The final step would be to determine which of the row, 2-column UDs in these set of candidates optimizes a chosen measure of the uniformity of scatter of its points. For 28 additional details on the construction of UDs using the glp method, see Fang and Wang, ; Zhang et al. When the number of design points is a prime p and the dimensionality of the experimental region s are both large, Fang and Wang and Fang et al. That is, they showed how to use 1, h2 , However, by limiting the search to designs generated by this method, generating vectors that produce the best UDs might be excluded.

Figure 2. We have only 9 distinct generating vectors although the Euler function in Table 2. The reason for that is that some of the generators are simply a rotation of the other. In this dissertation, in addition to the proposed methods of selecting the best UDs, we are also proposing methods of reducing the number of distinct generating vectors significantly which is what happened in this example.

Details for the proposed methods will be presented in Chapter 4. As can be seen from the plots, the NT-nets with generators 34;1,13 and 34;1,5 appear to have more uniformly scattered points than the other NT-nets. Good Points Method Hua and Wang and Fang and Wang presented different methods of generating points that can be applied to the generation of a set of space-filling design points.

The set of n design points in C s obtained by a good point method have 29 Figure 2. The good points generated by these methods are called good point or gp sets. As an example of these good point methods, the gp set generated by the cyclotomic field CF method will be presented. It appears that the 34 design points generated for the best designs using the glp method Figure 2. Fang and Wang recommended using the gp set found by the CF method when the number of experimental factors s is large. The Halton H Set The Halton or H method is another method of constructing a low discrepancy set of design points and is used as a building block for the Hammersley method that also provides a set of design points with low discrepancy.

The detailed mathematical formulation of the H method is available in Fang and Wang and Hua and Wang That is, the dimension of the Halton or H-set is simply the generalization of the Van der Corput sequence using the first s prime numbers as the base. Thus, the H-set is suitably used when s is small Fang and Wang, It can be seen from Figure 2. Other Methods There are several other methods of generating uniformly scattered points.

These include the square root sequence method and the Haber sequence method Fang and Wang, The square root sequence method works by taking the fractional parts of the scaled square roots of the first s primes. Measures of uniformity, such as discrepancy, are much better for the square root sequence method than for the Haber sequence method.

The Haber method, results not presented here produces designs that have worst measures of uniformity in comparison to the other methods used in this study. In the previous sections, methods of constructing uniformly scattered good space filling designs have been presented and discussed. However, those methods on their own do not provide a quantifiable measure of uniformity. Thus, the degree of uniformity has to be measured and quantified so that one can directly compare different designs.

Different measures of uniformity have been adopted by different authors. For example, Hickernell a,b modified the star L2 discrepancy measure and obtained the star L2 centered discrepancy and the star L2 wrap-around discrepancy measures of uniformity. Hickernell also provided analytic formulas for computing both of these discrepancy measures.

Fang and Wang also used the star discrepancy in measuring uniformity while Borkowski and Piepel used distance-based measures of uniformity for highly constrained mixture designs. In this study two commonly-used classes of measures of uniformity will be studied. These are the discrepancy measures used to study number theoretic methods NTMs Fang and Wang, and the metric distance approach proposed by Borkowski and Piepel for highly constrained mixture designs.

Finally, modifications of the discrepancy measure used by Fang and Wang will be proposed and studied. Both the discrepancy measures and metric distance approach methods proposed in this dissertation will also be modified for applications to experimental regions other than the unit cube C s. Thus, Fn x equals the proportion of points in P that lie in the hyperrectangle formed by the origin and point x.

Hickernell provided analytic formulas for these two discrepancies that are easy to compute. However, because they are modifications of the Lp -discrepancy, both can yield biased underestimates of Dp Pn. The application of discrepancy measures to number theoretic methods for generating designs will now be discussed. Or, in other terms, it is a good space-filling design.

Conversely, a large discrepancy measure indicates a poor space-filling design. For an overview of discrepancy measures, see Fang and Wang, ; Fang et al. Figure 3. Consider the two inner and outer rectangles displayed in Figure 3. Then 0. Thus, using STRD to compare the uniformity of scatter for competing designs can potentially lead to the selection of an inferior design. The concept of dividing the experimental region into 2s hyper-rectangles appears, at first glance, to be similar to the approach used by Hickernell in the star L2 centered discrepancy.

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In this study, we remove this single center point restriction and permit the use of random evaluation points to form the hyper-rectangles. Thus, as the dimension of the experimental region C s increases, the computational cost for calculating a measure of discrepancy increases. Because the computational demands for calculating a measure of discrepancy increases with dimension, the distance-based measures used in Borkowski and Piepel are recommended alternatives to the discrepancy-based measures.

In business, the maximin criterion can be described as a plan for allocating shops that would ensure no 2 shops are very close to each other within a fixed spatial region. Conversely, the minimax criterion would ensure that remotely located customers have one or more shops to choose from within a reasonable distance Johnson et al. In short, maximin can be considered as maximizing the minimum gain profit where as minimax as minimizing the maximum loss. Recently, Borkowski and Piepel have proposed and used an alternate version of the maximin distance criterion and two other criteria for selecting the best space-filling designs for highly-constrained mixture designs.

They can be classified as distance-based criteria. The details of the distance-based approach are now presented. Let x1 , That is, the N distances between a point x in C S and each of the N points in X are calculated, and then the minimum of these N distances is retained. That is, d x, X is the distance between a point x in C s and the nearest design point in X. The following selection criteria are measures of uniformity for space-filing designs that were used by Borkowski and Piepel : 1. That is, for a large random sample of N points x1 , x2 ,.

Conversely, large values of these criteria indicate that a design is poor with respect to its space-filling properties. The hope is to find a design that is superior for multiple criteria and performs well across all criteria. Figures 2. The plots, however, do not identify which design has the most uniformly scattered set of points across methods and criteria. Table 3. In computing the measures of uniformity evaluation points with points in four equal sized square strata are used. The bold-faced values in Table 3. For the good lattice point glp method and for all the measures of uniformity except STRD, the best design has generator 34;1, This suggests that different designs may be best when considering different evaluation criteria.

As discussed earlier in Chapter 45 2, the Hammersley method produces designs with points that are more uniformly scattered than those generated by the Halton method based on discrepancy. This can be seen in Table 3. A comparison of the measures of uniformity among the 4 methods of constructing uniformly scattered design points suggests the glp method is the best, and this is consistent with the fact that the glp method is good for small s.

This is because the glp method has multiple candidate design generators for each design size with at least one generator leading to good coverage of the experimental region. For large s, the glp method produces excellent designs, but it is computationally expensive and often infeasible to implement. Note also that in Table 3. The CF gp method is ranked third and the Halton method is ranked fourth as can be seen from the values in Table 3.

This finding also agrees with what was presented in Fang and Wang Clearly, the glp method performs best as can be seen from Figure 3. Hence, the glp method generates points that more uniformly cover the unit square C 2 experimental region. As expected, a general decreasing trend is observed for all four methods. That is, as the number of design points increases, the magnitude of the evaluation criteria tends to decrease.

However, a decrease or near-constant change in the magnitude of the distance-based evaluation criterion is observed for all methods of generating designs except for the glp method. The reasons for the observed patterns are: 46 1. For the CF gp method the prime number generators of the best designs are often the same for the neighboring number of design points. Hence, the expectation is a negligible change of the evaluation criteria for the metric distance approach when the generators remain the same.

On the other hand, for the glp method, increasing the design size by a single point can either decrease or increase the magnitude of the evaluation criteria. The evaluation criteria are highly dependent on the generator and how well the points are scattered in the entire region. This is evident in the plots for the glp method in Figure 3. Ham Ham 20 30 40 Design Points 50 glp method after considering only the nonequivalent generators which significantly reduces the number of distinct generators.

Thus, it is not unusual to observe such distinct patterns between the CF gp method and the glp method. It is important to remember that a downward trend is not necessarily seen for neighboring design points, but rather a general downward trend across the set of all design sizes n. This is similar to the idea of augmented design. For example, as can be seen from Figure 3. The difference might partly be due to the fact that different sets of evaluation points will yield different measures of uniformity as well. What may also affect the discrepancy values is the implementation of stratified sampling 49 yielding better discrepancy estimates in this study.

That is, the proposed Modified Star Discrepancy MSTRD together with the stratification of evaluation points that fill the experimental region yield better results. As previously stated, due to the method of constructing uniform designs, the choice of the generators and how the design points are scattered on the entire experimental region, one cannot always expect a smooth, monotonically decreasing trend for all the methods and for all the evaluation criteria.

One observation from Figure 3. The Hammersley method is the second next best after the glp method in all the evaluation criteria except the maximum distance md criterion. Overall in C 2 , the glp method is found to be the best method for producing low discrepancy sets of design points. Hence, it is worth investigating the relative performance of the other 3 design generating methods in C 2 relative to the glp method.

That is, the relative efficiency of the methods should be computed and plotted for each evaluation criteria and the different number of design points using the glp method as the baseline. This allows the experimenter to make an easier and proper comparison of the methods. With the exception of the RE for the maximum distance md criterion where the CF gp method was second with the other evaluation criteria, the Hammersley method is found to be the second best after the glp method. This is consistent with what was previously observed from Figure 3.

A summary measure of the relative efficiency is the average of the evaluation criteria for each method relative to the average for glp method. For example, for the rmsd criterion, the ratio of the average rmsd from the CF gp method from each of the different design points to the average rmsd from the glp method, the result is 1.

This measure is called the average relative efficiency ARE. Ham 2 1. The ARE comparison for each method in each evaluation criterion is presented in Table 3. The boldfaced values in Table 3. For example, the ARE for the rmsd criterion is 1. For example, the rmsd ARE of 1. Similarly, the CF gp is, on average, 6. Hence, based on the majority of the evaluation criteria presented in Table 3. In this chapter, an extension to computational reduction in higher dimensions will be discussed and presented for the glp method. To circumvent the computational requirements when using the glp method, two approaches are proposed and discussed in the next two sections.

Two NT-nets generated by the glp method are defined to be equivalent if the points in one NT-net can be generated by a column permutation of the other.

### Computational Methods for Combinatorial and Number Theoretic Problems

Therefore, because of the invariance under rotations, two equivalent NT-nets generated by the glp method will have equal values for the uniformity measures considered in this research. Hence, only measures of uniformity for NT-nets generated by 1, hi and not 1, hj need to be computed.

To see this, note the following: 1. Hence, only measures of uniformity for NT-nets generated by 1, hi , hj and not hk , 1, hl need to be computed. Hence, only measures of uniformity for NT-nets generated by 1, hi , hj and not hm , hq , 1 need to be computed. Hence, using equivalence can significantly reduce the computational demands of finding the best UD using the glp method. Table 4. Thus, these measure only need to be computed for the NT-net generated by 1,3. Thus, these measures only need to be computed for the NT-net generated by 1,3,5. After a check of equivalence, the generators that are in the first row of Tables 4.

After rearranging the order of the relative primes, Figure 4. Figure 4. The uniform scatter nature of the design points in both Figures show the computational task reduction method makes sense geometrically. Geometrically, as can be seen in Figure 4. Even with computational reduction using equivalence, the number of generating vectors to be considered will become very large as the dimension increases and especially when the number of design points is prime.

For higher dimensions, an alternative approach to using equivalence to reduce computations is considered and is now presented. The proposed method is called the projection method and is defined by the following procedure: 1. Find the best glp generating vectors 1, hi for C 2 or, when available, find the best 1, hi , hj for C 3. These vectors are easy to find because of the minimal computation required for low dimensions. Project the best UDs generated in Step 1 to the next higher dimension.

This is accomplished by creating a new set of generating vectors by adding an additional h value to the existing best generating vectors while retaining the sequence of 59 increasing values in the generating vectors. Retain the vectors that generated UDs having the smallest measure of uniformity. Iterate the procedure in Step 2 until the desired dimension is reached. Retain the UD having the smallest measure of uniformity.

Adding 9 to 1, 7 to create generator 1, 7, 9 was the optimal choice. This figure illustrates why this approach works well. That is, for any design having good spacefilling properties in 3 dimensions, it must also have good space-filling properties in 2 dimensional projections. As can be seen in Table 4. The bold-faced values in Table 4. Note that the values of the measures of uniformity in Table 4. The equivalence approach and the projection method approach are both effective ways to reduce of the computational demands for finding very good, if not the best, designs generated using the glp method.

The projection method, in particular, is simple and computationally thrifty, and is, therefore, recommended for generating good designs in higher dimensions. The number of design points that fall in each of those 8 hyperrectangles will be counted and the 8 proportions relative to the cube C 3 will be measured. Then each of these proportions is compared to the corresponding proportion of the volume of the hyperrectangles to the volume of C 3. The results are presented in Table 4. For the glp method there are candidate generators, but as seen in Table 4. Computational reduction using the equivalence method cut the required calculations by Thus, only results for the 35 nonequivalent generators are presented in Table 4.

The best generators are those that correspond to the bold-faced values in Table 4. For the glp method, 34; 1, 11, 27 is the best generators for the distance-based rmsd and ad criteria and the best generators based on discrepancy criteria STRD and MSTRD are 34; 1, 13, 25 and 34; 1, 9, 21 , respectively. This shows that different criteria can lead to different designs. For the glp method, the 34 design points generated by 34; 1, 11, 27 and by 34; 1, 9, 21 are displayed in Figure 4.

The 34 design points from Halton and Hammersley methods are displayed in Figure 4. This can be seen from the bold faced values in Table 4. The CF method has the lowest measure of uniformity after the glp method followed by the Hammersley and the Halton methods for the distance-based measures. However, in Chapter 3, Table 3. The space-filling nature of the designs can be clearly observed in Figures 4.

In general, however, the differences in the measures of uniformity between the two methods is very small. In the field of response surface methodology, extensive research exists for designs in spherical design regions. For example, model-based designs like the popular spherical central composite designs Box and Wilson, , the Box-Behnken designs Box and Behnken, , and the hybrid designs Roquemore, , are designs for which spherical design regions are assumed. However, despite the points of Qn being uniformly scattered in C 2 , this transformation does not preserve the uniformity of the scatter of the design points in B 2.

To achieve uniformity in B 2 a different 67 transformation is needed. Figure 5. However, note that the 8 glp designs in Figure 5. In this dissertation the measures of uniformity developed are invariant under 90o rotations in C 2 and, hence, there is no reason to have two or more equivalent generating vectors as candidate generators.

Details of the proposed methods that identify the equivalent and non-equivalent generators were presented in Chapter 4. Those that are in the first row of Table 5. In this dissertation the candidates or UD generating vectors in C s and B s are the same in the first stage. In other words, the first step is to transform the design points from the non-equivalent generators in C s to B s and identify the best UD generators using the measures of uniformities and then the search for the best UDs generators is restricted to the ones identified in the first step and their equivalent generators.

That is, for example, if the best glp has generator 1,2 for the rmsd criterion after transforming into B 2 , in step 2, it will then be compared to the glp with its corresponding equivalent generator 1,6 , and 70 then the best design generator that has smallest rmsd criteria is reported as the final best generator in B 2. In this subsection we present methods of measuring the uniformity of design points in B 2.

Analogously, let Pn be the set of points of Qn transformed into B 2 using 5. Thus, D P in 5. The supremum is estimated by taking the maximum of the absolute values over random evaluation points in B 2. The evaluation points are a stratified random sample of points in C 2 which are then transformed into B 2 using the above transformations. As alternatives to discrepancy measures, the rmsd, ad, and md distance criteria do not require converting the random evaluation points xi , yi into angles and radii. The 8 pairs of plots in Figure 5. Table 5. A total of evaluation points in B 2 were used.

The best measures of uniformity from the non-equivalent generators are shown by the bold-faced figures in Table 5. That is, after transformation from C 2 to B 2 , the glp generator 1,13 is found to be the best for the three distance-based criteria whereas the glp generator 1,15 is the best for the new discrepancy measure. Now these best generators have to be compared with their corresponding equivalent generators.

That is, 1,21 and 1,25 are the corresponding equivalent generators for 1,13 and 1,15 , respectively, and their measures of uniformity are presented in Table 5. Thus, the bold-faced values in Table 5. The best UDs generated by 1,21 and 1,25 are shown in Figure 5. Generator 1,3 1,5 1,9 1,11 1,13 1,15 1,27 1,33 Measures rmsd 0.

For the CF method and the Halton and Hammersley methods and the glp method the designs are plotted in Figures 5. In the glp method after transformation from C 2 to B 2 , the design generators selected as the best are different from those selected in C 2 meaning what is best in C 2 is not necessarily best in B 2. This is the main reason why the best non-equivalent generators have to be compared to their corresponding equivalent generators.

Looking at the magnitude of the evaluation criteria from Table 5. Is it always true that the rank will be preserved for any number of design points after transformation from C 2 to B 2?

Like the comparisons in C 2 in Figure 3. In order to investigate this further as in C 2 , in B 2 we also need to consider plotting the relative efficiency RE of each method for each evaluation criterion. Since the glp method is expected to be the best, the way RE is defined is the same as in C 2 except that now we have only 4 evaluation criteria: rmsd, ad, md, and Discrepancy D Pn.

Ham 0. Ham ad 0. Ham 50 CF glp Hal. Thus, this indicates that the performance rank of the methods in C 2 observed before transformation is preserved in B 2. As previously seen, the UDs constructed using the glp method 77 are the best. However, the number of candidate generating sets increases as the number of design points and dimensions increase, and thus it is computationally expensive. Then, the spherical coordinate transformation given by 78 Figure 5.

Ham 1. The reason is that the arctan function does not identify the quadrant of the inverse tangent whereas the arctan 2 uses the sign of x1 and x2 to determine the quadrant and thus gives the four-quadrant inverse tangent of x1 and x2 Aguilera and Aguila and Matlab A New Discrepancy Measure Of Uniformity For B 3 No statistical literature was found that addresses how discrepancy for uniformly scattered sets of design points can be measured for the set of transformed points in B 3. In this section, a new discrepancy measure is proposed and is applied to sets of points in B 3.

Although it is simple to find the area of the random sector in B 2 , finding the volume of a three-dimensional sector in B 3 is much more complicated. It depends 80 Figure 5. Thus, even though one can visualize a 3D sector in the interior of B 3 , there is no general closed-form formula to compute its volume because the height is not easily defined. Thus, a new alternative measure of discrepancy for B 3 is proposed. Unlike discrepancy in B 2 , the new discrepancy for B 3 is defined in terms of the proportion of points in a varying volume formed by a random ball cap, or more simply, a random cap.

In Figure 5. It is determined by the direction of the axis in which slicing is done. Before defining a new measure of discrepancy for designs in B 3 , we must first define discrepancy in terms of the proportion of design points in an upper or lower cap. Let Pn be the set of design points in B 3 formed by applying the spherical coordinate transformation in 5.

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The random points are stratified random points in C 3 and then transformed to B 3. At first different rotation combinations were considered but it was computationally expensive and hence we decided to use the 36 different rotation combinations. The running time was min and hence this was the main reason why the 36 angle rotation is considered. Note that the volume of the lower cap is subtracted from the volume of B 3 in 5. However, discrepancies defined in upper and lower caps or their complementary regions yield the same result. Thus, we only need to compute a total of 3 different discrepancies for each evaluation point.

The same procedure applies when finding the best UD in B 3 using any of the three distance-based criteria rmsd, ad, or md. The best measures of uniformity from the non-equivalent glp generators are shown by the bold-faced values. That is, 1,13,25 , 1,11,27 , 1,3,25 , and 1,3,11 are the best generators based on the rmsd, md, ad and the discrepancy criteria, respectively. However, these 84 best generators have to be compared with their corresponding equivalent generators. Comparison of the measures of uniformity is done among the equivalent glp generators shown in Table 5.

This is because 1,13,25 is equivalent to 1,15,21 and 1,15,25 , 1,11,27 is equivalent to 1,21,31 and 1,13,29 , and 1,3,11 is equivalent to 1,15,23 and 1,25, The measures of uniformities are presented in Table 5. Generator 1,3,5 1,3,7 1,3,9 1,3,11 1,3,13 1,3,15 1,3,19 1,3,21 1,3,25 1,3,27 1,3,29 1,3,31 1,3,33 1,5,7 1,5,9 1,5,11 1,5,19 1,5,21 rmsd 0.

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The boldfaced values indicate the best UDs generators. That means, the values appearing in Table 5. Note that if there is no remainder after dividing the number 85 Generator 1,13,25 1,15,21 1,15,25 1,11,27 1,21,31 1,13,29 1,3,25 1,23,31 1,11,25 1,3,11 1,15,23 1,25,31 Measures of Uniformity rmsd ad 0. For example, the first row of Table 5. Similarly, the md criterion for 1,13,25 is 0. Similarly, the bold faced values in Table 5. However, the pair of rotation angles are different for the distance-based rmsd and 86 ad and discrepancy D Pn criteria as shown in the last row of Table 5.

The best measures of uniformity for all the methods are shown by the bold faced values in Table 5.

### Number theoretic methods in designing experiments

The rank order of the methods in C 3 in Table 4. Based on rmsd, ad, and D Pn the rank order of the methods as shown in Table 4. Note that even though the glp method is best by all measures of uniformity, it is computationally expensive. The run-time for the Matlab codes for the glp method is about minutes, about 20 minutes for the CF gp method and about 6 minutes for the combined Halton and the Hammersley methods.

If the computational task reduction method was not introduced, then the estimated running time would have tripled to approximately 7. These times are hardware dependent and it may vary depending on the speed of the computer. In this research, a laptop that has a processor of 2. The built in functions plot3, scatter3 and surf of Matlab are used to plot Figures 5. Find an NT-net in the three-dimensional unit cube C 3. That is, find Pn. Compute the 3 distance-based rmsd,ad and md criteria for the design points from the generator for each rotation.

The maxima for each of the rotation- 90 generated distance criterion values from the 3 primary axes are saved, and the maximum of the 36 maxima is reported for each generator. Repeat steps for all candidate generators for each method. Finally, the generator that has the smallest measure of uniformity is considered to be the best for each method.

For future research discrepancy may be computed for regions other than cap such as spherical wedge. For example, in the building construction industry, concrete is made by mixing sand, water and cement. A second example is the formulation of a fruit juice blend, such as by mixing pineapple and orange juices. The overall aim of a mixture experiment is to search for a blended product or mixture that is cost effective, has superior quality, and is more desirable than a single ingredient product.

Similarly, the strength quality of the concrete depends on the proportions of sand, water and cement present.

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The quality and property of the blended juice depends on the proportion of each of the fruits present. These examples highlight why a mixture experiment is a special type of response surface experiment. Specifically, it is the proportion of the input variables ingredients or components that affect the response rather than the amounts of the ingredients in the mixture.

Excellent extensive reviews and discussion about mixture experiments can be found in Cornell and Smith Geometrically, the experimental region for the 2 and 3 component mixture experiments are presented in Figure 6. The fruit juice example is a 2 component mixture experiment with x1 denoting the proportion of orange juice and x2 denoting the proportion of pineapple juice.

For 92 Figure 6. For a 3 component mixture experiment of fruit juice, the pure blends occur at the 3 vertices of the simplex 1, 0, 0 , 0, 1, 0 and 0, 0, 1. A binary blend has one of the 3 components occur at a point on one of the three sides of the equilateral triangle, and the ternary blends that has a positive proportion for each of the 3 components occur in the interior of the triangle. If we count by threes, there is a remainder 2; if we count by fives, there is a remainder 3; if we count by sevens, there is a remainder 2.

Find the number of things. Answer : Method : If we count by threes and there is a remainder 2, put down If we count by fives and there is a remainder 3, put down If we count by sevens and there is a remainder 2, put down Add them to obtain and subtract to get the answer. If we count by threes and there is a remainder 1, put down If we count by fives and there is a remainder 1, put down If we count by sevens and there is a remainder 1, put down When [a number] exceeds , the result is obtained by subtracting If the gestation period is 9 months, determine the sex of the unborn child.

Answer : Male. Method : Put down 49, add the gestation period and subtract the age. From the remainder take away 1 representing the heaven, 2 the earth, 3 the man, 4 the four seasons, 5 the five phases, 6 the six pitch-pipes, 7 the seven stars [of the Dipper], 8 the eight winds, and 9 the nine divisions [of China under Yu the Great]. If the remainder is odd, [the sex] is male and if the remainder is even, [the sex] is female. Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods Apostol n.

Vinogradov's main attraction consists in its set of problems, which quickly lead to Vinogradov's own research interests; the text itself is very basic and close to minimal. Other popular first introductions are:. From Wikipedia, the free encyclopedia. Not to be confused with Numerology. Branch of pure mathematics. Further information: Ancient Greek mathematics.

Further information: Mathematics in medieval Islam. Main article: Analytic number theory. Main article: Algebraic number theory. Main article: Diophantine geometry. Main article: Probabilistic number theory. Main articles: Arithmetic combinatorics and Additive number theory. Main article: Computational number theory. This section needs expansion with: Modern applications of Number theory. You can help by adding to it. March Heath had to explain: "By arithmetic, Plato meant, not arithmetic in our sense, but the science which considers numbers in themselves, in other words, what we mean by the Theory of Numbers.

In , Davenport still had to specify that he meant The Higher Arithmetic. Hardy and Wright wrote in the introduction to An Introduction to the Theory of Numbers : "We proposed at one time to change [the title] to An introduction to arithmetic , a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book. This is controversial.

See Plimpton Robson's article is written polemically Robson , p. This is the last problem in Sunzi's otherwise matter-of-fact treatise. The same was not true in medieval times—whether in the West or the Arab-speaking world—due in part to the importance given to them by the Neopythagorean and hence mystical Nicomachus ca. See van der Waerden , Ch.

This notation is actually much later than Fermat's; it first appears in section 1 of Gauss 's Disquisitiones Arithmeticae. Fermat's little theorem is a consequence of the fact that the order of an element of a group divides the order of the group. The modern proof would have been within Fermat's means and was indeed given later by Euler , even though the modern concept of a group came long after Fermat or Euler. Weil goes on to say that Fermat would have recognised that Bachet's argument is essentially Euclid's algorithm.

There were already some recognisable features of professional practice , viz. Matters started to shift in the late 17th century Weil , p. Euler was offered a position at this last one in ; he accepted, arriving in St. Petersburg in Weil , p. In this context, the term amateur usually applied to Goldbach is well-defined and makes some sense: he has been described as a man of letters who earned a living as a spy Truesdell , p.

Notice, however, that Goldbach published some works on mathematics and sometimes held academic positions. The Galois group of an extension tells us many of its crucial properties. This is, in effect, a set of two equations on four variables, since both the real and the imaginary part on each side must match. As a result, we get a surface two-dimensional in four-dimensional space. After we choose a convenient hyperplane on which to project the surface meaning that, say, we choose to ignore the coordinate a , we can plot the resulting projection, which is a surface in ordinary three-dimensional space.

It then becomes clear that the result is a torus , loosely speaking, the surface of a doughnut somewhat stretched. A doughnut has one hole; hence the genus is 1. The term takiltum is problematic. Robson prefers the rendering "The holding-square of the diagonal from which 1 is torn out, so that the short side comes up Robson , p. Van der Waerden gives both the modern formula and what amounts to the form preferred by Robson. On Thales, see Eudemus ap. Proclus, Proclus was using a work by Eudemus of Rhodes now lost , the Catalogue of Geometers.

See also introduction, Morrow , p. Gifford — Book 10". See also Clark , pp. See also the preface in Sachau cited in Smith , pp. This was more so in number theory than in other areas remark in Mahoney , p. Bachet's own proofs were "ludicrously clumsy" Weil , p. The initial subjects of Fermat's correspondence included divisors "aliquot parts" and many subjects outside number theory; see the list in the letter from Fermat to Roberval, II, pp.

Numbers and Measurements. Encyclopaedia Britannica. II, p. I, pp. Euler was generous in giving credit to others Varadarajan , p. Early signs of self-consciousness are present already in letters by Fermat: thus his remarks on what number theory is, and how "Diophantus's work [ In Felix E. Browder ed. Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. American Mathematical Society. Andrews, American Mathematical Soc. Retrieved Apostol, Tom M. Introduction to analytic number theory.

Undergraduate Texts in Mathematics. Mathematical Reviews MathSciNet. Abteilung B:Studien in German. A History of Mathematics 2nd ed. New York: Wiley. University of Chicago Press. Colebrooke, Henry Thomas London: J. Davenport, Harold ; Montgomery, Hugh L. Multiplicative Number Theory. Graduate texts in mathematics. Edwards, Harold M. November Mathematics Magazine. Graduate Texts in Mathematics. Springer Verlag. Fermat, Pierre de Varia Opera Mathematica in French and Latin. Toulouse: Joannis Pech.

Historia Mathematica. In Christianidis, J. Classics in the History of Greek Mathematics. Berlin: Kluwer Springer. Disquisitiones Arithmeticae. Goldfeld, Dorian M. Goldstein, Catherine ; Schappacher, Norbert In Goldstein, C. The Shaping of Arithmetic after C. Gauss's "Disquisitiones Arithmeticae". Granville, Andrew The Princeton Companion to Mathematics.

Princeton University Press. Porphyry ; Guthrie, K. Life of Pythagoras. Alpine, New Jersey: Platonist Press. Calculus by Yuen Fong Book 10 editions published between and in English and Italian and held by WorldCat member libraries worldwide. China business culture : strategies for success by Karen Wang Book 14 editions published between and in English and held by 35 WorldCat member libraries worldwide. The Goldbach conjecture by Yuan Wang Book 11 editions published between and in English and Undetermined and held by 30 WorldCat member libraries worldwide.

Hua Luogeng de shu xue sheng ya by Yuan Wang Book 5 editions published between and in Chinese and held by 20 WorldCat member libraries worldwide Ben shu shi cong shu xue xue shu yan jiu jiao du chan shu hua luo geng yi sheng de shu xue gong zuo, Dao lu tan suo ji qi si xiang yu fang fa de zhong yao zhu zuo. Zai ken ding hua luo geng cheng gong de tian fen, Ke ku deng yin su wai, Ben shu zhe zhong chan shu le hua luo geng zai shu xue yan jiu shang chao fan de xue shu pin ge he te shu de fang fa yu ji qiao, Cong er wei quan mian le jie hua luo geng de shu xue si xiang ti gong le zhong yao de can kao.

International symposium in Memory of Hua Loo Keng by Sheng Gong 8 editions published between and in English and Undetermined and held by 15 WorldCat member libraries worldwide The international symposium on number theory and analysis in memory of the late famous Chinese mathematician Prof. The symposium was carried out in two separate sections: number theory and analysis. The distinguished list of main speakers and the contents of these two vol umes reflect the high level of the mathematical activity throughout the seven days.

Wang, G. The discussions among the mathematicians were always in a warm atmosphere. Our thanks go to professors Chern, Subbarao and Yau for their contributions to these proceedings. Xin yi by Yuanfeng Wang Book 3 editions published between and in Chinese and held by 11 WorldCat member libraries worldwide. Lecture notes in contemporary mathematics Book 2 editions published in in English and held by 10 WorldCat member libraries worldwide. Shu lun zai jin si fen xi zhong de ying yong by Luogeng Hua Book 4 editions published in in Chinese and held by 9 WorldCat member libraries worldwide.

Hua luo geng wen ji by Luogeng Hua Book 2 editions published in in Chinese and held by 7 WorldCat member libraries worldwide Ben shu jing xuan, Fan yi le hua luo geng zai ge ge shi qi shu lun fang mian de dai biao xing lun wen, Zhei xie lun wen shi guan yu hua lin wen ti, Tarry wen ti, Zhi shu he gu ji, Vinogradov zhong zhi ding li, Zheng shu fen chai, Pell fang cheng de zui xiao jie, Zui xiao yuan gen, Yuan nei ge dian deng zhong yao shu lun wen ti de yan jiu. Gong neng xing shi pin tan shui hua he wu by li ya de li si Bi Book 2 editions published in in Chinese and held by 7 WorldCat member libraries worldwide Ben shu jie shao le gong neng yin zi tan shui hua he wu zuo le jie shao, Bao kuo qi hua xue, Wu li, Jia gong te xing, Sheng chan, Sheng li xiao ying, An quan xing ji guan li.

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