.
In order to make the quantization error independent of the input signal, the signal is dithered by adding noise to the signal. This slightly reduces signal to noise ratio, but can completely eliminate the distortion. Quantization noise is a model of quantization error introduced by quantization in the analog-to-digital conversion ADC. It is a rounding error between the analog input voltage to the ADC and the output digitized value.
The noise is non-linear and signal-dependent. It can be modelled in several different ways. Moncef, N. ACI Materials Journal, 98 5 , Hover K. Concrete International. Ippei, M. Optimization Example. Thesis PhD. Department Nagabhushana and Sharada-bai. H Use of crushed rock of Architecture, Graduate School of Engineering, University powder as replacement of fine aggregate in mortar and con- of Tokyo, Japan.
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Structural Engineering Research Center. Chennai, India. Osama S. Kasperkiewicz J. Spreadsheet Package. Peter C. Khajehzadeh M and Eslami M. Gravitational search al- Education and Federal Highway Administration Office of gorithm for optimization of retaining structures.
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Tayfun, U. A New Approach to Deter- x18 number of available concrete test sample results mination of Compressive Strength of Fly Ash Concrete using x19 coefficient of variation in percentage Fuzzy Logic. Journal of Scientific and Industrial Research, x20 age at testing of concrete in days 65, Tesfamariam, S.
Journal of Materials in Civil Engineering, 19 7 , 23 surface condition of combined aggregates Teychenne D. Investigating the x use water absorption percentage of fine aggregate 26 of high performance concrete in partially prestressed beams x27 free surface moisture percentage of coarse aggregate and optimization of partially prestressed ratio. Indian Journal of Science and Technology, 5 7 , � Related Papers. Design of normal concrete mixes BRE. By Nor Hazurina Othman.
By HonEy ArAin. Chapter 22 Concrete mix design. By Hai Nguyen Tuong. Silica fume conrete. By nandish sajjan. By Natasya Intania. Download pdf. Whether they actually do will be left to future studies. For other contrasts vs. The formula most consistently rated as beautiful average rating of 0.
Other highly rated equations included the Pythagorean identity, the identity between exponential and trigonometric functions derivable from Euler's formula for complex analysis, and the Cauchy-Riemann equations Data Sheet 1: EquationsForm.
Formulae commonly rated as neutral included Euler's formula for polyhedral triangulation, the Gauss Bonnet theorem and a formulation of the Spectral theorem Data Sheet 1: EquationsForm. Low rated equations included Riemann's functional equation, the smallest number expressible as the sum of two cubes in two different ways, and an example of an exact sequence where the image of one morphism equals the kernel of the next Data Sheet 1: EquationsForm.
Post-scan understanding ratings. After scanning, subjects rated each equation according to their comprehension of the equation, from 0 no comprehension whatsoever to 3 Profound understanding. An excel file containing raw behavioral data is provided as Data Sheet 3: BehavioralData. Table 1: Pre-scan beauty ratings for each equation by subject. Table 2: Scan-time equation numbers by subject, session and trial. Table 3: Scan-time beauty ratings for each equation by subject.
Table 4: Scan-time beauty ratings by subject, session, and trial. Table 5: Scan-time beauty ratings by subject�Session and experiment totals. Table 6: Post-scan understanding ratings by subject. Table 7: Post-scan understanding ratings by subject, session, and trial. Table 8: Post-scan understanding ratings by subject�Session and experiment totals.
The pre-scan beauty ratings were used to assemble the equations into three groups, one containing 20 low-rated, another 20 medium-rated, and a third 20 high-rated equations, individually for each subject. These three allocations were used to organize the sequence of equations viewed during each of the four scanning sessions so that each session contained 5 low-rated, 5 medium-rated, and 5 high-rated equations.
Each subject then re-rated the equations during the scan as Ugly, Neutral, or Beautiful. In an ideal case, each subject would identify 5 Ugly, 5 Neutral, and 5 Beautiful equations in each session.
In fact, this did not happen. Figure 2A shows the frequency distribution of pre-scan beauty ratings for all 15 subjects; it is positively skewed, indicating that more equations were rated as beautiful than ugly. This is reflected in the frequency distribution of the scan-time beauty ratings Figure 2B which, again, shows a bias for beautiful equations.
Figure 2C shows the relationship between pre-scan and scan-time beauty ratings. These infrequent departures are not of great concern providing there was still a reasonable ratio of Ugly: Neutral: Beautiful scan-time designations for each session, which was the case.
Ideally this ratio would always be but, due to the predominance of Beautiful over Ugly scan-time ratings, we twice recorded and for particular sessions see Table 5 in Data Sheet 3: BehavioralData. Other sessions in general showed more equable ratios and, even with an extreme ratio such as , a relationship between Neutral and Beautiful equations could still be established.
Figure 2. Summary of Behavioral data. Behavioral data scores summated over all 15 subjects. A Frequency distribution of pre-scan beauty ratings. B Frequency distribution of scan-time beauty ratings. C Pre-scan beauty ratings plotted against scan-time beauty ratings. D Frequency distribution of post-scan understanding ratings. E Post-scan understanding ratings plotted against scan-time beauty ratings. Numbers in brackets give the count for each group.
Area of each circle is proportional to the count for that group. The frequency distribution of post-scan understanding ratings is given in Figure 2D , which shows that more of the equations were well understood, as would be expected from a group of expert mathematicians.
In this case, departures from a fully correlated relationship allow us to separate out effects of beauty from those of understanding, so that, for example, in the well understood category 3 the ratio of Ugly: Neutral: Beautiful is In order to analyze scanning data with regard to understanding ratings we would ideally have equal ratios of the four understanding ratios 0, 1, 2, and 3 in each scanning session.
We occasionally find missing categories in some sessions such as but we could still establish a relationship when one category is missing in a particular session. Mathematical subjects were as well given four questions to answer, post-scan. One subject did not respond to this part of the questionnaire, leaving us with 14 subjects. They also showed a very sophisticated knowledge of mathematics, by specifying equations that they considered particularly beautiful which they had known , as well as by the regret expressed at not finding, in our list, equations that they consider especially beautiful.
We also tried to gauge the reaction of 12 non-mathematical subjects to viewing the same equations. This was, generally, an unsatisfactory exercise because many had had some, usually elementary, mathematical experience [up to GCSE General Certificate of Secondary Education level, commonly taken at ages 14�16]. Reflecting this, the majority indicated that they had no understanding of what the equations signified, rating them 0, although some gave positive beauty ratings to a minority of the equations.
Overall, of the equations distributed over 12 non-mathematical subjects, Given this, we hypothesized that, when such non-mathematical subjects gave a positive beauty rating to the equations, they were doing so on a formal basis, that is to say on how attractive the form of the equations was to them.
This hypothesis receives support from the contrast to reveal the parametric relationship between brain activity and understanding in mathematicians see Results. Results should that activity that was parametrically related to the declared intensity of the experience of mathematical beauty was confined to field A1 of mOFC Ishizu and Zeki, , where there was a significant difference in the BOLD signal when viewing equations rated as beautiful on the one hand and as neutral and ugly on the other Figure 3.
Even though our subjects were experts, with an understanding of the truths that the equations depict, we had nevertheless asked them to rate, post-scan, how well they had understood the formulae, on a scale of 0 no comprehension to 3 profound understanding Data Sheet 2: UnderstandingForm.
As described above, separate parametric modulators were used for understanding and beauty ratings in the SPM analysis. This allowed us to model both understanding and beauty effects and examine responses to one that could not be explained by the other, thus separating out the two faculties in neural terms. Hence, crucially, the parametrically related activity in mOFC Figure 3 was specifically driven by beauty ratings, after accounting for the effects of understanding.
Figure 3. A Second level parametric analysis derived from 15 subjects, to show parametric modulation by scan-time beauty rating after orthogonalization to understanding rating. The location and extent of the cluster is indicated by sections along the three principal axes through the two hot-spots, pinpointed with blue crosshairs and superimposed on an anatomical image which was averaged over all 15 subjects.
Overall, as can be seen in B , these locations were deactivated relative to baseline, i. B A separate categorical analysis, based on scan-time beauty ratings alone, was used to generate contrast estimates for the three categories Ugly, Neutral, and Beautiful vs. Baseline at each of the locations in A.
Previous studies have nevertheless shown a number of areas that are active when subjects undertake mathematical tasks see Arsalidou and Taylor, for a meta-analysis and we observed some activity which did not reach significance in three regions which previous studies of mathematical cognition had reported to be active see Table 1B.
One of these is located in the left angular gyrus, one in the middle temporal gyrus and one in the caudate nucleus.
Although they did not attain significance, we nevertheless document them here and leave it to future studies to ascertain their possible role in the experience of mathematical beauty. A categorical analysis of Beauty ratings vs. Baseline is less sophisticated than a parametric analysis in two respects, for reasons given in the methods section.
Nevertheless, we thought it useful to employ such an analysis to examine the parameter estimates for Ugly, Neutral and Beautiful vs. Baseline at the locations in mOFC identified as significant in the parametric study.
Figure 3B shows the parameter estimates for the three beauty categories vs. It is evident that, at both, overall activity within this area of deactivation was greater for Beautiful than for Neutral or Ugly stimuli.
In neither case is a linear relationship particularly evident, probably due to the inferior sensitivity of the categorical analysis, for the reasons given above. There are cluster-level significant activations in mOFC, the left angular gyrus and the left superior temporal sulcus. The relative activation within mOFC occurs within a region of de-activation relative to baseline see section Cortical de-activations when viewing mathematical formulae. As with the previous study of Kawabata and Zeki , parameter estimates show that it is a change in relative activity within a de-activated mOFC that correlates with the experience of mathematical beauty.
Table 2. These included sites implicated in a variety of relatively simple arithmetic calculations and problem solving Dehaene et al. That these areas should have been active when subjects view more complex formulae suggests that they are also recruited in tasks that go beyond relatively simple arithmetic calculations and involve more complex mathematical formulations.
Table 3. Activations and de-activations for the contrast all equations vs. The most interesting of these is in mOFC. A Conjunction-Null analysis Nichols et al. The overlap between the activation and de-activation suggests that there may be separate compartments or sub-systems within field A1 of mOFC whose activities correlate with general cognitive tasks on the one hand and the more specific experience of beauty on the other.
Figure 4. Conjunction of activations with beauty rating and de-activations with equations. De-activations are shown in red, overlapping the area revealed by the parametric rating, shown in yellow.
Numerals refer to MNI co-ordinates. As described in the methods, we undertook a second parametric analysis, with Beauty rating and Understanding rating as first and second parametric modulators, respectively, to isolate activations due to understanding alone. The result, shown in Figure 5 , is that a large extent of the occipital visual cortex, comprising many of its subdivisions, was less active for well-understood equations or, put another way, more active for less understood equations.
The significance of this is discussed below under Beauty and Understanding. Figure 5. A Second level parametric analysis derived from 15 subjects, to show parametric modulation by understanding rating after orthogonalization to scan-time beauty rating.
The location of each peak is indicated by sections along the three principal axes, pinpointed with blue crosshairs and superimposed on an anatomical image which was averaged over all 15 subjects. Overall, as can be seen in B , these locations were significantly active above baseline.
B A separate categorical analysis based on understanding ratings alone was used to generate contrast estimates for the four understanding categories U0, U1, U2, and U3 vs. Yet both can provoke the aesthetic emotion and arouse an experience of beauty, although neither all great art nor all great mathematical formulations do so. The experience of mathematical beauty, considered by Plato a , b to constitute the highest form of beauty, since it is derived from the intellect alone and is concerned with eternal and immutable truths, is also one of the most abstract emotional experiences.
Although we approached the experiment with diffidence, given the profoundly different sources for these different experiences, we were not surprised to find, because of similarities in the experience of beauty provoked by the different sources alluded to above, that the experience of mathematical beauty correlates with activity in the same brain area s , principally field A1 of mOFC, that are active during the experience of visual, musical, and moral beauty.
That the activity there is parametrically related to the declared intensity of the experience of beauty, whatever its source, answers affirmatively a critical question in the philosophy of aesthetics, namely whether aesthetic experiences can be quantified Gordon, Viewed in that light, the activity in a common area of the emotional brain that correlates with the experience of beauty derived from different sources merely mirrors neurobiologically the same powerful and emotional experience of beauty that mathematicians and artists alike have spoken of.
The mOFC is active in a variety of conditions, of which experiences relating to pleasure, reward and hedonic states are the most interesting in our context. The relationship of the experience of beauty to that of pleasure and reward has been commonly discussed in the philosophy of aesthetics, without a clear conclusion Gordon, This is perhaps not surprising, because the three merge into one another, without clear boundaries between them; neurologically, activity in mOFC correlates with all three experiences thus reflecting, perhaps, the difficulty of separating these experiences subjectively.
The imperfect distinction between the three is also reflected in the positive post-scan answers given by the mathematical subjects to the question whether they experienced pleasure, satisfaction or happiness when viewing equations that they had rated as beautiful.
The mOFC is a relatively large expanse of cortex with several cytocrachitectonic subdivisions, including BA 10, 11, 12, and 32 see Kringelbach, for a review as well as BA 24 which may more properly be considered as part of the rostral anterior cingulate cortex and BA
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