Testing of Aggregates Analysis

Testing of Aggregates Analysis Numerous test has been developed that test toughness and abrasion resistance and durability and soundness of aggregates. This report analyses the testing of Aggregates using three Main tests to analyse the degradation of aggregates so best performance is observed in construction, asphalt, concrete or any other field. The aggregates must be abrasion resistant and weather durable to provide good base in pavements for roads. Detailed description of these tests is provided with their respective results. The results are then examined to see which of the following three test are more accurate to check the durability and abrasion of the rocks. Based on the Laboratory results and the Literature reviews, Los Angeles Abrasion test results are used more than others. Although most of the DOTs and construction companies use all three test prior to use of the materials. Soundness test has poor repeatability so it is often not considered as primary test. This research was conducted at Englobe Corp laboratory located in Dartmouth, Nova Scotia. It was performed under the direction of Mr. Mark Downie. The laboratory supervision was provided by Mr. Daniel McMorran. The quality of the aggregates in the construction materials, asphalt concrete is determined by various tests out of which Los Angeles, Micro Deval and Soundness are most widely used in construction industry. Los Angles Abrasion test and Micro Deval test involve the spinning of aggregates in a close vessel where there are spun in a medium of water or air. The vessel is filled with contact charges (Iron sphere) for a specific amount of time. The analysis of particle degradation using mechanical test can be classified in two class, fragmentation and wearing. The sample that has more wide range of grain size ( e.g. 1250 gm of ) indicates fragmentation and has a well graded distribution curve whereas the sample that has poor range of grain size( 5000gm ofà ¢Ã¢â€š ¬Ã‚ ¦) indicates wearing with a bad graded distribution curve. Soundness test involves testing the durability of the aggregates using sodium sulphate or magnesium sulphate solution. Samples of different grain sizes are washed and dried and kept in salt solution for 16 hours and then kept in oven to dry. This cycle is repeated for seven days and then the sample is weighed to see the loss of sample. This test usually received poor rating for its inconsistent repeatability and correlations. Standard Testing methods Los Angeles Abrasion Test AASHTO T96 ( ASTM C131) Micro Deval Test AASHTO T327 (ASTM D6928) Sodium and Magnesium Sulfate Soundness AASHTO T 104 (ASTM C88) As Per American Standard of Testing Materials following pass-fail criteria were used: LA abrasion: Passed if loss à ¢Ã¢â‚¬ °Ã‚ ¤ 40% Micro Deval: Passed if loss à ¢Ã¢â‚¬ °Ã‚ ¤ 18% Sodium Sulphate Soundness: Passed if à ¢Ã¢â‚¬ °Ã‚ ¤12% If the aggregates passed the above criteria, then they are durable.   Los Angeles Abrasion test involves the test to measure the degradation of the aggregates by creating actions like impact, abrasion, grinding and constant wear and tear inside a rotating steel drum. The steel drum spins for a specific time with a specific number of steel balls of specific weight to create an abrasion of aggregates. The number of steel balls and the amount of aggregates put in the steel drum depend upon the grading of the test sample. The steel sphere carries the aggregates and the steel balls creating a grinding effect and then drops it to the other side making a crushing effect. This cycle is repeated again and after certain number of revolution the sample is sieved too see the amount retained from the degradation and too see the percentage loss. Apparatus Los Angeles Machine with wall thickness of at least 12mm. The inside diameter should be 711 ±5mm and the length of 508 ±5mm. The rotating drum should be closed from all ends and should be set at a rotating speed of 30 ±3 rpm. ( ASTM C131) Sieve with 1.7mm (No.12) passing. An accurate scale with no more than 0.1% error of the test load. The charges or the steel balls. The number of steel balls used in the test depend on the gradation of the sample to be tested. The steel balls should have diameter between 46.038mm and 47.625, with mass between 400g and 440g each. A constant weight check should always be performed on the charges because this test is very aggressive and could lead in loss of weight of the charges. Table 1: Mass of Steel balls for LA Abrasion Test Grading Number of Spheres Mass of the charges, gm A 12 5000 ±25 B 11 4580 ±25 C 8 3330 ±20 D 6 2500 ±15 Table 2: Grading of the Test Sample for LA Abrasion Test Sieve Sizes (Square Opening) Mass of the aggregates, g Passing Through Screen Retained on Screen Grading A B C D 37.5 mm (11/2 in.) 25.0 mm (1 in.) 1250 ±25 25.0 mm (1 in.) 19.0 mm (3/4 in.) 1250 ±25 19.0 mm (3/4 in.) 12.5 mm (1/2 in.) 1250 ±25 2500 ±10 12.5 mm (1/2 in.) 9.5 mm (3/8 in.) 1250 ±25 2500 ±10 9.5 mm (3/8 in.) 6.3 mm (1/4 in.) 2500 ±10 6.3 mm (1/4 in.) 4.75 mm (No.4) 2500 ±10 4.75 mm (No.4) 2.36 mm (No.8) 5000 ±10 Total, g 5000 ±10 5000 ±10 5000 ±10 5000 ±10 Select the appropriate Grading according to the amount of aggregate available for test. It is recommended to go from higher to lower grade to gain accurate results. Wash and oven dry the sample at 110 ±5 °C (230 °F) to constant mass and then separate into individual sizes per their respective weights. Put the aggregates and the sample in the rotating drum. Close the Los Angeles Abrasion machine tightly and let it run for 500 revolutions at 30 to 33 rpm. After the drum stops take all the sample out and remove the steel balls from it. Now take the crushed aggregates and sieve it on 1.7 mm (No.12) sieve. Weigh the sample retained on the sieve and calculate the percentage loss. Micro Deval Micro-Deval is a test that involves measure of resistance of aggregates towards abrasion and test durability of the sample towards grinding of steel balls (ASTM D7428). The medium that is used here is water of room temperature. The sample and charges (steel balls) are kept in the Micro-Deval tank and then the apparatus is filled with water. The apparatus is rotated so that the aggregates undergo grinding and abrasion. The use of this test is mainly towards the aggregates that degrade more in presence of water than air. This test also gives a measure of how soft or shaley the sample is. The materials that give a high percentage loss degrade more during mixing or handling in industries. (ASTM D7428) Apparatus Micro-Deval Abrasion Tank with volume of 5.03 L and external diameter of 202mm and the internal height shall be from 170 mm to 177 mm. The stainless-steel tank comes with a rubber sealing to make it water-tight. The inner and outer surface of the tank should be smooth and ridge free. (ASTM D6928) Micro-Deval Abrasion machine is a rolling machine with an adjustable speed which rolls the tank at 100 ±5 rpm. Steel charges of diameter 9.5 ±.5mm are requires. The total mass of steel balls needed is 5000 ±5g. Sieves with 5mm and 1.25mm sizes are also required. An accurate scale with no more than 0.1% error of the test load. Table 3: Mass of Aggregates for Micro-Deval Test. Passing Retained Mass 20 mm 16 mm 375 g 16 mm 14 mm 375 g 14 mm 10 mm 750 g Take washed and oven dried sample so it loses the dust on it. Prepare a representative sample of 1500 ±5g put it in the Micro-Deval tank. Add 5000 ±5g of steel charges in it and 2.0 ±.05L of tap water in the Micro-Deval tank. Let this sit for 1 hour. After the sample, has been soaked tighten it up and put it on the Micro-Deval rolling machine to roll for 2 hours ±1 minute. After the machine stops rolling pour the sample on a stack of 5 mm and 1.25 mm sieve. Wash the remains of the sample in the tank on the sieve. Oven dry the sample at 110 ±5 °C and weigh it later. Calculate the percentage loss using the calculation sheet. Soundness Soundness test is a very crucial test in paving industries especially when making massive highways, bridges and dams. This test measure the amount of degradation caused by weathering freeze-thaw cycles. The aggregates that pass this test are more durable to be used and dont cause premature distress in pavements (http://www.pavementinteractive.org/article/durability-and-soundness/). The aggregates are kept in a sodium sulphate or a magnesium sulphate bath. The solutions is at a saturated state and causes salt crystals to be formed on the aggregates. This test is usually carried out for seven days which involves simultaneous wet-dry cycles. When the sample is submerged in the salt solution bath, salt crystals are formed in minute pores of the aggregates and causes internal forces that eventually lead crack in the aggregates. This gives us a replicated demonstration of how the substance will behave in natural habitat. This test has very poor repeatability so it is never considered as pr imary test to measure the degradation of the aggregates. Apparatus Sieves of different sizes- 5à ¢Ã‚ Ã¢â‚¬Å¾16 in., 3à ¢Ã‚ Ã¢â‚¬Å¾8 in., No. 50, 1à ¢Ã‚ Ã¢â‚¬Å¾2 in., 5à ¢Ã‚ Ã¢â‚¬Å¾8 in., No. 30, 3à ¢Ã‚ Ã¢â‚¬Å¾4 in., 1 in., No. 16, No. 8, No. 4. Metal baskets made of wire mesh or stainless steel that allows the aggregates to freely contact the solution and permit free drainage of the loss of sample. Temperature regulator to ensure that the temperature of the sulphate bath is constant at specified one. Balances with the accuracy of 0.1% are must for this test. Hydrometers are also needed for this test to measure the specific gravity within  ±0.001. Procedure Prepare the sodium sulphate solution that has specific gravity between 1.154 and 1.171. COARSE AGFREGATES Mass of the Sample Sieve Size Mass 2 in. (50 mm) 3000g 1.5 in. (37.5 mm) 2000g 1.0 in. 1000g 0.75 in. 500g 0.5 in. 670g 0.375 in. 330g No.4 300g Prepare Sample as per the table displayed above. The sample should be washed and dried at 110 ±5 °C. Mix the 2 inch and 1.5 inch retained material and place the 5000g sample in a container. Mix the 1 inch and 0.75 inch retained and place the 1500g sample in a separate container and mark the container by making a groove on it with a particular symbol so it does not get mixed up with the other containers. This way it is also more efficient to identify them when changing the cycles. Mix the 0.5 inch and the 0.375 inch retained samples and put the 1000g sample together in a container. After the test samples are ready place them in the solution prepared for 16 to 18 hours and then let them drain for 15 minutes. Let the sample oven dry at 110 ±5 °C for 4 hours and then let them cool down until they reach 20  °C to 25 °C. Again, immerse it in the solution and repeat this cycle 5 times. After the 5 cycles are done the aggregates are to be washed thoroughly so that all the salt on the surface is removed and then oven dried at 110 ±5  °C. Table 4: Sieve Sizes to be used to Measure Loss Aggregate Size Sieve Used >1.5 inch 1.25 inch 1.5 to 0.75 inch 5/8 inch 0.75 inch to 0.375 inch 5/16 inch 0.375 inch to No.4 No.5 Use the above given table to respective sieve the aggregates used in the test. Utmost care must be taken to sieve samples from each container separately. Take the sample that is retained on the sieve and weigh it to note it. The difference in the mass of the aggregates before and after the experiment gives us the amount lost due to the disintegration of the sample. To compare the results of the following three, test a study was examined in which 20 Lab results were considered. These tests were performed by Montana Department of Transportation either in the Montana State University soil laboratory or Montana Department of Transportation (MDT) Helena materials laboratory. The samples were obtained from various random pits and quarries across Montana by MDT personnel. (Western Transport Institute) To get a very good relation and a study between these test, 5 repeats were done on the Micro-Deval test and at least 3 repeats were done for L.A Abrasion test. This also provided a good study of the repeatability of the tests. There was no repeat test done on the soundness test since it has a very poor repeatability. Repeatability To analyse the repeatability of the tests, repeated Micro-Deval and Los Angeles Abrasion test were done on the same sample. The Coefficient of Variation was calculated to examine the variation in the test results for the same sample. There was no COV calculated for the Sodium Sulphate Soundness test because only one test result was provided by MDT. The Coefficient of variation is standardized measurement calculate by diving the standard deviation of a set of results by the average mean and then multiplying it by hundred to get a percentage value. By analyzing and comparing this value we can predict the repeatability of the test. If the COV is a lower number, then the test is less variable and hence it has a good repeatability. The COV calculated for L.A abrasion test came out to be 6.5% with standard deviation of 1.5 loss percentage. Similarly, the COV for Micro-Deval test came out to be 6.5% for a standard deviation of 0.7 percentage loss. Since both the Coefficient of variations are less than 10%, both the tests are considered to have good repeatability. Another evidence to support the repeatability of the test is that there is no significant difference in the COV of Micro-Deval and L.A abrasion which are 6.6% and 6.5% respectively. As per the comparative bar graph plotted below, majority of the coefficient of the variation fall between 5% to 15%. The COV of Lab number 861553 rocketed to 26.9% because the result was very small accounting to be 2.1% average loss. So, a small change in small result make a large COV. Figure 1: Graphical Representation of Coefficient of Variation for L.A Abrasion and Micro-Deval Tests. COMPARTITIVE RESULTS As per the specifications provided by American Standard for Testing Materials the aggregates are classified as durable if the loss percentage is less than the cut-off percentage and they are non-durable if the loss percentage of the aggregates is more than the cut-off percentage. The cut-off percentage that we have used for L.A Abrasion, Micro-Deval and Sodium Sulphate Soundness Tests are as follows: LA abrasion: Passed if loss à ¢Ã¢â‚¬ °Ã‚ ¤ 40% Micro Deval: Passed if loss à ¢Ã¢â‚¬ °Ã‚ ¤ 18% Sodium Sulphate Soundness: Passed if à ¢Ã¢â‚¬ °Ã‚ ¤12% To create a direct comparison in between these tests normalized value for each test is calculated. Normalized value is used for a direct comparison between Micro-Deval, L.A Abrasion and Sodium Sulphate Soundness test. Normalized value is average loss percentage divided by the cut-off for that test. (MDT paper) The ideal Normalized value is 1.0. If the calculated normalized value is greater than 1.0 it means that the test did not pass and the aggregate tested are not durable and if the value is less than 1.0 it means that the aggregates are durable and the test passed. To draw a direct comparison in between two, test a two-dimensional scattered graph is plotted with four quadrants. The top right (North-East) quadrant depicts the are where both the test failed and the aggregates are not durable. The top left (North-West) quadrant depicts the area where the test plotted on the X-axis passed but the one on Y-axis failed. The bottom right (South-East) quadrant indicates the area where the test plotted on the Y-axis passed but the test that was plotted on X-axis failed. The bottom left (South-West) quadrant depicts the region where both the tests passed and the aggregates are durable. The data points plotted in the top-right (NE) and bottom-left (SW) quadrants indicate that the tests are consistent as the aggregates were either durable for pass/pass or not durable for fail/fail. On the other hand, the data points plotted in the top-left (NW) and bottom-right (SE) quadrant indicate discontinuity and lack of coherence since one of the test would indicate that the aggregates passed the cut-off and are durable while the other would indicate that they didnt pass are not durable for use. L.A abrasion vs Micro-Deval. Figure 2. Graphical Representation of Comparison between Normalized Loss of L.A Abrasion and Micro-Deval Tests. The above shown graphical representation is the plot of comparison between the Micro-Deval and the L.A Abrasion test on 20 different samples that were tested by Montana Department of Transportation. The blue dotted line that runs at 45 ° along the centre of the graph indicates symmetry and a perfect correlation. The data points close to the line indicate a good co-relation between the test. There was only one result that had failed both the test and was considered to be non-durable which accounted 5% of the materials tested. There were five results (25% of the samples tested) that fall in the awkward category and had lack of coherence since, according to L.A abrasion test it passed as was considered durable but according to the Micro-Deval test it was considered to be non-durable. The relations between these two test is quite strong as 70% of the samples tested i.e. 15 out of 20 passed the test and fall in the bottom-left quadrant which indicates that the samples were durable accor ding to both the test. Sodium Sulphate Soundness Test Vs L.A Abrasion Test. Figure 3. Graphical Representation of Comparison between Normalized Loss of Sodium Sulphate Soundness and Micro-Deval Tests. Sodium Sulphate Soundness Test Vs Micro-Deval Test. Figure 4. Graphical Representation of Comparison between Normalized Loss of Sodium Sulphate Soundness and L.A Abrasion Tests. The co-op report guidelines suggest separating the Results and Discussion sections. I think that it is usually easier to present a discussion immediately after the results. But, you may choose to use the structure that makes the most sense for your report. Gregates typically encountered on Montana highway projects, and to determine if the MicroDeval test provides better, timelier, and more repeatable information about the quality of an aggregate than the Sodium Sulfate test. The laboratory testing program was structured to examine how well three aggregate durability test methods correlate for a sampling of Montana soils. Aggregate durability tests were conducted on 32 different soils using the Micro-Deval, L.A. Abrasion, and Sodium Sulfate tests. Multiple Micro-Deval and L.A. Abrasion tests were conducted on some of the soil samples to investigate the same-lab repeatability of the test methods. The methods differ in their treatment of the aggregate during testing; and consequently, each method produces a unique value of percent loss, which is used to distinguish between durable aggregate and non-durable aggregate. For the purposes of this study, the following percent loss pass-fail standards were used for each test: à ¢Ã¢â€š ¬Ã‚ ¢ Mic ro-Deval: passing (durable), if % loss à ¢Ã¢â‚¬ °Ã‚ ¤ 18%; à ¢Ã¢â€š ¬Ã‚ ¢ L.A. Abrasion: passing (durable), if % loss à ¢Ã¢â‚¬ °Ã‚ ¤ 40%; and à ¢Ã¢â€š ¬Ã‚ ¢ Sodium Sulfate: passing (durable), if % loss à ¢Ã¢â‚¬ °Ã‚ ¤ 12%. Because of the differences in percent loss criteria for each method, results from the suite of laboratory tests were normalized to facilitate direct comparisons between the three methods. Normalized results were obtained by taking the average percent loss for a particular soil and dividing it by the cutoff for that test. Table 16 summarizes the comparisons between each test using data collected during this study. Based on the metrics identified in the table, the MicroDeval and Sodium Sulfate tests had the best correlation, while the Micro-Deval/L.A. Abrasion and the L.A. Abrasion/Sodium Sulfate correlations were significant, but not as strong. Table 16. Summary Comparison of Test Methods Test Methods R2 Pass/Fail Agreement (%) Inconsistent Durability Determina tion* (%) M-D versus NaSO4 0.72 92.9 7.1 M-D versus L.A. 0.46 85.2 14.8 L.A. versus NaSO4 0.28 84.0 16.0 Perfect Correlation 1.0 100.0 0.0 *Note: Column 4 refers to the percentage of samples that passed one of the tests but failed the other test. This inconsistency is identified as a data point that plots in one of the cross-hatched zones identified in Figures 3, 4, and 5. Conclusions and Recommendations Western Transportation Institute 35 The percentages of inconsistent durability determinations (pass or fail inconsistencies) listed in column 4 of Table 16 are indicative of a discontinuity between tests and are probably the most important metric for the comparison study. In this case, one test characterized the material as durable aggregate, while the other test characterized the same material as non-durable aggregate. Qualitatively, the authors believe that an excellent correlation between two test methods is obtained when the percentage of inconsistent results is less than about 5%, values between 5 to 10% signify a good correlation, values between 10 to 20% signify a fair to poor correlation, and values above 20% signify a poor or unreliable correlation between tests. Multiple tests conducted on samples obtained from the same sources indicate similar values of same-lab repeatability for both the Micro-Deval and L.A. Abrasion tests. The coefficients of variation for the multiple tests were less than 10% for both methods. Considering the natural variability that occurs within an aggregate source, the measured variations were low, indicating good repeatability of the test methods. This conclusion has also been supported by others (Jayawickrama et al., 2006; Tarefder et al., 2003; and Hunt, 2001). Repeatability of the Sodium Sulfate test was not examined in this study. The relationship between Micro-Deval test results and field performance was not examined in this study; however, evaluations by Fowler et al. (2006), Rangaraju et al. (2005), Tarefder et al. (20 03) and Wu et al. (1998) indicate that Micro-Deval test results relate well with field performance. An excellent correlation between rutting performance and Micro-Deval test results were observed by White et al. (2006). They sug There are many reference styles available to choose from. It is not very critical which one you use, as long as you are consistent throughout the report. The APA reference style is an appropriate choice. You can quickly generation citations for your reference list using the online citation generator from RefME (RefME, 2016). Always review the text the generator has populated in the form for you as it is not always accurate. Entries in the reference list are sorted alphabetically. Some examples of common references for the reference list and in-text citations are shown below. Websites Robertson, J. (2016, August 8). Man solves Rubiks cube while free-falling. Retrieved August 9, 2016, from http://www.cbc.ca/news/canada/edmonton/learning-to-solve-a-rubik-s-cube-while-free-falling-1.3712116 RefME. (2016). FREE APA citation generator format. Retrieved July 30, 2016, from https://www.refme.com/ca/citation-generator/apa/ In-text Citation: (Robertson, 2016) In-text Citation: (RefME, 2016) Books Felder, R. M., Brent, R. (2016). Teaching and learning stem: A practical guide. United States: John Wiley Sons. In-line Citation: (Felder Brent, 2016) Journal Articles Lombardo, S. J. (2010). Teaching technical writing in a lab course in chemical engineering. Chemical Engineering Education, 44(1), 58-62. In-line Citation: (Lombardo, 2010)

The Mystery Behind Sudoku Mathematics Essay

The Mystery Behind Sudoku Mathematics Essay Puzzle games can be very enjoyable and is popular amongst kids as well as adults. Many of you may know the game Sudoku; where by the goal of the game is to fill in the remaining empty cells with each number from 1-9 appearing no more than once from each column, each row and each of the nine sub-grids. Sudoku is a type of logic-based numerical puzzle game that has a unique solution once completed. The most common form of a Sudoku is constructed as a 99 grid with nine 33 sub-grids and is primarily partially completed. Sudoku has become appealing among puzzle enthusiasts and involves complex thinking and practice. Available daily in newspapers, mobiles and many more, this addictive and brain-teasing puzzle game has become one of the most popular games to play since the time of the Rubiks cube. This dissertation discusses the mathematical side involved in Sudoku. There is no mathematics in actually solving a Sudoku but more of how it is used from a creators side. The 99 grid will be considered in the majority of the report; however a glimpse into other size grids will be discussed briefly also known as variants. Mathematicians have been questioning How many unique solutions are there in a Sudoku? Essentially meaning what are the possible ways of filling in an empty Sudoku grid so that each row, column and sub-grid contains the numbers 1 through 9. Your first thought of an answer may be a couple of thousands, but as you understand the concepts behind a Sudoku, you begin to grasp a whole new aspect. Combinatorics and permutation group theory are largely interwoven with analysing Sudoku. For that reason, I aim to explore these theories and understand how it applies to the methods of enumerating Sudoku grids. In particular I will be looking at Felgenhauer and Jarviss approach to enumerating all possible Sudoku grids where they employ several mathematical concepts. Furthermore I will uncover the importance of Latin squares and its use of constructing Sudokus. There are many constraints in regards to when are similar solutions considered different such as solutions of similar structure, symmetry etc. Preserving symmetries are known as relabeling symbols, band permutations, reflection, transposition and rotation. Burnsides Lemma theorem is one of their techniques in computing the number of essentially different solutions. Many difficult problems are of the type called nondeterministic-polynomial known as an NP-complete problem. This will direct me onto the debate on whether Sudoku is an NP-complete problem. Sudokus can take many forms and shapes. These are called Sudoku variants and consist of rectangular regions, Sudokus with a large region having no clues (numbers), an empty row, column or sub-grid and many more! Here I will research the logic behind irregular Sudokus as well as examining any occurring patterns or whether it has occurred by chance. 1.2 Latin squares and Sudoku Sudoku is also a special case of Latin squares. The Swiss mathematician, Leonhard Euler made many fundamental discoveries during 1782 including Latin squares. A Latin square is an N x N matrix where by a set of N characters are arranged such that each row and column contains one of each character. This is also in the case of a Sudoku, when complete, with an additional constraint that the nine sub-grids must hold the numbers 1-9. A reduction can be made to any Latin square by permuting the rows and columns. This arrangement is an aspect of combinatorics and is most commonly referred to as enumeration. Enumerative combinatorics is a classic area of Combinatorics and involves counting the number of infinite class of finite sets. Counting combinations and counting permutations are two of the most common forms. The number of valid Latin squares is known to be approximately 5.525 x 10 ²Ãƒ ¢Ã‚ Ã‚ ·. Write about Colbourns proof 1.3 Combinatorics and Permutation group theory Combinations and permutations have slightly different meaning. Combinations are the number of different ways of selecting n objects from a set but the order of events is not important. From a set of 3 objects, lets call these 1, 2 and 3. If for example I was asked to pick the number of ways of selecting 2 objects out of the 3, there would be three combinations 12, 23 and 13. 12 = 21 since the order of each pair is not important. A permutation on the other hand does consider the position. Therefore if I was to use the above example, there would be six permutations. A simpler way to calculate a larger set would be to use formula 1: Formula 1. = = Where is the combination formula, is the permutation formula, n is the total number of objects and r is the number to be arranged Both methods are one way of computing the number of possible Sudoku solutions and this will be looked at later in the report. Chapter 2 Enumerating possible Sudoku solutions 2.1 Distinct Sudoku solutions There are many approaches to enumerating possible Sudoku solutions. To enumerate every possible Sudoku solution, a Sudoku differs from another if they are not identical. Thus all solutions will be consider unless they are like for like. Felgenhauer and Jarvis was the first to enumerate the Sudoku grid solutions directly in 2005. There approach was to analyze the permutations of the top row used in valid solutions. Their knowledge of the complexity in computing the number of Latin squares has made them aware of how they should go about getting an answer with fewer computations. Hence by using relabeling this could shorten the number of counts. To make it easier, each sub-grid is given an abbreviation seen in figure 3. B1 B2 B3 B4 B5 B6 B7 B8 B9 Figure 1. Abbreviated sub-grid with top band (Felgenhauer and Jarvis, 2006) Firstly they consider every solution to filling in blocks B2, B3, given that B1 is in standard form. To work out every possible way of arranging B1 on its own would essentially be computing the number of permutations of 9 symbols. There are 9! of filling in B1. The main operation they use is called relabeling. 1 2 3 4 5 6 7 8 9 Figure 2. B1 in standard form (Felgenhauer and Jarvis, 2006) Felgenhauer and Jarvis have found that B2 and B3 is the same as the transpose of B2 and B3. Therefore the number of ways of arranging B1, B2 and B3 and B1, B2 and B3 to a complete grid is equally the same. This means that computing one set of possibilities will cut down the number of solutions. Inevitably, there are few pairs of B2 and B3 that needs to be worked out and as well as using reduction the number of possibilities for the top band of a Sudoku grid is 9! x 2612736 = 948109639680. The next section involves brute force computation. As running through all 2612736 possibilities would be exceedingly tedious for B2 and B3, Felgenhauer and Jarvis attempts to identify configurations of the numbers in these blocks which give the same number of ways of completing to a full grid. This in return, will cut down the number possibilities. Permuting B2 and B3 in every way such that the result gives a unique solution will preserve the number of complete grids. This is the same for B5 and B6, and B8 and B9. However this changes B1 from its standard form, so an additional relabeling of B1 needs to be performed. Another approach to reducing the number of possibilities is to permute the columns in each block and permute the rows of any block. Reducing the number of possible ways by permuting. Lexicographical reduction Permutation reduction Column reduction As a result of these methods, Felgenhauer and Jarvis have found that there are approximately 6670903752021072936960 à ¢Ã¢â‚¬ °Ã‹â€  6.671 x 10 ²Ã‚ ¹ Sudoku solutions. In light of this result, there are fewer solutions than Latin squares due to the fact that there is that extra restriction of 9 sub-grids. That being said, there will be no shortage of Sudoku puzzles any time soon. Verification of this result has been confirmed by several other mathematicians Ed Russell to be more precise. 2.2 Essentially different Sudoku grids Whether symmetrical Sudoku grids are considered as two separate solutions is another method of enumerating the possible solutions. In this case, the only solutions are ones that are essentially different. Lets say two Sudoku grids are equivalent if one is a transformation of the other by applying any number of symmetries. If however, no such chain of symmetries can occur between two grids, it is essentially different. Two Sudoku grids are the same if we can get from one to the other by applying some sort of symmetry. For instance, take figure 3 4 below; the set of 3s in the first grid can be interchanged by the placements of the set of 1s, effectively producing the second grid. Figure 3. Valid Sudoku grid Figure 4. Another valid Sudoku grid from Figure 1 As well as this, a solution is said to be the same as another if any two columns or rows are swapped. The first column and second column in figure 3 can be exchanged to give figure 5. The two solutions are said to be symmetrical because the transformation still produces a valid Sudoku grid. Figure 5. First and second column swapped from Figure 1. Another form of symmetries includes rotational grids. A rotation of Figure 3 by 90 degrees generates a new valid Sudoku grid shown in Figure 6. Figure 6. Rotational of 90 degrees from figure 1 Any of these operations performed on a valid grid maintains the property of it being valid and this is known as symmetries of a grid. When an object is subject to these operations, certain properties are preserved. An example would be if one performs symmetry on to a Sudoku grid and repeats this operation once more, the final transformation is itself symmetric. In addition a symmetrical object can be transformed back to its original state by another form of symmetry. Performing several symmetries on a Sudoku grid can also be achieved by grouping its neighbouring pair. So the first symmetry can be paired with the second or the second can be paired with the third and so on. The resulting transformation is nevertheless the same either way. From these properties, it is inevitable to say that the set of symmetries of any Sudoku grid form a group. A group is a set G if it satisfies the following properties: CLOSURE If f and g are elements of G, then f ·g is also an element of G. ASSOCIATIVITY If f, g, and h are elements of G, then f ·(g ·h)=(f ·g) ·h must satisfy. IDENTITY ELEMENT There is an element e in G such that g ·e=e ·g=g for all g in G. INVERSE For any element g of G, there is another element d of G such that g ·d=d ·g=e, where e is the identity element. (The element d = g-1.) The symmetry group is thus generated by the transformations of: re-labelling the nine digits, permuting the three stacks (3 vertical blocks of a Sudoku), permuting the three bands (3 horizontal blocks of a Sudoku), permuting the three columns within a stack, permuting the three rows within a band, and any reflection or rotation. These can be combined to form other elements of the group and together they comprise of the symmetry group G. Given that any element of G can be mapped so that it takes one grid to another, we can say that the set of valid Sudoku grids has a finite number of elements. Thus G has finitely many symmetries. The association between symmetrical Sudoku grids are in fact an equivalence relation and satisfies the following three properties: for grids A, B and C in set G Reflexivity A = A Symmetry If A = B then B = A Transitivity If A = B and B = C then A = C Let A be any valid Sudoku grid, we must consider all the grids that are equivalent to a valid Sudoku grid A. To do this, we firstly have to group together grids that are essentially the same so that we can partition the set of grids. This will break the set of Sudoku grids into subsets, with groups that contain no relating elements within each other. The term subset can be called equivalence classes and is denoted by X/G. In any equivalence class, there are elements that are equivalent to each other by symmetry. The total number of elements in X/G is equal to the number of essential Sudoku grids. To calculate the number of essentially different Sudoku grids, we shall look at all the symmetries neglecting the re-labelling of the nine digits for the time being. The number of distinct symmetries founded by Russell and Jarvis (2006) is said to contain 3359232 (pg 4). In this finite group H, we need to take the average number of grids fixed by an element of H, up to re-labelling. Next we need to verify the number of fixed points of all elements in H. Russell and Jarvis have found that there are 275 classes of symmetries using a software package called GAP. It is interesting to note that some of the elements in H have the same number of fixed grids. In other words, we only need to count the number of fixed points for one symmetry for each of the 275 classes. However there exist symmetries in H that have no fixed points. Subsequently, there is no need to count the number of fixed grids for those that have no fixed points. That being said, there are only 27 out of 275 classes that co ntain fixed points, meaning fewer computations. Rotman. J. J (1995) demonstrate that if X is a finite G-set and |X/G| is the number of G-orbits of X, then Formula 2 holds where, for gцG, X is the number of xцX fixed by g (pg 58-61). Using this notion, we have established that the number of valid Sudoku grids is of a finite set and X/G is the number of essentially different Sudoku grids, so we can obtain the number of essentially different Sudoku grids by using the Burnside Lemma Theorem. Formula 2. Burnside Lemma Theorem (Rotman, 1995) Burnside Lemma Theorem is a useful tool when dealing with symmetry with a set of countable objects. When used to enumerate the essentially different Sudoku grid, the set of equivalent grids form an orbit of the symmetric group. The number orbits are essentially the number of different grid solutions. This may sound slightly (ALOT) trickier to compute, nonetheless Russell and Jarvis have shown that the number of essentially different Sudoku grids is 5,472,730,538 with the implementation of Burnsides Lemma Theorem. Chapter 3 Nondeterministic polynomials 3.1 NP-complete and Sudoku Sudokus may relate to a variety of problems, in particularly, whether Sudoku is an NP-complete problem. It is known that NP-complete problems are one of the most complicated cases in NP, also referred to as nondeterministic-polynomial. Its rival, P problems relates to NP as both being in the same complexity class. Mathematicians have yet to solve whether NP-complete problems can be solved in polynomial time or more commonly whether P = NP. Consequently being one of the greatest unsolved mathematical problems. The majority of computer scientists believe that P à ¢Ã¢â‚¬ °Ã‚   NP, as a result would mean that NP-complete problems are significantly trickier to compute than to verify. Unfortunately, nobody has yet found an efficient algorithm, not even with the use of computers available today. A problem is said to be NP-complete when its solution can be proved in polynomial time. And if that problem can be solved in polynomial time, all problems in NP can be solved too. An interesting characteristic of NP-complete problems is that the time frame to solve the problem increases rapidly as the size of the problem gets larger. If that is the case and Sudokus are NP-complete, solving a Sudoku of higher order (say 17 ² x 17 ²) will become increasingly challenging algorithmically then the standard 3 ² x 3 ² version were talking trillions of years. It has been shown that Sudoku does belong to the category of NPC problems by Takayuki Yato of the Univeristy of Tokyo (2003). An exchange for the notation ASP-completeness (shorthand for Another solution problem), led the proof of NP-completeness of ASP. Their proof uses reduction in order to obtain the required polynomial-time ASP from the problem of Latin squares by Colbourn (1984) who has verified, the NP-completeness of ASP of Latin square completion Another accountable source by Provan states that, It is known that solving general-sized Sudoku puzzles is NP-hard, even for square grids with blocks consisting of the sets of rows and columns (Latin Squares) or for p2 x p2 grids with blocks consisting of rows, columns, and the p2 partitioned p x p subsquares. Mathematical programmes such as the 0-1 linear programming and the knapsack problems are also cases of NP-complete problems. A full list of other problems that are NP-complete can be found in Garey and Johnson (1979). Chapter 4 Sudoku Variants 4.1 Variation The classic form of a 99 Sudoku are polyominoes. There are other variations of Sudokus that can be applied to the rules of Sudoku. There are puzzles of the size 66 with 23 regions or a 1212 grid of 43 regions. More so, there are other fascinating Sudoku variants such as Greater than Sudoku. Chapter 5 Personal Critical Review The progress I have made during the duration of this project, have been fairly slow but surely getting there. Having said this on many occasions, I have still not conquered my time management skills! The project started very slow which meant I was behind schedule. Nevertheless my organisational skills have kept me on balance. The GANT chart has been of great help in doing so. What has kept me going throughout this project in particular would be self discipline and motivation. This project has proven that I am capable of working to my own initiative, but also well within a group; my time during the group project. Furthermore, my time on this project has definitely promoted a better mentality of my future ambitions. I have learnt that it is crucial to read a lot, as well as reading as broadly as I can. This in turn have aided in the running of my project. With other coursework deadlines, I made that a priority and had no time to meet with my supervisor. I understand that meeting with my supervisor is equally important because a supervisor is there to encourage and to advice on any difficult obstacles I may encounter. An area of interest to proof whether NP-complete problems can be solved in polynomial time, was left open as future work. This could be the next step of extending this report that little bit further. Chapter 6 Conclusion A challenging problem for further research is to proof whether NP-complete problems can be solved in polynomial time. This has yet to be solved and anyone who has a formal proof will be rewarded $1 million dollars by The Clay Mathematics Institute.

In this essay I will compare the presentation of family in digging

In this essay I will compare the presentation of family in digging with at least one other poem in identity. I have chosen to select Follower. The title of the poem Digging could refer to turning over soil for planting or harvesting, or digging deeper to uncover some sort of treasure. Alternatively the poet could be thinking of digging up the past, or uncovering some secret hidden in the past. The poem is written from the poet’s perspective and there is no doubt that this poem is about a writer for in the opening lines we learn that: ‘Between my finger and my thumb The squat pen rests...’ The poet is writing in his room which is upstairs and overlooks the garden. His attention is caught by the ‘clean rasping sound’ of a spade digging into the ‘gravelly ground’. The poet looks out and sees his father digging as he has done for twenty years. Heaney describes his father with great admiration for his strength and skill as a farmer. The poet reflects ruefully on the skill that his father and grandfather possessed with a spade. He is slightly in awe of them as he celebrates their skills and he regrets his own inability to wield a spade. The careful, deliberate, way his father cuts into the earth with his spade makes digging for peat sound like a skilled craft: ‘The course boot nestled on the lug, the shaft.’ When the poet describes his father uncovering the potatoes he uses alliteration again in ‘tall tops’ and ‘buried the bridge edge deep’ to capture the sharp, precise sound of the spade entering the soil. When the poet hears the sound of his fathers spade digging he lets us hear it to in the word ‘rasping’, an onomatopoeia, and in the hard alliterative sound of ‘gravelly ground’. In digging the dis... ... him the child: ‘Fell sometimes on the polished sod; sometimes he rode me on his back.’ This conveys to the reader just how close the relationship was between father and son but it also shows how following him was not always easy on the rough ground. Now he is grown up though it is his father who is the follower and he now keeps stumbling and ‘will not go away’ Even though the word ‘love’ is never used in the poem, it is obviously the word that best describes the basis of the relationship existing between Heaney and his father. The poem is very much a personal experience, but it has a much wider significance relating to any kind of hero – worship by a ‘follower’. Now that he is himself an adult, Heaney acknowledges that the father he hero worshipped as a young boy has grown old and needs as much tolerance and patience as he himself once showed his son.

Essays --

Moral Implications in the Cuban Missile Crisis Morality has been a constant in the foreign policy of the United States of America (U.S.A.). When it comes to the affairs of the world the U.S.A usually tries to make things right and stop evil. Some of the examples of morality playing a role in America’s decisions in foreign policy are the decision to combat the Barbary pirates, the Spanish-American War, World War I, World War II, the Korean War, the Vietnam War, and most recently, the wars in Iraq and Afghanistan. One thing that all of these situations have in common is that it involved the United States basically saying that she would not stand for immorality. One story that has not been investigated is the â€Å"morality element† card and the role of that variable in the choosing of a strategy in the Cuban Missile Crisis. This paper will examine the â€Å"morality element† and how by concentrating on that element the U.S.A successfully brought the Cuban Missile Crisis to an end. Cold War Reaches Boiling Point From 1945 to 1961, this â€Å"war† had some tense moments, but the Cold War hit its boiling point in 1962 with the Cuban Missile Crisis. During this time, John F. Kennedy was the leader of the United States while Nikita Khrushchev was the leader of the Soviet Union. When tensions began to rise in October 1962 these two men would decide whether the world would live to see another day or the world cease to exist. In regards to the Soviet Union and the threat of nuclear war, Kennedy stated that â€Å"[t[o be an American in the next decade will be a hazardous experience. [This nation] will live on the edge of danger. Freedom and communism are locked in a deadly embrace.† Kennedy would also state in his Inaugural Address that the United States w... ...iet Union now had missiles in countries that were very close to them. If the U.S. made the Soviet Union angry or vice versa then the world could be in danger of ceasing to exist. The next part of the Cuban Missile Crisis was very crucial to saving the future of the world. The Soviets Are Caught Red Handed As stated previously, the U.S. did not know about the missiles in Cuba, but that would change very quickly. One of the most crucial days of the Cuban Missile Crisis was October 13th, 1962. On this day, a U-2 plane flew over Cuba and took some photographs before returning safely to Florida where these photographs were developed, so that the U.S. could find out exactly what the Soviet Union was doing in Cuba. The information in these photographs was stunning. Basically what these photographs revealed was the missiles that had been put in Cuba by the Soviet Union.

Adam Smith :: Biography

Adam Smith Adam Smith was born in Kirkcaldy, Fife, Scotland. His exact date of his birth is unknown but he was baptized on June 5, 1723. At the age of fifteen, Smith began attending Glasgow University where he studied moral philosophy. In 1748 he began giving lectures in Edinburgh where he discussed rhetoric and later he began to discuss the economic philosophy of the â€Å"simple system of natural liberty† which he later proclaimed in his Inquiry into Nature and Causes of the Wealth of Nations. In 1751, Smith was appointed professor of logic at Glasgow university, transferring in 1752 to the chair of moral philosophy. His lectures covered the field of ethics, rhetoric, jurisprudence and political economy. In 1759 he published his Theory of Moral Sentiments, embodying some of his Glasgow lectures. This work was about those standards of ethical conduct that hold society together, with emphasis on the general harmony of human motives. Smith moved to London in 1776, where he published An Inquiry into the Nature and Causes of the Wealth of Nations, which examined in detail the consequences of economic freedom. It covered such concepts as the role of self-interest, the division of labor, the function of markets, and the international implications of a laissez-faire economy. The Wealth of Nations established economics as an autonomous subject and, launched the economic doctrine of free enterprise. In the western world, it is the most influential book on the subject. When the book, which has become a classic manifesto against mercantilism, appeared in 1776, there was a strong sentiment for free trade in both Britain and America. This new feeling had been born out of the economic hardships and poverty caused by the war but the British public and parliament still clung to mercantilism for many years to come. Smith laid the intellectual framework that explained the free market and still holds true today.

I felt a Funeral in my Brain Essay

Dickinson uses the metaphor of a funeral in her poem â€Å"I felt a Funeral in my Brain† to describe abstract emotions related to stressful situations that could destroy or take away everything that an individual values in life. The poem is quite permissive in the sense that it allows multiple readings where the reader may ascribe the emotions described in the poem to various situations that are difficult to cope with in life. Of course, to add dimension to the poem, Dickinson uses numerous poetic devices to enhance the experience of reading this particular piece. The piece features alliterations such as in the line, â€Å"I felt a Funeral, in my Brain,†. (1) Here there is a repetition of the consonant ‘f’ sound in the words ‘felt’ and ‘funeral’. Again, the same poetic device is seen in the line, â€Å"And I, and Silence, some strange Race†. (15) where the consonant ‘s’ sound is repeated in the words, ‘some’ and ‘strange’. Another one of these lines featuring an alliteration is the line, â€Å"And I dropped down, and down –â€Å". 18) While alliteration has no contextual function in these lines, the device serves to give the poem some sort of regular rhythm so that reading the poem is not just an intellectual experience, but also an experience that is enhanced by sound. The poem is given an interesting beat to it because of these alliterations. Another device intended to give the poem a more rhythmic quality is the assonance which is found in t he line, â€Å"Kept treading – treading – till it seemed†(3) where the vowel sound ‘ea’ is repeated in the word ‘treading’ and the vowel sound ‘I’ is repeated in the words ‘treading’,. ‘Til’, and ‘it’. In the same manner, the line, â€Å"And I, and Silence, some strange Race† (15) also demonstrates the use of assonance in the repetition of the vowel sound ‘I’ in the words ‘I’ and ‘silence’ and the repetition of the sound ‘ae’ in the words ‘strange’ and ‘race’. There are also metaphors in this very brief yet multi-dimensional poem. For instance, to demonstrate the frailty of reason, Dickinson writes, â€Å"And then a Plank in Reason, broke†. (17) Here one sees how reason is likened to a plank which breaks; a very clever way of illustrating the common human reactionary idiom which is ‘breaking under pressure’. Another metaphor in the poem is â€Å"With those same Boots of Lead, again† (11) which describes the heavy steps that a bothered or disturbed person makes. In effect, this particular line is an effective metaphor that illustrates the footfalls of a troubled or stressed person; perhaps, an experience that is familiar to most people as not having the energy to walk when stressed or trouble, hence heavy feet. Aside from metaphors, the very brief poem also manages to throw in a few similes, such as the line, â€Å"A Service, like a Drum –â€Å"(6), which likens the voices of the people in the funeral service to the incessant beating of a drum. So, the sound referred to in this line is an annoying or repetitive sound that seems to nag at the voice’s mind. Then, there is the line, â€Å"As all the Heavens were a Bell / And Being, but an Ear† (13-14) where the voice tries to illustrate how noisy or boisterous prayers are in the context of a person who is currently troubled or is dealing with problems that would seem insurmountable. These lines express the desperation of the state of the voice in that it describes how prayers are often empty and without significance or can do nothing to alleviate one’s sorry state. It is amazing how Dickinson is able to pack so many poetic devices into one very short poem. This demonstrates the brilliance of this poet in the art of poetry because it simply shows how expertly crafted and well-thought of the poem truly is. There is also a hyperbole in the poem in the lines, â€Å"And I dropped down, and down – / And hit a World, at every plunge† (18-19) describing the gravity or the seriousness of the state of the voice; hereto expressed as the feeling of falling down and hitting a world at every plunge. In effect, what the poet is saying here is that each disappointment or stressful situation puts the voice in a situation where he/she has to face the consequences presented or put before him/her by a confusing, unyielding, and indifferent society, here referred to as a â€Å"world†. Not to be left out in the poem is the personification which is found in the line, â€Å"And I, and Silence, some strange Race† (15) While personification is not immediately evident or apparent in this line, closer analysis of it would reveal, that in fact, personification is used to represent an abstract concept. When the voice puts silence in equal footage with himself/herself by treating silence as some sort of companion, personification is achieved. So, in this line, the voice of the poem claims that he/she and silence are both from a strange race; what this does is it gives the reader the illusion that apart from the ‘I’ who is a tangible concrete element of the poem, ‘silence’ which is an abstract concept also becomes a concrete element as it is personified by treating it as a companion of the ‘I’. This poem by Emily Dickinson simply demonstrates how craftsmanship is often very important in the creation of a poem. The intentions of a poet are mostly to convey emotion and to embody such emotion in words and the effective use of poetic devices achieves this particular poetic intention or objective. When a poem is able to effectively use poetic devices such reflects on the expertise and genius of the poet because while only some poetic devices can be found in some other singular poems, this poem packs in a large number, if not all poetic devices in the span of twenty lines spread throughout five stanzas. Such a feat is amazing in itself.

Palliative Care Essay

Culture is a fundamental part of one’s being which along with spirituality play a significant role in a person’s journey through life. Health beliefs may be strongly tied to a person’s cultural background and spiritual or religious affiliation. Palliative care is the active holistic care of terminally ill patients which demands to maintain the quality of life addressing physical symptoms as well as emotional, spiritual and social needs. This very nature of the palliative care poses challenges to health care workers when addressing a culturally diverse population. Australia is the most multicultural country in the world where its population ranges from the descendants of Aboriginal and Torres Strait Islander people to migrants or of descendants of migrants from more than 200 countries. The aim of this essay is to discuss the importance of providing spiritually and culturally competent care for a person and their family receiving palliation. This essay also discuss es how importance is to focus these principles to the culture of Aboriginal and Torres Strait Islander people with regards to death and dying. World Health Organisation defines palliative care as â€Å" an approach that improves the quality of life of patients and their families facing with life threatening illness through the prevention and relief of suffering by means of early identification and impeccable assessment and treatment of pain and other problems, physical, psychosocial and spiritual†(WHO,2009) . According to Matzo & Sherman (2010) the culture and spirituality are among the most important factors that structure human experience, values and illness patterns and determine how a person interact with the healthcare system. Moreover a person’s beliefs, values, rituals, and outward expressions can impact palliative care either positively or negatively. According to Brown & Edwards (2012) people experiencing the inevitability of death are in need of care givers who are knowledgeable about personal issues and attitude that affect the end of life experience. An adequate understanding of cultural and spiritual matters is vital and beneficial when focussing on dying person’s family needs and wants. According to Geoghan (2008) perception differ among culture in issues such as use of medication, personal space and touching, dietary issues, whether to be cared at home or seek health care facility. Long (2011) states that when determining the decision making and disclosures culture has a significant role to play with spiritual or religious implications. Brown & Edwards (2012) states that culturally expressions physical symptoms especially pain differs in different cultures and leads to ethnic minority groups are often being undertreated in terms of pain medication. Moreover, nonverbal cues such as grimaces, body positions and guarded movements also significance in providing culturally competent care.Ferrell &Coyle (2010) states language has an important role in streaming communication patterns and style between health providers and patients and lack of effective communication may mean less than satisfactory exchanges between health providers, patients and their families in a multicultural society. According to Matzo & Sherman(2010) spirituality is a way to be connected with God as well as to self, fellow human beings and to nature. Moreover, when the terminally ill patients go through critical life adjustments, spirituality considered to be as a domain of palliative care which serves as the binding force for physical, social, and psychological domains of life. According to Ferrell &Coyle (2010) majority of the palliative patients may experience a growth in spirituality and considers spirituality to be one of the most important contributors to quality of life and frequently used as helpful coping strategies for their physical illness. Furthermore, the family caregivers of seriously ill patients also find comfort and strength from their spirituality and considerably assist them in coping .At the same time many of such patients with their uncertainty of life, long term nature of illness, potential for pain, altered body image and confrontation of death may lead to spiritual distress as well (Matzo & Sherman, 2010). Spiritual care is an important factor for both those expressing spiritual wellness and those experiencing spiritual distress during their period of illness (Amoah, 2011). Matzo & Sherman (2010) states that spirituality facilitate coping with chronic pain, disability, sense of illness and provides strength and self-control and thus reduce the anxiety and depression. Furthermore, those who participate in religious services and ceremonies experience a relief from their loneliness and isolative life style and such practices may generate significant support and peace in difficult times for the patient and their family. Brown & Edwards (2012) states that assessment of spiritual need in palliative care is a major factor because spiritualty is not necessarily equate religion and a person do not have particular faith or religion may have deep spirituality. According to Brown & Edwards (2012) awareness and sensitivity to cultural beliefs and practices regarding death and dying is vital when caring end of life patients, especially in a multicultural societies such as Australia. Ferrell & Coyle (2012) states that in spite of strong government initiatives, Aboriginal and Torres Strait Islander people remain a marginalised group with health status significantly below that of other Australians. Consequently Aboriginal and Torres Strait Islander people have high rate of mortality and premature death (McGrath & Philips, 2009). Furthermore, Ferrell & Coyle (2012) states that while addressing palliation of Aboriginal and Torres Strait Islander people, culturally responsive model of palliative care to be delivered and traditional practices that surrounds care of dying people and death are understood, respected and incorporated in to care. According to Thackrah & Scott (2011) an understanding of cultural, practices, protocols and customs with rega rds loss and grief is of topmost priority when dealing with traditional aboriginal men and women. According to Queensland Health (2013) Aboriginal and Torres Strait Islander have strong cultural and spiritual concepts about the cause of diseases and death which may conflict with Western explanations and diagnosis of illnesses. According to McGrath & Philips (2008), though the Indigenous Australians adopted many aspects of the non-Indigenous culture over the years, their expectations and rituals around end of life still mostly well connected to their land, culture and tradition. Most of the Aboriginal people have a strong wish to die at home with family, surrounded by their ‘Country’ and in their own community where their spirit belongs (O’Brien &Bloomer, 2012). According to Thackrah & Scott (2011), most people dislike the hospital environment because they believe in the hospital they may experience isolation, structural racism and disempowerment. Since death in a hospital can create stress in the family along with fear and disputes, family protocols to be strictly followed and they have to be given space to finish up with dignity and compassion. During a situation of an expected death of an Aboriginal or Torres Strait Islander person, there is usually a gathering of immediate and extended family and friends which are a mark of respect of the patient. Based on the belief that life is a part of a greater journey, it is cultural practice to prepare the person for the next stage in their journey and often the extent of gatherers correlates with the patient’s value to the community. The passing of an elder may induce immense grief and mourning upon the whole community, hence expect many visitors and a grand funeral ceremony that reflects the respect. According to Thackrah & Scott(2011),When a death occurs in traditional indigenous communities in Australia, community members and visiting relatives from elsewhere move away from settlements into a special place called ‘‘sorry camp’’. Also the Indigenous way of grieving is a long process with different phases not only consoling each other but by tradi tional ways of harming themselves. Palliative care is truly a holistic care delivered to patients and family members with life threatening illness by providing physical as well as emotional and spiritual support. It is evident that culture and spirituality are central to palliative care which must be given due consideration at every point of assessing and planning care for patients and families. Although spiritual beliefs might help most people to cope well in the face of illness, for other people such beliefs may be ineffective or problematic. Since health beliefs are strongly associated with culture and spirituality it is critical that healthcare professionals understand and implement best practices in attending to cultural and spiritual needs during their illness journey. The death and dying in Aboriginal and Torres Strait Islander people have a crucial cultural significance and health professional must be competent in religious and cultural practices when addressing palliative care. Reference List AmoahC.F. (2011).The central importance of spirituality in Palliative care. International Journal of Palliative Nursing, 17,353-358.Retreived from http://ea3se7mz8x.search.serialssolutions.com.ezproxy.uws.edu.au/?V=1.0&pmid=21841704 Brown, D., and Edwards, H. (2012).Lewis’s Medical-surgical nursing (3rd ed.). Chatswood, Elsevier Australia. Ferrell, B.R., & Coyle, N. (2010).Oxford textbook of Palliative Nursing. NewYork, Oxford university press. Geohan, D.A. (2008).Understanding palliative care nursing. Journal of Practical Nursing, 58.6 Retrieved from http://search.proquest.com.ezproxy.uws.edu.au/docview/228052494/fulltextPDF?accountid=36155 Long, C.O. (2011).Cultural and spiritual considerations in palliative care. Journal of Paediatr Hematol Oncol, 33, S-96-101 doi: 10.1097/MPH.0b013e318230daf3. Matzo,M.,& Sherman,D.W.(2010). Palliative care nursing-Quality of care to the end of life (3rd ed.) New York, Springer. McGrath, P. & Phillips, E. (2008). Insights on end-of-life ceremonial practices of Australian Aboriginal peoples. Collegian, 15, 125 – 13.Retreived from http://www.nursingconsult.com.ezproxy.uws.edu.au/nursing/journals/1322-7696/full-text/PDF/s1322769608000243.pdf?issn=1322-7696&full_text=pdf&pdfName=s1322769608000243.pdf&spid=21611429&article_id=708072 McGrath, P. & Phillips, E. (2009). Insights from the Northern Territory on Factors That Facilitate Effective Palliative Care for Aboriginal Peoples. Australian Health Review, 33,636-644.Retreived from http://search.informit.com.au.ezproxy.uws.edu.au/documentSummary;dn=301730371388187; res=IELHEA> ISSN: 0156-5788 O.brien,A & Bloomer,M.(2012). Aborginal palliative care and mainstream services.Australian Nursing Journal ,20,39.Retrieved from http://search.informit.com.au.ezproxy.uws.edu.au/fullText;dn=993462370312798;res=IELHEA Queensland Health. (2013). Sad news, sorry business- Guideline for caring Aboriginal and Torres Strait Islander people through death dying Retrieved from http://www.health.qld.gov.au/atsihealth/documents/sorry_business.pdf Ulrik, J., Foster, D., &Davis, V. (2011). Loss, Greif, Bad luck and sorry business. In R.Thackrah, &Scott (1st Ed.) Indigenous Australian health and Culture (190-2008). Frenchs Forrst,Pearson Australia. World Health Organization. (2009). WHO definition of palliative care, Retrieved from http://www.who.int/cancer/palliative/definition/en/