Choosing GPU for mining is now easier than ever. Clear linking rules are abided to meet reference reputability standards. Only authoritative sources like academic associations or journals are used for research references while creating the content. If there's a disagreement of interest behind a referenced study, the reader must always be informed. Hello and welcome to this guide on choosing the best GPU for mining.

My point was since he paid for his pint, he finally drank it. So the barman had topped for 6 pints, hence the equation. William Chau :. More squares in a circle: find an expression in n for the ratio of area of 2n-1 squares to the area of a circle that contains the squares fitted perfectly on the diameter where n is an odd number Similar to the last problem: Find the area of the enclosed space between the 4 circles, each with a radius of 1 unit :.

Paul Beirne:. Lau Vincent:. The pole is located in the bottom left of the field 3 m out and 2 m up. What is the grazing area for the sheep give your answer to the nearest square metre? Ah getting better i Let me try the "gnomon" way is gnomon also an english word? Now I take a cube 4 points wide. I can consider this cube as 4 square layers of 4x4 points. From each layer I can "cut" 2 T3 triangles out of a rectangle of 3x4 points. Since I have 4 layers I overall get my 4x2xT3 points.

On each layer I'm left with 4 points on a single line, so for the 4 layers, I get a square made up of the 4 single lines, i. So I've got all the extra points I needed and I reach T4. The same can be done for Tn. I thought about working the problem by induction, but it isn't all that elegant.

Say S n is the sum of integers up to n and T n is the sum of cubes up to n. Mary Tierney:. Last night I was thinking of the "Go First" dice There is a much easier system for 2 players. So can you design 2 dice with the following conditions: 1 There MUST be a winner on a single event an event being 2 people throwing the dice and the winner is the person with the greater number.

Basically 3 numbers greater and 3 numbers less Christopher Duff: If the dice are not similar if think they should not would something like 1 3 5 9 11 13 and 2 4 6 8 10 12 work? Which one has a greater quality? Re-write the numbers 1 1 2 2 3 3 4 4 so that they meet the following conditions. The opposite angles, A, are equal.

What is the length of the diagonal d? A man walks due south to location B, where there is a road due east from B to E 12 km long. He then walks half the distance from B to C to arrive at D which also has a road to F going through E. The line FG is due east of point A.

How far is it from F to G? I decide to race him over the m. Now I'm not crazy well I am a little bit so I decide to race in a car. The car I'm in is pretty rubbish and accelerates from rest to 60 miles per hour in 12 seconds assuming uniform acceleration. So assuming Bolt runs the m in the same time as his olympic time and we both start from rest, would I beat Usain Bolt in the car?

What is the area of his grazing area? Let's also assume that the earth is million km from the centre of the sun and that mars is million km from the centre of the sun. Let's also assume it takes days for the earth to orbit the sun and days for mars to orbit the sun. They orbit the sun in the same direction and on the same plane.

In Season 6 Episode 15 "Benderama" The Professor invented a machine called the Banach-Tarski Dupla-Shrinker which is taken from the surprising Banach-Tarski paradox which states you can "cut" up a sphere into finitely many pieces and form 2 other spheres of the same size as the original. Bender of course abuses this machine in order to reduce his workload.

He creates 2 smaller copies of himself and those 2 create 4 more, even smaller, copies of themselves, and then those 4 create 8 smaller versions of themselves and so on Taking the first 2 copies of Bender as the first generation of "Benders" and taking Bender's mass to be M.

They play a game where there are 3 winners and 1 loser after the game. The loser must double the amount of discs for each of the 3 other players. After 4 games they have all lost 1 game each and they now have the same amount of money. I am interested in the Volume of the snub cube as it is related to the Tribonacci numbers which is similar to the Fibonacci numbers except the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms e.

The ratio of two adjacent Fibonacci numbers tends towards phi, the golden ratio, BUT the ratio of two adjacent Tribonacci numbers tends to the tribonacci constant which you can look up , as well as the volume of the snub cube. The 4 vertices of the tetrahedron are touching the centres of 4 of the cubes faces.

What is the ratio of the volume of the Cube to the volume of the embedded tetrahedron? Specifically double-6, double-9, double, double and double forms. They come as a complete set, i. All possible combinations of dominoes are in the set. The game is played so that the end of one domino is joined to another domino For what values of n 6, 9, 12, 15 and 18 and for any general n, can we take the complete set of dominoes and lay them out according to those rules without being left with any dominoes?

Source: Adapted from a problem posed by Orly Terquem in Continuing on my love for the platonic solids: Kepler also showed it was possible to nest an Icosohedron in a Dodecahedron so that each vertex of the Icosahedron was in the centre of a face on the dodecahedron.

Now what is the ratio of the volume of the Dodecahedron to the Icosahedron? Assuming it forms a convex polyhedron, prove that no matter how many hexagons there are, there must ALWAYS be 12 pentagons, no more, no less : William Chau: Let m and n be the number of hexagons and pentagons, respectively. The distances of the planets were known relative to the distance from the earth to the sun 1 astronomical unit This was known as a result of Kepler's 3rd law.

However the distance to the sun was not calculated until Venus transits were analysed which uses Kepler's law and parallax. Venus had a transit last night and this morning. To this end it seems fitting to give a problem related to Venus and Parallax The actual calculations involve the time taken for the transit and recording over different parts of the earth then using parallax and Kepler's law and we need not go into here, this is a simple parallax problem.

A man measures the angle to the middle of Venus to be Another man km away assuming a straight horizontal distance measures the angle to the middle of Venus to be Calculate the perpendicular distance from the earth to Venus. This is certainly odd behaviour indeed, however Jimmy didn't mind as he only ever had 1 pint and was quite happy to drive home, also he saved quite a bit of money.

He was stopped by the gardai and they told him he had had 7 pints in total and was way over the legal limit. As he was explaining himself to the police, he remembered that the bar man turned his back at least once but less than the number of times he didn't but couldn't remember how many times he did. This was perplexing, can you tell how many times the bar man turned around. Christopher Duff: My point was since he paid for his pint, he finally drank it. In Rugby union a team can score a drop goal 3 points , a try 5 points and a conversion after a try 7 points Between 1 and what scores are impossible to get?

A perfectly cylindrical glass 10cm high is filled to the brim with coke. A perfectly cylindrical stirrer is also placed in the glass, resting at the bottom corner and coming out the top as shown in the diagram.

The coke obviously overflows but all the coke is collected in an identical glass. The level of coke in the new glass is 1cm high. NOTE: You may assume the stirrer forms a cylindrical in shape even at an angle at the surface of the water. What is the longest pole we can put in the van assuming it is not impeded by any other pole?

The pendulum is set swinging at a relatively small angle. The period of the pendulum was noted. The length of the pendulum was increased by a factor of 3 and it was noted that the period of the pendulum increased by 1 second. What was the original length of the pendulum to the nearest 10th of a cm? What is the area enclosed by the 3 circles the shaded area? What, in terms of N and pi, is the area of the region enclosed by the N circles? The wall is 3 m away from the rope when it is hung at rest.

How high is the ceiling from the floor? For that, t his study assumes a new alternative mode as an improved train system because no other ground modes are able to provide a speed of or more miles. Based on the findings, this study suggests viable options for planners and decision makers to plan for long distance transportation in the future.

I t should be noted that highway delays near urban areas caused travelers to waste an average of 34 hours resulting in 3. T hese problems are expected to be worse in the future because the U. DOT, p. In order to address growing congestion and delay problems federal, state and local governments have traditio nally focused on the capacity expansion of existing transportation infrastructure through significant capital outlays However, even these capacity additions will not be able to f ully accommodate the predicted highway s and airports demand at an adequate se rvice level.

Instead, h ighway congestion is rather PAGE 14 14 expected to spread to medium sized cities small cities and even suburban and rural fringe over the next 10 to 15 year s U. Meanwhile, providing additional capacity of highways near ur ban area s is increasingly difficult not only because highways are becoming increasingly expensive to build, but also because h ighway construction cost around urban area s 8 10 million per mile of 4 lane highway, is nearly double compare to that of rural an d suburban areas, 4 6 million per mile ARTBA, 0.

It is also estimated that a greater number of large hub airports and their associated metropolitan areas are expected to face capacity constraints by and 0. Besides, they will not be able to inc rease capacity beyond what is currently p rovided because some areas have limited geographical capability to increase runways The MITRE Corporation, p. A variety of alternative options have been evaluated over the past decade s and c onsiderable attention has been directed toward new ground transportat ion mode such as high speed rail HSR Maglev rail, and upgraded rail service s T he US Department of Transportation DOT has attempt ed to provide a full array of affordable and practical mode choice option s to reduce road travel U.

DOT, 6 ; T he Federal Aviation Administration FAA has emphasize d the importance of exploring alternative means of transportation that can substitute short to medium distance air routes U. In th e same context, t he FAA recommended the development of an PAGE 15 15 efficient and effective intermodal system to mitigate capacity problem s at airports U.

The HSR system is also expected to improve attractiveness and the potential economic development of the regions by promoting accessibility and connectivity. However, it has been difficult to implement the high speed rail HSR proposals in the US because the s uccess of the HSR option has been in question Much of the criticism of HSR is based on the concerns about its cost effectiveness in the near to medium distance range.

One of the major challenge s geography with lo w er population density in urban areas compared to urban areas in Asia and Europe Peterman et al More importantly, against p ersonal automobiles puts the success of the system in question.

Critics argue that HSR will not be able to compete with personal cars in the US because the 58 percent of the long distance trips are not that long, less than miles or km in ro und trip distance U. These travel PAGE 16 16 patterns transportation because previous studies in Europe and Asia have suggested that HSR can have strong potential at a one way s ervice range between mile and mile Given these promises and concerns, it is needed to evaluate whether or not a new alternative high speed train system would be a viable option in the US.

In the same context, it is necessary to understand whethe r there is such a desired service qualit y for a new alternative mode to become an effective and efficient option. However, t here ha ve not been sufficient efforts to evaluate the potentiality of HSR system in the US even though such supports are essential.

In particular, academic studies have not been active in developing a broad spectrum of measures to propose a viable option for the US. Statement of Problems When a new alternative mode of transportation enter s into an existing transportation market, it wi ll inevitably compete with other modes of transportation in the market, and consequently a portion of other mode use will shift to a new mode T h at portion of shift is critical to evaluate the adequacy and efficiency of a new transportation investment in a wide transportation system In this respect, the focus has to be o n the understanding of travelers mode choice behavior because travel demand of certain mode of transportation depends largely on the individuals choice decisions o f mode s However, previous studies of HSR system have been faced criticisms with regarding to methodologies, data, and reliability of ridership estimations.

First of all, the method that was used to forecast the demand for HSR systems has raised several valid issues. HSR line have developed disaggregate demand models to forecast t he travel demand of HSR system s in a specific corridor. However, m ajority of these studies formulated mode choice as separate binary diversion models in which percentage of automobile, air, and bus passengers are diverted to HSR.

It is the simplest specification to evaluate the impact of a new alternative mode on the com peting mode in the market, and to utilize the stated preference SP data that is collected for a HSR project in a given corridor. However, m o st survey data sets collected for HSR proj ects are considered to include a large portion of samples from wrongly defined population s For example, SP data for California HSR surveyed largely from air transportation instead of personal car users that accounts for nearly 93 percent of total interreg ional travel Hence, the estimated models can produce distorted mode choice decision of the average intercity travelers Brownstone, et al.

In addition, the SP data is susceptible to serious problem s of survey respondent bias because respondents h ave the tendency to stick with their current mode choice regardless of the alternatives attributes Roth, PAGE 18 18 Thus, the estimated models are likely to contain unobserved serial correlation making the parameter estimates in consistent. Third, most of pre vious ridership studies are commercially confidential studies, and thus demand forecasts are often significantly optimistic and different ridership forecasting methods have yielded uncertain and inconsistent results U.

GAO, In addition, ridership studies have encountered criticism because parameters of the mathematical models were considerably adjusted from assigned values in the models, and thus the forecasts of demand have a strong likelihood of very large error bound from one level that can sup port the implementation of HSR investment to the other level that show significantly low demand Brownstone et al.

Th e reason is that most of ridership studies for HSR projects are so far, conducted by private consultants who are working under a v ariety of pressures and for different types of clients that are directly related to the implementation of HSR projects. The study noted that rail travel must be extremely competitive in factors such as speed or cost to attract automobile travelers because automobile travel is different from air and rail tra vel in that it involves door to door transportation, provides higher flexibility in time of departure and does not require individuals to share space with others U.

PAGE 19 19 Therefore, it is needed to develop reasonable and applicable measures that can provide answers for p olicy makers and private industries to the questions: what are the existing patterns and trends of long distance travel in the US and what are the factors that affect people to choose one transportation mode or another one?

Furthermore, it is essential to lay the foundations of understanding: what are desired service qualit ies that may change long distance travelers mode choice behavior, and what are viable strategies for the future long distance transportation plans? Purpose of Study The overall goal of this study is to enhance fundamental understanding of long distance travel p atterns and characteristics of the US.

T his study also aims to evaluate the potentiality of an alternative option for long distance passenger s ervice in the US and provide strategic policy options for transportation planning in the future. For that t his study analyzes current long distance travel patterns and trends in the US using the National Household Travel Survey NHTS dataset estim ate mode choice models for long distance travel, and predict the probability of choosing a new alternative option.

In order to estimate mode choice models, this study develops methods to estimate synthetic travel time and costs for all alternative modes in the market. Since the NHTS provide information of travel time of mode used, this process is essential to develop mode choice models. Therefore, the NHTS is useful in examining the patterns and trends of long distance travel.

Previously conducted descriptive studies have focused on the fundamental questions: who is traveling, where are people traveling on long distance trips, what are the modes used, and why are people traveling long distance s? M oreover, this study prese nts origins and destinations of long distance travel.

The descriptive analysis is expected to enhance the understanding of c urrent patterns and trends of long distance travel from both national and regional perspectives Second, t his study develops the multinomial log it MNL model s adopting the mathematical function of the conditional logit CL model. The MNL model have re latively simple and closed form mathematical structure s and thus are straight forward to estimate and interpret the interactions of choice behaviors with the explanatory variables.

In the mathematical function of the CL model the probability of choosing a particular alternative mode is expressed as a function of the alternative specific variables and thus provide s direct measures to evaluate the impacts of changes in a specific alternative mode on the probability of choosing that mode. This study uses the choices of individuals among four modes such as personal cars, bus, airplane and train as dependent variable.

T hese factors are considered as the key variables that enable a particular mode to gain a comp ara tive advantage over other alternative modes of transportation in the market. Therefore i t is needed to know travel time and costs of all available modes for each household. However, the NHTS dataset provides limited information of travel time of the mode used for that trip. Given these conditions, this study estimates synthetic travel ti me and costs for all alternative modes such as personal cars, bus, airplane, and train.

Since this study can obtain census block group level spatial information on the households, it is possible to identify the intercity terminals that might be used by eac h household assuming people use the closest facilities, and in turn to calculate approximate distance and associated access time and costs to intercity terminals. The empirical models are expected to provide useful information to ex plain in the relationships with the unique characteristics of travel mode s and travel ers.

In particular the results are expected to provide useable variation s among individuals and households and present the potential variations among MSAs or regions This study tests various scenarios of travel time and cost combinations to predict the changes in the probability of choosing a specific mode of transportation. For that, this study assumes an improved train system that can provi de a speed of or more miles per hour The estimated probabilities of choosing an improved train system under PAGE 22 22 scenarios are expected to provide valuable information to evaluate the possibility of an improved train system in the US.

In addition, this study examines whether the probabilities of choosing an improved train system would vary across geographical locations. T he results are expected to help policymakers and researchers to predict the potential role of an alternative mode in the transportation market It w ould also be possible to decide th e level of service improvement to existing public long distance travel modes to reduce the use of personal car s and airplanes.

Scope of Study The first section of the Chapter 2 summarizes previous studies that developed mode choice models for long distance or intercity travel In Chapter 3, this study explains the theoretical frameworks of logistic regression models and then presents the specific model that will be developed in this study.

This will be followed by the section about data in which the NHTS data and states add on data This study focuses on long distance trips among trips by definitions. The Chapter 4 describes the processes of choice model estimation comprising four sub sections. T he first se ction explains explanatory variables that might be included in this study, and this study presents hypothetical relationships of explanatory variables to the decisions on mode choices.

T he third section presents the methods to calculate synthetic travel t ime and costs, and access time and costs. Since, the NHTS presents travel time of the used mode for that trips, this study estimates the travel costs of mode used and both travel time and costs of alternative modes for the given travel distance. The former model s are necessary where demand forecasts are required for all modes or those modes which are of inter est to the stud ies These models have predicted the market share for each alternative mode using the estimated mode choice models, and then apply the estimated mode share of each alternative mode to identify total demand for intercity travel.

Meanwhile th e latter models are a dopted where researchers are interested in how many travelers would shift from existing alternative mode to the new alternative mode. For example, many studies have actively measured the impacts of a new intercity mode on existing tran sportation market as many countries in both Europe and Asia have developed broad HSR networks By estimating discrete choice models, those studies have presented the potential competitiveness of HSR against air, passenger car bus and convention al rail ser vice.

M any of mode choice models have, more importantly, attempted to identify critical points of travel time, cost, or distance where travelers may change their mode choice decisions. In e stimating the potential changes in long distance travelers mode choice decisions mode choice models have adopted various factors that represent travel characteristic such as travel time, cost, distance, trip purpose and frequency , travelers characteristics such as income, gender, age and group size , spatial characteristics in both origin and destination such as population density, size of metropolitan area and PAGE 25 25 public transit service quality , and subjective factors such as comfort, convenience, safety, reliability, and privacy.

Am ong the range of variables that have been examined in the previous models, both travel time and cost consist of the key variables to all the models. Travel time is often split into in vehicle time and out of vehicle time of which the latter includes access and egress time, waiting time, terminal time, and transfer time.

Travel time and cost are also important variables to predict the impact of a new alternative mode on existing modes in the transportation system. The demand model estimates the mod e choices of passengers assuming that once a person decides to make the journey, the available alternatives and their characteristics such as travel time and cost will For example, m any co untries in both Europe and Asia have PAGE 26 26 present ed the impacts of a new intercity mode on existing transportation market and the potential competitiveness of HSR against air, passenger car bus and convention al rail service as they devel oped broad HSR networks Gonzalez Savignat , Lopez Pita and Robuste , Roman et al.

Ivaldi and Vibes investigated intermodal competition between aviation and HSR in Europe travel market. Kim et al. Meanwhile, Chang and Chang , Zhang and Xiao Li , and Ortuzar and Simonetti estimated the potential mode share of HSR system in competition with aviation, personal car, and conventional train. In those models, travel time and costs are commonly considered as variabl es that affect travelers decisions.

The travel distance is taken into account of its potential influence on the unobserved perception of comfort and convenience of the ground transportation modes Grayson, ; Koppelman ; Koppelman and Sethi, ; Abdelwahab Innes and Stevens ; Ashiavor Baik and Trani , Wilson Damodaran and Innes 19 In these models, travel distance presents the likelihood of choosing surface modes such as car, bus, or rail relative to air.

Travel dis tance is tested in regard of potential s on the same trip is another variable that is directly connected to travel cost, and thus many previous studies included it in the models Morris on and Winston, ; Bhat, 7a ; Koppelman ait Sethi, ; Mandel et al, ; Swait, ; Wardman et al, ; LaMondia Snell PAGE 27 27 and Bhat This variable assumes that travelers who travel alone likely prefer expensive but fast and comfort mode mor e than travelers in a group.

Moreover, purpose of trip is considered to have significant impacts on mode choice Morrison and Winston, ; Wardman et al, ; Carlsson ; Limtanakool Dijst, and Schwanen Since travelers preferences vary w ith the purpose of their trip, the different trip purpose could be an important issue for mode choice decisions.

Previous studies have shown that business travelers and leisure travelers are expected to be different in their sensitiv ity to travel time and cost because business travelers have subsidization of travel cost while leisure travelers pay for themselves Service frequency, one of frequently employed variables, is mostly defined in terms of departures by time interval or headways Algers, ; Mand el et al, ; Kitagawa et al ; Winzer Pidcock and Johnson ; Wardman et al, ; Vrtic and Awhausen, But it is difficult to exactly measure service frequency because it varies depending on travel demand.

In addition to travel attributes, attributes such as income, education car availability, age, gender and educati on are also employed in many mode choice models. Among various forms of traveler related variables, income has been the most widely used in the models Bhat, 7a ; Grayson, ; Koppelman, ; Koppelman and Sethi, ; Swait, ; Limtanakool et al, ; Abdelwhab et al, ; LaMondia et al, Higher income travelers are generally assumed to choose an alternative mode that provides fast and convenient service even though it is more expensive.

It should be noted that some studies focused PAGE 28 28 more on the impacts of these socioeconomic and demographic variables than travel attributes in order to explain mode choice behavior For example, Limtanakool, Dijst and Schwann examined the effects of age, gender, education, household type and car availab ility on mode choice decision in addition to income.

Bhat 7b tested whether gender has impact on mode choice decisions, while McFadden predicted the potential impact of race, occupation and ratio of cars to workers in the household on mode choi ce decision for shopping trips. Notably, there are researches that attempted to identify the interrelations between spatial attributes and travel behavior by means of measuring differences of travel patterns in different types of urban form size or dens ity and supply of public transportation services or infrastructure.

With regard to spatial characteristics, some of these studies have suggested that the travelers in dense and compact cities with mixed land use use comparatively more public transportat ion for a large part of their daily trips Frank and Pivo ; Timmermans et al. In a similar vein higher population densities are expected to associate with higher demand for transport, and thus they likely facilitate well developed public transportation networks resulting in smaller shares for automobile and larger proportions of public transportati on trips.

In this respect it is important to measure impacts of spatial PAGE 29 29 characteristics on travelers mode choice decisions. It is also worthy of note that spatial characteristics and travel behavior can mutually affect each other. T ravel behavior might b e a critical factor for individuals or households to make location decisions while urban form at the place of residence affect travel behavior Scheiner and Holz Rau, Comfort is expected to have impacts on whether or not travelers have higher probability to choose the luxury service class and to avoid bus a less comfortable alternative.

Each utility that decision maker n obtains from alternatives allows researchers to rank a series of alternatives and identify the alternative that has the highest utility. Thus, it has PAGE 31 31 motivat ed researchers to develop numerous mathematical model structure s by applying a different set of assumptions to the distribution of the error components of the utility function for each alternative.

The MNL model gives the choice probabilities of alternative as a function of the deterministic portion of the utility of all the alternatives. In the MNL model, t he deterministic or observ ed portion V n ,t of the model is represented by a linear additive function that parameters, t and explanatory variables, X. In estimating the MNL models, this study employs mathematical function of the conditional logit CL model in which the probability of choosing a particular alternative mode is expressed as a function of the alternative specific variables.

The Conditional Logit Model According to Rodriguez , McFadden proposed the CL model in which the expected utilities U n t in terms of characteristics of the alternatives rather than attributes of the individuals. In other words, the estimated model can present a sep arate coefficient on each independent variable for each possible outcome. Consequently, the impact of a unit change of explanatory variable is assumed to be constant across alternatives, and a variable Z is appeared to have no impact on choice probability if it has with no variation across alternatives Hoffman and Duncan, In the CL model, t he error terms follow independently and identically an extreme value distribution.

Therefore, the difference of two error terms follows a logistic distribution a s in the MNL model. Given these assumptions the choice probability of the individual n to choose the alternative i can be expressed as equation 3 9 : 3 9 Since the CL model assumes that the choice of mode depends only on the differences of variables in the utility function, it is appropriate to measure the impacts of a unit change in each explanatory variable on the probability of choosing a particular alte rnative.

Therefore, the CL model is useful to evaluate how government polic y affects to the attractiveness of an alternative mode Hoffman and Duncan, In theory, the CL model is assumed to estimate the probability of choosing an alternative mode us ing only the differences in the value of characteristics of the alternatives for example, travel time and costs Thus, it may be different from the MNL model which depends on individual characteristics to estimate the probability of choosing a specific m ode.

Yet in reality, many studies have d eveloped the choice models to examine how both the characteristics of an alternative mode and the PAGE 34 34 characteristics of individual affect the probability of choosing specific alternative mode Abdekwahab Innes and Stevens, ; Koppelman and Bhat, ; Koppelman and Sethi, ; LaMondia, Snell and Bhat, ; Swait, ; Wardman and Toner, ; Winzer, Pidock and Johnson, Moreover, there are studies that have developed the methods to use both alternative specific attributes and the characteristics of individuals in the m odeling framework of the CL model indicating that the CL model is just slightly different form of the exact same model as the MNL model Hoffman and Duncan, ; Rodriguez, 20 12 ; So and Kuhfeld, Model Specification Thi s study estimates the utility of choosing an alternative mode as a function of the alternative specific attributes and the characteristics of individual s.

In estimating mode choice model, this study phases in these explanatory variables T his study, first, enters only travel time and travel cost s of each travel mode into the mode choice model These attributes are unique service attributes that are different among travel modes, and thus are expected to affect differently people s choice of travel mode.

Since different characteristics of among individuals may have different effects on mode choice decisions, it is also valuable to understand the impact of individual characteristics on the probability of choosing travel mode. With these explanatory var iables, the utility of an alternative is expressed as function of the attributes of the alternative modes and the characteristics of the traveler. For example, this study constructs personal car users age variable by applying the each age of an individual n to each personal car user if the person used car and zero otherwise.

The NHTS included more samples in both number of household and number of person terms than the NHTS The NHTS is therefore, expected to increase the availability of a data base with a n ational coverage of long distance travel I t is also expected to allow develop ing mode choice models that can test the potential variation among various geographic sectors.

The NHTS officially provides household, person, vehicle and daily trip level d atasets for public use US DOT FHWA, of which this study focuses on the daily trip level dataset The daily trip dataset provides fully disaggregated information of daily trips for each person of sample household such as purpose of trip, mode of transportation, travel time, number of people in the vehicle and the most importantly travel miles taken for a given trip The dataset also provide s information of traveler s socio demographic characteristics such as age, income, gender, MSA category, and existence of heavy rail service.

More importantly, it is possible to use the spatial information of sample households in a census block group scale. Thus, this study can calculate the shortest distance to each intercity terminal, and in turn estimate acce ss time and costs In analyzing the daily trip datasets t his study focus es on the long distance tr avel among various travels by definition.

Thus, this study n arrow s the standard of long distance tr ip by counting a trip as long distance trip only if a single trip segment is 50 or more miles This confines the long distance trips mostly to intercity trips. By the narrowed standard of long distance trip in this study, t he NHTS includes 28 segments that are 50 or more miles.

These trip s account for 2. Each household generated an average of 0. Operationaliz ing Daily Trips in to Household Level Data This study operationalizes the daily trips in the NHTS into house hold level data by applying four steps: 1 transform 1,, daily trips into , individual level data, 2 sort out 17, individuals who have one or more trip segments that are 50 or more miles 3 identify trip segments that are involved in actual long distance trips, and 4 transform 17, individual level data into 12, household level data.

For step 2, this study counts a trip as long distance trip if at least one segment in all tri p segments is 50 or more miles. In step 3, this study ident ifies actual trip segments that comprise long distance trips among daily trips segments. In other words, this study identifies true origin destination or origin intermediate stop s destination from all trip segments made by an individual in a given survey date For example, four trip segments out of ten daily trip segments could be involved in making a single long distance trip.

In most cases, intermediate destination s includes stop s for gas, rest or meal on the way to destination. Table 3 2 shows the results from data operationalization processes, and table 3 3 presents number of samples by census division. It is also worthy to note that WNC, ESC, WSC, and MT divisions show n early no record of train use possibly because people in these areas live far away from train stations compared to other divisions In detail the average distances to train stations in these divisions of Buses are appeared across the divisions even though there are variations among states.

The average distance to bus terminals at Airplane s hows the longest average travel distance about 1, miles with relatively large standard deviation among samples. This implies that air transportation covers various service distance range s. In addition, there are large variations in average distances of air travel. For example, the average trip distances of both WNC Interestingly, the two PAGE 41 41 former divisi ons have no large scale airports, while the three latter divisions have one or more of airports that provide nationwide services.

Table 3 4 presents descriptive statistics of long distance travel by mode, and Table A 2 in Appendix A shows the same feature s comparatively among divisions Figure 3 1. MA and P AC divisions show relative large share of trip distance between 50 and 99 miles, w hile MT division records the smallest share at 45 percent in the same distance range. These patterns may be associated with the share of travel mode used. For example, MT division was the lowest for personal cars use, while airplane s share was the largest among census divisions.

Table 3 5 and Figure 3 2 present these patterns of mode share by distance group. In contrast, a irplane actively appeared over miles travel distance range and its share increases as travel distance increases. Airplane accounts for higher than personal cars from miles travel distance.

Both bus and train s shares are less than 3 percent in all distance ranges. These patterns are expected to appear across the census divisions even though there would be variations. Figure 4 1 illustrates the share s of mode s used for long distance travel by distance. Figure 3 3 Share of Travel Mode by Distance Group As shown in table 3 6, people make long distance travel as they have social and recreational demand among purposes.

It accounted for Meanwhile, medical and dental trips show the shorted standard deviation of Table 3 6. Home 1 In detail, m ore than 60 percent of long distance trips have home as their origins, while nearly 13 percent ended at home. Work place s accounted for 6. Table 3 7. Assuming that business travelers are less sensitiv e to travel cost but more sensitive to travel time, these mode shares are acceptable. In particular, train s share of 2. Both airplane and bus s shares are similar with train.

These patterns result in the relatively low share of personal cars in location s with heavy rail services. T ravelers in MSA s of 1 million or more with heavy rail service are favorable to train, while residents who live in MSA s of 1 million or more but ha ve no heavy rail service are likely to be similar in choosing train.

Instead, these MSAs show higher share of airplane in making long distance trips. Personal cars account for more than 90 percent in non MSA areas These patterns would be interesting to examine in the mode choice model. In addition, s tandard deviation of Interestingly, train appears slightly higher at high income group. Therefore, it would be interesting to see whether these differences can be proved by the choice model.

Figure 3 4 Share of Long Distance Travel Mode by Income Group E xistence of child seems to have no effect on the average travel distance and the capability of traveling distance range. In addition, the standard deviations of This is double of airplane s share for travelers traveling with child.

The average number of people traveling together might be responsible for this gap. People traveling without child have an average of 1. It should be noted that t his gap between groups is similar with the average number of travelers for personal cars at 1. However, tota l travel costs are different because personal cars costs are the same regardless of the number of travelers on that trip, while air travelers costs are expected to increase as number of people increase Table 3 12 and Figure 3 5 display descriptive statistics of long distance travel depending on existence of child.

Similar with existence of child, traveling with the elderly this includes travels made by the elderly seems to have no effect on traveling patterns such as average travel distance and the range of traveling d istance. As shown in table 3 13 and figure 3 6, the average distances of However, pe ople show different mode choice behavior, if they are traveling in company with the elderly.

Travelers are likely to choose more personal cars when they travel with the elderly in the group, while people traveling without the elderly show relatively higher share of airplane. In order to simplify model development process this s tudy concentrate s on the attributes that can be quantitatively measured among various set of potential explanatory variables In estimating the model, t his study is particularly, interested in the impact of travel time and cost on mode choice decision s for long distance travel In case of public transportation modes, both travel time and costs include access time and cost s to t he closest intercity terminal.

Then, this study attempts to test the impacts of traveler s characteristi cs on mode choice decisions by adding individual ch aracteristics such as age related attributes, purpose of trips and income group to explanatory variables For that, this study transforms individual attributes into alternative specific variables.

In addition, this study applies dummy variables that repre sent the geographical characteristics of residence location. Furthermore, this study attempts to examine whether the impacts may vary across smaller geographical locations because these MSA categories are large.

These sub categories include state, combination of census division, MSA sta tus and presence of a heavy rail system, and major transportation corridors. Table 4 1 presents potential explanatory variables that may be applied to the mode choice models. Meanwhile, t ravel cost commonly m eans cost of driving automobiles or fare level of public modes such as air, train and bus. All these subcategories of travel time and costs are also expected to have negative sign in the estimated models.

It should be noted that the ratio of coefficient of travel cost over travel time implies the monetary value of travel time b y which traveler makes trade offs between v arious travel modes. In general, it is known that public intercity mode bus or train users are more sensitive to changes in travel fares tha n travel time, while air passengers a re not very sensitive to travel cost and are highly sensitive to travel time having the highest value of time Ashiabor, Baik, and Trani, ; Bhat, b; Carlsson, The conditional models are expected to provide the travel time and cost combinations to make a new alternative intercity mode attractive.

Travel distance can be used as an alternative indicator of travel time and cost because both time and cost are highly related to travel distance. T rip distance is expected to show negative sign and this implies that longer trip distances are associated with more time, more expense and less frequency. Therefore, this is also negatively related to individual s utility.

In particular, distance variable is expected to interact negatively with automobile users utility while, airplane user s utility is expected to increase as distance increase. Many previous studies have suggested that there exist thre mode choices vary Bel, ; Hensher, ; Jorensen and Preston, ; Kitagawa, Terabe, and Sarachai, ; Wardman, In addition to the attributes of the alternatives characteristics are also frequently emplo yed in many mode choice models Thus, this study includes th es e variables into the model by transforming personal characteristics into alternative specific variables.

In specific, t his study include s income trip purpose, and age related attributes i n the model. Hence, it is expected to have a positive sign for air travel, while a negative sign for pub lic modes such as train and bus. Trip purpose s such as work related trips and social and recreational trips are employed in the models as dummy variable s that present the reasons why people travel. Among trip purposes, work related variable is expected to positively affect to business travelers, particularly those who use airplane, since it is traditionally known PAGE 55 55 that business passengers prefer air travel while non business passengers tend to choose tr ain or automobile for intercity travel.

This is because most business passengers have no burden to pay for their trips. In addition social and recreational trips are expected to have positive and significant effects on personal car and ground intercity mode users utilities. Social and recreational trips are known to be less sensitive to travel time, and large in the average number of travelers.

In regard of age effects on mode choice decisions, this study uses dummy variables that reflect existence of child or the elderly in the travel group. Age is considered to affect to the mode choice decisions However, t his study cannot directly examine the impacts of age on mode choice decisions because household level samples can include many travelers, and thus it is not possible to apply a single age into the model.

Given this condition, this study examines whether people traveling with child or the elderly are different in their mode choice decisions. Existence of child implies increase in number of travelers on that trip, and thus increases in trav el costs except personal cars. Meanwhile, existence of the elderly may imply both increase in number of traveler and more demands on comfortable and convenient modes With these variables, this study expect s that traveling without child have positive effec ts on the probability of choosing airplane, while traveling with the elderly could increase the probability of choosing personal cars.

Notably, there are studies that emphasize the impacts of spatial attributes on travelers mode choice decisions. MSA size, as a n indicator of large PAGE 56 56 population and more public transportati on service is expected to have a positive sign for train and bus because a large metropolitan area. Existence of heavy rail service is also understood in the same context as MSA size. Higher demand of transport ation will facilitate public transportation n etwork, and thus large population will be positive ly related with the probability of choosing bus and train.

T he number of traveler s on the trip is another variable that is expected to interact negatively with utility of certain modes including air. As number of travelers increase to tal travel cost increases and thus travelers may choose personal automobile instead of airplane because the larger party size the less a pers on is able to afford an expensive alternative Capon et al. Figure 4 1 presents these relations in graphic.

Figure 4 1 Hypothetical relationship of explanatory variables to travelers utility PAGE 57 57 Estimating Synthetic Travel Times and Costs In order to develop the mode choice models, travel time and costs for each alternative mode is essential.

T he NHTS data only provides actual travel time for each trip segment, and thus this study can calculate total travel time for a given long distance travel made by the mode used. However, the NHTS does not provide travel costs of the mode used, and travel times and costs for other alternative modes that can be potentially used for each trip. Therefore, this study develops synthetic travel time and costs for all alternative modes using average driving costs per mile by passenger car type, air passeng ers fare and flight distance survey, bus and train fare and travel time tables.

In estimating synthetic travel time and costs, this study tak es into account one way trips assum ing that people use the same mode of transportation for their returning trips. Automobile Travel Times and Costs Automobile travel time mainly includes driving time and rest time for night. In order to calculate d riving times this study applies average travel speed of 60 miles per hour to travel distance and then inflates calculated travel time by distance ranges: T ravel distance less than miles: 10 percent T ravel distance between and miles: 7 percent Travel distance over miles: 5 percent This method is simple, but effective to calculate driving time.

In addition to driving time this study applies time for intermediate overnight stay after every 10 hour driving. According to Ashiabor, Baik and Trani the Virginia Tech travel surveys re ported that travelers make overnight stay after 8 hours and 10 hours trips for business and PAGE 58 58 non business trips, respectively. However, this study simple applies non business standard because business trips accounted for less than 25 percent of total long distance trips.

The total travel times are sum of driving time a nd overnight time. The driving costs are calculated similar to the procedure of driving time. The driving costs are calculated by multiplying an average driving cost per mile to travel distance reported in the NHTS. In calculat ing driving cost of the personal cars, this study use s the average driving costs per mile that has been issued by the American Automobile Association AAA in every year AAA, The NHTS data does not present exact model and maker of personal cars used, but it provides information about modes such as car, mini van, sports utility vehicle SUV and pickup truck and other truck.

By matching these classifications to the vehicle categorie s of the AAA such as small sedan, medium sedan, large sedan, SUV, and minivan, this study calculates an approximate driving cost of each long distance trip. The average driving costs are given as: Average of Sedan: It would be ide al to increase the lodging costs depending on number of people on that trip, but this is not considered in this study.

In addition, these fare and flight time data can be separately obtained for origin and desti nation flights. So, it is possible to appl y different fares depending on incoming or out going travel. In theory, air travelers prefer airports with low fares, high departure frequencies, and a large number of connections to other airports Ashiabor, Baik and Trani, This may be true because air passengers for example, pay 55 cents per mile at Orlando International Airport traveling to destination within miles, while people pay 90 cents per mile to make the similar flight from Gainesville Regiona l Airport.

Moreover, Orlando International Airport provides wide range of flight destinations with frequent schedule than Gainesville Regional Airport. However, this study simply assumes that air travelers c hoose nearest airport from home. In order to ide ntify the cl osest airport from each sample, this study uses the location information of census block group where each household is located.

T his study associates the average fare and flight time from the DB1B data with the nearest airport to each sample, and then calculates air fare and flight time for each sample at given travel distance. PAGE 60 60 In addition to air fare and flight time, this study adds access costs and time to airport using the distance from sample household to the neares t airport As shown in table 4 2, the average distance s to airport are relatively short at 1 4.

In contrast, ESC records the longest average distance of Interestingly, ENC is the second longest in average distance to airport at In estimating access time and costs to airport, this study applies the same procedure with driving t ime and cost calculation.

T his study assumes that people uses personal cars to access airport Table 4 2. It should be noted that total travel costs can be calculated by multiplying air fares with number of people on that trip. Unlike personal cars, air fares increase depending on number of travelers, and thus using total travel costs is reasonable.

Individual s who are traveling alone may have PAGE 61 61 greater tendency to choose common carriers while, people travel ing with family members may prefer automobile to lower the travel cost per person. Bus and Train Travel Time and Costs B oth travel time and costs for bus and train are obtained through similar procedure as that of air travel time and costs. In both modes, travel time consists of include in vehicle time and access time, while travel costs comprises fares and access costs.

Therefore, the total travel time and cos ts are sum of these two components. First, this study builds up travel time and fares data for both modes in reference to Amtrak and Greyhound s real service operation s In order to identify patterns of travel time and fare for bus and train, this study collects actual fare and travel time from randomly selected origin destination pairs in regard of distances and location characteristics of terminals Using these times and fares, this study estimates the travel distance per minute mile per minute and travel costs per mile cents per mile by distance group.

In reality, b oth bus and train s fares do not monotonically increase as travel distance increase s but increase stepwise Yet, this study allows them to increase proportionally within distance group Secondly, this study estimates the shortest distance from each household to bus terminal and train station like as this study did for airport. In calculating the shortest distances, this study uses 1, Greyhound terminals and train stations. Table 4 3 presents the average distances by census division.

In case of train station, people in both WNC and ESC divisions are the farthest in an average distances at 75 miles and PAGE 62 62 Table 4 3. Like air fares, t his study reflects number of travelers to the total fares of both bus and train. Results of Model Estimations As explained above, this study intends to test the impacts of alternative specific characteristics o n the probability of choosing a specific mode.

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They exchange in a format as shown in the diagram, i. But they make a rule that once a person has exchanged with another person they CANNOT exchange with that person again. After they all get their new cards they suddenly realise they don't want their new cards but they want their original ones back. But they are now faced with their original rule and can't exchange with the person who gave them a card. Can you work out a system :. Call the players A, B, C and D and assume they have x discs left.

Let's also assume they lose in the following order, A, B, C and D. Aayush Kumar :. On July 4 th Cern held a press conference. Conclusion, they have found a particle consistent with the Higgs Boson. Higgs Boson is a quantum particle that fits into the Standard model.

It is the "force carrier" of the Higgs field similar to a photon to the electromagnetic field Elementary particles interact with the Higgs field, the more interaction they have the more mass they have So basically the HB particle has the responsibility of giving mass to elementary particles not to be confused with giving macro object mass, however now for the mindbender And here is some basic maths The mass of the HB particle they "found" is approx. So for the chain, 2 numbers will be unpaired when there are still n-2 numbers to be laid down.

Build a graph with each node representing one number among 0, Each domino that relates 2 numbers is then represented by an arc between the nodes representing the 2 numbers that can be identical. If n is even, the arity of all nodes is even, so there exists an Eulerian path. Once this path is determined, lay the domino in the same order as that of the arcs edges to build the chain. Hence there is an Eulerian path.

Let's assume that a is the side of the Outer Dodecahedron. Let's find the length of the Icosahedron in terms of a. Firstly we need the dihedral angle of the dodecahedron, i. The dihedral angle is Now the side of the icosahedron runs from the centre of on of the pentagonal face to the centre of an adjacent pentagonal face. Let m and n be the number of hexagons and pentagons, respectively.

Let V, F, and E be the number of vertices, faces, and edges of the polyhedron, respectively. What is the ratio of the cube's volume to the inscribed Octahedron where each vertex of the octahedron is in the centre of each face of the cube Kepler's dual polyhedrons? Let C be the point where the "altitude" intersects AB.

My point was since he paid for his pint, he finally drank it. So the barman had topped for 6 pints, hence the equation. William Chau :. More squares in a circle: find an expression in n for the ratio of area of 2n-1 squares to the area of a circle that contains the squares fitted perfectly on the diameter where n is an odd number Similar to the last problem: Find the area of the enclosed space between the 4 circles, each with a radius of 1 unit :.

Paul Beirne:. Lau Vincent:. The pole is located in the bottom left of the field 3 m out and 2 m up. What is the grazing area for the sheep give your answer to the nearest square metre? Ah getting better i Let me try the "gnomon" way is gnomon also an english word? Now I take a cube 4 points wide. I can consider this cube as 4 square layers of 4x4 points. From each layer I can "cut" 2 T3 triangles out of a rectangle of 3x4 points. Since I have 4 layers I overall get my 4x2xT3 points.

On each layer I'm left with 4 points on a single line, so for the 4 layers, I get a square made up of the 4 single lines, i. So I've got all the extra points I needed and I reach T4. The same can be done for Tn.

I thought about working the problem by induction, but it isn't all that elegant. Say S n is the sum of integers up to n and T n is the sum of cubes up to n. Mary Tierney:. Last night I was thinking of the "Go First" dice There is a much easier system for 2 players. So can you design 2 dice with the following conditions: 1 There MUST be a winner on a single event an event being 2 people throwing the dice and the winner is the person with the greater number.

Basically 3 numbers greater and 3 numbers less Christopher Duff: If the dice are not similar if think they should not would something like 1 3 5 9 11 13 and 2 4 6 8 10 12 work? Which one has a greater quality? Re-write the numbers 1 1 2 2 3 3 4 4 so that they meet the following conditions. The opposite angles, A, are equal. What is the length of the diagonal d?

A man walks due south to location B, where there is a road due east from B to E 12 km long. He then walks half the distance from B to C to arrive at D which also has a road to F going through E. The line FG is due east of point A.

How far is it from F to G? I decide to race him over the m. Now I'm not crazy well I am a little bit so I decide to race in a car. The car I'm in is pretty rubbish and accelerates from rest to 60 miles per hour in 12 seconds assuming uniform acceleration. So assuming Bolt runs the m in the same time as his olympic time and we both start from rest, would I beat Usain Bolt in the car?

What is the area of his grazing area? Let's also assume that the earth is million km from the centre of the sun and that mars is million km from the centre of the sun. Let's also assume it takes days for the earth to orbit the sun and days for mars to orbit the sun. They orbit the sun in the same direction and on the same plane.

In Season 6 Episode 15 "Benderama" The Professor invented a machine called the Banach-Tarski Dupla-Shrinker which is taken from the surprising Banach-Tarski paradox which states you can "cut" up a sphere into finitely many pieces and form 2 other spheres of the same size as the original. Bender of course abuses this machine in order to reduce his workload.

He creates 2 smaller copies of himself and those 2 create 4 more, even smaller, copies of themselves, and then those 4 create 8 smaller versions of themselves and so on Taking the first 2 copies of Bender as the first generation of "Benders" and taking Bender's mass to be M. They play a game where there are 3 winners and 1 loser after the game. The loser must double the amount of discs for each of the 3 other players.

After 4 games they have all lost 1 game each and they now have the same amount of money. I am interested in the Volume of the snub cube as it is related to the Tribonacci numbers which is similar to the Fibonacci numbers except the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms e. The ratio of two adjacent Fibonacci numbers tends towards phi, the golden ratio, BUT the ratio of two adjacent Tribonacci numbers tends to the tribonacci constant which you can look up , as well as the volume of the snub cube.

The 4 vertices of the tetrahedron are touching the centres of 4 of the cubes faces. What is the ratio of the volume of the Cube to the volume of the embedded tetrahedron? Specifically double-6, double-9, double, double and double forms. They come as a complete set, i. All possible combinations of dominoes are in the set. The game is played so that the end of one domino is joined to another domino For what values of n 6, 9, 12, 15 and 18 and for any general n, can we take the complete set of dominoes and lay them out according to those rules without being left with any dominoes?

Source: Adapted from a problem posed by Orly Terquem in Continuing on my love for the platonic solids: Kepler also showed it was possible to nest an Icosohedron in a Dodecahedron so that each vertex of the Icosahedron was in the centre of a face on the dodecahedron. Linear Congruences.

Diophantine Equations. The Theorems of Fermat, Wilson and Euler. The Divisor Functions. The Prime Divisor Functions. Certified Signatures. Primitive Roots. Knapsack Encryption. Quadratic Residues. Fast Transformations and Kronecker Products.

Quadratic Congruences. Pseudoprimes, Poker and Remote Coin Tossing. Generating Functions and Partitions. Cyclotomic Polynomials. Linear Systems and Polynomials. Polynomial Theory. Galois Fields. Spectral Properties of Galois Sequences. Random Number Generators.

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