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ing for replenishable and seasonal products
Retail price modeling for replenishable and seasonal products
January 30, 2019 • 31 min read
Pricing is one of the key factors that drives a successful retail business model, and it is also one of the most difficult to manage. Retailers made lots of improvements on their pricing strategies
over the last decade, moving from cost plus or value pricing to modern day AI-driven price management. In this article, we will discuss the common process with which retailers set their pricing, the
challenges that they are facing in the area of price management, and how they incorporate big data analytics and machine learning into their pricing strategy.
1. The retail pricing process
Retail has a complicated pricing process that involves almost all of the functions in the retail company – from Finance, Merchandising, and Inventory Management to Pricing, Marketing and Store
Operations. We start with a high-level description of this process to set the stage for further discussion of pricing models, and then explain how price modeling fits into the big picture and helps
to improve retail profitability.
1.1 Plan-to-sell process
A common practice of the retail pricing process is to start by setting up a strategic or financial goal. The financial goal is usually based on historical data, for example, a 5% increase of year
over year sales and margin, or $20 billion sales and $8 billion in margin. Once the goal has been set, multiple teams build a plan to help achieve that goal. This process, known as the plan-to-sell
process, is led by a merchant team, with Assortment Planning, Inventory, Pricing and Marketing teams helping the merchants manage their respective processes and achieve the financial goal, as
illustrated in Figure 1.
Figure 1. An overview of the plan-to-sell process.
The plan-to-sell process is generally focused on developing two groups of assets: a Buy Plan that specifies what to buy, and a pricing plan and models that specify how to sell, as shown in Figure 2.
Figure 2. Assets developed in the scope of the plan-to-sell process.
The plan-to-sell process starts with development of a Buy Plan by the merchandising team. This plan typically includes the following:
• Composition of the buy: What items do we want to carry in our stores? Size, color, and all other attributes, including a price banding for each program (program is a set of SKUs, e.g Levis Boys
Jeans of different fit, style, color, and size can be a program).
• Size of the buy: How many units do we need for each item/program/category?
• Lifecycle of the buy: Usually only applies to seasonal items.
Once the merchant team has established the initial buying strategy, the pricing plan can be developed to specify how the merchandise will be sold. This step is known as pricing optimization planning,
and we describe it in more detail in the next section.
1.2 Price optimization planning
Price optimization planning aims to create a high-level pricing strategy, and it often uses rougher pricing models. The actual price setting is done later at the selling stage using more fine-grained
and accurate pricing models.
Just like any optimization problem, price optimization planning requires the definition of target metrics that need to be maximized, variables that can be changed to maximize the target metrics, and
constraints that have to be taken into account as we search for optimal values of the variables. For most retail companies, these aspects of the price optimization planning look as follows:
• Variables. The structure of retail pricing starts with cost, and the listed retail price or regular price is obtained by adding a markup to the cost. Promotional offers or markdowns can be
applied on top of the regular price to obtain the selling price, also known as out-the-door (OTD) price. Price optimization can focus on regular prices or OTD prices, depending on the price
communication strategy (everyday low pricing or high-low pricing). Without loss of generality, we will assume that the goal of price optimization is to optimally adjust the OTD price, and that
these price adjustments are communicated to customers via promotions and discounts.
• Metrics and objectives. The fundamental goal of price optimization is to maximize profits. However, the exact metric to be optimized can differ across different product types. For replenishable
items, since inventory can be replenished and there are no constraints for the inventory ownership, the goal is to optimize profit for every selling period. For seasonal items, because the
inventory is limited and fixed, the goal is to maximize sales for the entire item life cycle.
• Constraints. Price optimization can be constrained by the available inventory, as well as retailer’s pricing rules and policies. For price optimization planning purposes, it is typically assumed
that the inventory will match the sales goals.
The price optimization planning usually starts with the following input data:
• Sales history and trend
• Cost components
• Promotion and events
• Elasticity of demand estimated with sales data
• Product and store attributes
• Seasonality and holiday impact
• Competitive pricing
These inputs are used to roughly estimate optimal pricing parameters. The pricing models used at this stage are pretty high-level: they usually stay at category level (i.e. use the average price
elasticity of the entire category) of the entire geographic location, and assume that the available inventory will be enough to achieve the sell-through goal in the plan. The pricing model uses
historical sales data to calculate the own-item price elasticities of demand, and assumes elasticities are constant with no geolocation adjustment or price discount adjustment. Or in other words,
elasticity for one particular item will remain the same no matter whether you are selling it in a store in Hawaii or Alaska, or whether the price is 20% off or 80% off. For cross-item effects,
competitor pricing is adjusted on an ad-hoc approach if necessary. For example, a private label fashion dresses will not have cross-item effects and competitor price effects since every category of
dresses is not the same, and a private label is only carried in one retail brand. But for brand name jeans, like Levis, some ad-hoc adjustment is necessary.
The output of the pricing model run is a complete pricing plan that optimizes the pricing strategy of the product over an entire life cycle. A plan for seasonal products will include both an
in-season promotion pricing plan and a post-season clearance markdown pricing, as the pricing goal for the seasonal plan is to achieve maximum sales revenues for the entire item life cycle or entire
season. A plan for replenishable items is usually much simpler since it aims to maximize margin in each selling period.
This pricing plan will also get fed back to the Buy Plan to adjust the size of the buy. For example, if the original Buy Plan does not have enough unit buy to achieve the financial goal after
exhausting all the pricing options, then that information needs to feed back to the Buy Plan for increasing the unit buy.
2. Challenges of the retail pricing process
The previously described plan-to-sell model that is currently used in most retail businesses is more of a strategy and people-driven process than a data and technology-driven process. The lack of
predictive analytics, gaps in process automation, and the limitations of price management systems can and often do significantly decrease pricing efficiency — and eventually harm profits. In this
section, we describe several typical examples of such issues.
2.1 Inaccurate forecasting and lost sales
Price planning often uses simple demand models, so that the forecasting is based on trends. In other words, the plan-to-sell model looks at what we sold in the past and estimates future demand based
on these sales numbers.
One of the most common issues with this approach is that lost sales are ignored in these estimates. The true level of demand is a sum of actual sales and lost sales, so a demand forecast that relies
solely on sales number and does not account for lost sales is not accurate. This issue can be alleviated by using more advanced demand models that account for inventory and estimate true demand.
2.2 Inaccurate modeling of the demand heterogeneity
The second challenge is that pricing planning modeling typically stays at too high a level, although it uses different techniques for seasonal and replenishable items. It often assumes constant
elasticity and assumes that all geo locations have the same selling pattern and trend. It also assumes no inventory constraints, which is definitely not true, especially for seasonal items.
One common issue with this approach is that models with high level of aggregation cannot properly account for demand heterogeneity. For instance, Figure 3 shows a real-life example of a
season-to-date sell-through chart for three price bands ($2.99, $4.99, and $6.99). There are four sell-through percentage groups in the chart (less than 50% of units sold, 50% to 70% of units sold,
70% to 99% of units sold out, and 100% sold out or out of stock), and each bar shows the percentage of purchased merchandise (in terms of SKUs) in the given price band that fall into the
corresponding sell-through bucket. For context, the chart shows the situation on the 30th week of sales for a product with a life cycle of 40 weeks.
As you can tell for the $6.99 priced items, 48% of total purchased inventory units are sold out in their geolocations, while only 2% of units sold less than 50%. Given that this is a snapshot of the
situation at the end of product life cycle, it tells us that we priced too low for those sold out locations, and priced too high for the low sell-through areas. Therefore, if we price all store
locations with the same discount, we would leave lots of money on the table.
Figure 3. A real-life example of sell-through rates variation.
In order to resolve this issue, we have to go to a very granular level to build our pricing model, and feed the model with real time inventory and selling data to make the price recommendation more
optimal. The level that a retailer can use to run the pricing model depends on the price execution limit. For example, if Levi’s Boys Jeans are displayed on the same fixture, then the model can only
pick one price for the entire program by store. If Ninja Blenders have one facing for each model on the shelf, then the model can price those blenders at the item/store level. How frequently the
retailer can update the model depends on the size of the data set, IT resources, advertising frequency and price change labor in the stores. It typically runs once a week, while online store pricing
can run much more frequently.
2.3 Operational challenges
Advanced modeling techniques can help resolve issues with unaccounted lost sales and heterogeneity of demand, but there are other challenges that lead to suboptimal pricing which are very difficult
or impossible to resolve by just improving the pricing model.
For example, a seasonal buy usually has a long lead time. It often takes one year from planning the buy until you can start selling the products. Such a long lead time frequently results in not
buying enough units in certain areas. Inventory allocation is another challenging task to get done just right, as retailers always end up not having enough inventory in certain stores, but too much
in others. Retailers also face other operational challenges which constrain the price optimization, such as store execution and advertising execution. They wish that they can use real time pricing to
change prices several times a day, but this would cost too much store labor and would be confusing for customers. Retailers also wish that they could have different versions of advertising by store
clusters or geo locations, but it costs too much to alter advertising with such multiple versions.
In the retail world, “pricing touches everything, and everything touches pricing”. Price management creates cross-functional chaos, so a price solution that addresses only pricing itself will not
work for a retailer. That is one of the key reasons why most of the off-the-shelf price management software does not work well in the complex retail environment.
3. Basic price modeling
In the previous two sections, we outlined the context in which retail price modeling and optimization happens, and described how it relates to some other enterprise processes. In the remainder of
this article, we drill down into the mathematical details of price modeling. We introduce several basic methods and techniques in this section, and then develop more specialized methods separately
for replenishable and seasonal products in the subsequent sections.
3.1 Linear demand model
The most basic price optimization model can be defined using the following simple equation that ties together an item’s cost, demand, price, and profit:
In order to evaluate the basic profit equation for a given price, one needs to specify and estimate the demand function that quantifies the dependency between price and demand. The most basic choice
for the demand model is a constant-elasticity model, also referenced to as a loglinear or simply linear demand model. This model is based on the assumption that the demand decreases by a fixed
percentage in response to each percent of price increase:
where elasticity is a coefficient with an estimate based on the sales data. Given that this coefficient and current demand are known, one can straightforwardly evaluate price increase and decrease
scenarios using the basic profit equation above (by changing the price by a certain percentage and demand by the same percentage times elasticity). It can also be shown that the constant elasticity
assumption is equivalent to the following relationship which we will use later in this article:
In the coming sections we will discuss the limitations and trade-offs of this simplistic model, as well as processes needed to apply it in practice.
3.2 Limitations of the linear model
The linear demand model uses sales history and price change history data to estimate price elasticity, and assumes that price is the only factor that drives demand change. The linear model is
typically appropriate for the pricing planning stage, since the level of the planning stays at the category level. This has enough data to normalize the demand and to not have much impact on
cannibalization and affinity. But once the retailer passes the planning stage and goes into the selling stage, a linear demand model will not be enough to catch all the moving pieces, since pricing
is not the only parameter to impact the demand. The following parameters also need to be considered in the comprehensive modeling process:
• Promotions and events. Retailers use multiple traditional marketing and digital marketing channels to promote their products. They also use different kinds of offers to get to optimal prices,
such as percent-off offers, buy one get one free, coupons, stacking offers, etc. It is important to figure out the promotion lift by marketing channel and offer type, since they have different
impacts on the promotion lift. For example, selling the same Barbie doll for $20 via an ad vs. $20 not via an ad would yield different units.
• Price perception and competitive pricing. Some items are highly price sensitive, while others are not. For example, everybody knows how much detergent from Tide should cost, but very few people
know how much a private labor cocktail dresses should cost.
• Cannibalization and affinity. When one item has been promoted, negative demand effects cross items, and positive demand effects attached items.
• Seasonality. The demand for certain items has seasonal trends that need to be incorporated into the models for accurate forecasting.
• Volume impact. The linear demand model works well when the data has been normalized, or in other words, when every geo-product sells the same. But when the selling pattern is different by time
period or by location, the model accuracy will not be very good.
• Store attributes. Elasticity will be different in different stores, depending on the demographic of those stores and their distance from competitors.
• Out of stock or size breakage. If you are running out of stock or desirable sizes in the selected location, no matter how low the price is, the unit lift just will not be the same.
• Zero sales. The traditional linear demand modeling ignores weeks with zero sales, biasing estimates of price sensitivity. Consider a sales chart example in Figure 4. The prices in this chart are
the listed promo prices for a certain product, which go up and down by week depending on how a retailer promotes this product, and sales units which represent the actual sold units. The circled
weeks are the cases when this store had zero sales after a price increase. Traditional models would remove both the unit and price change history for zero sales, which (1) biases elasticity
estimates toward appearing less price sensitive, and (2) optimizes prices based on these flawed estimates.
Figure 4. A real-life example of zero sales caused by a price increase.
We will discuss how these various factors can be incorporated into the pricing process in the following sections, which are dedicated to replenishable and seasonal items.
3.3 Choosing the level of aggregation
There is always a trade-off in terms of which level of aggregation the retailer picks to run the pricing model. There are two dimensions of level aggregation. One is at the geolocation level, from
one store all the way to all of the store locations. The other is at the merchandise hierarchy level, from one kids apparel item with a specific size, color and style all the way to the entire Kids
department. Inventory sell-through and elasticity work better if you go down to the granular level like item/store, since the elasticity is different from one store to the other, and from one item to
a group of items.
Most importantly, the inventory sell-through percentage also varies from one store to the other, even with the exact same item. From example, the same down coat could be sold down to 1 in the Chicago
store vs. still have 10 left in the Nashville store. Obviously, we need to price these two stores differently to maximize the margin. However, the lower level data always come with tons of noise and
outliers. The goal is to find the correct aggregation level that gets the outputs most closely fit to the actual demand curve. There are multiple ways that retailer uses to resolve this issue, but
the simplest way is just to move up the aggregation level. If the item/store can not yield a good fit, then we would just move to an item/store group level for aggregation. Or, if that doesn’t work,
we can move up to the item/climate zone level. The best way to resolve this is to use machine learning techniques to get rid off the noise and outliers in the data, and find the nearest neighbor that
provides better approximation to the actual patterns of demand.
3.4 Modeling and optimization workflow
The traditional retail price modeling workflow has two parts: analytics, and optimization. Analytics use historical data to develop a baseline forecast – what would happen if we take no pricing
action and just sell everything at regular prices. As we discussed in the previous sections, this forecast can be created using very basic demand models for preliminary planning purposes, and then
refined using more advanced techniques to incorporate seasonality, promotional support, and cross-item effects. The forecasting model quantifies the dependency between the demand/profit and price,
and thus enables what-if analysis of different pricing scenarios.
Optimization is the process that runs every possible combination of prices for every available price change date by allowed geo-product combination, and then chooses the option that provides the most
revenue, or the margin which meets the business objectives and constraints. This process also estimates the amount of inventory needed to fulfill the demand at the recommended price. This price
modeling workflow is summarized in Figure 5.
Figure 5. A price modeling workflow.
3.5 Model diagnostics
Optimal pricing from the pricing model generates millions of dollars in profit for retail companies. Therefore, it is very important to make sure the pricing model is recommending the correct price
to maximize the profit.
The common way to diagnose the pricing model health is by running a hold out forecast. In this approach, one runs the model to forecast sales for the previous couple of weeks using its recommended
prices, and then compares the forecasted sales with the actual sales to see how big the difference is. The hold out forecast usually forecasts both inventory constrained sales and unconstrained
sales, then uses the mean absolute percentage error (MAPE) to measure the difference between the forecast and actual numbers. The acceptable MAPE ranges from 0% to 30%. If the error is above 30%,
then the model setting should be changed, and the hold out forecast should be re-ran until the MAPE comes down to the acceptable level. This self-learning process can be automated to improve the
It can also be the case that the model performs well on average and achieves good MAPE, but has large prediction errors (outliers) for certain products or dates. This issue can be mitigated by
running hold out forecast at a very granular level, both in terms of product hierarchy and time.
4. Pricing model for replenishable items
The basic pricing model described in the previous section is acceptable for preliminary planning, but the actual price optimization for the selling phase requires the development of more accurate
models that incorporate a larger set of signals and effects. These more advanced models are typically more specialized as well, as they are built for specific product categories. The first such
specialization we will consider is a pricing model for replenishable items.
The best approach to pricing replenishable items is to combine machine learning and decision automation with human inputs. Modern day machine modeling helps to process big data, but it also leverages
human inputs, and has governance of business rules. Additionally, it enables the input of clean attributes of the products, which is critical: the model output will not be useful without good product
attributes. Business rules also act as a guardrail to the pricing model outputs, helping to prevent erroneous decisions and keep micro-decisions consistent with the overall strategy.
The pricing model for replenishable items has to account for several factors that are summarized in Figure 6. Own price elasticity is the most important effect in the pricing modeling, and the other
effects are secondary. Cannibalization and affinity is usually the least relevant for pricing among all the secondary effects. We will further discuss below how these different effects can be
quantified and incorporated into the optimization model.
Figure 6. The main factors and effects that need to be accounted for in a pricing model for replenishable items.
4.1 Own item elasticity vs. cross item elasticity
Own item price elasticity is the effect of price changes of a focal product on the sales of a focal product. Cross item effects or a comparable product index are the effects of price changes of other
products on the sales of a focal product. Simultaneously modeling all own- and cross-item effects in a single, comprehensive demand model is the best way to price basic replenishable items.
The cross effects between multiple items that share similar product attributes can estimated by running a linear regression. In this case, the cross elasticity will be quantified by a regression
coefficient for the price of one item in the demand model for another item. Figure 7 shows an example of cross-elasticity analysis for four similar items in the coffee maker category. In this
example, private label coffee maker pricing may be a lever for moving branded items, because customers appear to trade into/out of branded items as similar private label coffee maker prices rise/
fall. However, the opposite is not true, and the demand for private label coffee makers is relatively insensitive to branded prices.
Figure 7. A real-life example of cross elasticity analysis for four similar items in the coffee maker category.
The example in Figure 7 can be extended for an arbitrary number of items, but retailers typically avoid carrying a large number of similar items that have a cannibalization impact on each other, as
an over-assortment of merchandise can be a big hit to margins. In the coffee maker example, you might see a retailer carrying 20 different coffee makers, but not all those 20 items will have a
cannibalization impact from each other. For instance, a $20 coffee maker will not compete with a $600 one.
4.2 Competitive pricing
A competitive pricing strategy is not all about web scraping, price matching, and undercutting competitor prices. Otherwise, a retailer would end up with the everyday lowest price on the market,
which is a big margin hit, and something that most retail companies would not be able to afford. Competitive pricing is about setting the price based on what the competition is charging, and using
the competition’s prices as one of the pricing model parameters to determine the optimal prices. The following two steps are what retailers often use to incorporate the competition prices into the
pricing modeling.
First, a retailer runs analytics to figure out the impact from competitor’s prices on their own sales. In order to run the correct analysis, retailers need to have a thorough study not only on its
internal company data, but also on competitors’ activities (such as pricing, promo etc.). It is also very important to have a profound knowledge of the market and the retailer’s position in it. This
analysis needs to combine human knowledge and machine knowledge, and is a combined effort from multiple teams within the retailer. Merchants know their products best, so they can list the items that
they feel are important to stay competitive on the market. The marketing team understands who their competitors are in each specific category, so they can provide knowledge on what other retailers
their own customers also shop at, and work with 3rd party marketing data providers to get the detail pricing promo activities from their competitors. The pricing/data science team can use this
information to run models and quantify the exact impact from the competition to the retailer’s own sales.
The impact from the competition to own sales is generally quantified using the cross item price elasticity analysis that we discussed in the previous section, but with regard to competitor’s prices:
If the cross elasticity is positive, that means we need to consider the competitive pricing when we price our own items. Since our own sales units reduce when a competitor reduces their prices, our
own sales increase when a competitor increases their prices. Competitive price cross elasticity can be measured at the individual item level or with a group of items. The level of measurement is
defined by the business rules, and depends on how competitors are promoting their items. For example, if retailers usually promo their underwear products by brand with multiple items during the back
to school season (like all Hanes underwear being on sale for one week), then the competitor analysis needs to be done at the brand level instead of the individual item level.
The second step is to generate operational pricing rules based on the cross elasticity parameters estimated in the previous step. Once retailers have figured out the exact impact from competitors,
they will add business rules in the modeling section to guardrail the model price outputs. For example, let’s look at one $79.99 brand name toast oven. If there is no impact on our own sales as long
as we stay at +10% of the competitor prices in the New York market, and +6% of Chicago market, then a corresponding business rule should be added to the modeling section of this item.
4.3 Cannibalization, affinity, and pull-forward sales
The last group of effects that are commonly incorporated into the model are the effects related to demand redistribution between products, or between current and future time periods.
Cannibalization usually happens within the same category for two or more competing items when those items share lots of similar product attributes, but differ by one or few other attributes, like
brand, color, flavor, price banding, etc. As cannibalization happens within the same category, we need to run the analysis on the sales of the entire category to determine the cannibalization from
one item to the rest of the similar items (i.e. we calculate the total category sales change with each item price change). Normally, when you put the brand name item on promo sales, the
cannibalization is much higher than when you put an unknown/private label brand on sales for these two reasons:
1. Brand name items have much high customer loyalty.
2. Brand name items are more expensive than similar items that are private label, so once they are on sale, the price difference gets smaller. Therefore, customers are more willing to trade up.
The affinity or halo effect occurs when promoting one item has a positive influence on other items. For example when you promote grills, grill accessories will also get a sales lift. Or, when you
promo Coke, you might see more sales from other snacks. A basket analysis will be able to determine the affinity and halo effect. We need to look at historical transactional data across multiple
promotional periods, analyze all baskets that contained the item promoted, and determine among all those baskets which were the common items that were not on promotion. What you should get is finding
that something like 50% of grill baskets had a grill accessories purchase. Since grill accessories were not promoted, any lift we see in sales to grill accessories during that promotional period can
be attributed as an affinity to grill. This information is vital, as it allows us to determine the affinity effect of the promotion at the same time. This informs us to not price down correlated
items at the same time, as we would be giving up margin unnecessarily.
In practice, a retailer can run multiple promotions at the same time, so that there is a complex interplay between multiple affinity effects. A retailer can use regression analysis to disentangle
this mix and isolate the effect for an individual promotion. Consider an example data set in Figure 8: it includes sales and promotion data for three products over five months. The impact of each
promotion is quantified using single-output or multi-output regression analysis where promotion flags are inputs, and sales uplifts are the target label.
Figure 8. An example data set for affinity effects analysis. Sales uplift is the change in the number of baskets with a given product compared to the baseline forecast.
The pull-forward effect, also known as the pantry load effect, only happens in consumable categories. The pull-forward effect occurs when consumers see a commodity they regularly purchase, which has
a long shelf life (e.g. bathroom tissue, detergent), go on promotion. When the item goes on promotion, the customers tend to buy a significantly larger volume than they usually purchase. They
stock-up on the product for a period of time, and therefore will not buy the item at their regular frequency until they are out of stock.
To calculate pull-forward, you first need to determine the baseline purchase frequency of those items, then look at historical promotional data, and determine the effect on sales post promotion. So,
if an item was promoted for a month, and if the baseline purchase frequency for this item is purchased every 3 months by the consumer, and it has a 4 month shelf life, one would look at 4 month of
sales data past the promotional date to see whether the item’s weekly sales are below the baseline. If it is below, it would be considered a pull-forward effect, provided another item in the category
is not promoted and did not record a lift. One needs to consider pull-forward as an impact not only to item sales, but also to category sales, as the promoted item that is being stocked by the
consumer can impact the overall category performance for a period of time too.
For a replenishable item, since the inventory is unconstrained, the following relationship is generally true:
4.4 Price optimization for replenishable items
It seems challenging for retailers to combine all the above factors together into pricing modeling for replenishable items — so how do they do it? This process takes two steps:
1. Run historical data to calculate own item elasticity and figure out the relationship between items. The output of this step is a set of parameters that quantify which item sets will cannibalize
each other or have affinity for one another, what the cross elasticity is between items, which items will be impacted by competitor pricing, and by how much.
2. Run what-if analysis for all possible price change scenarios, and find the scenario that will yield the highest total sales/margin for the items set.
Let us consider an example shown in Figure 9. The historical data analysis found that when item A (Tostitos Chips) go on sale, item B (Crunchy Cheetos) sales will go down (cannibalization), but item
C (Tostitos Salsa) sales will increase (halo). Additionally, item A has a pull forward sales impact for two months, so items A+B+C+(A pull forward) will be combined as an item set for the price model
Figure 9. An example of price optimization for a replenishable item that combines multiple effects together. What-if analysis is done for two scenarios for Item A – Price 1 ($3.00) and Price 2 ($
In the above example, the sales difference between Price 1 and Price 2 is $7950-$7150-$300 = $500, and hence Price 2 (a promotional price of $2.50 for item A) is a good price recommendation.
Retailers typically do not directly dial competitor pricing into price modeling. Instead, they use it as a business rule to guardrail the price change choices in the what-if analysis above. For
example, if the historical data analysis shows that the sales impact is negative once we priced +10% or more to our competitor pricing, we would cap our price choices for the what-if model at +10% of
our competitor’s prices.
An alternative approach to the optimization process described above is building a high-capacity predictive model (using methods like boosted decision trees or neural networks) that uses the
coefficients estimated by econometric models (cannibalization, halo, etc.) and other signals as features, and forecasts profits or revenues. This approach is more flexible and accurate given that the
outputs of econometric models are combined with sales time series data, inventory data, and other features that help to improve demand forecasting.
5. Pricing model for seasonal items
A pricing model for seasonal items is different than that for replenishable items for a couple reasons. First of all, seasonal buy has a long lead time, inventory is committed, and there is not much
flexibility to adjust the inventory quantity after the purchase. Therefore, as retailers always consider seasonal items to have sunk costs, the goal for pricing the seasonal items is to maximize
revenue instead of margin or profit. This can be illustrated by the following toy example. Consider a retailer that has two units of some product in stock, and the cost of each unit is $10. Compare
the following two scenarios: a retailer sets the unit price to $15 and sells both units by the end of the season, or sets the unit price to $30, manages to sell just one unit, and throws away the
second one. In these two scenarios, the margins are different ($5 and $20, respectively), but revenues and total profits will be the same, $30 and $10, respectively.
Second, seasonal items have a much shorter life cycle than the replenishable items. For example, most of the winter goods usually get delivered to stores in late October, go on clearance after
Christmas, and liquidate in March to free up store space for spring goods. Therefore, the life cycle for winter goods only lasts for five months, with a peak selling period of two months. Thus, it is
more challenging for retailers to price seasonal items than replenishable items.
5.1 Seasonality
Seasonality is the most important parameter for the seasonal pricing model. Seasonality usually consists of periodic, repetitive patterns in demand, usually caused by weather or holidays. For
example, the winter seasonal goods uplift is caused by cold winter weather, and the Christmas season uplift is caused by the holiday of Christmas. Seasonality does not include non-recurring events
that impact demand, such as snow storms, hurricanes, marketing promotion events, etc. It also does not include price-driven changes in demand.
Retailers usually estimate seasonality at the national level when they are in the planning stage to compose the seasonal buy, and go down to the climate zone level when they are selling the products.
Season code and sub season code are other key product attributes that retailers use to estimate seasonality. For example, fall is one season code, and has three different sub season codes: the
regular fall sub season; back to school sub season, and Halloween sub season. The demand of regular fall sub season goods is driven by weather, and depends on how cold or how early winter comes. Back
to school demand is driven by the date that schools open in the area. Halloween is a floating holiday, so the demand for Halloween goods changes based on the actual day of the week that Halloween
falls on every year. As you can tell, although all three sub seasons belong to the fall season code, the seasonality of them varies drastically. Therefore, it is important to estimate the seasonality
by the sub season code.
In statistics, there exist a large number of methods for seasonality detection, measurement, and adjustment (i.e. removal of the seasonality components). In the next two sections, we discuss two
relatively basic techniques for seasonality measurement that are known to be efficient for retail pricing applications. Both techniques are based on the nearest neighbor approach – they try to
estimate the future seasonality trend by finding a sample in historical data that is similar to the current situation. The difference between them is how the similarity (distance) metric is defined.
5.2 Seasonality estimation using the difference between time series
The first method we consider uses the correlation between time series as a distance metric to find a historical sample similar to the current situation, and estimate the seasonality uplift for the
next month(s) from it. The methodology is as follows:
1. Pull 3-4 years of historical data for weekly sales units and revenues at promo prices or regular prices for a given climate zone and program.
2. Use elasticity to remove the price impact. This can be done based on the definition of the constant-elasticity demand function provided in the beginning of the article:
1. Find the most similar weekly sales pattern in the previous years using the correlation coefficient between time series as a similarity metric, and use that year’s sales pattern to forecast the
seasonality of the current year.
For example, in order to find the seasonality of Feb 2019 for winter season coded items, you need to run the correlations of week-over-week sales unit changes from Oct 2018 to Jan 2019 versus the
previous 4 year weekly sales unit changes during the same period, and find the year that best fits the same pattern as the current year. If the 2016 winter season selling pattern is the best fit,
then you would use Feb 2016 to estimate Feb 2019’s seasonality multiplier (uplift). This example is illustrated in Figure 10.
Figure 10. Seasonality estimation using nearest neighbor search in time domain. The sales quantity for week t is denoted as qt and the seasonality multiplier is denoted as kt
5.3 Seasonality estimation using a custom distance metric
Ideally, the seasonality computation should be done at the climate zone, sub season code, and program level (e.g. tropical area, sub season back to school, and Disney character boys t-shirt
respectively). Unfortunately, it is not always possible to find a historical sample that is highly correlated with the current situation for a given climate zone and program, as the number of samples
is limited by just the past 3-4 years. At the same time, one can argue that it makes sense to not only look at the past samples for a given pair of climate zone and program, but also at related or
similar zones, sub seasons, and programs. This idea essentially means that we can introduce a more elaborated distance metric that accounts not only for correlation between time series, but also for
distances between sub seasons and programs. Once this metric is defined, we can search for a nearest neighbor in a larger set of samples to find the seasonality multiplier.
Let us consider an example illustrated in Figure 11. We define the distance metric between two samples as the distance between a two-dimensional vector. The first element of this vector is the sub
season numerical code. The codes are assigned to sub seasons sequentially, so the difference between the codes naturally quantifies the temporal distance between the sub seasons. The second element
of the vector is defined as the average seasonality multiplier (month-over-month sales change) for the past three months. These two values are computed for all sub seasons and programs, and thus each
pair of a sub season and program is mapped to a point in two-dimensional space. We then search for the nearest neighbor in this space and estimate the seasonality multiplier for the next time period
based on this sample. Dissimilar to the time series correlation method described in the previous section, the sample may or may not belong to the same sub season or program.
Figure 11. Seasonality estimation using a distance metric with sub season and program components.
The nearest neighbor analysis with custom distance metrics is very useful for seasonal pricing modeling. Retailers purchase most of their seasonal items once a year, and as they usually add new
products or new categories to new store locations to expand their business, it is difficult to find meaningful data in the historical data set. The nearest neighbor analysis helps to solve this
5.4 Dog and winner approach (slow seller vs. quick seller)
As we mentioned before, seasonal items have a short selling window with a fixed amount of inventory units. Therefore, the goal for pricing the seasonal items is to maximize the sales for the entire
life cycle instead of maximizing the sales/profit for every single transaction. If we are running out of stock before the season ends, that means we priced too low and left money on the table. If we
are selling too slowly, that means we priced too high and will lose money in the end due to a high leftover inventory that will be sold below cost after the season.
It is very important to differentiate slow sellers (dogs) from quick sellers (winners), and price them differently. Typically, dogs remain dogs and winners remain winners for the entire season, since
the reason for a dog is due to the product itself, like a color, style, or fit that is less appealing to customers. It is not due to the weather or any other outside reasons. Therefore, identifying
the dogs in the early season will help retailers address the problematic items earlier, and avoid a margin loss post-season.
The most important metric to identify dogs is the sell-through rate, but we should not use a straight sell-through category rate because it varies across locations and products. For example, although
a lightweight jacket and a heavy winter coat belong to the same outerwear category, their demand curve is absolutely different during the entire season by different locations. A good methodology to
identify the dogs and winners includes the following steps:
1. Use the demand forecast to build a weekly sell-through target by product group by location.
2. Compare that with the actual sell-through rate and calculate the ratio between actual and target sell-through rates.
3. If the ratio is below 0.9, then those product group will be tagged as dogs, if the ratio is above 1.1, then they are winners.
The chart in Figure 12 illustrates a poor seasonal buy that contains a lot of dog items.
Figure 12. A real-life example of a seasonal buy that fell behind the sell-through target.
The benefit for a retailer of identifying the dogs and winners in the early season is to make changes for their promotion offer levels that fit the sell-through rate. Consider a retailer that runs a
promotion at the brand-category level (for example, all Tahari women’s sweaters will be priced with the same discount). This retailer can do the above sell-through analysis for each brand-category
group early in the season, and take action depending on the number of dogs and winners as follows:
• If only a few sizes and colors are dogs in this group, then a retailer can select those items to go on clearance first, and eliminate the need to go with deeper discounts for the entire promotion
offer group.
• If one of the styles is dog, then a retailer can break that style into different promotion offers and price them differently.
5.5 Breakage caused by size variation
Breakage refers to the situation when some item variants (certain sizes, colors, etc.) go out of stock while other variants are still in stock. Breakage can be caused by uneven inventory levels for
different variants, uneven demand, or both.
The breakage factor can be forecasted based on the total number of variants, the initial stock levels for each variant, and the total forecasted inventory on hand. For example, if we have a product
with 5 variants and the initial inventory is 1 unit per variant (so 5 units in total), then the initial breakage factor is 1.0 (all variants are in stock). Once we sell one item and on hand inventory
decreases to 4 units, the breakage factor also decreases to 0.8 (one of 5 units will be out of stock). Once we sell two items and the on hand inventory goes down to 3 items, the breakage factor will
be 0.6, and so on. If the initial inventory is more than one unit per variant, the dependency will be more complex. For instance, if we have 5 units per variant (25 units in total), then the breakage
factor will stay equal to 1.0 until we sell at least five units, and then will slowly decrease. Once the 5th unit is sold, there are only 5 chances out of 125 that some item will go out of stock
(i.e. all five units sold will be of the same variant) if the demand for all variants is assumed to be the same. An example of breakage functions is shown in Figure 13. These curves can be computed
either analytically or using Monte Carlo simulations for different initial parameters and demand distributions.
Figure 13. Examples of breakage curves for an item with 5 size variants. Each curve corresponds to a certain initial level of inventory (from 1 to 5 units per size variant). The demand for all
variants is assumed to be the same. Breakage also depends on the number of possible sizes: one out of stock size matters more for a product that has just 5 sizes compared to a product that has 10
Incorporation of the breakage factors into the seasonal pricing model is very important, especially in a business that is size intensive, like apparel or footwear, as it allows us to differentiate
the content of inventory units. The breakage factor is typically incorporated as a demand multiplier.
5.6 Price optimization model for seasonal items
The factors that we discussed above are the main components that a retailer uses to build a pricing model for seasonal items and, as you can tell, is different compared to the pricing model for
replenishable items. It does not usually consider cross effects like cannibalization or affinity, since it is very unusual for retailers to price seasonal items individually. Promotion offers usually
stays at the rack level, and each rack contains different styles, colors, and sizes. Also, seasonal items are more of a fashion style than the basic style of replenishable items, so the substitution
impact is much smaller.
Two unique factors for seasonal model are seasonality and breakage. Seasonality is important because the selling period for seasonal items is much shorter than the basic replenishable items, and the
demand for seasonal items changes a lot from one week to the other. With limited inventory on hand, breakage is another important factor to discount your normal demand because you can only sell what
you have in stock. All these factors are summarized in Figure 14.
Figure 14. The main factors and effects that need to be accounted for in a pricing model for seasonal items.
The method to incorporate the above factors into the optimization model is as follows:
1. Develop a baseline demand for the entire selling period by week with no discount prices.
2. Incorporate elasticity to estimate demand with the price change/discount.
3. Multiply that demand by a weekly seasonality factor, holiday factor, and breakage factor.
4. Run what-if modeling for every possible weekly price combination and pick the highest sales revenue option for the entire life of the product, taking inventory constraints into account.
5. Build business rules to guardrail the discount options. For example, the deepest discount allowed before Christmas is 70% off for high-low products, and 50% off for mid-low products. Another
example is that an item can only be on sale for 6 weeks out of 9 weeks based on the legal requirement.
Similar to modeling for replenishable items, modeling for seasonal items can take advantage of machine learning methods and train a non-parametric revenue model. This can incorporate econometric
parameters like breakage, and learn some parameters like seasonality directly from the sales data.
Another important area where machine learning can be helpful is in the planning of flash sales (short-term sales events). Such events have many common features with seasonal sales (inventory
constraints, breakage, etc.), but often go at a much faster pace (weeks or even days). Therefore, online learning and reinforcement learning methods that can estimate price-demand functions based on
very few samples are usually beneficial.
6. Conclusions
Compared to the traditional cost plus or value-based pricing methods, sophisticated modeling to optimize prices can help retailers improve their profits substantially. The pricing impact is
immediate, and is the most vital component when it comes to making money for the retailers.
A pricing strategy is also the most complex strategy that retailers can execute successfully, since pricing touches everything in the retail operation, and everything touches pricing as well. Despite
recent advances in analytics, decision-support tools, and methodologies, retailers are finding that off-the-shelf pricing optimization products are still not enough to keep pace. Not all retailers
are the same: they all carry different products and have unique ways of managing their products, so it is very hard to fit a one-size-fits-all kind of pricing modeling. Some large retailers own more
than 10 off-the-shelf price optimization products that they use for different product lines, and they still have to spend millions of dollars to customize the tools to make them fit their own needs.
A pricing strategy is also the most complex strategy that retailers can execute successfully, since pricing touches everything in the retail operation, and everything touches pricing as well. Despite
recent advances in analytics, decision-support tools, and methodologies, retailers are finding that off-the-shelf pricing optimization products are still not enough to keep pace. Not all retailers
are the same: they all carry different products and have unique ways of managing their products, so it is very hard to fit a one-size-fits-all kind of pricing modeling. Some large retailers own more
than 10 off-the-shelf price optimization products that they use for different product lines, and they still have to spend millions of dollars to customize the tools to make them fit their own needs.
A custom price optimization suite is the key to ensure that retailers stay competitive and successful in this tough environment. In fact, the new digital era stemming from big data, mobile commerce,
and the explosion of omnichannel retailing has meaningfully changed the retail environment. In this era, a set of robustic pricing optimization products to make the retailer profitable is not an
advantage, but a requirement. | {"url":"https://test.stage.griddynamics.net/blog/retail-price-modeling-for-replenishable-and-seasonal","timestamp":"2024-11-11T22:26:30Z","content_type":"text/html","content_length":"428333","record_id":"<urn:uuid:69c62fdf-73b4-4e19-8939-7a729ebd18f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00038.warc.gz"} |
What is differential amplifier formula - Home Automation Technology
A differential amplifier is an electronic circuit that amplifies the difference between two input signals. It is a type of analog circuit that is used in many applications, such as audio preamps,
filter circuits, and motor control systems.
The differential amplifier formula is used to calculate the gain of a differential amplifier. The gain of a differential amplifier is the ratio of output voltage to input voltage, and it is expressed
as a constant for a given circuit.
At its simplest, the differential amplifier formula looks like this:
Gain (A) = (Vout)/(Vin)
Where Vout is the output voltage and Vin is the input voltage. The gain of the differential amplifier will be positive if the output voltage is greater than the input voltage, and negative if the
output voltage is less than the input voltage.
When calculating the gain of a differential amplifier, it’s important to take into account any additional components that are connected to it, such as resistors or capacitors. These components will
affect the gain of the amplifier, so they must be included in the equation. For example, if there are two resistors connected in series with the amplifier, then you would have to add their
resistances together when calculating the gain.
The differential amplifier formula can also be used to calculate other parameters such as input impedance, output impedance, and power dissipation. In addition, it can help determine stability and
phase margin of an amplifier circuit. To calculate these parameters, you will need to use more complex equations that involve additional components and parameters.
The differential amplifier formula is an important tool in understanding how different types of amplifiers work and how they can be used in various applications. It can help you calculate gains and
other parameters accurately, giving you a better understanding of how these circuits work.
How do you calculate dB gain
dB Gain is a measure of the power level of an electrical signal in comparison to an original reference signal. It is used to quantify the amount of amplification or attenuation applied to a signal,
and is expressed in decibels (dB).
The basic formula for calculating dB gain is:
dB Gain = 10 x log10 (Power Out / Power In).
Power Out is the output power level of the amplifier or device under consideration, while Power In is the input power level of the same device. Log10 is simply the logarithm base 10.
Let’s look at an example of how to calculate dB gain. Say you have an amplifier with an output power of 1 watt, and an input power of 0.1 watts. To calculate the dB gain, you would use the formula
dB Gain = 10 x log10 (1 Watt / 0.1 Watt) = 20 dB Gain
Note that if you are dealing with Voltage and not Power, you would use:
dB Gain = 20 x log10 (Voltage Out / Voltage In).
Both calculations result in identical gain values in dB.
Another important consideration for calculating dB gain is that it is typically expressed using relative values—the ratio between two signals—rather than absolute values. This means that if you
want to measure gain accurately, you must take into account any losses that occur within your system. For instance, if your amplifier has a power output of 1 watt but there is a 0.5 watt loss between
the amplifier’s output and its destination, then you must subtract this 0.5 watt from your output power level before calculating gain:
dB Gain = 10 x log10 (1 Watt – 0.5 Watt / 0.1 Watt) = 15 dB Gain
So as you can see, calculating dB gain requires taking into account both input and output levels as well as any losses that occur within your system or circuit. With this knowledge, you should be
able to accurately measure dB gain in any system or device!
What is the formula of power gain in an amplifier
Power gain is a measure of the amplification of an electrical signal, usually expressed in decibels (dB). It is an important parameter in the design of amplifiers and other electronic circuits. The
formula for power gain is typically expressed as:
Power Gain (dB) = 10 log (Output Power/Input Power)
This equation can be used to calculate the power gain of a given amplifier or circuit. To start, you need to know the output power and input power of the amplifier or circuit. Output power is
generally stated as the total power delivered to a load (speaker, antenna, etc.), while input power is the total power required from a source (power supply, battery, etc.).
Once you have these two values, you can plug them into the formula and calculate the power gain. For example, if an amplifier has an output power of 50 Watts and an input power of 10 Watts, then the
power gain would be 20 dB.
Power gain is an important measure of how well an amplifier can amplify a given signal. The higher the power gain, the more powerful the amplifier and the better it performs at boosting weak signals.
It’s also important to note that different types of amplifiers will have different levels of power gain. For example, Class A amplifiers tend to have lower levels of power gain than Class B or Class
AB designs.
When selecting an amplifier for a particular application, it’s important to consider both its total output power and its power gain. The higher the total output power and the higher its power gain,
the better it will perform in boosting weak signals or driving large loads.
What is the equation of gain
The equation of gain is a mathematical expression used to calculate the ratio between the output power of an electronic device and the input power it receives. It is an important aspect to consider
when designing electronic devices since the output power can be significantly less than the input power if the gain is not correctly calculated.
The equation of gain is expressed as:
Gain = Output Power / Input Power
In other words, the numerical value of this equation represents how much larger the output power is than the input power. This can be expressed in decibels (dB) which is a logarithmic unit of
measurement that expresses a ratio between two numbers in terms of powers of ten. A gain of 10 dB means that the output power is 10 times greater than the input power.
When designing electronic devices, it is important to consider the equation of gain in order to ensure that the output power will be sufficient for its intended purpose. If the gain is too low, then
not enough power will be delivered to get the desired results and if it is too high then too much power will be delivered which can cause damage to components.
It is also important to consider other factors such as noise and distortion when designing electronic devices since these can also have an impact on the overall performance. The equation of gain can
help provide insight into how these factors may affect a device’s performance. | {"url":"http://homeautotechs.com/What-is-differential-amplifier-formula/","timestamp":"2024-11-12T20:15:49Z","content_type":"text/html","content_length":"60116","record_id":"<urn:uuid:f35decbc-af1b-4b99-b3c7-efef2f7f7512>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00891.warc.gz"} |
From Encyclopedia of Mathematics
A non-damped oscillation in a non-linear dynamical system, whose amplitude and frequency can remain constant during a long period of time, are largely independent of the initial conditions, and are
determined by the properties of the system itself (see also Oscillations, theory of). The term "auto-oscillation" was introduced by A.A. Andronov (see [1]).
Dynamical systems capable of performing oscillations are known as auto-oscillating systems. They include clocks, generators of electric vibrations, electric bells, blow and bow musical instruments,
etc. Under certain conditions auto-oscillations can also be produced in dynamical systems which normally function without auto-oscillations (e.g. auto-oscillations in the front suspension of a car-
"noise" , the flutter of an aircraft wing, or auto-oscillations in automatic control and monitoring systems). The simplest auto-oscillating system can be represented as consisting of a constant
source of energy, a mechanism regulating the supply of energy to the oscillating system, and the oscillating system itself. An essential feature of such a system is its feedback nature: on the one
hand, the regulating mechanism controls the motion of the oscillating system but, on the other, it is the motion of the oscillating system that influences the operation of the regulating mechanism
[6]. From the mathematical point of view, autonomous auto-oscillating systems with one degree of freedom may be defined as systems whose equations of motion have one or more limit cycles in the phase
plane (cf. Limit cycle).
An important typical property of auto-oscillations is that their amplitude is largely independent of the initial conditions, i.e. the existence of one or more ranges of initial conditions such that
to any initial set of conditions within any one of these ranges there corresponds the same auto-oscillation amplitude. This means that oscillating systems may contain several stationary processes
with different amplitudes; each one of these becomes realized in the system, depending on a choice of a region of initial conditions. This property constitutes the basic difference between periodic
motion in an auto-oscillating system and periodic motion in a conservative system. Another typical property of auto-oscillating systems is the fact that the periods of the auto-oscillations are
determined by the properties of the system itself and are not imposed from outside. This is the basic difference between auto-oscillations and forced oscillations.
An essential feature of auto-oscillations consists in the fact that the loss of energy must be compensated by a constant source of energy. Since in an autonomous system the forces do not explicitly
depend on time, this source of energy must produce a force which is not a given function of time and which is determined by the system itself. Constant sources of energy include the winding mechanism
in a clock, a shaft rotating at a constant angular velocity, an endless belt moving at a constant rate, or a jet of liquid or gas having a constant velocity; in systems in which the motive force is
an electric flow, an electric battery may serve as such a source.
An example of a very simple auto-oscillating system is a pendulum in a viscous frictional medium, acted upon by a force of constant magnitude which always acts in the direction of the motion. The
differential equation of this dynamical system is
where [3].
A typical property of all auto-oscillating systems is the connection between the constant source of energy and the system, which is such that the energy given by the source varies periodically, the
period of this variation being determined by the properties of the system. This may be expressed by saying that an auto-oscillating system is a system which executes a periodic process at the expense
of a non-periodic energy source. The shape of auto-oscillations can be close to sinusoidal, but can also be very different. The latter type of auto-oscillations is known as relaxation oscillation.
Auto-oscillations with a shape close to sinusoidal usually occur in systems in which the loss in energy during one period is small, and consequently, the amount of the energy fed into the system is
also small. They may also occur when the energy losses in the system during one period are large, but if the parameters are properly chosen, the losses are compensated not merely for each period, but
also for each small fraction of a period. Such systems include the so-called RC-generators of sinusoidal oscillations [4]. During relaxation oscillations energy losses are usually large and are
compensated for after one period by almost the entire energy of the oscillating system.
The generation of auto-oscillations may be "soft" or "hard" [2]. In the former case, variation of some parameter of a system in a stable equilibrium results in the generation of auto-oscillations
whose amplitude steadily increases from zero as the parameter is continuously varied. If the parameter is now varied in the opposite direction, the amplitude of the auto-oscillation steadily
decreases to zero, and the system returns to a stable equilibrium. If the generation of the auto-oscillations is hard, the system passes from a state of stable equilibrium to an auto-oscillation with
finite amplitude as one of its parameters is slowly and continuously varied; as the parameter is further varied, the rate of increase of the amplitude becomes constant. If the parameter is varied in
the reverse direction, the amplitude continuously decreases and, after it has attained a certain value, the system returns to the state of stable equilibrium. It is important to note in this
connection that the system passes from stable equilibrium to auto-oscillations and from auto-oscillations back to stable equilibrium at different values of the parameter.
Auto-oscillating systems display an interesting and important property — the effect of enforced synchronization, which is sometimes called "capturecapture" or "mode lockingmode locking" . This takes
place when the difference between the frequency of the auto-oscillating system and that of the external force acting on it is sufficiently small. The steady periodic motion of the system then assumes
the frequency of the external force, i.e. the external force imparts its own frequency to the auto-oscillating system [3], [5].
[1] A.A. Andronov, , Collected work , Moscow (1956) (In Russian)
[2] A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Pergamon (1966) (Translated from Russian) MR0198734 Zbl 0188.56304
[3] N.V. Butenin, "Elements of the theory of non-linear oscillations" , Blaisdell (1965) (Translated from Russian) MR0176160
[4] G.S. Gorelik, "Oscillations and waves" , Moscow-Leningrad (1950) (In Russian)
[5] A.A. Fel'dbaum, "Introduction to the theory of non-linear chains" , Moscow-Leningrad (1948) (In Russian)
[6] A.A. Kharkevich, "Auto-oscillation" , Moscow (1953) (In Russian)
The general mathematical definition of "autonomous auto-oscillating systems" would be: vector fields having attractors which are non-constant periodic solutions. The "ranges of initial conditions"
are the basins of attraction of these solutions. In conservative systems, periodic solutions cannot be attractors and, if all solutions in an open set are periodic, the amplitude and period will
usually depend on the initial conditions. The "soft auto-oscillations" can be identified with the Hopf bifurcation in dynamical systems.
In [a1], especially pages 517 and 504 are useful.
[a1] R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin/Cummings (1978) MR0515141 Zbl 0393.70001
How to Cite This Entry:
Auto-oscillation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Auto-oscillation&oldid=24370
This article was adapted from an original article by N.V. Butenin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
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Calculating Bonuses
“Very well. Keep your secrets.”
This article/section contains unofficial information, concepts, or terminology derived from or based on community discussion, invention, or knowledge. It may be subjective and contain information or
terminology that is not used by
Digital Extremes
or in official
communications, and may not be an officially recognized concept.
When modding anything in WARFRAME, the order in which bonuses are applied follows a pattern. With the exception of mods that provide elemental damage bonuses, the order in which mods are installed
does not matter; the resultant stats will always be the same regardless of the mod configuration.^[1] The same goes for temporary or conditional buffs and debuffs from abilities, mods, Arcane
Enhancements, and enemies.
Last updated: Sat, 27 Jul 2024 06:50:29 +0000 (UTC) by User:Headbox8424
Semantics[ ]
Percent Bonuses vs. Multipliers[ ]
In most cases in WARFRAME, percent bonuses to stats (not to be confused with flat percentage points) can also be represented as stat multipliers using the following conversion:
{\displaystyle \begin{aligned} \text{Net Multiplier} = 1 + \frac{\text{Net Percent Bonus as a percentage}}{100} = 1 + \text{Net Percent Bonus as a decimal} \end{aligned} }
Converting percent bonuses to multipliers.
For simplicity, calculations in this article (and throughout the wiki) will represent percent bonuses as decimals and will use 1 + Net Percent Bonus as a decimal expression instead.
Complementary Percentages[ ]
Many stat modifiers in WARFRAME are expressed in terms of percentages, but there may be more than one way to describe the same stat modifier.
For example, Damage Vulnerability can be expressed in terms of Damage Reduction. If an enemy is 25% more vulnerable to taking Impact damage (e.g. +25% Impact damage bonus), one can also say that
the enemy:
Overview[ ]
Types of Bonuses
Operation On Base Bonus Stacking Internal Name Typical Context Example(s)
Stat Behavior (OperationType)
Entropy Burst's status chance bonus
Final Status Chance = Base Status Chance + 20
Addition Additive Stacking ADD Flat value or percentage point bonuses: Laetum's Elemental Excess perk (-10 critical chance
percentage points)
Final Critical Chance = Base Critical Chance + -10
Serration's and Heavy Caliber's damage bonus
Multiplication Additive Stacking STACKING_MULTIPLY Most common bonus, applies to almost all percentage-based bonuses:
Final Damage = Base Damage * (1 + 1.65 + 1.65)
Antitoxin's and Toxin Resistance's damage resistance to
Multiplicative Damage Taken = Initial Toxin Damage * ( 0.55 * 0.85)
Multiplication Stacking MULTIPLY
Affinity bonuses
Total Affinity = Initial Affinity * 2 * 1.3 * 1.25
Override N/A SET Very rare bonus type used to set a stat at a particular value, ignoring all negative
or positive modifiers
• Note that there are known instances where a particular stat can have flat value and percentage-based bonuses. For example, Stinging Truth provides a flat increase to Magazine Capacity for the
Viper while Slip Magazine provides a +30% Magazine Capacity bonus at max rank. In these scenarios, typically percentage-based bonuses are applied to the base stat first then flat value
increases. For example:
Final Magazine Capacity = 14 * (1 + 0.3) + 40
Additive Stacking[ ]
Percent Bonuses[ ]
{\displaystyle \begin{aligned} \text{Resultant Stat Value} = \text{Base Stat Value} \times (1 + \text{Additive Stat Bonus 1} + \text{Additive Stat Bonus 2} + \cdots) \end{aligned} }
Generic formula for additive stacking percent bonuses.
All sources that add a percent bonus of the same type will typically have their bonuses added together. This commonly referred to as additive stacking (internally represented as STACKING_MULTIPLY
operation type). For example, if a primary weapon has Serration (+165% damage) and Heavy Caliber (+165% damage) installed, it will receive a total of +330% bonus base damage or 4.3x the base
This is true of all percent bonuses, not just damage. For example, combining Speed Trigger (+60% fire rate) and Shred (+30% fire rate, +1.2 punch-through) will give a total of +90% bonus fire rate.
Flat Value and Percentage Point Bonuses[ ]
{\displaystyle \begin{aligned} \text{Resultant Stat Value} = \text{Base Stat Value} + (\text{Additive Stat Bonus 1} + \text{Additive Stat Bonus 2} + \cdots) \end{aligned} }
Generic formula for additive stacking flat value and percentage point bonuses.
Sources that grant flat value or percentage point increases are typically applied after all other bonuses are applied (internally represented as ADD operation type). For example, say a melee weapon
with 20% critical chance and True Steel equipped has 0.2 × (1 + 1.2) = 44% critical chance. Since a max ranked Arcane Avenger grants a flat 45% critical chance when triggered, the melee weapon will
have 0.44 + 0.45 = 89% critical chance as a result.
Note that flat bonuses are additive with each other. Using the previous example, if the player's Adarza Kavat's Cat's Eye buff triggers (adding a flat 60% critical chance), then the resultant
critical chance will be 0.44 + 0.45 + 0.6 = 149%.
Multiplicative Stacking[ ]
{\displaystyle \begin{aligned} \text{Resultant Stat Value} = \text{Base Stat Value} \times (1 + \text{Multiplicative Stat Bonus 1}) \times (1 + \text{Multiplicative Stat Bonus 2}) \times \cdots \end
{aligned} }
Generic formula for multiplicative stacking bonuses.
Sources that affect the same fundamental stat but have different conditions for granting its bonuses will typically have their bonuses multiplied together (but not always such as Chroma's Vex Armor
additively stacking with base damage mods). This is commonly referred to as multiplicative stacking. For example, a primary weapon with Serration (+165% damage) and Bane of Grineer (+30% damage to
Grineer) would result in a [(1 + 1.65) × (1 + 0.3) - 1] = 244.5% bonus in base damage against Grineer enemies. Note that multiplicative stacking effects grant greater bonuses than additive stacking.
For example, if Serration and Bane of Grineer were to additively stack, it would only grant 1.65 + 0.3 = 195% bonus base damage.
Examples of sources of damage bonuses/multipliers that will multiplicatively stack with each other:
Exponential Stacking[ ]
In rare instances, some bonuses are applied to itself multiplicatively and is multiplicative with all other bonuses. This is often referred to as exponential stacking. For example, Arca Titron has a
unique mechanic where each successive kill adds a charge that multiplies the next slam radial damage bonus by 2 using the following equation:
Total Radial Damage = Base Radial Damage × 2^n (where n is the number of charges)
Opportunity Cost In Modding[ ]
Additive Stacking[ ]
Players may refer to additive stacking having "diminishing returns" (despite being linear increases; a more accurate term would be "opportunity cost"), meaning that at higher percent bonuses, the
additional value gained per additional percent bonuses is less, relatively speaking. For instance, if one were to equip a secondary that dealt 100 base damage:
1. Adding a maxed rank Hornet Strike (+220%), it would deal 320 base damage, a 2.2 / 1 = 220% relative increase in base damage.
2. Adding a maxed rank Magnum Force (+165%), it will deal 485 base damage, a 1.65 / (1 + 2.2) = 51.5625% relative increase in base damage over just equipping Hornet Strike.
3. Adding a maxed rank Augur Pact (+90%), it will deal 575 base damage, a 0.9 / (1 + 2.2 + 1.65) = 18.5567% relative increase in base damage over just equipping Hornet Strike and Magnum Force.
4. Adding an unmodded Rank 3 Vex Armor at max buff (+275%), it will deal 850 base damage, a 2.75 / (1 + 2.2 + 1.65 + 0.9) = 47.8261% relative increase in base damage over just equipping Hornet
Strike, Magnum Force, and Augur Pact.
□ Notice that despite the larger base damage bonus of +275%, Vex Armor only provided a smaller relative increase in base damage.
5. Adding a maxed rank Anemic Agility (-15%), it will deal 835 base damage, a -0.15 / (1 + 2.2 + 1.65 + 0.9 + 2.75) = −1.7647% relative increase (or 1.7647% relative decrease) in base damage
over just equipping Hornet Strike, Magnum Force, Augur Pact, and with Vex Armor active.
□ Notice that base damage bonus penalties are also relatively low at higher base damage bonuses.
Because of this, sometimes it is better to not equip mods that provide relatively lower additive bonuses due to opportunity cost. That mod slot is better reserved for bonuses that multiplicatively
stack with other bonuses.
Armor[ ]
Although most Armor bonuses are percentage based (thus having "diminishing returns" strictly speaking on relative armor point increases) and that the relative increase of Damage Reduction percentage
is smaller at higher armor values (e.g. the difference between 100 and 200 armor is 15% while the difference between 400 and 500 is 5.3571%), the actual number of effective health points (EHP) gained
per additional point of armor does not have diminishing returns and remains constant (linear change). Every 300 armor points added will provide an additional 100% nominal health to EHP. This means
that at higher armor values, the difference between EHP and nominal health is greater as seen in the above graph.
{\displaystyle { \begin{aligned} \text{EHP} &= \text{Nominal Health} \cdot \frac{\text{Net Armor} + 300}{300} \\ &= \text{Nominal Health} \cdot \frac{(\text{Base Armor} * \text{Armor Bonus
Multiplier}) + 300}{300} \\ &= \text{Nominal Health} \cdot \text{Nominal Health Multiplier} \end{aligned} } }
Nominal health refers to listed health points as displayed in-game; in other words, it is the total health after mods and buffs are applied
Simplified EHP calculation to demonstrate how damage reduction from armor can be expressed as a multiplier to nominal health. Every 300 armor points gained will increase the health multiplier by
Multiplicative Stacking[ ]
Note that multiplicative stacking does not have "diminishing returns". For example, for a secondary that deals 100 base damage:
Like multiplicative stacking, exponential stacking also has no "diminishing returns" since the next stat bonus always give the same relative additional value gained per additional percent bonuses as
the previous bonus. For instance, the Arca Titron has a unique passive where kills increases the damage of the next Slam Radial Attack by 100%, stacks multiplicatively with itself up to 10 times.
With an unmodded Arca Titron with a base radial damage of Electricity 360:
• At one charge, the total radial damage will be Electricity 720, a 2 / 1 = 2x increase in base radial damage.
• At two charges, the total radial damage will be Electricity 1440, a (2 * 2) / (1 * 2) = 2x increase in base radial damage over one charge.
• At three charges, the total radial damage will be Electricity 2880, a (2 * 2 * 2) / (1 * 2 * 2) = 2x increase in base radial damage over two charges.
• And so on.
Order of Operations[ ]
To summarize, the order in which bonuses for the same fundamental stat applies as follows:
1. Bonuses that additively stack with each other
2. Bonuses that multiplicatively stack with each other and the previous additive bonuses
3. Bonuses that grant a flat number or percentage points, additively stacking with each other
{\displaystyle { \begin{aligned} \text{Resultant Stat} &= [ \text{Base Stat Value} \times (1 + \text{Add Bonus 1} + \text{Add Bonus 2} ) \times \\ &\qquad(1 + \text{Separate Add Bonus 1} + \text
{Separate Add Bonus 2} ) ] + \text{Flat Bonus 1} + \text{Flat Bonus 2} \end{aligned} } }
Example equation demonstrating the above order.
{\displaystyle { \begin{aligned} \text{Resultant Stat} &= [ \text{Base Stat Value} \times \prod (1 + \sum \text{Additive Stacking Bonuses} ) ] + \sum \text{Flat Bonuses} \end{aligned} } }
Expressing above equation as a product of summations since multiplicatively stacking bonuses can have additively stacking components.
Weapons[ ]
“It's taking longer than I calculated.”
This page is actively being worked on
and may not be completely correct. Please assist in making this page accurate. See
WARFRAME Wiki:Research
on ways to perform research on this game.
Click here to add more info
Not accounting for Kuva/Tenet weapons with an innate primary damage type that can also have a primary elemental progenitor bonus
“It's taking longer than I calculated.”
This page is actively being worked on
and may not be completely correct. Please assist in making this page accurate. See
WARFRAME Wiki:Research
on ways to perform research on this game.
Click here to add more info
Also missing elemental damage bonus from non-Mod sources like Warframe abilities (readers may ask what is the element type priority in these cases)
Types of Damage Bonuses[ ]
Mods primarily increase damage in one of four different ways:
The Damage Application Order[ ]
1. When calculating damage, first base damage bonuses are added together and applied. For example, the Karak has a base damage of 29. Equipping a max rank Serration (+165% base damage) adds 1.65 ×
29 = 47.85 additional damage for a total of 76.85 damage (the arsenal will round this number to 76.9). The added damage will be of the same damage type distribution the weapon innately deals.
2. Then, all elemental and physical damage bonuses are calculated based on the modified base damage. For example, adding a Hellfire (+90% Heat damage) to a Karak that already has Serration
equipped will add 90% of 76.85 for a total of 69.165 Heat damage. This damage is added to the base, and the Karak now deals a total of 146.015 damage if you add together all damage types.
3. After that, faction damage bonuses are applied to all damage types. Note that faction damage bonuses will not be accounted for in the arsenal stats. A Karak equipped with Serration, Hellfire,
and Bane of Corpus (+30% damage against Corpus) will deal an extra 0.3 × 76.85 = 23.055 physical damage and an extra 0.3 × 69.165 = 20.7495 Heat damage for a total of 189.8195 damage of all
Once damage is calculated, it may be affected by Critical Hit mechanics or modified based on the opponent's armor or sources of damage reduction. For detailed calculations of how various damage types
affect different types of enemies, see the Damage page.
Multishot[ ]
For weapons that fire multiple projectiles like shotguns or Cernos Prime, the damage calculated is the damage dealt by each projectile. In other words, if the weapon only fires one projectile at a
time, all of the damage is the base damage per projectile.
If the weapon has multishot from a mod like Split Chamber, it will have a percent chance to fire additional projectiles per shot, each of which will deal the full modded damage as if one projectile
was shot. The arsenal will display multishot bonuses as percent increases in total damage when in fact this is not true in reality. The total damage stat shown in the arsenal actually reflects the
average damage dealt per shot in this case. For example, a Karak with only Split Chamber equipped will see a 29 × 0.9 = 26.1 increase in damage for an average of 55.1 damage per shot.
For weapons that only fire a single projectile, Split Chamber's 90% multishot will cause each shot to randomly have a 90% chance to fire two projectiles instead of one. Since Cernos Prime already
fires 3 arrows per shot, adding Split Chamber will make it fire 3 × 0.9 = 2.7 additional arrows, so each shot will fire at least 5 arrows, and 70% of shots will fire 6.
On continuous beam weapons like the Glaxion, multishot adds a chance to increase damage on a tick. For example, adding Split Chamber to a Glaxion will give a 90% chance for a damage tick to double
as if two projectiles were shot.
Calculating Physical Damage[ ]
Physical damage mods apply only to base damage of the same type. For example, at max rank Fanged Fusillade increases Slash damage by +120%. The Karak does 29 damage split into 13 Impact, 8.7
Puncture, and 7.3 Slash. A Karak equipped a max rank Fanged Fusillade will gain 7.3 × 1.2 = 8.76 additional Slash damage for a total of 16.06 Slash and 37.76 total damage.
If a physical damage mod is added to a weapon that does no physical damage of the corresponding type, the mod will have no effect. For example, the Amprex does entirely Electricity damage when
unmodded, so Fanged Fusillade will provide no Slash damage bonuses.
Calculating Elemental Damage[ ]
Elemental damage mods apply to all damage done by a weapon. For example, a max rank Hellfire adds 90% Heat damage to a rifle. If a Karak was equipped with a max rank Hellfire, it would gain 29 ×
0.9 = 26.1 Heat damage.
Each two different elemental mods will be added up into a secondary element, with has different damage type modifiers against certain enemy health, shields, and armor classes. For combined elements,
the slots are ordered from left to right, top row then bottom row, with any inherent elemental damage (from the weapon) added last.
For example, if both Hellfire (+ Heat damage) and Cryo Rounds (+ Cold damage) are equipped on an Amprex, they will combine to add Blast damage to the weapon's base Electricity damage. If only
Hellfire is used, it will combine with the Amprex's base Electricity damage to make Radiation damage instead.
If multiple mods of the same element are added, only the first is used when making combinations. For example, if Hellfire, Cryo Rounds, and Thermite Rounds are added to an Amprex, no matter what
the order is the weapon will deal Blast and Electricity damage since the Heat element has already combined with Cold, so the second Heat mod simply increases the amount of Blast damage dealt.
Combining Physical and Elemental Damage[ ]
Both physical and elemental damage are combined in the same step and are based only on the weapon's base damage and any mods that effect base damage.
In terms of raw damage, this makes a +90% elemental damage bonus superior to +90% physical damage bonus unless the weapon deals all of its damage as a single physical damage type. In reality, the
total effect varies based on the health, shields, abd armor classes of enemy being damaged due to damage type vulnerabilities and resistances.
Damage Calculations[ ]
Patch History[ ]
Update 34.0 (2023-10-18)
Base vs Final Stats in Modding - Health / Energy / Shield / Armor Stat Overhaul
If you’ve spent any time invested in the deeper nuances of Modding, you may be familiar with “Warframe Math” - math that upon first glance doesn’t really make sense, but once you learn the inner
workings of the game, it all comes together. While we can appreciate the value that complex systems offer to a certain subsect of players, there are other aspects of the game that should have clear
and understandable outcomes. Namely: Shield, Health, Energy, and Armor Modding.
Pop quiz: what is 300 + 440%? If you answered 740, you may just be an Excalibur player.
Vitality (+440% Heath), Redirection (+440% Shields), Flow (+150% Energy), and Steel Fiber (+110% Armor) come with large modifier values that don’t seem to match their outcome in-game. This is because
these Mods apply their multiplier to the base stats of the Warframe - i.e., the stats you have at Rank 0. In the Excalibur example, a Rank 30 Excalibur’s Health stat of 300 earns an additional 440
Health from max rank Vitality (+440% Heath) since it applies to his base rank Health stat of 100, resulting in 740 total health.
In this update, we have removed this obfuscation by having Health, Shield, Energy, and Armor Mods apply to the stats of Warframes at their current rank. Continuing our Excalibur example, instead of
Vitality always applying to Excalibur’s base rank 100 Health, it would apply to his Health stat based on his rank - namely, the stat you can actually see in your Arsenal. If your Excalibur were Rank
30, his Health stat would be 300, which means Vitality’s multiplier would be calculated off of 300.
With previous Health and Mod values, additional adjustments are needed to make this revision work while maintaining game balance. By only changing where the multiplier applies, a Rank 30 Excalibur
would receive an extra 1,320 Health from max rank Vitality, resulting in a total health stat of 1,600. This outcome is a significant buff, which is not the intention of this system change.
To remedy this, we approached this problem in two ways:
1 - We reduced the overall multiplier for Health, Shield, Energy, and Armor Mods.
Since these now affect Max Rank Warframe stats, these Mods need to scale differently to maintain the status quo. Additionally, we wanted these new values to be as clear and understandable to all
players as possible! Here are a few examples of these value changes:
• Vitality: Reduced from +440% to +100% Health
• Redirection: Reduced from +440% to +100% Shield Capacity
• Steel Fiber: Reduced from +110% to +100% Armor
• Flow: Reduced from +150% to +100% Energy Max
Note: These are not all of the Mods affected by this change. We share the comprehensive list further down in this section of the update notes.
Doing some quick math, this means that a Rank 30 Excalibur (300 Health) with a reworked Vitality Mod (+100% Health, applied to the final Health stat) would receive 300 extra Health, for a total of
600. That, in contrast, is a nerf, which we also don’t want to do.
So, our next step:
2 - We adjusted Warframe Health, Shield, Energy, and Armor values to keep the end result of the revised Mods as close to the original values as possible.
With this change, Excalibur’s Rank 30 Health stat is 370. With +100% Health from a max Vitality Mod, his resulting Health stat would be 740, which matches what it was originally.
While this path to the same result may seem a little complicated, the outcome matches our intention: we want players to be able to look at their Health, Shield, and Armor Mods, and be able to
understand how they affect the stats they see in their Arsenal.
In addition to everything above, we also increased the base stat values for Warframes so that these revised Mods offer similar value for lower-ranked Frames. To do so, we reduced the amount of Health
/Shield/Energy that Warframes earn per rank in half, and transferred the sum of that value to their base stats.
For Armor, this is the one stat that does not increase with your Warframe’s level (with some exceptions). Armor values across the board have been slightly increased to compensate for the Mod changes.
Not to beat a dead Kaithe, but Mods will now be applying to the Max Rank stat instead of the Base Rank. You may look at these numbers and think “nerf” or “buff” depending, but the outcome is that
total Modded values are the same, if not a little higher in some cases.
Game System Mechanics
Currencies Credits • Orokin Ducats • Endo • Platinum • Aya • Regal Aya • Standing
Basics Arsenal • Codex • Daily Tribute • Empyrean • Foundry • Market • Mastery Rank • Nightwave • Orbiter • Player Profile • Reset • Star Chart
Lore Alignment • Fragments • Leverian • Quest
General Factions Corpus • Grineer • Infested • Orokin • Sentient • Syndicates • Tenno
Social Chat • Clan • Clan Dojo • Leaderboards • Trading
Squad Host Migration • Inactivity Penalty • Matchmaking
Player Housing Clan Dojo • Dormizone • Drifter's Camp • Orbiter
Basics Affinity • Buff & Debuff • Death • Hacking • Invisible • Maneuvers • One-Handed Action • Open World • Pickups • Radar • Stealth • Tile Sets • Void Relic • Waypoint
Damage Mechanics Critical Hit • Damage • Damage Redirection • Damage Reduction • Damage Reflection • Damage Type Modifier • Damage Vulnerability • Health • Status Effect
Enemies Bosses • Death Mark • Enemy Behavior • Eximus (Overguard) • Lich System
Gameplay Mission Arbitrations • Archon Hunt • Break Narmer • Empyrean • Invasion • Sortie • Tactical Alert • The Circuit • The Steel Path • Void Fissure
Activities Captura • Conservation • Fishing • K-Drive Race • Ludoplex • Mining
PvP Duel • Conclave (Lunaro) • Frame Fighter
Other Gravity • Threat Level
Modding and Arcanes Arcane Enhancements • Archon Shard • Fusion • Mods (Flawed, Riven) • Polarization • Transmutation • Valence Fusion
Warframe Attributes (Armor, Energy, Health, Shield, Sprint Speed) • Abilities (Augment, Casting Speed, Helminth System, Passives, Duration, Efficiency, Range, Strength)
Weapons Accuracy • Alternate Fire • Ammo • Area of Effect • Attack Speed • Bounce • Critical Hit • Damage Falloff • Exalted Weapon • Fire Rate • Hitscan • Holster • Incarnon •
Equipment Melee • Multishot • Noise • Projectile • Projectile Speed • Punch Through • Recoil • Reload • Ricochet • Trigger Type • Zoom
Operator Amp • Focus (Madurai, Vazarin, Naramon, Unairu, Zenurik) • Lens
Drifter and Duviri Decrees • Drifter Combat • Drifter Intrinsics • Kaithe
Other Archwing • Companion • K-Drive • Necramech • Parazon • Railjack
General AI Director • Drop Tables • HUD • Key Bindings • Material Structures • PBR • Rarity • RNG • Settings • String Interpolation • Text Icons • Upgrade
Software, Networking, Cross Platform Play • Cross Platform Save • Dedicated Servers • EE.cfg • EE.log • File Directory • Fonts • Network Architecture • Public Export • Public Test Cluster •
Technical and Services Stress Test • Warframe Arsenal Twitch Extension • World State
Audio Mandachord • Music • Shawzin • Somachord • Sound
Mathematical Calculating Bonuses (Additive Stacking, Multiplicative Stacking) • Condition Overload (Mechanic) • Enemy Level Scaling • Maximization • User Research | {"url":"https://warframe.fandom.com/wiki/Calculating_Bonuses","timestamp":"2024-11-09T19:51:08Z","content_type":"text/html","content_length":"424695","record_id":"<urn:uuid:4657d30f-6bee-4ae7-9482-c84599431334>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00374.warc.gz"} |
Algorithms for operations on probability distributions in a computer algebra system
Document Type
Degree Name
Doctor of Philosophy (Ph.D.)
Applied Science
Lawrence M Leemis
In mathematics and statistics, the desire to eliminate mathematical tedium and facilitate exploration has lead to computer algebra systems. These computer algebra systems allow students and
researchers to perform more of their work at a conceptual level. The design of generic algorithms for tedious computations allows modelers to push current modeling boundaries outward more quickly.;
Probability theory, with its many theorems and symbolic manipulations of random variables is a discipline in which automation of certain processes is highly practical, functional, and efficient.
There are many existing statistical software packages, such as SPSS, SAS, and S-Plus, that have numeric tools for statistical applications. There is a potential for a probability package analogous to
these statistical packages for manipulation of random variables. The software package being developed as part of this dissertation, referred to as "A Probability Programming Language" (APPL) is a
random variable manipulator and is proposed to fill a technology gap that exists in probability theory.;My research involves developing algorithms for the manipulation of discrete random variables.
By defining data structures for random variables and writing algorithms for implementing common operations, more interesting and mathematically intractable probability problems can be solved,
including those not attempted in undergraduate statistics courses because they were deemed too mechanically arduous. Algorithms for calculating the probability density function of order statistics,
transformations, convolutions, products, and minimums/maximums of independent discrete random variables are included in this dissertation.
Recommended Citation
Evans, Diane Lynn, "Algorithms for operations on probability distributions in a computer algebra system" (2001). Dissertations, Theses, and Masters Projects. William & Mary. Paper 1539623382. | {"url":"https://scholarworks.wm.edu/etd/1539623382/","timestamp":"2024-11-07T10:38:44Z","content_type":"text/html","content_length":"38658","record_id":"<urn:uuid:65ec283f-4b9f-49d4-9524-ab480368af33>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00354.warc.gz"} |
Talisman Square
An Array of the integers from 1 to such that the difference between any one integer and its neighbor (horizontally, vertically, or diagonally, without wrapping around) is greater than or equal to
some value is called a -talisman square. The above illustrations show (4, 2)-, (4, 3)-, (5, 4)-, and (6, 8)-talisman squares.
See also Antimagic Square, Heterosquare, Magic Square, Talisman Hexagon
Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 110-113, 1979.
Mathematica notebook MagicSquares.m.
© 1996-9 Eric W. Weisstein | {"url":"http://drhuang.com/science/mathematics/math%20word/math/t/t022.htm","timestamp":"2024-11-03T21:30:36Z","content_type":"text/html","content_length":"4214","record_id":"<urn:uuid:e515d652-2ab7-4e67-bcb6-d7594acd2cb2>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00577.warc.gz"} |
Vertical lines drawn from the centre of buoyancy at small angles of heel will intersect at a point called the metacentre (M). The metacentre can be considered as being similar to a pivot point when a
vessel is inclined at small angles of heel.
The height of the metacentre is measured in metres from the reference point (K) and is therefore called KM.
A vessel is in stable equilibrium if returns to the upright after being inclined. This only occurs if the centre of gravity (G) is below the metacentre (M).
A stable vessel when upright is said to have a positive metacentric height (GM) when the metacentre (M) is found to be above the centre of gravity (G). It is also said that the vessel has a positive
GM or a positive initial stability. The distance between G and M is either called metacentric height or initial stability.
If the centre of gravity (G) of the vessel is above the metacentre (M) the vessel is said to have a negative GM or a negative initial stability. A vessel in this state has a loll, i.e. she floats at
an angle from the upright to one side or the other and there is a danger that she may capsize.
When weight is added to a vessel, the centre of gravity (G) of the vessel always moves in the direction of the added weight.
Weight added at the deck results in a rise of the vessel’s centre of gravity (G). That causes a decrease in the vessel’s metacentric height (GM) and thereby her stability. The time that it takes for
a vessel with little metacentric height to roll from side to side is comparatively long and is said to be a TENDER VESSEL.
Weight added low down in the vessel lowers the vessel’s centre of gravity (G). That causes an increase in the vessel’s metacentric height (GM) and thereby also an increase in her stability. The time
that it takes for a vessel with large metacentric height to roll from side to side is comparatively short and is said to be a STIFF VESSEL.
Heavy weights, such as catch and fishing apparatus, should not be situated at the deck, because the vessel’s centre of gravity (G) will rise and the metacentric height (GM) will decrease which will
increase the likelihood of a capsize of the vessel. A stiff vessel tends to be comparatively difficult to heel and will roll from side to side very quickly. A tender vessel will be much easier to
incline and will not tend to return quickly to the upright. The time period taken to roll from side to side will be comparatively long. This condition is not desirable, and it can be corrected by
lowering the vessel’s centre of gravity (G ). | {"url":"https://www.plato.is/stability_of_fishing_vessels/equilibrium/","timestamp":"2024-11-06T13:38:13Z","content_type":"application/xhtml+xml","content_length":"13758","record_id":"<urn:uuid:07de230d-3506-4e6a-bd34-14ccb20b985f>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00120.warc.gz"} |
Spectrum Module and Cyclical Analysis
Items covered in this section
This lesson explores projection models based on fixed cycles. The idea here is that the stock market follows specific cycles. If these cycles can be detected, it is possible to make a forecast based
on them. You will learn here how to find these cycles and create a projection line.
Let start with the basics of cyclical analysis. If you are already familiar with it, you may skip this part and move here to learn how ideas of cyclical analysis are applied in Timing Solution
Basics of cyclical analysis
A cycle is something that repeats itself in time. The simplest example is a workday routine: you come to work at 9AM, do your duties, have lunch at 1PM, go home at 4:30PM; same schedule day to day.
Another example is an annual cycle - from January 1 to December 31; same order of days, weeks and months, year by year. Cycles as a routine take place also inspace (consider a bus route - from
station A to station B, again and again).
We can talk about different cycles indefinitely. The good thing with cycles is that they make a forecast possible. If we know that some cycle exists, we can find out where we are in this cycle: and
then we can tell what happens in a minute, in an hour, next year, next kilometer, etc.
Scientists and engineers add to the cycle's idea a rational dimension, trying to figure out what is common to all cycles and whether it is possible to apply some quantitative approach. As a result,
many cycles can be described by equations and functions. It means that once a cycle is studied, we are able to program it and forecast some activity related to this cycle. This is why scientists love
The subject of this lesson is cycles that can be described by harmonic functions, i.e. can be represented by some combination of sine/cosine waves.
This is a classic sample of sine (cosine) wave:
Any sine wave has three parameters: period - the length of the cycle, amplitude - the strength of the cycle (it is represented on the diagram as the height of the sine curve) and phase - the angle
that defines the start of the cycle (the start point on the diagram is marked as "A").
It is possible to combine various cycles together (combine two different sine waves as an example). This procedure is called "a superposition of the cycles". Its result is a cycle as well. The
resulting cycle may look like this one:
In this case there is a superposition of 80, 145 and 170-days cycles. It is displayed together with Dow Jones Industrial Index chart (a black line). In regards to making a forecast, superposition of
the cycles is a very useful thing as normally we have many different cycles working at the same time.
Cycle Weights
In the previous example the same weights were used for all three cycles. Suppose one of the cycles is more important than others. We can double the weight of that cycle (keep the same period and
phase and double the amplitude), changing the appearance of the resulting curve. Thus, modifying the cycle by assigning different weights to its components, we can significantly improve our
superposition curve. Let us modify the cycles from the example above: increase the weight of 145-days cycle 5 times leaving other cycles as they are. Now the superposition fits the price curve better
than the previous one, thus giving us a better projection line.
Another way to improve the cycle is using its overtones. What are they?
Touch the guitar string; it starts vibrating. How many different sounds do you hear?
First of all, you hear the main vibration of the string. On the picture below this is the upper diagram; it shows the vibration that involves the whole string length. Next, you hear the vibration of
half of the string length; it is the next "octave" of the main sound. Then you hear vibrations caused by 1/3 the length of the string, 1/4, 1/5, etc. These "additions" to the main sound are called
overtones. Some overtones are louder while others are quieter; it is the reason why every musical instrument makes its unique sound.
Overtones can be used not only in music. They can add something meaningful to any cyclic process. See the difference between a pure sine wave calculated by Timing Solution software and the same wave
with overtones - the enriched wave.
This is a 145-days pure sinus wave:
This is the same wave with two overtones:
And below is the same wave with eight overtones:
Nyquist frequency
Scientists and engineers knew the importance of finding a cycle/cycles in a certain process. There was a Swedish-American engineer Harry Nyquist; he has spent a lot of time determining the minimal
time interval where it is possible to see a cycle. Nyquist frequency is named after him. The rule is that, based on a type of data you use, the minimal cycle that you may find there is at least
twice. It means that if you are working with daily data, the cycle you may find there would be at least two-days cycle; if you work with 15 minutes of data, the shortest cycle to be considered is 30
minutes. Based on the research of Timing Solution team, it is recommended to use 5-7 time intervals, i.e. for daily data use the cycles with a period of 5-7 days, for 5 minutes of data 25-35 minutes
cycles, etc. Nyquist frequency is a kind of a door to a kingdom of Chaos. Keep this door locked for now.
When working with financial instruments, some cycles fit the price chart better than others. It is possible to collect all these cycles and compare how they fit the data. This is represented by a
graph that shows the length of the cycle versus how well it fits the financial instrument. A graph like this is called a spectrogram.
A sample of a spectrogram produced by Timing Solution software is shown below:
This spectrogram is also called a periodogram. The X axis on this diagram corresponds to the period of the cycle, while Y is the strength of this cycle (more exactly, spectrum density). The highest
peaks on this chart indicate cycles that are the strongest in this data set (in other words, they fit the analysed data the best way). The spectrogram simplifies our life: when we have it, our task
is only picking up these top cycles and asking the program to make a projection line based on these selected cycles.
Cyclical analysis in action
To find out the most important cycles inside some stock market data set, Spectrum Analyzer (or Spectrum) module has been developed. The following is a description of how to work with this module.
There are some steps to do.
Calculating periodogram
Download some price history data (as an example, it is corn futures EOD continuous data) and click "Spectrum" button:
In seconds, the program makes the diagram similar to this one:
This is a periodogram. It helps to figure out what cycles are present inside this data set and which ones are more prevalent than others.
The horizontal X axis shows periods of analyzed cycles; in this example the periods are 30, 50, 100, 20 days, 1 year and 2 years (Your results might be slightly different due to different data
interval being loaded):
The vertical Y axis shows the importance of the cycle, or, if you prefer, the energy accumulated in this cycle. Accordingly, the peaks of this diagram correspond to the strongest cycles. These cycles
will be used to create a model for the projection line.
Picking up cycle
Let's look at one of these cycles closer. There is a peak around 500 days period. Click on this cycle; the program extracts it for the forecast (it is placed in the field just below the Spectrogram):
The program also calculates precisely the period of this cycle. In this example, the actual period of the cycle is 496.89 days (not 500). Now look at the Main screen:
Here the red wave shows how this 496-days cycle works in time. It is a fixed cycle, and its wave can be prolonged for as long as needed.
Playing with overtones
In this example the amount of overtones is set to 1 to calculate a pure sine wave:
This overtones parameter allows to enrich the main sine wave. Here is the enriched by 2 overtones 496.8-days wave:
3 overtones
The pictures above show that the amount of overtones equal to one gives a pure sine curve; when the amount of overtones is increased, the shape of that 496.89 day cycle wave becomes more detailed,
more complex. It resembles the original price chart better than just the sine curve. Still, this curve alone is not enough to cover important characteristic points of the price chart. Maybe, the
result would be better if more overtones are used. From the other side, using too many overtones opens the door of Chaos: instead of accuracy, fast harmonics may bring more noise. Look at the same
496.89-days cycle with 12 overtones:
Playing with FSM "life expectancy"
It is good to know that some cycle exists, being able to make its projection. But for how long will this cycle be active? You may have seen already that even a slight change of initial data set could
bring to the surface other cycles and dismiss those that you have used as the most important ones. Based on the research of Timing Solution team, it is recommended to pay attention to FSM parameter
that represents the time span within which this cycle is active, i.e. the "life expectancy" of the analyzed cycle:
Let us look at FSM parameter closer. FSM stands for Forecast Stock Memory. It is the expected "life time" of some cycle. What does that mean? The cycle is there all the time, though in regards to the
effect of this cycle on the stock market one and the same cycle may appear active/"live" or not relevant, at different periods of time. The cycle's "life expectancy" is the time span where this cycle
appears being active. That is why we can use this cycle to forecast future stock market moves. We introduce this parameter as a result of the research conducted by Timing Solution team.
In the example below, FSM=3. It means that the cycle is present in the analysed data set (it is confirmed by the spectrogram), and we have observed this cycle working within the time span in the most
recent data that equals at least to the last 3 cycles. Here, as an example, we have found 100-days cycle. We expect it be working for the last 300 days of price history:
For FSM=4, it is the last 400 days:
Usually, for any stock market data set, cycles do not "live" forever; they appear being active for some restricted period of time. With FSM parameter, we can specify the "life expectancy" of the
analyzed cycles. As it is mentioned earlier, do not set FSM too big. Based on our research, usually it is less than 7.
Picking up several cycles
Till now, you have tried just one cycle, the top one on the spectrogram. Let us pick more cycles (166 and 263 days cycles above). All three cycles are shown now in the Main screen. They are color
coded for your convenience. Here the 499-day cycle is shown in red, the 166-day cycle is shown in bluem while 264-day cycle is shown in green:
You can explore each one of these cycles separately and you can see how they work together.
Superposition of several cycles
To see the mutual performance of these cycles in regards to the analysed data, create a superposition of these three cycles. It is very simple to do: drag and drop these cycles from the Spectrum
window onto the Main screen. Or click the Wave button in the Spectrum window.
Now the superposition curve reflects the initial price chart better than any of those three cycles on their own. Prolonging these cycles superposition as far as you like to the future, you can have a
nice projection line based on these three cycles.
Customize the projection line
To your convenience while working with some projection line, you can customize it. You may change its color and the line's thickness through Main Window View.
In order to do that, you should make the RIGHT mouse click in the Main window and then, in the pop-up menu, choose "Main Window View" item or click this button:
To modify a color and thickness of the superposition projection line, please use these controls:
This option allows to display projection line in separate panel or overlay of price chart:
To delete this projection line from the Main Window, click this
It can be done as well by making a RIGHT mouse click in the Main window and choosing "Delete ULE event" in the pop-up menu.
What cycle to choose?
You know now that combining several cycles may provide a better result than just one cycle. You can create a projection line based on as many cycles as you like. And you know also that too many
cycles lead to getting a noise instead of a valid forecast. How to choose the best cycles?
There are two criteria and one recommendation for that:
1. Choose the highest peaks on the periodogram;
2. The width of a chosen peak should be as narrow as possible.
The higher the peak, the bigger the amplitude of this cycle. The narrower the cycle, the more energy is concentrated in this cycle.
Take a look at this example:
The peaks at this periodogram marked by red circles (1, 3, 5) are "good" cycles; i.e. these peaks are high and the width of these peaks is narrow. There are also other peaks; they are marked by blue
circles (2 and 4), these cycles are not as good because these peaks are not as narrow. It means that the energy of these cycles is distributed in a wider range. It makes these cycles less precise.
Timing Solution team did a research for many different financial instruments trying different combinations of cycles. The recommendation is: do not use too many cycles, several cycles (1-5 cycles)
are enough. You should remember that the models that use too many cycles are very good in explaining PAST price movements while they are not so good in FORECASTING future movements. Adding just one
bad/unimportant cycle to your cyclical model may spoil the whole model. Be very picky while choosing your cycles.
Dominant cycles versus permanent cycles
The spectrum analysis is performed for fixed cycles. You may have seen as well the mention of "dominant" and "permanent" cycles. What are they? These are just terms. And sometimes the use of these
terms in other software or in the literature is confusing. Let's discuss it a bit.
Permanent cycles are more obvious. According to their name, they are there all the time. Actually, it may be just another name for cycles in general.
In Timing Solution, the distinction between dominant and permanent cycles is in the way how to take their presence in data.
By default Timing Solution is oriented on search of dominant cycles, the cycles that have an important role for some time and then disappear. They disappear not in a sense of not existing any more.
Instead, their effectiveness changes over time: for a while they would play a big role and then they would become less effective. Fixed cycles are always present, just like a sine curve that is
unlimited on both sides, past and future. At any given moment, cycles accumulate or lose some portion of energy becoming more or less apparent. They do not work in the same manner all the time.
Multiframe technology is developed to catch these cycles. When the new portion of the price data is coming, the cyclical portrait of the financial instrument is changing as well due to the changes in
the periodogram. Cycles used initially to create the forecast model may become not as important while other cycles ignored previously may start being important. It means that from time to time it is
recommended to recalculate the spectrum by clicking "Recalculate" button.
To find permanent cycles (i.e. the cycles that always work the same way), set this parameter in Spectrum module:
As an example, let us try to find a permanent cycle for DJII from 1885 to 2014.
This is the spectrum for Dow Jones; it shows the peak around the period of 40 months. It is a cycle well known to economists, so called Kitchen inventory cycle:
In most cases we work with dominant cycles. They are more typical for financial data, more tradable. Permanent cycles are mostly used for economical analysis, as these cycles are believed to work in
the same manner now as they worked 10, 50, 100 etc. years ago.
Detrending, or what to forecast?
Most financial instruments have some trends. You have heard a saying: "Trend is a friend". It may be true at certain moments for a trader, but it is not true from the point of view of Cyclical
Analysis theory. When scientists evaluate data sets, the first step is to get rid of a trend and anything that can distort a statistical picture. Therefore, it is mathematically necessary to modify
price data before doing anything. We have to do that as our goal is to model the market behavior and make a forecast as close as possible to the functions used in forecasting, i.e. to sin curves. To
reach this goal, we do not want the general trend of the stock to be used in the forecast, so we do not use the price itself to calculate the spectrum diagram. Instead, we use the detrending
indicators. One of the most popular is the relative price oscillator (or percentage price oscillator). As an example, look at this indicator with the period=100 bars:
The parameters of this indicator are defined manually in relation to swings that the program tries to catch. Other indicators may serve as a forecast target as well: RSI, ADX, Volatility, etc.
In other words, the program performs the spectrum analysis not for original data (as it was done for Dow Jones Industrial index in the example above), with its up and down trends, but for one of its
Oscillators are much more convenient for the cyclical analysis.
Intraday Data and Turbo Cycles
While working with intraday data, it is better to calculate the spectrum using price bar metric:
Also for intraday it is recommended to use Turbo cycles module. It extracts the most important cycles automatically and calculates the projection line based on these cycles. This projection line is
updated in real time when the extra price history data is downloaded. You will find more info about this module here.
Working with Q-Spectrum module - going beyond "curve fitting"
Q-Spectrum module sets the new standard in financial analysis. In this module we have incorporated the methods of Walk Forward Analysis (WFA), a standard in finance, together with standard procedures
of classical cyclical analysis.
Classical cyclical analysis (Fourier transform) is good when we try to describe the past assuming that the future simply repeats the past movements. This is true for many physical/natural phenomena
... but not for stock market and any other activity that involves humans. Classical cyclical analysis applied to the stock market brings "curve fitting" effect. It is a situation when some projection
line perfectly describes the past but fails to forecast the future. Q-Spectrum module allows to select only those cycles that are suitable for forecasting and, accordinly, suitable for trading.
This is how the periodgram for corn futures calculated with Q-Spectrum looks:
As with the classical spectrum, the highest peaks here correspond to the most important cycles. Exactly in the same manner, as we did for classical spectrum, we can pick up these cycles. When picking
up a cycle in Q-Spectrum module, it is recommended to pay attention to the cycles that are confirmed by cycles with multiple periods. As an example, 100-days cycle should be confirmed by the presence
in the same periodogram of the cycles with periods of 200, 300, 400 etc. days. We still do not quite understand the nature of this phenomena, so we take it as phenomenological fact.
Here is a practical example:
Q-Spectrum reveals here 273.3-days cycle; it is confirmed by the presence of the cycle with doubled period of 546.6 (2 x 273.3) days cycle though it is just a small peak on periodogram. This cycle is
also confirmed by the presence of 1093-days cycle (4 x 273.3). Thus, 273.3-days cycle (seen as the most important on the periodogram) is confirmed by two larger cycles.
Another cycle that we can pick up the same way is 478.8-days cycle.
In the Main screen you can see the superposition projection line based on these selected cycles. You can modify FSM and the amount of overtones in the "Modify Cycles" tab:
You can change FSM and the amount of overtones for each cycle separately when "One" button is pressed. In this case, the program makes a change to the highlighted cycle only:
Here is video recording that shows how to build Q-Spectrum projection line for the same financial instrument, Corn futures (data till the mid of 2013). This class is updated in December 2019, so we
can compare how this forecast works:
Advanced forecasting techniques
Timing Solution software has more features that apply different methods of Cyclical Analysis. Below is a short description of those features.
Wavelet Module: this module allows to see the cyclical phenomena in dynamic environment, i.e. you can see how the cycles appear-live-and disappear. More information about this module is contained
Committee: the same cycle/cycles can generate several versions of projection lines, this module allows to see all these projection lines together. Documentation for it can be found here.
Q-Spectrum module: this is a unique module that applies methods of Walk Forward Analysis (the standard financial analysis technology) to financial data. Its description is here.
Visual cyclical analysis
Behind very complicated math methods used in Timing Solution software (like Spectrum or Wavelet or some other method) lies a very simple and clear idea. We try to find the track of regular waves in
financial data. All mentioned mathematical powerful tools are designed for this simple task of catching a regular wave in financial data as early as possible. Very often the simplest visual analysis
of the price chart helps to reveal these cyclical patterns. Specially developed Timing Solution charting tools are very helpful for this task.
That is why we recommend, before working with some sophisticated tools, to apply a simplest visual cyclical analysis. It is just looking at your price chart attentively and identifying some regular
patterns there.
The charting tools in the "Wave" section help you to do that:
Let's try these charting tools:
Harmonic Wave
Suppose that you see a two-wave regular pattern in your price chart, like this:
To model it, we build a sine wave overlay of price chart using "Harmonic wave" charting tool:
Adjust the overtones, try and calculate half-length of the wave, one third of it, and so on. Sometimes (as shown on the picture below) they work:
For your convenience it is recommended to disable snapping mode by pushing this button (otherwise the program automatically catches the nearest highs/lows):
Fourier String (1 wave)
Take a look at A-B wave between mid of 2010 and end of 2011:
We can model this A-B wave using Fourier analysis and prolong it into the future. Choose "Fourier String (1 wave)" charting tool; drag the mouse cursor from the beginning of this wave (point A) to
its end (point B). You will see this wave prolonged into the future.
This is another example. Here there are two waves in the price chart between the beginning of November 2013 and the beginning of March 2014.
To model and prolong this two-wave pattern into the future, apply "Fourier String (2 waves)" charting tool:
and drag the mouse cursor to cover two-wave pattern:
It is recommended to experiment with the amount of overtones parameter to enrich/simplify the projection line:
These are just some charting tools based on classical Cyclical Analysis methods. The software has more charting tools, and they are discussed here.
This concludes the tutorial about the basic cyclical analysis and the use of Spectrum Module. Next lesson covers Astronomy Module.
Additional study regarding Spectrum and Cyclical analysis
Very basics of harmonic analysis - a detailed explanation of basic definitions and methods used in harmonic analysis (periodical functions, waveforms, Fourier analysis, overtones ...)
Cyclical analysis in essence in 33 pictures - a general explanation of different cyclical models that are present now in Timing Solution software (classical cyclical analysis, wavelet analysis, astro
cycles, analysis of price patterns).
Wavelet analysis: Wavelet analysis is a part of cyclical analysis especially important for financial data where practically any cycle does not represent itself forever; cycles have some restricted
"living" time. Wavelet analysis issues are discussed in these articles below:
Wavelet analysis - cycles early warning system - a basic explanation regarding wavelet analysis
Wavelet Cycle Hunter - thoughts on application of wavelet analysis for financial data
Fading cycles of the stock market - the techniques of classical cyclical analysis applied in physics not always may be applied for financial data. In this article you will find the explanation why...
Q-Spectrum: Q-Spectrum is a new powerful module that allows to reveal cycles using a unique combination of methods of classical cyclical analysis and classical financial analysis (walk forward
analysis). These articles illustrate the idea:
Q-Spectrum - an explanation of Q-spectrum features
Anti-Information - a discussion of new issues that Q-spectrum brings to our attention
Turbo cycles: This module automatically calculates spectrum, extracts the most important cycles and calculates the projection line based on these cycles. This is a quick way to do cycles analysis
which is especially important for intraday chart; the module perfectly works in real time mode. These articles are recommended:
Turbo Cycles module - a detailed explanation how Turbo cycles module works
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What is the equation of the line between (-17,14) and (19,6)? | HIX Tutor
What is the equation of the line between #(-17,14)# and #(19,6)#?
Answer 1
$y = - \frac{2}{9} x + \frac{92}{2}$
First, we find the slope #m# of the line. The slope of the line is the change in #y# per unit of change in #x#. Equivalently, this means that a line with slope #a/b# will rise #a# units as #x#
increases by #b# units. Then, we can find the slope from two points with the following formula:
#m = ("change in "y)/("change in "x) = (y_2-y_1)/(x_2-x_1)#
In this case, that gives us
#m = (6-14)/(19 - (-17)) = -8/36 = -2/9#
Now, we can write the equation using the point-slope form of a line. #y - y_1 = m(x - x_1)#
Picking either of the points will work, so let's use #(19, 6)# (as an exercise, verify that this gives the same result if you use the other point). This gives us the equation
#y - 6 = -2/9(x - 19)#
If we wish to put that into the more common slope-intercept form, we can just multiply it out and solve for #y#.
#y - 6 = -2/9x + 38/9#
# y = -2/9x + 92/2#
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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6.17: Discrete Random Variables (2 of 5)
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Learning Objectives
• Use probability distributions for discrete and continuous random variables to estimate probabilities and identify unusual events.
Probability Distribution for Discrete Random Variables
In this section, we work with probability distributions for discrete random variables. Here is an example:
Consider the random variable the number of times a student changes major.
(For convenience, it is common practice to say: Let X be the random variable number of changes in major, or X = number of changes in major, so that from this point we can simply refer to X, with the
understanding of what it represents.)
Here is the probability distribution of the random variable X:
Here is what it tells us:
For a randomly selected student, we cannot predict how many times he or she will change majors, but there is a predictable pattern described by the probability distribution (or model) above. So this
is a random variable for which we are assuming the values range from 0 to 8. (In reality, a negligible proportion of students change majors more than 8 times.) The table provides a way to assign
probabilities to outcomes. Note that if we add up the probabilities of all possible outcomes (0.135 + 0.271 + … + 0.002), we get exactly 1, which is not surprising (because one of the possible
outcomes 0, 1, … , 8 will occur for sure).
Another way to represent the probability distribution of a random variable is with a probability histogram.
The horizontal axis accounts for the range of all possible values of the random variable (in our case, 0–8), and the vertical axis represents the probabilities of those values.
The heights of the bars add to 1, which is not surprising since the heights represent probabilities.
Let’s summarize the features of a probability distribution:
• The outcomes described by the model are random. This means that individual outcomes are uncertain, but there is a regular, predictable distribution of outcomes in a large number of repetitions.
• The model provides a way of assigning probabilities to all possible outcomes.
• The probability of each possible outcome can be viewed as the relative frequency of the outcome in a large number of repetitions, so like any other probability, it can be any value between 0 and
• The sum of the probabilities of all possible outcomes must be 1.
Where do these probability distributions come from? Recall that probability distributions can come from data, such as the distribution of boreal owl eggs. Scientists observe thousands of nests and
record the number of eggs in each nest. Then they calculate the relative frequency of each outcome. The relative frequency of each outcome represents the empirical probability for that outcome.
We can also use a mathematical formula to represent a probability distribution. In this case, we make assumptions about how outcomes will be distributed. In other words, we use a mathematical formula
to describe the predicted relative frequencies for all possible outcomes. We do not look at mathematical formulas for probability distributions in this course, but we want you to be aware that not
all probability distributions come from data.
Recall the probability distribution of the random variable X = number of changes in major:
Let’s see what kinds of probability questions we can answer using it.
1. What is the probability that a college student will change majors at most once?
The phrase “at most once” means either the student never changes majors (X = 0) or the student changes majors once (X = 1). Therefore, to find this probability, we need to add the probabilities that
are highlighted in the table:
P(a college student changes majors at most once) = P(X = 0) + P(X = 1) = 0.135 + 0.271 = 0.406
The probability that a randomly selected college student will change majors at most once is about 0.406. We can also say that about 40.6% of the time, a randomly selected college student will change
majors at most once.
2. John’s parents are concerned that he has decided to change his major for the second time. John claims that he is not unusual. What is the probability that a randomly selected college student will
change his major as often as or more often than John?
To answer the question about John, we need know the probability that a randomly selected student will change his major 2 or more times. We need to add together the probabilities shaded in the table.
P(change major 2 or more times) = P(X = 2) + P(X = 3) + … + P(X = 8) = 0.594
Here is another way to figure this out. We can use the idea that all of the probabilities together make up 100% of the possibilities. So if we add up all the probabilities in the table we should get
1. Now if we figure out the probability that someone changes majors 0 or 1 times, we can just subtract this from 1 to find the probability that someone changes majors 2 or more times. As we learned
previously, this is the complement rule.
P(change major 2 or more times) = 1 – [P(X = 0) + P(X = 1)] = 1 – [0.135 + 0.271] = 0.594
Do you think John has given a convincing argument that he is not unusual? Yes! Fifty-nine percent of the time, a college student will change majors as often as or more often than John did. Stating
this same result in terms of probability, we might say, “There is a 59% probability that a randomly selected college student will change majors 2 or more times while in college.”
We found that changing a major 2 or more times is not very unusual, since it happens about 59% of the time. So…
3. How often would John need to change his major to be considered unusual?
One way to answer this question is to just a make a judgment call about what we might consider “unusual” based on the table. For example, we might notice that the probability that a student will
change majors 5 or more times is about 5%.
P(change majors 5 or more times) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)
= 0.036 + 0.012 + 0.003 + 0.002 = 0.053
An event that occurs only 5% of the time is pretty unusual.
Are there other ways to more definitively determine what might be considered unusual? Well, we might use a measure of center, such as the mean, to determine a “typical” number of times that students
change majors. Values that are 2 standard deviations above the mean could be used to identify unusual behavior. We will come back to this question after we have developed an understanding of mean and
standard deviation for a probability distribution.
Contributors and Attributions
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How To Calculate The Moment Of Inertia
mipan/iStock/Getty Images
In physics, the amount of matter that an object has is reflected in its mass, which largely determines its resistance to changes in motion — or inertia. For things that rotate or spin, however, the
picture becomes more complicated; instead of mass, physicists talk about an object's moment of inertia. An object's shape strongly affects the moment of inertia, as does the location of the center of
rotation. Although calculating the moment of inertia can be very complicated, shapes such as spheres, rods and discs simplify the math considerably.
Rolling Rod, Cylinder or Disc
Step 1
Measure the radius of the object from the center to the edge in centimeters; enter this figure into the calculator. Square it by pressing the "x^2" button or by multiplying the figure by itself. For
example, a cylinder weighing 5,000 grams rolls across the floor. Its radius is 5cm. Five squared is 25.
Step 2
Multiply the previous result by the mass. In this example, 25 times 5,000 is 125,000.
Step 3
Divide by two; this gives the moment of inertia. Continuing the example, 125,000 / 2 equals 62,500. Units are in grams times centimeters squared.
Rolling Solid Sphere
Step 1
Measure the radius of the sphere from the center to the edge in centimeters; enter this figure into the calculator. Square it by pressing the "x^2" key or by multiplying the figure by itself. For
example, a sphere weighing 5,000g rolls across the floor. Its radius is 10cm. Ten squared is 100.
Step 2
Multiply the previous result by the mass, then multiply by 2. In the example, 100 times 5,000 is 500,000, and 500,000 times 2 is 1,000,000.
Step 3
Divide by 5, giving the moment of inertia. Continuing the example, 1,000,000 / 5 equals 200,000. Units are in grams times centimeters squared.
Rolling Thin Spherical Shell
Step 1
Measure the radius of the sphere from the center to the edge in centimeters; enter this figure into the calculator. Square it by pressing the "x^2" key or by multiplying the figure by itself. For
example, a basketball weighing 200g rolls across the floor. Its radius is 10cm. Ten squared is 100.
Step 2
Multiply the previous result by the mass, then multiply by 2. In the example, 100 times 200 is 20,000, and 20,000 times 2 is 40,000.
Step 3
Divide by 3, giving the moment of inertia. Continuing the example, 40,000 / 3 equals 13,333.33. Units are in grams times centimeters squared.
Cite This Article
Papiewski, John. "How To Calculate The Moment Of Inertia" sciencing.com, https://www.sciencing.com/calculate-moment-inertia-5161917/. 24 April 2017.
Papiewski, John. (2017, April 24). How To Calculate The Moment Of Inertia. sciencing.com. Retrieved from https://www.sciencing.com/calculate-moment-inertia-5161917/
Papiewski, John. How To Calculate The Moment Of Inertia last modified March 24, 2022. https://www.sciencing.com/calculate-moment-inertia-5161917/ | {"url":"https://www.sciencing.com:443/calculate-moment-inertia-5161917/","timestamp":"2024-11-15T04:52:30Z","content_type":"application/xhtml+xml","content_length":"73189","record_id":"<urn:uuid:8fe3f06f-5c55-4c88-832f-6b4a76c7a4cd>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00539.warc.gz"} |
Solve the following linear equations and verify yo... | Filo
Question asked by Filo student
CHECKPOINT 1 Solve the following linear equations and verify your result: 1. 2. 3. 4. 6. ?. 8. 9. . olving an Equation When Variables are on both Sides the above section, we discussed that a term in
an equation can be transpo is method is used widely in cases where the variables are present on both en a term is shifted to another side after changing its sign, the equality of hanged. The shifting
of the term from one side of the equation to another ile transposing the term, its sign is changed as follows: + changes to - - - changes to + changes to + rugh this process, we move all the
variables on one side and constants ther side. We then proceed with solving the equation, as discussed in ple 3: Solve the equation . To get all the variables on one side and constants on ter side of
the equation, we transpose from the RHS to LHS. Its sign changes from negative to positive . from the LHS to RHS. Its sign changes from negative o positive (1). [Dividing both sides by 6 ] re is the
solution of the given equation.
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CHECKPOINT 1 Solve the following linear equations and verify your result: 1. 2. 3. 4. 6. ?. 8. 9. . olving an Equation When Variables are on both Sides the above section, we discussed that a
term in an equation can be transpo is method is used widely in cases where the variables are present on both en a term is shifted to another side after changing its sign, the equality of
Question hanged. The shifting of the term from one side of the equation to another ile transposing the term, its sign is changed as follows: + changes to - - - changes to + changes to + rugh this
Text process, we move all the variables on one side and constants ther side. We then proceed with solving the equation, as discussed in ple 3: Solve the equation . To get all the variables on one
side and constants on ter side of the equation, we transpose from the RHS to LHS. Its sign changes from negative to positive . from the LHS to RHS. Its sign changes from negative o positive
(1). [Dividing both sides by 6 ] re is the solution of the given equation.
Updated Feb 7, 2023
Topic All topics
Subject Mathematics
Class Class 9
Answer Video solution: 1
Upvotes 126
Video 3 min | {"url":"https://askfilo.com/user-question-answers-mathematics/checkpoint-1-solve-the-following-linear-equations-and-verify-34313433303237","timestamp":"2024-11-14T00:45:48Z","content_type":"text/html","content_length":"275924","record_id":"<urn:uuid:725090a3-9839-40fd-878c-54ee6cec2eb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00116.warc.gz"} |
Theoretical physics II
This unit provides part of a major in theoretical physics. It consists of two 12-lecture sub-units, Advanced Quantum Mechanics and Computational Physics and a 12-hour seminar sub-unit. The key areas
of study are:
1. Advanced Quantum Mechanics: spin angular momentum, perturbation theory, scattering theory and the quantum theory of radiation;
2. Computational Physics: discrete arrays to model the space and time evolution of functions or physical systems; a hands-on approach is used throughout to develop confidence and competency in using
a computer to solve physical problems; includes a computer based assignment and short computational physics project; and
3. Theoretical Seminar: seminar participation in theoretical problems, projects and presentations.
On completion of this unit students will be able to:
1. Recall fundamental concepts from the sub-unit of Advanced Quantum Mechanics, which include Approximate Methods in Quantum Mechanics I: stationary (time-independent) perturbation theory, first and
second order perturbation of a non-degenerate state, Higher order perturbation theory, Perturbation of a degenerate state and applications in atomic and nuclear physics, Time-dependent
perturbation theory, Fermi's golden rule, The Ritz variational method, Semi-classical (WKB) approximation, Scattering Theory, Stationary scattering states, The Born approximation, Partial wave
expansions, Phase shifts, Scattering of identical particles, The optical theorem, Introduction to Green function techniques, Charged Particles in an Electromagnetic Field, Gauge potentials and
the electromagnetic field, Hamiltonian of a particle in an electromagnetic field, The Quantum Theory of Radiation and the interaction of radiation with atomic systems, Transition rates ,
Multipole transitions. Quantum electrodynamics, Geometric phases in quantum mechanics, Berry's phase, The Aharonov-Bohm effect, Path Integrals, and Free space propagator;
2. Use a high level computer language such as Matlab to solve computation problems, and model systems, applicable to theoretical physics which include Numerical differentiation and integration,
Finding roots, Special functions, Change of basis, Reduction to dimensionless forms, Discretization of quantum mechanical operators, Stationary and time-dependent Schrodinger equation, 1D
scattering, Quantum harmonic oscillator, Eigenvalue problems, Bose-Einstein condensation and the Gross-Pitaevskii equation, Quantized vortices, Stochastic methods, Pseudo-random numbers, Monte
Carlo method, Metropolis algorithm, 2D Ising model, Signal and Image Processing using the DFT and FFT, Convolution theorem, Filtering in 1D and 2D, Non-linear filtering and mathematical
morphology, and Radon transform and tomography;
3. Solve new problems in physics related to the core concepts of the unit by drawing on the theoretical underpinnings that illustrate the physics;
4. Research topics in contemporary physics, and present critically assessed summaries as scientific reports and visual presentations;
5. Apply educated reasoning to provide approximate solutions to scientific questions and advanced problems (Fermi questions).
Examination (2 hours): 23%
Assignments and computational projects: 43%
Seminar contributions: 34%
An average of 2 hours lectures, one 1-hour tutorial and one 1-hour seminar per week
See also Unit timetable information | {"url":"https://www3.monash.edu/pubs/2015handbooks/units/PHS3142.html","timestamp":"2024-11-04T20:17:12Z","content_type":"application/xhtml+xml","content_length":"30764","record_id":"<urn:uuid:74632802-8455-407c-be7c-fbea12d58f56>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00210.warc.gz"} |
Minimizing node expansions in bidirectional search with consistent heuristics
A* is optimally effective with regard to node expansions among unidirectional admissible algorithms-those that only assume that the heuristic used is admissible. Among bidirectional algorithms the
Fractional MM algorithm is optimally effective (given the correct parameters) among admissible algorithms. This paper generalizes the bidirectional result to more complex settings where more
information on the problem domain can be exploited: (1) When the cost of the minimal edge ? is known. (2) When the algorithm knows that the heuristics are consistent. This characterization uses a
novel algorithm called MT. MT is similar to Fractional MM and is also optimally effective, but simpler to analyze.
Original language English
Title of host publication Proceedings of the 11th International Symposium on Combinatorial Search, SoCS 2018
Editors Vadim Bulitko, Sabine Storandt
Publisher AAAI press
Pages 81-89
Number of pages 9
ISBN (Electronic) 9781577358022
State Published - 1 Jan 2018
Event 11th International Symposium on Combinatorial Search, SoCS 2018 - Stockholm, Sweden
Duration: 14 Jul 2018 → 15 Jul 2018
Publication series
Name Proceedings of the 11th International Symposium on Combinatorial Search, SoCS 2018
Conference 11th International Symposium on Combinatorial Search, SoCS 2018
Country/Territory Sweden
City Stockholm
Period 14/07/18 → 15/07/18
ASJC Scopus subject areas
• Computer Networks and Communications
Dive into the research topics of 'Minimizing node expansions in bidirectional search with consistent heuristics'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/minimizing-node-expansions-in-bidirectional-search-with-consisten","timestamp":"2024-11-10T18:25:03Z","content_type":"text/html","content_length":"56952","record_id":"<urn:uuid:b55387ea-e54f-4c24-83fc-7b1754aa3260>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00709.warc.gz"} |
Random paths
A random walk consists of a sequence of positions, each one obtained from the previous one by making a random step. In this exercise we consider random walks in the plane, starting at (0, 0). Let k
be a strictly positive natural number. Each step will be an increment between −k and k, random and independent, of the two coordinates.
Hence, we need a source of randomness. The usual way to simulate it consists in generating pseudorandom numbers. These numbers are the result of an algorithm, so they are not really random, but they
look random enough. The linear congruential generators are defined with four natural numbers m (module), a (multiplier), b (adder) and s (initial seed). The generated sequence is
x[1]=(a*s+b) mod m, x[2]=(a*x[1]+b) mod m, x[3]=(a*x[2]+b) mod m, …
For instance, if m = 9, a = 2, b = 7, s = 3, then we get
x[1]=(2*3+7) mod 9=4, x[2]=(2*4+7) mod 9=6, 1, 0, 7, 3, 4, 6, …
These numbers are between 0 and m − 1, but in this exercise we need numbers between −k and k. The easiest way to achieve this is with the following code; use it just like that:
int random(int k, int m, int a, int b, int& s) { s = (a*s + b)%m; return s%(2*k + 1) - k; }
Following with the example, for k = 2 the sequence of increments is
4 mod 5−2=2, 6 mod 5−2=−1, 1 mod 5−2=−1, −2, 0, 1, 2, −1, …
and, if we increase the first coordinate before the second one, the steps are
(0,0), (2,−1), (1,−3), (3,−3), …
Write a program to compute the first n steps of a sequence of random walks defined by k, m, a, b and s.
Input is all natural numbers, and consists of several cases, each one in two lines. The first line contains n and k. The second line contains m, a, b and s. All n, k and m are strictly positive, and
a, b and s are less than m.
For each case of the input, first print its number starting at 1, followed by the walk of n steps defined by k, m, a, b and s. If some position gets repeated, indicate it and stop the walk as it is
shown in the example. Print an empty line at the end of each case. | {"url":"https://jutge.org/problems/P33549_en","timestamp":"2024-11-10T11:28:47Z","content_type":"text/html","content_length":"28830","record_id":"<urn:uuid:ce8f27f8-9006-4597-85b8-b65e836c11c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00721.warc.gz"} |
Computing likelihoods
Computing likelihoods#
In general, the likelihood function is defined as the probability (or probability density) assigned to an observed outcome by a particular model. It is often alternately referred to as the sampling
probability, and is a cornerstone of many statistical inference procedures.
msprime provides functions for evaluating two sampling probabilities: that of a stored tree sequence for a given diploid population size \(N_e\) and per-link, per-generation recombination probability
\(r\) under the standard ancestral recombination graph (ARG); and that of a pattern of mutations given a tree sequence and per-site, per-generation mutation probability \(\mu\) under the infinite
sites model.
Generic analysis tools for arbitrary tree sequences, allowing efficient calculation of genetic diversity, allele frequency spectra, F statistics, and so on, are provided within tskit and described in
the statistics documentation. The likelihood functions below are provided by msprime because they are specific to the ARG and infinite sites models.
Evaluating these log likelihoods requires knowledge of a whole ARG, rather than just the more compact tree sequence representation. Additionally the log likelihood implementations are only available
for continuous genomes. Hence, the underlying tree sequence has to conform to the record_full_arg = True and discrete_genome = False options of the sim_ancestry() function.
Quick reference#
Log likelihood of an ARG topology and branch lengths
Log likelihood of a pattern of mutations arising from a given ARG
ARG sampling probability#
The following example simulates an ARG with 5 diploid samples and evaluates the likelihood of the realisation for four combinations of parameters. Note that one combination coincides with the
parameters used to simulate the ARG, while the other combinations are quite different.
import msprime
ts = msprime.sim_ancestry(
5, recombination_rate=1, record_full_arg=True,
sequence_length=1, discrete_genome=False, random_seed=42)
print(msprime.log_arg_likelihood(ts, recombination_rate=0, Ne=1))
print(msprime.log_arg_likelihood(ts, recombination_rate=0.1, Ne=1))
print(msprime.log_arg_likelihood(ts, recombination_rate=1, Ne=1))
print(msprime.log_arg_likelihood(ts, recombination_rate=10, Ne=10))
In this example, the simulated ARG contains at least one recombination, which is an event of probability 0 when the recombination_rate = 0. Hence, the log likelihood for recombination rate zero
returns a numerical representation of negative infinity, i.e. the logarithm of zero. The other three combinations of parameters all result in positive likelihoods, or finite log likelihoods.
The data was generated from a recombination rate of one and a default population size of one, and these parameters give rise to a relatively high log likelihood. The same ARG would have been an
unlikely realisation under very different parameters, which thus result in more negative values of the log likelihood.
Mutation sampling probability#
The next example adds random mutations to the tree sequence generated above in ARG sampling probability and evaluates the unnormalised log probability of the mutation realisation, given the tree
sequence and a prescribed mutation rate.
ts = msprime.mutate(ts, rate=1, random_seed=42)
print(msprime.log_mutation_likelihood(ts, mutation_rate=0))
print(msprime.log_mutation_likelihood(ts, mutation_rate=0.1))
print(msprime.log_mutation_likelihood(ts, mutation_rate=1))
print(msprime.log_mutation_likelihood(ts, mutation_rate=10))
Since there is at least one mutation in the realisation, its probability given mutation_rate = 0 is 0, resulting in a log likelihood of negative infinity. The mutation realisation is a typical
outcome when mutation_rate = 1, which is the value used to simulate it, and which thus results in a relatively high log likelihood. Mutation rates which are significantly higher or lower result in
more negative log likelihoods because generating the same realisation using those rates is unlikely. | {"url":"https://tskit.dev/msprime/docs/latest/likelihoods.html","timestamp":"2024-11-10T09:51:26Z","content_type":"text/html","content_length":"29422","record_id":"<urn:uuid:bf18af60-8149-42ae-8760-672df828178f>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00573.warc.gz"} |
Measurement, instrumentation, and techniques
Exploring Recent Advances in the Physics of Biofluid Locomotion
This Special Topics Edition of the JPSJ describes the latest advances in the field of biofluid locomotion, shedding light on the underlying physics behind the movement of organisms that swim and fly.
Cross-disciplinary physics and related areas of science and technology
Electromagnetism, optics, acoustics, heat transfer, and classical and fluid mechanics
Mathematical methods, classical and quantum physics, relativity, gravitation, numerical simulation, computational modeling
Measurement, instrumentation, and techniques
Statistical physics and thermodynamics
Structure and mechanical and thermal properties in condensed matter | {"url":"https://jpsht.jps.jp/article/fields/199/","timestamp":"2024-11-03T13:24:05Z","content_type":"text/html","content_length":"69410","record_id":"<urn:uuid:f8e299ac-4b5b-42fd-bf6a-92b38552f088>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00468.warc.gz"} |
What is a nibble in computers and digital technology? – TechTarget Definition
What is a nibble?
In computing and digital technology, a nibble is four consecutive binary digits or half of an 8-bit byte. When referring to a byte, it is either the first four bits or the last four bits, which is
why a nibble is sometimes referred to as a half-byte. The term nibble also carries on the "edible data" metaphor established with bit and byte. Due to its byte connection, a nibble is occasionally
spelled nybble or nyble.
Because a nibble is made up of binary data, each of the four digits is either a 0 or 1, in any combination, as in 0010, 0110, 1011 or 1111. The total number of possible combinations is 16, calculated
as 2^4. A nibble can also be represented by a hexadecimal digit. Hexadecimal is a base-16 numbering system that uses the digits 0 through 9 and the letters A through F to represent data, including
nibbles and bytes. Figure 1 shows each possible bit combination in a nibble, along with its hexadecimal and decimal equivalent.
Figure 1 illustrates each possible bit combination in a nibble (binary), along with its hexadecimal and decimal equivalent.
Two-digit hexadecimal numbers are used to represent bytes, which are made up of two consecutive nibbles. Figure 2 shows the digital data from a small text file based on American Standard Code for
Information Interchange (ASCII) text encoding. Figure 2 includes both the binary data and the corresponding hexadecimal digits (in the rightmost column). Each row, except the last, contains four
bytes, separated by spaces.
Figure 2 illustrates the digital data from a small text file based on the ASCII text encoding, both the binary data and the corresponding hexadecimal digits (in the rightmost column).
The first byte (01010100) in the first row is highlighted, as is its corresponding hexadecimal code (54). The highlighted byte is the ASCII letter T (uppercase), which is ASCII code 084. The first
nibble in the byte, 0101, is represented by the hexadecimal number 5, and the second nibble in the byte, 0100, is represented by the hexadecimal number 4, resulting in a byte hexadecimal value of 54.
What is a nibble in communications?
In communications, a nibble is sometimes referred to as a quadbit. As with any nibble, the quadbit is 4 bits and has 16 possible combinations. A signal might be encoded in quadbits rather than one
bit at a time. Nibble interleaving, a process used in multiplexing, takes a quadbit from a lower-speed channel as input for a multiplexed signal on a higher-speed channel.
Figure 3. Nibble interleaving is a process used in multiplexing communications. With nibble interleaving, a quadbit is taken from a lower-speed channel as input for a multiplexed signal on a
higher-speed channel.
See also: most significant bit, bitwise, bit stuffing, bit rot, qubit and classical computing.
This was last updated in November 2022
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Some class of iterative TVD scheme for solving variational inequality appear in Elasto-hydrodynamic lubrication problems
Dr. Peeyush Singh, IIT, Kanpur
Speaker Dr. Peeyush Singh, IIT, Kanpur
When Sep 08, 2017
from 04:00 PM to 05:00 PM
Where LH 006
Add event to calendar vCal
Abstract: This talk is motivated by presenting some class of iterative TVD scheme for solving variational inequality appear in Elasto-hydrodynamic lubrication problem in Tribology.
Discussion will be start by briefly examine a linear quadratic problem of the following form
\(\min J(y)\) = \(\frac{1}{2}\)\(\langle y, Ay \rangle -\langle f, y \rangle\)
\(y \ge \phi.\)
Further study involves in developing a solution procedure and convergence analysis of the problem and giving an algorithm for solving our model problem of the type (2)-(7)
\(\frac{\partial }{\partial x}\) \(\Big(\epsilon^{*}\) \(\frac {\partial p}{\partial x}\Big)+\) \(\frac{\partial }{\partial y} \)\(\Big(\epsilon^{*}\)\( \frac {\partial p}{\partial y}\Big)\)\(\le \
frac {\partial (\rho h)}{\partial x},\)
\(p\ge 0,\)
\(p.\Big[\frac{\partial }{\partial x}\) \(\Big(\epsilon^{*}\) \(\frac {\partial p}{\partial x}\Big)+\) \(\frac{\partial }{\partial y} \)\(\Big(\epsilon^{*}\)\( \frac {\partial p}{\partial y}\Big)\)\
(-\) \(\frac {\partial (\rho h)}{\partial x}\) \( \Big] \) = 0,
where p,\(\rho\) are pressure and density of the lubricant and diffusive coefficient \(\epsilon^{*}=\frac{\rho h^{3}}{\eta \lambda}\)
\(\eta(p) = \eta_{0}e^{l_{2}p},\epsilon^{*} = \frac{\rho h^{3}}{\eta \lambda}, \lambda = \frac{12\mu v(2R)^{3}}{\pi E}.\)
Above nonlinear variational inequality (2)-(4) is defined in a bounded, but large domain \(\Omega\) with natural boundary condition
\(p= 0 \quad \text{on} \quad \partial \Omega.\)
Dimensionless film thickness h(x,y) is written as follows
\(h(x,y) = h_{00}+\frac{x^{2}}{2}+\frac{y^{2}}{2}\)+\(\frac{2}{\pi^{2}}\) \(\int_{\Omega}\) \(\frac{p(x^{'},y^{'})dx^{'}dy^{'}}{\sqrt{(x-x^{'})^2\)+(y-y^{'})^2}},\)
The dimensionless force balance equation is defined as follows
\(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}p(x',y') dx'dy' = \frac{3\pi}{2}.\) | {"url":"https://www.math.tifrbng.res.in/events/some-class-of-iterative-tvd-scheme-for-solving-variational-inequality-appear-in-elasto-hydrodynamic-lubrication-problems","timestamp":"2024-11-01T23:54:14Z","content_type":"application/xhtml+xml","content_length":"35618","record_id":"<urn:uuid:3d83d560-9241-4202-b468-8fdfc231dd47>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00866.warc.gz"} |
xor_cipher | Dart package
xor_cipher 1.0.0+1
xor_cipher: ^1.0.0+1 copied to clipboard
The XOR Encryption algorithm is a effective and easy to implement method of symmetric encryption.
Symmetric XOR cipher library #
About XOR cipher #
XOR Encryption is an encryption method used to encrypt data and is hard to crack by brute-force method, i.e generating random encryption keys to match with the correct one. The XOR Encryption
algorithm is a very effective yet easy to implement method of symmetric encryption. Due to its effectiveness and simplicity, the XOR Encryption is an extremely common component used in more complex
encryption algorithms used nowadays. In cryptography, the simple XOR cipher is a type of additive cipher, an encryption algorithm that operates according to the principles:
A XOR 0 = A,
A XOR A = 0,
A XOR B = B XOR A,
(A XOR B) XOR C = A XOR (B XOR C),
(B XOR A) XOR A = B XOR 0 = B
where XOR denotes the exclusive disjunction (XOR) operation. This operation is sometimes called modulus 2 addition (or subtraction, which is identical). With this logic, a string of text can be
encrypted by applying the bitwise XOR operator to every character using a given key. To decrypt the output, merely reapplying the XOR function with the key will remove the cipher.
The XOR operator is extremely common as a component in more complex ciphers. By itself, using a constant repeating key, a simple XOR cipher can trivially be broken using frequency analysis. If the
content of any message can be guessed or otherwise known then the key can be revealed. Its primary merit is that it is simple to implement, and that the XOR operation is computationally inexpensive.
A simple repeating XOR (i.e. using the same key for xor operation on the whole data) cipher is therefore sometimes used for hiding information in cases where no particular security is required. The
XOR cipher is often used in computer malware to make reverse engineering more difficult.
If the key is random and is at least as long as the message, the XOR cipher is much more secure than when there is key repetition within a message. When the keystream is generated by a pseudo-random
number generator, the result is a stream cipher. With a key that is truly random, the result is a one-time pad, which is unbreakable in theory.
The XOR operator in any of these ciphers is vulnerable to a known-plaintext attack, since plaintext ^ ciphertext = key. It is also trivial to flip arbitrary bits in the decrypted plaintext by
manipulating the ciphertext. This is called malleability.
Usage example #
import 'package:xor_cipher/xor_cipher.dart';
void main() {
const source = 'Hello 🦊 world!!!';
const secret = 'Top 😺 secret';
'Source: $source\n'
'Secret: $secret',
final encrypted = XOR.encrypt(source, secret, urlEncode: true);
print('Encrypted: $encrypted');
final decrypted = XOR.decrypt(encrypted, secret, urlDecode: true);
'Decrypted: $decrypted\n'
'Identical: ${identical(source, decrypted)}',
Coverage #
Changelog #
Refer to the Changelog to get all release notes.
Maintainers #
License # | {"url":"https://pub.dev/packages/xor_cipher","timestamp":"2024-11-12T03:17:00Z","content_type":"text/html","content_length":"26130","record_id":"<urn:uuid:988127cc-f27c-4c6d-b673-e236f3e6e731>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00690.warc.gz"} |
Prime Factorization
I got two new great-nephews in 2022!
They are both second sons of their families. I didn’t knit their older brothers’ new blankets because I had barely finished my nephew Martin’s blanket, and they came so close together, I settled for
two little cardigans.
But for Kellen and Tobi, I decided it didn’t matter if I was late finishing (and I was) — they needed Prime Factorization Blankets!
The idea is the same as my first Prime Factorization Blanket for Arianna:
Rows of Entrelac squares (or diamonds), going from 1 to 99. 1 is white and I put rows of white squares in between the rows with other numbers. After 1, every prime number gets its own color. For
composite numbers, I put sections of the colors for each factor. So 4 gets two sections of 2, 6 gets a section of 2 and a section of 3, and so on, all the way up to 99, which gets two sections of the
color for 3 and one section of the color for 11.
Kellen is modeling his blanket in the picture above, and here are some more pictures of it.
First the blanket as a whole. I knew he was a boy, so I used lots of blues, with 2 being yellow.
The corner at the start with the missing square for zero:
And the right bottom corner with some numbers labeled:
I don’t think I knew Tobi’s gender when I started his blanket, and I decided to try for bright colors instead of pastels, so 2 was red. Here’s the whole blanket:
Detail for the lowest numbers:
Detail for the highest numbers:
And some primes at the top of the blanket:
So much fun! (Tobi’s parents, if you read this, I need more pictures of Tobi modeling his blanket!)
And yes, I’m happy to report that my youngest sister is now expecting a baby, and he’s going to get a prime factorization blanket, too! I learned tonight that he’s a boy, so 23 is going to be blue.
Babies and math are beautiful!
Mathematical Colors and Codes, Episode Six — Binary Codes and Booktalks
Episode Six of Mathematical Colors and Codes, my Virtual Program Series for the library is up!
Episode Six now looks at the Base Two number system, binary, and puts that into a code. To finish up the series, I talk about more books that play with mathematical ideas.
Like all the other videos in the series, this one has a downloadable coloring page. This one has a chart for a Binary Code.
Here’s this week’s video:
Here are links to the entire Mathematical Colors and Codes series:
Episode One, Prime Factorization
Episode Two, Prime Factorization Codes
Episode Three, Nondecimal Bases
Episode Four, Color Codes with Nondecimal Bases
Episode Five, More Codes with Nondecimal Bases
Episode Six, Binary Codes and Booktalks
Mathematical Colors and Codes, Episode Five — More Codes with Nondecimal Bases
Episode Five of Mathematical Colors and Codes, my Virtual Program Series for the library is up!
Episode Five looks at more ways you can use nondecimal bases to make coded messages.
This video, like all the others has a downloadable coloring page. This one has charts for a Base Six Code and a Base Five Code.
Here’s this week’s video:
Here are links to the entire Mathematical Colors and Codes series:
Episode One, Prime Factorization
Episode Two, Prime Factorization Codes
Episode Three, Nondecimal Bases
Episode Four, Color Codes with Nondecimal Bases
Episode Five, More Codes with Nondecimal Bases
Episode Six, Binary Codes and Booktalks
Mathematical Colors and Codes, Episode Four — Color Codes with Nondecimal Bases
Episode Four of Mathematical Colors and Codes, my Virtual Program Series for the library is up!
Episode Four now takes the Nondecimal Base systems we talked about in Episode Three and uses them to make coded messages.
This video, like all the others has a downloadable coloring page. This one has a chart for choosing your own colors and making your own coded messages with nondecimal bases.
Here’s this week’s video:
Here are links to the entire Mathematical Colors and Codes series:
Episode One, Prime Factorization
Episode Two, Prime Factorization Codes
Episode Three, Nondecimal Bases
Episode Four, Color Codes with Nondecimal Bases
Episode Five, More Codes with Nondecimal Bases
Episode Six, Binary Codes and Booktalks
Mathematical Colors and Codes, Episode Three – Nondecimal Bases
Episode Three of Mathematical Colors and Codes, my Virtual Program Series for the library is up!
Episode Three is the longest episode. (They do get shorter!) I talk about various bases and look at them together with prime factorization color charts. I’m hoping it gives kids a feel for how other
bases work.
This video, like all the others has a downloadable coloring page. [Right now this is the incorrect link. I’ll fix it with the correct one tonight.] This one will let you see for yourself how prime
factorization patterns change in other bases, as well as giving you a feel for how counting works in other bases.
Here’s this week’s video:
Here are links to the entire Mathematical Colors and Codes series:
Episode One, Prime Factorization
Episode Two, Prime Factorization Codes
Episode Three, Nondecimal Bases
Episode Four, Color Codes with Nondecimal Bases
Episode Five, More Codes with Nondecimal Bases
Episode Six, Binary Codes and Booktalks
Mathematical Colors and Codes, Episode Two: Prime Factorization Codes
Episode Two of my Mathematical Virtual Program Series is up!
In Episode Two, I talk more about prime factorization and ways to show it with colors. Then I show how you can use that idea to make a prime factorization code.
This video has a downloadable coloring page to help you make your own prime factorization code.
Here’s this week’s video:
Here are links to the entire Mathematical Colors and Codes series:
Episode One, Prime Factorization
Episode Two, Prime Factorization Codes
Episode Three, Nondecimal Bases
Episode Four, Color Codes with Nondecimal Bases
Episode Five, More Codes with Nondecimal Bases
Episode Six, Binary Codes and Booktalks
Mathematical Colors and Codes
My Mathematical Virtual Program Series is up!
This program is a series of six videos with downloadable coloring pages. New videos will post on Mondays at 3 pm.
They will show kids how to use math to make colorful patterns and coded messages, learning about prime factorization and nondecimal bases along the way.
They’ll post on Fairfax County Public Library’s website, but I’ll post them here as well.
These will be best for kids who already understand multiplication.
And this week, Episode One is up! It covers Prime Factorization, with an explanation of my Prime Factorization Sweater. And it explains how you can color your own chart, using this downloadable
coloring page.
I hope you enjoy it!
Here are links to the entire Mathematical Colors and Codes series:
Episode One, Prime Factorization
Episode Two, Prime Factorization Code
Episode Three, Nondecimal Bases
Episode Four, Color Codes with Nondecimal Bases
Episode Five, More Codes with Nondecimal Bases
Episode Six, Binary Codes and Booktalks
Normal Distribution Scarf
Today I finished a second Normal Distribution Scarf.
The first one I made was designed to highlight outliers to show that outliers are what makes the world beautiful.
For this one, I only wanted to show the Normal Distribution. I decided to knit it the long way so this time I wouldn’t have to sew any ends in.
I took colors from light to dark, in shades of pink. Colors B and C were a little closer than I wanted them to be, but it still gave the idea.
I generated numbers from a normal distribution and made a big list. For positive values, I purled the row, and for negative values, I knitted — so those values should be about even, making random
For the color, I used the absolute value, from light to dark. Since the normal distribution is a bell curve, there should be many more values in the lighter colors.
For 0 to 0.5, I used White.
0.5 to 1.0 was Victorian Pink.
1.0 to 1.5 was Blooming Fuchsia (only a little darker than Victorian Pink).
1.5 to 2.0 was Lotus Pink — a bright, hot pink.
Above 2.0 was Fuchsia — a dark burgundy.
Naturally, I used a lot more of the lighter colors. So for my next project after my current one, I think I’m going to do another normal distribution scarf, but this time reversing the values. So the
new scarf would be mainly dark colors with light highlights.
In fact, if I weren’t using pink (maybe purple or blue), it would be fun to make scarves for a couple this way. Use dark, staid, sedate colors for the man, with light highlights. Use pastel shades
for the woman — with dark highlights. [Hmmm. If I knit a scarf for a boyfriend before he exists, would the boyfriend jinx not apply?]
In this version, the lighter colors were more prominent.
Here’s a view of the scarf draped over my couch, showing both sides.
The different look has to do with where the knits and purls were placed and which side has a ridge and which is smooth.
Here’s a closer look:
I like the way the color combinations turned out so pleasing.
The only real problem is that the scarf is made out of wool, and it was almost 100 degrees outside today. So for now, I’m going to have to enjoy it draped over my couch rather than wearing it. I’ll
look forward to this winter!
Update: I made an opposite scarf to this one, also generating random numbers and using the same exact yarn, but going from dark to light. Together, they make a matched set, so I gave them to my
daughter and her wife-to-be!
Prime Factorization Coloring Sheets
I’ve posted several Prime Factorization Coloring Sheets on my Sonderknitting page lately.
I decided I should try coloring them myself, so I could post a thumbnail of each one. I had a lot of fun doing it, and was reminded of lots of cool properties I discovered from knitting my prime
factorization sweater and looking at these charts.
I have a manuscript for a math-related children’s nonfiction book about using math to make codes with colors. Originally, I put several of these charts into the book — but I eventually decided it was
a distraction and decided to put them on my website instead.
But they show all sorts of cool things!
First, there’s the ten-by-ten prime factorization chart using ordinary, decimal numbers.
Coloring this chart gives you a great feeling for factorization and multiples. I posted about watching a second grader color it. I think of it as more for older kids, who are learning about primes
and multiples, or indeed adults, in keeping with the adult coloring book craze. But watching a second grader color it assured me that it can give insights to anyone. (I made the instructions such
that you don’t even have to know how to multiply. Just color every second square the color for 2, every third square the color for 3, and so on.)
Now, in my original sweater, I put rows of 8 on the back and rows of 2 and rows of 3 on the sleeves. The prime factorization charts in different bases are the same idea.
First, they give you a feeling for how different bases work.
Here’s the sheet for octal, base 8:
You can color it exactly the same way as you did the ten-by-ten chart. Color every second square with the color for 2, every third with the color for 3, and so on. If you take the time to do that,
you’ll grasp how the numbers count up to 7 and then use the next digit, since place value in octal gives the ones digit, the eights digit, and the sixty-fours digit.
The chart also makes a good way to translate between octal and decimal. (Though you can just multiply the eights digit times eight and add the ones digit.)
But I enjoy some of the other patterns.
The first, most obvious pattern is that in the decimal chart, the multiples of 5 and the multiples of 2 line up vertically (as well as the multiples of 10, which are both). That’s because 10 = 2 x 5.
In the octal chart, the multiples of 2 line up vertically, since 8 = 2 x 2 x 2. So do the multiples of 4 — each with two factors of 2, and the multiples of 8 — each with three factors of 2.
In the Base 6 chart, as you’d expect, the multiples of 2 and the multiples of 3 line up vertically. (And the multiples of 6, with a factor of 2 and a factor of 3, do as well.)
But it’s also fun what happens to the color for Base Plus One and Base Minus One.
In the 10×10 chart, look at what happens to the color for 11, orange, and the multiples of 11. They go diagonally to the right up the chart: 11, 22, 33, 44, . . .
In the 10×10 chart, 9 is represented by two sections of blue, for 3 x 3. These colors go diagonally up the chart in the opposite direction: 9, 18, 27, 36, . . .
In the 8×8 chart, the octal number 11 is the decimal number 9 — so it is still represented by two sections of blue. But since 9 is one bigger than our base in that chart, the two sections of blue go
diagonally up the chart to the right — just like 11 in the decimal chart.
In the octal chart, the color for 7, purple, goes diagonally up the chart to the left, with the octal numbers 7, 16, 25, 34, . . . .
In the 6×6 chart, we’ve got the same patterns, this time with 7 (which is 11 in base six) and 5.
7 (purple) goes diagonally right up the chart, and 5 goes diagonally left up the chart.
And we’ve got the same patterns in a 7×7 Base Seven chart:
Notice that since 7 is prime, no colors line up except purple, the color for 7.
And the colors for 8 and 6 go diagonally up the chart.
The Hexadecimal chart in base 16 is even more interesting:
Notice how all the multiples of 2 line up vertically, with multiples of 4, 8, and 16 also lined up.
11 in Base 16 is decimal 17, which is brown, and it acts like all the other 11s, going diagonally up and to the right.
1 less than 16 is F = 15, and the blue and green colors for F go diagonally up and to the left.
Before I finish I want to mention one more pattern I noticed from looking at these charts. It’s the familiar trick in Base 10 of the rule for figuring out if any number is a multiple of 9: Just add
up the digits, and they will be a multiple of 9.
The reason this works is that 10 is congruent to 1 mod 9.
In base 10, each decimal place represents a number multiplied by a power of 10.
In base 9, that’s going to be the same as multiplying by 1 — so if you add up the digits, you get what the number is congruent to mod 9.
If none of that made any sense to you, just know this:
If you add up the digits of a base 10 number (and if you get a number bigger than 9, add them up again), your result is the remainder you’ll get if you divide the number by 9.
Since multiples of 9 have no remainder when divided by 9 — the digits of multiples of 9 in base 10 always add up to multiples of 9. (And by the same reasoning, the digits of multiples of 3 in base 10
always add up to multiples of 3.)
But you might have noticed when looking at the diagonal colors:
In Base 8, the digits of multiples of 7 always add up to multiples of 7.
In Base 6, the digits of multiples of 5 always add up to multiples of 5.
In Base 7, the digits of multiples of 6 always add up to multiples of 6.
And the digits of multiples of 2 always add up to multiples of 2.
And the digits of multiples of 3 always add up to multiples of 3.
(Use the colors to tell which numbers these are in Base 7.)
In Base 16, the digits of multiples of F (15) always add up to multiples of F.
And the digits of multiples of 5 always add up to multiples of 5.
And the digits of multiples of 3 always add up to multiples of 3.
(Use the colors to tell which numbers these are in Base 16.)
Forgive me, but I think these patterns are Awesome!
Let’s face it, you’ll see them much more clearly if you color the charts yourself!
Download the coloring charts at Sonderknitting!
Happy Coloring!
Coloring to Learn Math Concepts!
I’m super excited about something I’ve been working on lately — posting Mathematical Coloring Sheets on my Sonderknitting webpage.
Why Sonderknitting? Because the ideas in the coloring pages come from my mathematical knitting projects, which all began with my Prime Factorization Sweater.
I wore the sweater to the library today, for our Family Math Games event. (We have lots of board games and card games that build math skills and ask only that parents play with their kids.) I also
printed out some copies of the Prime Factorization Coloring Sheet — the one that matches my sweater — and brought some crayons.
A girl named Ana who is a regular at our Crazy 8s Math Club was there. She got tired of playing games with her little brother, and her Mom showed Ana the coloring sheet, and Ana became the first
actual child to color one!
I explained the idea to Ana, using my sweater as a visual aid.
There are different ways you can approach it, but what I suggested was to choose a color for 2, then color a section of every second number. Then choose a color for 3 and color a section of every
third number. Then I had to explain you use the color for 2 again to color a second section in the square for 4, then give every 4th number a second section of the color for 2. Then you choose a new
color for 5, and she quickly caught on that all the multiples of 5 were in columns….
I can’t tell you how happy it made me to hear what she’d say as she was understanding how to do it (“Oh, I see!”) and seeing the patterns come out.
I think Ana’s in 2nd grade (Crazy 8s is for Kindergarten to 2nd grade.), so she can’t have studied much multiplication in school yet. So it made me all the happier to see the wheels turning and the
connections forming.
But my favorite thing she said? “I like this! This is fun!” | {"url":"https://sonderbooks.com/blog/?cat=206","timestamp":"2024-11-03T10:46:26Z","content_type":"text/html","content_length":"127660","record_id":"<urn:uuid:2f9cec56-7cdf-4793-b16c-5c008bd810f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00210.warc.gz"} |
What Is Gravity, and How Does It Work?
Credit: ThomasVogel/E+ via Getty Images
Legend has it that Isaac Newton had the moment of inspiration that would lead to his theory of gravity when, on a warm afternoon, he saw an apple fall from a tree and wondered why it should fall down
instead of up. (Some versions of the story have the apple falling and hitting poor Newton on the head.) He named his new hypothesis after the Latin gravitas, for "weight." The tale of Newton's
bruised noggin may be apocryphal, but his interest in what makes things move—and especially what makes them fall—was very real. Newton had come home to stay when his university closed down due to a
bout of bubonic plague, and he was looking for something to occupy his mind. In 1665, he found it, apple or no apple.
It took Newton, one of the world's greatest mathematical minds and, by all accounts, an irascible jerk, 20 years to articulate his thoughts on gravity to his satisfaction. His biggest problem? The
question of whether the Earth's gravitational influence could extend all the way to the Moon.
Two hundred years later, the calculations that took humanity to the Moon were based on Newton's mechanics. But how does gravity work in the first place? What did Newton understand that was such a
Gravity: The Basics
Can someone just... explain gravity to me?
Let's start with a definition. Gravity, or gravitational attraction, is the tendency of mass to gather toward itself, drifting together even across great distances due to curvature in spacetime. This
tendency allows the formation of stars, planets, galaxies, and black holes.
Standing on Earth's surface, the planet's mass creates a gravitational force sufficient to accelerate any object downward (toward the core of the planet, or perpendicular to the planet's surface) at
9.8 m/s²—that is, an additional 9.8 meters per second, each second. Everything experiences the same amount of gravity, but if you take a hammer in one hand and a feather in the other and drop them at
the same time, the hammer will hit the ground first. Why? Drag (air resistance) opposes acceleration due to gravity. In a vacuum, such as on the Moon, both objects will hit the ground at the same
How does gravity work?
Gravity applies an effective force of mutual attraction to things with inertial mass, including physical matter and photons. The force of gravity is transmitted through spacetime at the speed of
light, which creates wavefronts we can detect with special equipment like the LIGO gravitational wave detectors.
In classical (or "Newtonian") mechanics, which describes the motion of macroscopic objects (i.e., things larger than an atom, such as planets), gravity is sometimes called a central force. A central
force is directed towards or away from a point called the center of force. Gravity, electrical charge, and magnetism are three examples of central forces. Centers of electromagnetic energy radiating
inward or outward are known as poles.
Newton used a mathematical approach to gravity not unlike Coulomb had done with electrostatics, with a field that falls off as the inverse square of the distance between two objects. Looking at it
this way, the gravitational force at a point can be expressed as a vector, with magnitude and direction.
How is gravity transmitted?
Christian Huygens, a contemporary of Isaac Newton, discovered that light carries energy. This suggests a force-carrying graviton as an obvious theoretical parallel to a photon. But where photons are
the force carrier of the electromagnetic field, relativity frames gravity as an emergent consequence of the way inertial mass warps spacetime. Instead of requiring a force carrier, according to
general relativity, what we think of as gravity is more like the idea of "downhill."
Massive or high-energy objects warp the mesh of spacetime, dragging it in toward themselves and creating a gravitational field of influence or "gravity well" from which it can be difficult to escape.
Central Forces
When matter is collected in one place, it forms a center of mass from which its inward-pointing field of gravitational influence extends. The force of attraction between two objects falls off as
distance increases from the center of mass.
Objects under the influence of a gravitational field will move toward the field's center of gravity. Sometimes, as with the Sun and Jupiter, their mutual center of gravity or barycenter lies slightly
outside one of the bodies; Jupiter is large enough that it drags the Sun in a little circle, centered slightly outside of the Sun's radius, as Jupiter makes each orbit.
On Earth, we have to contend with our own gravity well when we launch rockets and spacecraft; if a rocket isn't powerful enough to escape its gravity well, it will fall back to Earth. As matter
becomes more and more dense, that effect becomes more pronounced. Black holes create a gravity well so deep that there's a threshold around black holes called an event horizon, a boundary in space
marking the point of no return. Nothing inside the event horizon can escape from a black hole. Indeed, it's thought that the only thing that can ever escape a black hole's gravity is a frisson of
virtual particles called Hawking radiation, thrown off every so often when subatomic symmetries align.
Sacred Geometry
Astronomers in ancient Greece noticed that the planets sometimes seem to move in retrograde across the sky, backward with respect to their normal orbits. This offended some astronomers' sense of
cosmos, the orderliness of the universe. In a universe perfectly ordered by the hands of their gods, there was little room for irrational numbers or eccentric orbits.
In their attempt to reconcile their geocentric models with their empirical observations, they proposed the idea of epicycles: complex orbits that were neither circular nor elliptical, with planets
dancing around the Earth in paths that look like geometric lace.
Geocentric models resorted to convoluted orbits to explain the apparent motion of planets through the sky. Credit: Public domain
Geocentrism reigned unchallenged for more than two thousand years. Despite the repeated proposal of a heliocentric solar system over the millennia by scholars as respected as Leonardo da Vinci,
heliocentrism wasn't taken seriously until the medieval era. However, the scientific consensus began to change in the 1500s. Nicolaus Copernicus developed a heliocentric model, backing up his
argument with astronomical observations—and predictions that would confirm his model as correct or invalidate it. Galileo Galilei, using the newly invented refracting telescope, made and published
observations showing that the planet Venus went through phases just like the Moon, and that Jupiter was orbited by its own moons.
Music of the Spheres
Then, Johannes Kepler put forth a solution to the problem of retrograde planets that would have satisfied even the strictest Pythagorean. Even with their orbits taking the "imperfect" shape of an
ellipse, Kepler showed that a planet swept out an equal geometric area of its orbit over the same length of time, no matter where in its elliptical orbit it might be, nor how eccentric that ellipse.
Kepler was a big believer in musica universalis, the music of the spheres; the idea that an inaudible mathematical harmony existed between the orbits of the planets was central to his Mysterium
Johannes Kepler's nesting Platonic solids, as depicted Credit: Johannes Kepler. From Kepler's "Mysterium Cosmographicum", Tübingen 1596, Tabula III: Orbium planetarum dimensiones, et distantias per
quinque regularia corpora geometrica exhibens.
In 1687, Isaac Newton published his opus, Philosophiæ Naturalis Principia Mathematica (Latin for Mathematical Principles of Natural Philosophy, but many affectionately call it just Principia for
short), which combined his laws of motion with a new mathematical analysis—calculus!—that could replicate Kepler's empirical observations of the planets and their moons. In Principia, Newton proposed
a law of universal gravitation that now bears his name. Newton's law of universal gravitation holds that any two bodies, no matter how far they may be separated in space, are attracted by a force
proportional to their mass and inversely proportional to the square of the distance between them.
'Spooky Action at a Distance'
Yet the question remained: How could one planet affect another at such a great distance? Newton considered action at a distance to be, in his own words, "so great an Absurdity that I believe no Man
who has in philosophical Matters a competent Faculty of thinking can ever fall into it." Familiar as he was with electrostatics, Newton's theory of gravitation didn't require what he viewed as an
exotic, unnecessary transmission mechanism when the inverse-square law modeled gravitational attraction entirely well enough.
Newton was by no means in the scientific minority on the topic of action at a distance. Albert Einstein operated under some assumptions of aether theory when developing his theory of relativity.
Einstein would eventually dismiss the notion of quantum entanglement between two particles as spukhafte Fernwirkungen (translated as "spooky action at a distance"). Likewise, Newton and others of his
day believed there must be a transmission medium, such as the luminiferous aether, through which electromagnetic or gravitational forces could exert a force on bodies separated in space.
Quantum Gravity
What Einstein knew that Newton didn't is that the universe is permeated not by an aetheric substance made of molecules of some kind but by an invisible warp and weft of field lines along which forces
such as gravity are transmitted. No aether is required to produce the effects described in Maxwell's laws of electromagnetism or to produce gravity as we understand it. Today, it looks like the
graviton will go the way of the aether.
On the deepest level, our cosmos is governed by four fundamental forces or fundamental interactions: electromagnetism and gravity, whose reach appears unlimited, and the weak and strong nuclear
forces, which constrain themselves to the smallest scale, the inner workings of the atom. We call them fundamental forces because when we try to answer how spacetime works under the hood, these four
forces appear not to be reducible to simpler interactions.
The human understanding of gravity still has some problems. Chief among them is the difficulty of applying current models to the subatomic scale or to extremely high-energy environments such as black
holes or the very early universe.
As technology advances, scientists' understanding of what the laws of physics will permit stays in a state of flux. Once fusion, which must surmount the Coulomb barrier, was an exotic fiction; now,
it's an engineering problem. It may be the same with gravity and other phenomena on the quantum scale. Until then, we can all enjoy the view from where we stand—on the shoulders of giants. | {"url":"https://www.extremetech.com/science/what-is-gravity-and-how-does-it-work","timestamp":"2024-11-07T00:44:18Z","content_type":"text/html","content_length":"84071","record_id":"<urn:uuid:93bc87f4-1f51-40ca-b25b-4aebadea1e9d>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00215.warc.gz"} |
Nerdfighteria Wiki - Probability Part 2: Updating Your Beliefs with Bayes: Crash Course Statistics #14
Probability Part 2: Updating Your Beliefs with Bayes: Crash Course Statistics #14
YouTube: https://youtube.com/watch?v=oZCskBpHWyk
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Today we're going to introduce bayesian statistics and discuss how this new approach to statistics has revolutionized the field from artificial intelligence and clinical trials to how your computer
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Hi, I’m Adriene Hill, and Welcome back to Crash Course Statistics.
We ended the last episode by talking about Conditional Probabilities which helped us find the probability of one event, given that a second event had already happened. But now I want to give you a
better idea of why this is true and how this formula--with a few small tweaks--has revolutionized the field of statistics.
INTRO In general terms, Conditional Probability says that the probability of an event, B, given that event A has already happened, is the probability of A and B happening together, Divided by the
probability of A happening - that’s the general formula, but let’s give you a concrete example so we can visualize it. Here’s a Venn Diagram of two events, An Email containing the words “Nigerian
Prince” and an Email being Spam. So I get an email that has the words “Nigerian Prince” in it, and I want to know what the probability is that this email is Spam, given that I already know the email
contains the words “Nigerian Prince.” This is the equation.
Alright, let’s take this part a little. On the Venn Diagram, I can represent the fact that I know the words “Nigerian Prince” already happened by only looking at the events where Nigerian Prince
occurs, so just this circle. Now inside this circle I have two areas, areas where the email is spam, and areas where it’s not.
According to our formula, the probability of spam given Nigerian Prince is the probability of spam AND Nigerian Prince which is this region... where they overlap…divided by Probability of Nigerian
Prince which is the whole circle that we’re looking at. Now...if we want to know the proportion of times when an email is Spam given that we already know it has the words “Nigerian Prince”, we need
to look at how much of the whole Nigerian Prince circle that the region with both Spam and Nigerian Prince covers. And actually, some email servers use a slightly more complex version of this example
to filter spam.
These filters are called Naive Bayes filters, and thanks to them, you don’t have to worry about seeing the desperate pleas of a surprisingly large number of Nigerian Princes. The Bayes in Naive Bayes
comes from the Reverend Thomas Bayes, a Presbyterian minister who broke up his days of prayer, with math. His largest contribution to the field of math and statistics is a slightly expanded version
of our conditional probability formula.
Bayes Theorem states that: The probability of B given A, is equal to the Probability of A given B times the Probability of B all divided by the Probability of A You can see that this is just one step
away from our conditional probability formula. The only change is in the numerator where P(A and B) is replaced with P(A B)P(B).
While the math of this equality is more than we’ll go into here, you can see with some venn-diagram-algebra why this is the case. In this form, the equation is known as Bayes’ Theorem, and it has
inspired a strong movement in both the statistics and science worlds. Just like with your emails, Bayes Theorem allows us to figure out the probability that you have a piece of spam on your hands
using information that we already have, the presence of the words “Nigerian Prince”.
We can also compare that probability to the probability that you just got a perfectly valid email about Nigerian Princes. If you just tried to guess your odds of an email being spam based on the rate
of spam to non-spam email, you’d be missing some pretty useful information--the actual words in the email! Bayesian statistics is all about UPDATING your beliefs based on new information.
When you receive an email, you don’t necessarily think it’s spam, but once you see the word Nigerian you’re suspicious. It may just be your Aunt Judy telling you what she saw on the news, but as soon
as you see “Nigerian” and “Prince” together, you’re pretty convinced that this is junkmail. Remember our Lady Tasting Tea example... where a woman claimed to have superior taste buds ...that allowed
her to know--with one sip--whether tea or milk was poured into a cup first?
When you’re watching this lady predict whether the tea or milk was poured first, each correct guess makes you believe her just a little bit more. A few correct guesses may not convince you, but each
correct prediction is a little more evidence she has some weird super-tasting tea powers. Reverend Bayes described this idea of “updating” in a thought experiment.
Say that you’re standing next to a pool table but you’re faced away from it, so you can’t see anything on it. You then have your friend randomly drop a ball onto the table, and this is a special,
very even table, so the ball has an equal chance of landing anywhere on it. Your mission--is to guess how far to the right or left this ball is.
You have your friend drop another ball onto the table and report whether it’s to the left or to the right of the original ball. The new ball is to the right of the original, so, we can update our
belief about where the ball is. If the original is more towards the left, than most of the new balls will fall to the right of our original, just because there’s more area there.
And the further to the left it is, the higher the ratio of new rights to lefts Since this new ball is to the right, that means there’s a better chance that our original is more toward the left side
of the table than the right, since there would be more “room” for the new ball to land. Each ball that lands to the right of the original is more evidence that our original is towards the left of the
table. But, if we get a ball landing on the left of our original, then we know the original is not at the very left edge.
Again, Each new piece of information allows us to change our beliefs about the location of the ball, and changing beliefs is what Bayesian statistics is all about. Outside thought experiments,
Bayesian Statistics is being used in many different ways, from comparing treatments in medical trials, to helping robots learn language. It’s being used by cancer researchers, ecologists, and
And this method of thinking about statistics...updating existing information with what’s come before...may be different from the logic of some of the statistical tests that you’ve heard of--like the
t-test. Those Frequentist statistics can sometimes be more like probability done in a vacuum. Less reliant on prior knowledge.
When the math of probability gets hard to wrap your head around, we can use simulations to help see these rules in action. Simulations take rules and create a pretend universe that follows those
rules. Let’s say you’re the boss of a company, and you receive news that one of your employees, Joe, has failed a drug test.
It’s hard to believe. You remember seeing this thing on YouTube that told you how to figure out the probability that Joe really is on drugs given that he got a positive test. You can’t remember
exactly what the formula is...but you could always run a simulation.
Simulations are nice, because we can just tell our computer some rules, and it will randomly generate data based on those rules. For example, we can tell it the base rate of people in our state that
are on drugs, the sensitivity (how many true positives we get) of the drug test... and specificity (how many true negatives we get). Then we ask our computer to generate 10,000 simulated people and
tell us what percent of the time people with positive drug tests were actually on drugs.
If the drug Joe tested positive for--in this case Glitterstim--is only used by about 5% of the population, and the test for Glitterstim has a 90% sensitivity and 95% specificity, I can plug that in
and ask the computer to simulate 10,000 people according to these rules. And when we ran this simulation, only 49.2% of the people who tested positive were actually using Glitterstim. So I should
probably give Joe another chance...or another test.
And if I did the math, I’d see that 49.2% is pretty close since the theoretical answer is around 48.6%. Simulations can help reveal truths about probability, even without formulas. They’re a great
way to demonstrate probability and create intuition that can stand alone or build on top of more mathematical approaches to probability.
Let’s use one to demonstrate an important concept in probability that makes it possible to use samples of data to make inferences about a population: the Law of Large Numbers. In fact we were
secretly relying on it when we used empirical probabilities--like how many times I got tails when flipping a coin 10 times--to estimate theoretical probabilities--like the true probability of getting
tails. In its weak form, Law of Large Numbers tells us that as our samples of data get bigger and bigger, our sample mean will be ‘arbitrarily’ close to the true population mean.
Before we go into more detail, let’s see a simulation and if you want to follow along or run it on your own - instructions are in the description below. In this simulation we’re picking values from a
new intelligence test--from the normal distribution, that has a mean of 50 and a standard deviation of 20. When you have a very small sample size, say 2, your sample means are all over the place.
You can see that pretty much anything goes, we see means between 5 and 95. And this makes sense, when we only have two data points in our sample, it’s not that unlikely that we get two really small
numbers, or two pretty big numbers, which is why we see both low and high sample means. Though we can tell that a lot of the means are around the true mean of 50 because the histogram is the tallest
at values around 50.
But once we increase the sample size, even to just 100 values, you can see that the sample means are mostly around the real mean of 50. In fact all of the sample means are within 10 units of the true
population mean. And when we go up to 1000, just about every sample mean is very very close to the true mean.
And when you run this simulation over and over, you’ll see pretty similar results. The neat thing is that the Law of Large numbers applies to almost any distribution as long as the distribution
doesn’t have an infinite variance. Take the uniform distribution which looks like a rectangle.
Imagine a 100-sided die, every single value is equally probable. Even the sample means that are selected from a uniform distribution get closer and closer to the true mean of 50.. The law of large
numbers is the evidence we need to feel confident that the mean of the samples we analyze is a pretty good guess for the true population mean.
And the bigger our samples are, the better we think the guess is! This property allows us to make guesses about populations, based on samples. It also explains why casinos make money in the long run
over hundreds of thousands of payouts and losses, even if the experience of each person varies a lot.
The casino looks at a huge sample--every single bet and payout--whereas your sample as an individual is smaller, and therefore less likely to be representative. Each of these concepts can help us
another way ...another way to look at the data around us. The Bayesian framework shows us that every event or data point can and should “update” your beliefs but it doesn’t mean you need to
completely change your mind.
And simulations allow us to build upon these observations when the underlying mechanics aren’t so clear. We are continuously accumulating evidence and modifying our beliefs everyday, adding today's
events to our conception of how the world works. And hey, maybe one day we’ll all start sincerely emailing each other about Nigerian Princes.
Then we’re gonna have to do some belief-updating. Thanks for watching. I’ll see you next time.
Reply to | {"url":"https://nerdfighteria.info/v/oZCskBpHWyk/","timestamp":"2024-11-12T18:45:13Z","content_type":"application/xhtml+xml","content_length":"86835","record_id":"<urn:uuid:50b14644-7f97-4883-aeab-d70449dc6c7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00420.warc.gz"} |
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Question #913e1 | Socratic
Question #913e1
1 Answer
An interesting approach to have here would be to convert the density of mercury from grams per milliliter to kilograms per milliliter by using the fact that
$\textcolor{b l u e}{\underline{\textcolor{b l a c k}{\text{1 kg" = 10^3color(white)(.)"g}}}}$
So, you know that mercury has a density of ${\text{13.546 g mL}}^{- 1}$, which means that every $\text{1 mL}$ of mercury has a mass of $\text{13.546 g}$.
Use the aforementioned conversion factor to find the density of mercury in kilograms per milliliter
${\text{13.546 g mL"^(-1) = (13.546 color(red)(cancel(color(black)("g"))))/"1 mL" * "1 kg"/(10^3color(red)(cancel(color(black)("g")))) = "0.013546 kg mL}}^{- 1}$
So if $\text{1 mL}$ of mercury has a mass of $\text{0.013546 kg}$, you can say that $\text{37.0 mL}$ of mercury will have a mass of
$37.0 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{mL"))) * "0.013546 kg"/(1color(red)(cancel(color(black)("mL")))) = color(darkgreen)(ul(color(black)("0.501 kg}}}}$
The answer is rounded to three sig figs, the number of sig figs you have for the volume of the sample.
Impact of this question
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CourseNana | ELEC576/COMP 576: Introduction to Deep Learning Assignment 1 - Backpropagation
ELEC 576 / COMP 576 – Fall 2024 Assignment 1
Due: Oct 10, 2024 11 a.m. via Canvas
Submission Instructions
Every student must submit their work in a zip file in the following format: netid-assignment1.zip. You should also provide intermediate and final results as well as any necessary code. Submit your
zip file on Canvas.
GPU Resource from AWS
To accelerate the training using GPU, you can optionally use Amazon Web Services(AWS) GPU instance using AWS Education credits. You can also get additional AWS credits from Github Student Developer
After having an AWS account, You can either create a fresh ubuntu instance and install software dependencies by yourself or use off-the-shelf TensorFlow ready image from AWS Marketplace.
1 Backpropagation in a Simple Neural Network
In this problem, you will learn how to implement the backpropagation algorithm for a simple neural network. To make your job easier, we provide you with starter code in three layer neural network.py.
You will fill in this starter code to build a 3-layer neural network (see Fig. 1) and train it using backpropagation.
a) Dataset
We will use the Make-Moons dataset available in Scikit-learn. Data points in this dataset form two interleaving half circles corresponding to two classes (e.g. “female” and “male”). In the main()
function of three layer neural network.py, uncomment the “generate and
visualize Make-Moons dataset” section (see below) and run the code. Include the generated figure in your report.
# generate and visualize Make-Moons dataset
X, y = generate_data()
plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)
b) Activation Function
Tanh, Sigmoid and ReLU are popular activation functions used in neural networks. You will implement them and their derivatives.
1. Implement function actFun(self, z, type) in three layer neural network.py. This function computes the activation function where z is the net input and type ∈ {‘Tanh’, ‘Sigmoid’, ‘ReLU’}.
2. Derive the derivatives of Tanh, Sigmoid and ReLU
3. Implement function diff actFun(self, z, type) in three layer neural network.py. This function computes the derivatives of Tanh, Sigmoid and ReLU.
c) Build the Neural Network
Lets now build a 3-layer neural network of one input layer, one hidden layer, and one output layer. The number of nodes in the input layer is determined by the dimensionality of our data, 2. The
number of nodes in the output layer is determined by the number of classes we have, also 2. The input to the network will be x- and y- coordinates and its output will be two probabilities, one for
class 0 (“female”) and one for class 1 (“male”). The network looks like the following.
Mathematically, the network is defined as follows.
z1 =W1x+b1 (1) a1 = actFun(z1) (2) z2 = W2a1 + b2 (3) a2 = yˆ = softmax(z2) (4)
where zi is the input of layer i and ai is the output of layer i after applying the activation function. θ ≡ {W1, b1, W2, b2} are the parameters of this network, which we need to learn from the
training data.
If we have N training examples and C classes then the loss for the prediction yˆ with respect to the true labels y is given by:
Figure 1: A three-layer neural network
L(y,yˆ)=−N1 XXyn,ilogyˆn,i (5) n∈N i∈C
Note that y are one-hot-encoding vectors and yˆ are vectors of probabilities.
1. In three layer neural network.py, implement the function feedforward(self, X, actFun). This function builds a 3-layer neural network and computes the two probabilities (self.probs in the code or a
2 in Eq. 4), one for class 0 and one for class 1. X is the input data, and actFun is the activation function. You will pass the function actFun you implemented in part b into feedforward(self, X,
2. In three layer neural network.py, fill in the function calculate loss(self, X, y). This function computes the loss for prediction of the network. Here X is the input data, and y is the given
d) Backward Pass - Backpropagation
It’s time to implement backpropagation, finally! 1. Derive the following gradients: ∂ L , ∂ L , ∂ L ,
2. In three layer neural network.py, implement the function backprop(self, X, y). Again, X is the input data, and y is the given labels. This function implements backpropagation (i.e., computing the
gradients above).
e) Time to Have Fun - Training!
You already have all components needed to run the training. In three layer neural network.py, we also provide you function visualize decision boundary(self, X, y) to visualize the
decision boundary. Let’s have fun with your network now.
1. Train the network using different activation functions (Tanh, Sigmoid and ReLU). Describe and explain the differences that you observe. Include the figures generated in your report. In order to
train the network, uncomment the main() function in three layer neural network.py, take out the following lines, and
run three layer neural network.py.
plt.scatter(X[:, 0], X[:, 1], s=40, c=y, cmap=plt.cm.Spectral)
2. Increase the number of hidden units (nn hidden dim) and retrain the network us- ing Tanh as the activation function. Describe and explain the differences that you observe. Include the figures
generated in your report.
f) Even More Fun - Training a Deeper Network!!!:
Let’s have some more fun and be more creative now. Write your own n layer neural network.py that builds and trains a neural network of n layers. Your code must be able to accept
as parameters (1) the number of layers and (2) layer size. We provide you hints below
to help you organize and implement the code, but if you have better ideas, please
feel free to implement them and ignore our hints. In your report, please tell us why
you made the choice(s) you did.
1. Create a new class, e.g DeepNeuralNetwork, that inherits NeuralNetwork in
three layer neural network.py
2. In DeepNeuralNetwork, change function feedforward, backprop, calculate loss and fit model
3. Create a new class, e.g. Layer(), that implements the feedforward and back- prop steps for a single layer in the network
4. Use Layer.feedforward to implement DeepNeuralNetwork.feedforward
5. Use Layer.backprop to implement DeepNeuralNetwork.backprop
6. Notice that we have L2 weight regularizations in the final loss function in ad- dition to the cross entropy. Make sure you add those regularization terms in DeepNeuralNetwork.calculate loss and
their derivatives in DeepNeuralNetwork.fit model.
Train your network on the Make Moons dataset using different number of layers, different layer sizes, different activation functions and, in general, different network configurations. In your report,
include generated images and describe what you ob- serve and what you find interesting (e.g. decision boundary of deep vs shallow neural networks).
Next, train your network on another dataset different from Make Moons. You can choose datasets provided by Scikit-learn (more details here) or any dataset of your interest. Make sure that you have
the correct number of input and output nodes. Again, play with different network configurations. In your report, describe the dataset you choose and tell us what you find interesting.
Be curious and creative!!! You are exploring Deep Learning. :)
2 Training a Simple Deep Convolutional Network on MNIST
Deep Convolutional Networks (DCN) have been state-of-the-art in many perceptual tasks including object recognition, image segmentation, and speech recognition. In this problem, you will build and
train a simple 5-layer DCN on MNIST Dataset. We provide you with starter in the attached .py file on the Canvas assignment page You will fill in this starter code to complete task (a), (b), and (c)
below. Also, since one of the purposes of this assignment is to get you familiar with Pytorch, please review this online tutorial . You are encouraged (but not required) to re-organize the starter
code but be sure to explain your code in the report.
MNIST is a dataset of handwritten digits (from 0 to 9). This dataset is one of the most popular benchmarks in machine learning and deep learning. If you de- velop an algorithm to learn from static
images for tasks such as object recognition, most likely, you will want to debug your algorithm on MNIST first before testing it on more complicated datasets such as CIFAR10 and SVHN. There are also
modified versions of MNIST, such as permutation invariant MNIST, which will come in handy for benchmarking at times.
More details, the MNIST data is split into three parts: 55,000 data points of training data (mnist.train), 10,000 points of test data (mnist.test), and 5,000 points of validation data
(mnist.validation). The digits have been size-normalized and centered
in a fixed-size image. MNIST images are of size 28 x 28. When loaded in Tensorflow, each image is flattened into a vector of 28x28=784 numbers. Each MNIST image will have a corresponding label which
is a number between 0 and 9 corresponding to the digit that is drawn in that image.
a) Build and Train a 4-layer DCN
The architecture of the DCN that you will implement is as follows.
conv1(5-5-1-32) - ReLU - maxpool(2-2) - conv2(5-5-32-64) - ReLU - maxpool(2-2)
- fc(1024) - ReLU - DropOut(0.5) - Softmax(10)
More details on the architecture can be found in this tutorial Deep MNIST for Expert.
Follow the tutorial Deep MNIST for Expert to fill in dcn mnist.py. Particularly,
1. Read the tutorial Deep MNIST for Expert to learn how to use Tensorflow.
2. Complete functions weight variable(shape), bias variable(shape), conv2d(x, W), max pool 2x2(x) in dcn mnist.py. The first two functions initialize the weights and biases in the network, and the
last two functions will implement convolution and max-pooling operators, respectively.
3. Build your network: In dcn mnist.py, you will see ”FILL IN THE CODE BELOW TO BUILD YOUR NETWORK”. Complete the following sections
in dcn mnist.py: placeholders for input data and input labeles, first convolutional layer, convolutional layer, densely connected layer, dropout, softmax.
4. Set up Training: In dcn mnist.py, you will see ”FILL IN THE FOLLOWING CODE TO SET UP THE TRAINING”. Complete section setup training in dcn mnist.py.
5. Run Training: Study the rest of dcn mnist.py. Notice that, different from the tutorial Deep MNIST for Expert, I use the summary operation (e.g. summary op, summary writer, ...) to monitor the
training. Here, I only monitor the training loss value. Now, run dcn mnist.py. What is the final test accuracy of your net- work? Note that I set the batch size to 50, and to save time, I set the
max step to only 5500. Batch size is the number of MNIST images that are sent to the DCN at each iteration, and max step is the maximum number of training iter- ations. max step = 5500 means the
training will stop after 5500 iterations no matter what. When batch size is 50, 5500 iterations is equivalent to 5 epochs.
Remind that, in each epoch, the DCN will see the whole training set once. In this case, since there are 55K training images, each epoch is consisted of 55K/50 = 1100 iterations.
6. Visualize Training: In your terminal, type tensorboard --logdir=path/to/results where path/to/results is result dir in dcn mnist.py. Follow the instruction in your terminal to visualize the
training loss in the training. You will be asked to navigate to a website to see the results, e.g. http://172.28.29.81:6006. Include the figures generated by TensorBoard in your report.
b) More on Visualizing Your Training
In part (a) of this problem, you only monitor the training loss during the train- ing. Now, let’s visualize your training more! Study dcn mnist.py and this tutorial TensorBoard: Visualizing Learning
to learn how to monitor a set of variables during the training. Then, modify dcn mnist.py so that you can monitor the statistics (min, max, mean, standard deviation, histogram) of the following terms
after each 100 iterations: weights, biases, net inputs at each layer , activations after ReLU at each layer, activations after Max-Pooling at each layer. Also monitor the test and validation error
after each 1100 iterations (equivalently, after each epoch). Run the training again and visualize the monitored terms in TensorBoard. Include the resultant figures in your report.
c) Time for More Fun!!!
As you have noticed, I use ReLU non-linearity, random initialization, and Adam train- ing algorithm in dcn mnist.py. In this section, run the network training with different non- linearities (tanh,
sigmoid, leaky-ReLU, MaxOut,...), initialization techniques (Xavier...) and training algorithms (SGD, Momentum-based Methods, Adagrad..). Make sure you still monitor the terms specified in part (b).
Include the figures generated by TensorBoard and describe what you observe. Again, be curious and creative! You are encouraged to work in groups, but you need to submit separate reports.
Collaboration Policy
Collaboration both inside and outside class is encouraged. You may talk to other students for general ideas and concepts, but individual write-ups must be done independently.
Plagiarism of any form will not be tolerated. You are expected to credit all sources explicitly. | {"url":"https://coursenana.com/programming/assignment/elec576-comp-576-introduction-to-deep-learning-assignment-1-backpropagation","timestamp":"2024-11-12T19:14:04Z","content_type":"text/html","content_length":"99913","record_id":"<urn:uuid:3f2314c1-e5bf-4988-b355-11500b3ce179>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00466.warc.gz"} |
Topics: Rindler Space
In General > minkowski space.
* Idea: Minkowski spacetime with coordinates adapted to a boost Killing vector field, i.e., to a uniformly accelerated observer.
* Coordinates: If (X, T) are the Minkowski coordinates, Rindler coordinates (x, t) are defined on the right wedge (X > 0, |T| < |X|) by
X = g^−1 e^gx cosh gt , T = g^−1 e^gx sinh gt ,
and coordinates (x', t') on the left wedge (X < 0, |T| < |X|) are defined by
X = −g^−1 e^gx' cosh gt' , T = −g^−1 e^gx' sinh gt' ;
In either case, the inverse transformation is given by
t, t' = g^−1 tanh^−1(T/X) , x, x' = (2g)^−1 ln[g^2(X^ 2 − T^ 2)] ;
The lines t = constant are straight half-lines, while x = constant are hyperbolae of acceleration g e^−gx.
* Line element: Given by
ds^2 = e^2gx (−dt^2 + dx^2) ,
so proper time is related to coordinate time by τ = e^gx t.
@ General references: Born AdP(09) [precursor]; Rindler AJP(66)dec; Felix da Silva & Dahia IJMPA(07) [non-Euclidean geometry of spatial sections].
@ Related topics: Kowalski-Glikman PRD(09)-a0907 [deformed, κ-Rindler space]; Daszkiewicz MPLA(10)-a1004 [twisted]; Chung PRD(10) [asymptotic symmetries]; Bianchi & Satz PRD(13)-a1305 [mechanical
laws of the Rindler horizon]; Araya & Bars PRD(18)-a1712 [infinite stack of identical Minkowski geometries as a multiverse model]; > s.a. black-hole geometry [interior]; modified theories of gravity
[Rindler force]; tests of general relativity [Rindler-type acceleration].
> Online resources: see Wikipedia page; 't Hooft page with animated gif on Rindler coordinates.
And Classical Field Theory > see dirac fields.
And Quantum Theory > s.a. gravitational thermodynamics.
* Idea: The Minkowski vacuum looks like a thermal state in Rindler space, for an observer moving along x = constant, with temperature depending on its acceleration; This makes it useful for mimicking
black-hole radiation.
@ Thermal properties: Fulling PRD(73); Unruh PRD(76); Lapedes JMP(78); Dray & Manogue pr(87); Laflamme PLB(87); Nikolić MPLA(01)gq [criticism of use]; Xiang & Zheng IJTP(01) [horizon entropy];
Socolovsky a1304 [application to the Unruh effect]; Kolekar & Padmanabhan PRD(14)-a1309 [Rindler-Rindler spacetime]; Chowdhury et al PRD-a1902 [and thermal bath]; Padmanabhan a1905 [simple
derivation]; > s.a. radiation; quantum field theory in curved backgrounds.
@ Quantum mechanics: Dai PLA(16)-a1609 [hydrogen atom energy eigenvalues and wave functions].
@ Quantum field theory: Michel a1612 [quantization of scalar and gauge fields]; > s.a. mirrors.
@ Related topics: Balasubramanian et al JHEP(13) [entropy of a "spherical Rindler space" hole in spacetime].
> Related topics: see quantum technology [communication].
main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 22 may 2019 | {"url":"https://www.phy.olemiss.edu/~luca/Topics/st_types/rindler.html","timestamp":"2024-11-07T20:45:43Z","content_type":"text/html","content_length":"9011","record_id":"<urn:uuid:81f545a9-b00d-4b80-9d04-26d194c45e88>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00743.warc.gz"} |
What is the kinetic energy of an object with a mass of 7 kg that has been in freefall for 3 s? | HIX Tutor
What is the kinetic energy of an object with a mass of #7 kg# that has been in freefall for #3 s#?
Answer 1
Kinetic Energy $K E = 3025.26 \text{ }$joules
The given data are as follows mass #m=7" "#kg time #t=3" "#seconds
Assuming it starts from rest at #v_0=0#
#v_f=-29.4" "#m/sec
#KE=3025.26" "#joules
God bless...I hope the explanation is useful.
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Answer 2
The kinetic energy of the object can be calculated using the formula:
[ KE = \frac{1}{2} m v^2 ]
• ( KE ) is the kinetic energy,
• ( m ) is the mass of the object, and
• ( v ) is the velocity of the object.
In freefall near the surface of the Earth, the velocity of the object can be approximated using the equation of motion:
[ v = g t ]
• ( g ) is the acceleration due to gravity (approximately ( 9.8 , \text{m/s}^2 )), and
• ( t ) is the time in seconds.
Substituting the values, we get:
[ v = (9.8 , \text{m/s}^2) \times 3 , \text{s} = 29.4 , \text{m/s} ]
Now, substituting the mass and velocity into the formula for kinetic energy:
[ KE = \frac{1}{2} \times 7 , \text{kg} \times (29.4 , \text{m/s})^2 ]
[ KE \approx 1029.3 , \text{J} ]
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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MCNEMAR TEST Definition in Psychology
McNemar’s Test is a statistical test used to compare two related proportions, or proportions of related groups, in a single sample. It can be used in a variety of situations, such as analyzing the
outcomes of a medical treatment, or determining whether a marketing campaign is successful. The test is also referred to as the “Paired Binary Test” or the “Paired Difference Test.”
McNemar’s test is based on the binomial distribution, which is the probability distribution of the number of successes in a sequence of independent events. It can be used to test the null hypothesis
that the proportions of successes in two samples are equal. The test statistic is the difference between the two proportions, and the test is conducted by calculating the probability of obtaining a
result at least as extreme as the observed result.
McNemar’s Test is well suited to situations where two related proportions are being compared, such as whether a medical treatment is more effective than a placebo or whether a marketing campaign is
successful. In such cases, it is important to consider the relationships between the two samples, as the difference in proportions may be due to the influence of the other sample. The test is also
useful in situations where there are a large number of subjects, as it can be used to compare proportions of related groups.
McNemar’s Test is a powerful statistical tool for assessing differences between proportions in a single sample. It is used in a variety of applications, from medical research to marketing campaigns,
and is well suited to situations where two related proportions are being compared. The test is based on the binomial distribution, and the test statistic is the difference between the two
Kelly, C. (2017). McNemar’s Test. Retrieved from https://statistics.laerd.com/statistical-guides/mcnemars-test-statistical-guide-2.php
Kosinski, M., & Kosinski, P. (2020). McNemar’s Test. Retrieved from https://www.statisticshowto.datasciencecentral.com/mcnemars-test/
Harvey, J. (2015). McNemar’s Test for Paired Nominal Data. Retrieved from https://www.statisticshowto.datasciencecentral.com/mcnemars-test/#:~:text= | {"url":"https://encyclopedia.arabpsychology.com/mcnemar-test/","timestamp":"2024-11-07T17:17:10Z","content_type":"text/html","content_length":"135277","record_id":"<urn:uuid:25ed37f2-53b5-4042-b227-f83620c12cb0>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00094.warc.gz"} |
Sequential Model-based Optimization Algorithm
A Sequential Model-based Optimization Algorithm is an Optimization Algorithm that sequentially selects the next point to evaluate based on a model of the objective function.
• Context:
□ It aims to balance exploration and exploitation by using the model to guide search.
□ It builds a surrogate model of the objective function and uses it to select the next point to evaluate.
□ The model is updated as new points are evaluated, allowing it to improve over time.
□ It can use models such as: Gaussian Processes, Random Forests, and Bayesian Neural Networks.
□ It evaluates points one at a time (or in small batches), sequentially updating the model.
□ It aims to minimize simple regret over the sequence of evaluations.
□ It Uses a model of the objective function to guide search.
□ It Balances exploration and exploitation.
□ It Evaluates points sequentially and updates model.
□ It Optimizes over the sequence of evaluations.
□ ...
• Examples: | {"url":"http://www.gabormelli.com/RKB/Sequential_Model-based_Optimization_Algorithm","timestamp":"2024-11-01T22:54:08Z","content_type":"text/html","content_length":"39193","record_id":"<urn:uuid:1d6d2dda-8d8a-4f02-846d-035eee49c2a0>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00149.warc.gz"} |
Eureka Math Grade 8 Module 1 Lesson 6
Engage NY Eureka Math 8th Grade Module 1 Lesson 6 Answer Key
Eureka Math Grade 8 Module 1 Lesson 6 Exercise Answer Key
Exercise 1.
Show that (C) is implied by equation (5) of Lesson 4 when m>0, and explain why (C) continues to hold even when
Equation (5) says for any numbers x, y, (y≠0) and any positive integer n, the following holds: (\(\frac{x}{y}\))^n=\(\frac{x^{n}}{y^{n}}\). So,
(\(\frac{1}{x}\))^m = \(\frac{1^{m}}{x^{m}}\) By (\(\frac{x}{y}\))^n= \(\frac{x^{n}}{y^{n}}\) for positive integer n and nonzero y (5)
= \(\frac{1}{x^{m}}\) Because 1^m =1
If m= 0, then the left side is
(\(\frac{1}{x}\))^m =(\(\frac{1}{x}\))^0
=1 By definition of x^0,
and the right side is
\(\frac{1}{x^{m}}\) = \(\frac{1}{x^{0}}\)
= \(\frac{1}{1}\) By definition of x^0
Exercise 2.
Show that (B) is in fact a special case of (11) by rewriting it as (x^m)^-1 = x^(-1m) for any whole number m, so that if b=m (where m is a whole number) and a=-1, (11) becomes (B).
(B) says x^-m = \(\frac{1}{x^{m}}\).
The left side of (B), x^-m is equal to x^(-1)m.
The right side of (B), \(\frac{1}{x^{m}}\), is equal to (x^m)^-1 by the definition of (x^m)^-1 in Lesson 5.
Therefore, (B) says exactly that (x^m)^-1 = x^(-1)m.
Exercise 3.
Show that (C) is a special case of (11) by rewriting (C) as (x^-1)^m = x^m(-1) for any whole number m. Thus, (C) is the special case of (11) when b=-1 and a=m, where m is a whole number.
(C) says (\(\frac{1}{x}\))^m = \(\frac{1}{x^{m}}\) for any whole number m.
The left side of (C) is equal to
(\(\frac{1}{x}\))^m =(x^-1)^mBy definition of x^-1 ,
and the right side of (C) is equal to
\(\frac{1}{x^{m}}\) =x^-m By definition of x^-m, and the latter is equal to x^m(-1). Therefore, (C) says (x^-1)^m) =x^m(-1) for any whole number m.
Exercise 4.
Proof of Case (iii): Show that when a<0 and b≥0, (x^b )^a=x^ab is still valid. Let a=-c for some positive integer c. Show that the left and right sides of (x^b )^a=x^ab are equal.
The left side is
(x^b )^a=(x^b )^-c
= \(\frac{1}{\left(x^{b}\right)^{c}}\) By x^-m = \(\frac{1}{x^{m}}\) for any whole number m (B)
= \(\frac{1}{x^{c b}}\) . By (x^m)^n=x^mn for all whole numbers m and n (A)
The right side is
x^ab = x^(-c)b
= \(\frac{1}{x^{c b}}\) . By x^-m = \(\frac{1}{x^{m}}\) for any whole number m (B)
So, the two sides are equal.
Eureka Math Grade 8 Module 1 Lesson 6 Problem Set Answer Key
Question 1.
You sent a photo of you and your family on vacation to seven Facebook friends. If each of them sends it to five of their friends, and each of those friends sends it to five of their friends, and
those friends send it to five more, how many people (not counting yourself) will see your photo? No friend received the photo twice. Express your answer in exponential notation.
The total number of people who viewed the photo is (5^0+5^1+5^2+5^3 )×7.
Question 2.
Show directly, without using (11), that (1.27^-36 )^85= 1.27^-36∙85.
(1.27^-36 )^85= (\(\frac{1}{1.27^{36}}\))^85 By definition
= \(\frac{1}{\left(1.27^{36}\right)^{85}}\)By (\(\frac{1}{x}\))^m =\(\frac{1}{x^{m}}\) for any whole number m (C)
= \(\frac{1}{1.27^{36 \cdot 85}}\)By (x^m)^n=x^mn for whole numbers m and n (7)
= 1.27^-36∙85 By x^-m = \(\frac{1}{x^{m}}\) for any whole number m (B)
Question 3.
Show directly that (\(\frac{2}{13}\))^-127∙(\(\frac{2}{13}\))^-56=(\(\frac{2}{13}\))^-183.
Question 4.
Prove for any nonzero number x, x^-127∙x^-56=x^-183.
x^-127∙x^-56 =\(\frac{1}{x^{127}}\) ∙\(\frac{1}{x^{56}}\) By definition
= \(\frac{1}{x^{127} \cdot x^{56}}\) By the product formula for complex fractions
=\(\frac{1}{x^{127+56}}\) By x^m ∙x^n=x^m+n for whole numbers m and n (6)
= \(\frac{1}{x^{183}}\)
= x^-183 By x^-m = \(\frac{1}{x^{m}}\) for any whole number m (B)
Question 5.
Prove for any nonzero number x, x^-m ∙x^-n=x^-m-n for positive integers m and n.
x^-m∙x^-n= \(\frac{1}{x^{m}}\)∙\(\frac{1}{x^{n}}\) By definition
= \(\frac{1}{x^{m} \cdot x^{n}}\) By the product formula for complex fractions
= \(\frac{1}{x^{m+n}}\) By x^m ∙x^n=x^m+n for whole numbers m and n (6)
=x^-(m+n) By x^-m = \(\frac{1}{x^{m}}\) for any whole number m (B)
Question 6.
Which of the preceding four problems did you find easiest to do? Explain.
Students will likely say that x^-m ∙x^-n=x^-m-n (Problem 5) was the easiest problem to do. It requires the least amount of writing because the symbols are easier to write than decimal or fraction
Question 7.
Use the properties of exponents to write an equivalent expression that is a product of distinct primes, each raised to an integer power.
Eureka Math Grade 8 Module 1 Lesson 6 Exit Ticket Answer Key
Question 1.
Show directly that for any nonzero integer x, x^-5∙x^-7 = x^-12.
x^-5∙x^-7 =\(\frac{1}{x^{5}}\) ∙\(\frac{1}{x^{7}}\) By x^-m = \(\frac{1}{x^{m}}\) for any whole number m (B)
=\(\frac{1}{x^{5} \cdot x^{7}}\) By the product formula for complex fractions
=\(\frac{1}{x^{5+7}}\) By x^m ∙x^n=x^m+n for whole numbers m and n (6)
= x^-12 By x^-m = \(\frac{1}{x^{m}}\) for any whole number m (B)
Question 2.
Show directly that for any nonzero integer x, (x^-2 )^-3=x^6.
(x^-2)^-3 = \(\frac{1}{\left(x^{-2}\right)^{3}}\) By x^-m = \(\frac{1}{x^{m}}\) for any whole number m (B)
= \(\frac{1}{x^{-(2 \cdot 3)}}\) By case (ii) of (11)
= x^6 By x^-m = \(\frac{1}{x^{m}}\) for any whole number m (B) | {"url":"https://eurekamathanswerkeys.com/eureka-math-grade-8-module-1-lesson-6/","timestamp":"2024-11-14T03:42:44Z","content_type":"text/html","content_length":"44590","record_id":"<urn:uuid:f9ce32a2-0026-47e3-ba0b-b6d523bb4c0f>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00674.warc.gz"} |
Radial basis function
From Scholarpedia
Martin Buhmann (2010), Scholarpedia, 5(5):9837. doi:10.4249/scholarpedia.9837 revision #137035 [link to/cite this article]
Radial basis functions are means to approximate multivariable (also called multivariate) functions by linear combinations of terms based on a single univariate function (the radial basis function).
This is radialised so that in can be used in more than one dimension. They are usually applied to approximate functions or data (Powell 1981,Cheney 1966,Davis 1975) which are only known at a finite
number of points (or too difficult to evaluate otherwise), so that then evaluations of the approximating function can take place often and efficiently.
Primarily in computational applications, functions of many variables often need to be approximated by other functions that are better understood or more readily evaluated. This may be for the purpose
of displaying them frequently on a computer screen for instance, so computer graphics are a field of practical use. Radial basis functions are one efficient, frequently used way to do this. Further
applications include the important fields of neural networks and learning theory. Since they are radially symmetric functions which are shifted by points in multidimensional Euclidean space and then
linearly combined, they form data-dependent approximation spaces. This data-dependence makes the spaces so formed suitable for providing approximations to large classes of given functions. It also
opens the door to existence and uniqueness results for interpolating scattered data by radial basis functions in very general settings (in particular in many dimensions).
Indeed, one of the greatest advantages of this method lies in its applicability in almost any dimension (whence its versatility) because there are generally little restrictions on the way the data
are prescribed. Examples below include positive definite kernels where there are no restrictions on the data except that they need to be at distinct points. This should be contrasted to, e.g.,
multivariable polynomial interpolation (but see de Boor and Ron 1990 for an especially flexible approach) or splines. A further advantage is their high accuracy or fast convergence to the
approximated target function in many cases when data become dense.
For applications it is indeed desirable that there are few conditions on the geometry or the directions in which the data points have to be placed in space. No triangulations of the data points or
the like are required for radial basis function algorithms, whereas for instance finite element (Brenner and Scott 1994,Ciarlet 1978) or multivariate spline methods (de Boor 1993,Lai and Schumaker
2007) normally need triangulations. In fact, the advance structuring of the data that some other approximation schemes depend on can be prohibitively expensive to compute in applications, especially
in more than two dimensions. Therefore our approximations here are considered as meshfree approximations, also for instance to be used to facilitate the numerical solution of partial differential
equations (Fasshauer 2007). On the other hand, as will be mentioned below, advanced numerical methods for computing the radial basis function approximations are required when the data set is large
while more standard software is required for instance for finite elements.
Definition of the method
The given data
The method usually works in \(n\) dimensional Euclidean space which we call \( R ^n\) fitted with the Euclidean norm \(\|\cdot\|\ .\) Other choices of spaces are possible. There are \(m\) points in
this space at which the function to be approximated is known, call them \(x_1, x_2, \ldots, x_m\ .\) These points are usually assumed to be all different from each other, otherwise the problem will
become singular when interpolation is used. Both \(n\) and \(m\) are positive integers.
The given values at the points may be called \(f(x_1), f(x_2), \ldots, f(x_m),\) the idea being that they come from a function \(f: R ^n\to R \) which is evaluated at the respective points. This
function may be completely unknown except at those \(m\) points or, for example, too difficult or time-consuming to evaluate otherwise. Although this is a useful ansatz, other approaches without an
underlying function are possible which allow coalescing points with different values, using the idea of spline smoothing (Wahba 1990).
Functions \(f: R^n\to R^k \ ,\) where \(k\) is a positive integer as well, are also admitted in the concept of radial basis functions; approximations are then carried out componentwise in the \(k\)
components of \(f\ :\) \(f=(f_1,f_2,\ldots,f_k), f_i:R^n\to R, 1\leq i\leq k,\) each. Generalising the radial basis function approach to matrix-valued kernels, alternative ideas are also available
(e.g., Lowitzsch 2005).
The approximation scheme
Given this information, we create the sought approximant by a sum
\[\tag{1} s(x) = \sum_{j=1}^m \lambda_j \phi (\| x-x_j \|),\qquad x\in R^n, \]
• the mentioned \(x_j\) are the data points, at which we know \(f\ ,\) which lie in \( R ^n\ ,\)
• the \(x\) is a free variable at which we wish to evaluate our approximant later,
• the \(\phi\) is a univariate, normally continuous function \(\phi:R_+\to R\ ,\) namely the radial basis function (examples will be given below),
• the \(\|\cdot\|\) denotes a norm \(\|\cdot\|: R ^n\to R \ ,\) normally the Euclidean norm but there are more general approaches, and
• the \(\lambda_j\) are scalar parameters.
In fact, norms other than Euclidean are possible, but rarely chosen, and at any rate the individual terms would then of course no longer be radially symmetric about the \(x_j.\)
The scalar parameters are chosen so that \(s\) approximates \(f\) in a useful way at all desired points other than at the \(x_j\ ,\) where it is known anyway.
Thus, in general, the unknown (or difficult to evaluate) function \(f\) is approximated by a linear expression, which remains conceptually simple even if \(m\) or \(n\) become very large. The
flexibility of the approach is also based on the radial symmetry of each term (although not, of course, of the whole expression (1)) since its definition essentially depends only on a univariate
Choice of interpolation as approximation method
Most often, radial basis function approximations are used in combination with interpolation, i.e. the scalar parameters \(\lambda_j\) are chosen, if possible, such that \(s\) matches \(f\) exactly at
the given \(m\) points \(x_j\ .\) This is called interpolation and can be defined by the conditions
\[\tag{2} s(x_j) = f(x_j),\qquad j=1,2,\ldots,m. \]
These, in combination with the form (1), result in a square, \(m\times m\) linear system of equations in the \(\lambda_j\) which may or may not be uniquely solvable. Its interpolation matrix is the
square symmetric matrix \[\tag{3} A=\Bigl(\phi(\| x_j-x_\ell \|)\Bigr) \]
whose non-singularity will guarantee the unique existence of the coefficients \(\lambda_j\ .\) They are then to be computed in standard ways of solving linear systems (using matrix decompositions) if
\(m\) is small or in non-standard ways (see below) if it is large. When the kernel function in form of the radial basis function is strictly positive definite, the interpolation matrix is a positive
definite matrix and non-singular (positive definite functions were considered in the classical paper Schoenberg 1938 for example). Positive definite functions, and their generalisations conditionally
positive definite functions, see below, are closely related to reproducing kernel Hilbert spaces (see literature under further reading).
Indeed, for those radial basis functions which are most often used in practice, the linear system is often hard to solve numerically because of high condition numbers but there exists specialised
software to compute the sought coefficients even for very large \(m\ .\)
Examples of radial basis functions
Clearly, a good choice of the \(\phi\) is important for the quality of the approximation and for the existence of the interpolants. Many choices guarantee the unique existence of (1) satisfying (2)
for all \(m\) and \(n\) solely under the condition that the data points are all different (Micchelli 1986). This is the case for
• linear radial basis function \(\phi(r)=r\ ,\) so long as \(m>1\ ,\)
• (Hardy) multiquadrics radial basis function \(\phi(r)=\sqrt{r^2+c^2}\ ,\) which contains another scalar parameter \(c\) which may be adjusted to improve the approximation, where the choice \(c=0
\) gives the previous example,
• Gaussian kernel \(\phi(r)=\exp(-c^2 r^2) \ ,\) which also contains another scalar parameter \(c\neq0\) which may be adjusted to adapt the approximation, or finally
• (Hardy) inverse multiquadrics radial basis function \(\phi(r)=1/\sqrt{r^2+c^2}\ ,\) which contains another scalar parameter \(c\neq0\) which provides further flexibility.
Further example with linear additional terms
Sometimes, the unique existence of interpolants can be guaranteed with a small variation on the concept by adding low order polynomials to \(s\) and some mild extra conditions. For example
\[ s(x) = \sum_{j=1}^m \lambda_j \phi (\| x-x_j \|)+a+b^T x,\qquad x\in R^n, \]
where \(a\) is a scalar and \(b\) is a vector in \( R ^n\) will give unique existence of interpolating \(s\) using thin-plate splines \(\phi(r)=r^2\log r\ ,\) with its value at the origin declared to
be zero, so long as the \(x_j\) are not collinear and the extra conditions
\[\tag{4} \sum_{j=1}^m\lambda_j=0,\qquad \sum_{j=1}^m\lambda_j x_j=(0,0,\ldots,0)^T, \]
are fulfilled. Here, \((0,0,\ldots,0)^T\) denotes the zero vector in \(n\) dimensions. Thus, under these conditions, the scalars \(\lambda_j, a\) and the vector \(b\) can be solved for uniquely.
These extra conditions (4) take up the new degrees of freedom that come in with \(a\) and \(b\ .\) Many further examples of radial basis functions \(\phi\) exist which are equally useful and
sometimes require that polynomials of degree more than one are added and suitable extra conditions similar to (4) are imposed. But the polynomials are normally not of very high degree, constant to
cubic being typical. The side conditions (4) have to be adjusted to different order conditions when polynomials of degree other than one are used. Also the geometric conditions (centres not being
collinear in the case (4)) will have to be strengthened accordingly (Duchon 1976). Such kernels are no longer positive definite as mentioned above, but conditionally positive definite due to the
aforementioned side conditions.
Properties of the method
In spite of the simplicity of the idea even in high dimensions, good convergence properties have been observed when the \(x_j\) become dense, for example in compact sub-sets of the space \(R^n\ .\)
In particular Duchon has studied the thin-plate splines and related radial basis functions when the scattered data points are becoming dense, other examples of radial basis functions that are
included in his theory being pseudo cubics \(\phi(r)=r^3\) and \(\phi(r)=r^4\log r\) (Duchon 1976), (Powell 1994), or with multiquadrics (Madych and Nelson 1992).
For the convergence analysis, one assumes sometimes that the data points are on equispaced grids in \( R ^n\ ,\) so that infinitely many data are given. The spacing being denoted by \(h\ ,\) one then
lets \(h\to0\ .\) We find in cases that include most of the radial basis functions mentioned above that the uniform difference between \(s\) and \(f\) (the error) goes to zero at the same rate as
some power of \(h\) (Buhmann 2003, Wendland 2005). One also looks specifically for the best possible powers there (saturation orders) when \(f\) satisfies suitable conditions (Johnson 2000).
An example of a uniform convergence result in \(n\) dimensions states that multiquadric interpolation on an infinite uniform grid of spacing \(h\) provides convergence of \(O(h^{n+1})\) to
sufficiently smooth \(f\) with certain partial derivatives bounded, i.e., the uniform error is bounded above by a fixed multiple of \(h^{n+1}\ .\) It should be noted here that the exponent of \(h\)
increases with the dimension.
More general convergence theory is given for instance in (Wu and Schaback 1993,Narcowich, Ward and Wendland 2005).
Computation of the approximations
An unwelcome aspect appears when the linear systems are solved for large \(m\) since for most radial basis functions the matrix is full and ill-conditioned (Narcowich and Ward 1991). An exception is
provided by the radial basis functions of compact support described below.
A typical case is the multiquadric function, where the interpolation matrix has spectral properties which depend both on the parameter and on the distances of the data-points. They can lead to large
condition numbers, and of course the matrix is not sparse.
Some preconditioning and iterative methods are to be applied (for an early approach see Dyn and Levin 1983). The case when multiquadrics are used is very important since they are most often used in
applications, other important choices the aforementioned thin-plate splines and exponential functions. One class of particularly successful methods for computing interpolants with many centres are
Krylov space methods (Powell 1994), others contain particle methods and far field expansions (Beatson et al. 1998, see also the article of Beatson and Greengard in Levesley et al. 1997). General
preconditioning methods are also useful, especially if \(m\) is not too large (Fasshauer 1999).
Other approaches which avoid the difficulty of ill-conditioned interpolation matrices include the idea of quasi-interpolation (e.g. see Buhmann 2003 for a number of useful examples of
quasi-interpolation), or the aforementioned spline smoothing.
Aspects of the parameters in multiquadrics and exponentials
In applications, the parameters \(c\) in multiquadrics, inverse multiquadrics and exponentials play an important role. The aforementioned ill-conditioning problems become very severe if the
parameters go to limits (e.g., \(c\to\infty\) in (inverse) multiquadrics, or to zero in Gaussian kernels, where the entries of the matrix become constant asymptotically). Nonetheless, methods have
been developed to handle this situation and it has even been observed that sometimes the whole approximants \(s\) tend to polynomial limits in those cases (Larsson and Fornberg 2005).
Compactly supported radial basis functions
Compactly supported radial basis functions have been invented for the purpose of getting finite-element type approximations (Brenner and Scott 1994). They give rise to sparse interpolation matrices
and can be used to solve numerically partial differential equations (Fasshauer 1999). Some of them are piecewise-polynomial as a one-dimensional function \(\phi\) (usually only two pieces) (Wendland
1995 where there are useful lists of examples provided together with the theory). Under suitable conditions on degree and dimension \(n\ ,\) they give rise to positive definite interpolation matrices
\(A\) that are banded, therefore sparse, and then of course also regular (for further choices see Buhmann 2001).
A little less flexibility stems from restrictions on \(n\) which may not be arbitrarily large anymore, but there are still no upper bounds on \(m\ .\) Also, the positive definiteness of the
interpolation matrices (similarly as with Gaussian kernels and inverse multiquadrics) makes the radial basis functions useful in statistics.
Applications are manifold, they include
Further reading
• Buhmann, Martin (2003). Radial basis functions: theory and implementations Cambridge University Press, Cambridge. ISBN 978-0-521-10133-2.
• Fasshauer, Greg (2007). Meshfree Approximation Methods with Matlab World Scientific, Singapore.
• Powell, M J D (1981). Approximation Theory and Methods Cambridge University Press, Cambridge.
• Wendland, H (2005). Scattered data approximation Cambridge University Press, Cambridge.
Internal references
• David Gottlieb and Sigal Gottlieb (2009) Spectral methods. Scholarpedia, 4(9):7504.
• Beatson, R; Cherrie, J and Mouat, C (1998). Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration Advances in Computational Mathematics 11: 253-270.
• de Boor, C (1993). Multivariate piecewise polynomials Acta Numerica 2: 65-109. doi:10.1017/s0962492900002348.
• de Boor(1990). On multivariate polynomial interpolation Constructive Approximation 6: 287-302. doi:10.1007/bf01890412.
• Brenner, S and Scott, L (1994). The mathematical theory of finite elements Springer, New York.
• Buhmann, Martin (2001). A new class of radial basis functions with compact support Mathematics of Computation 70: 307-318. doi:10.1090/s0025-5718-00-01251-5.
• Cheney, E W (1966). Introduction to Approximation Theory McGraw-Hill, New York.
• Ciarlet, P (1978). The finite element method for elliptic problems North-Holland, Amsterdam.
• Cucker(2002). On the mathematical foundations of learning Bulletin AMS 39: 1-49. doi:10.1090/s0273-0979-01-00923-5.
• Davis, P J (1975). Interpolation and Approximation Dover, New York.
• Dyn(1983). Iterative solution of systems originating from integral equations and surface interpolation SIAM Journal of Numerical Analysis 20: 377-390. doi:10.1137/0720026.
• Duchon, Jean (1976). Sur l'erreur d'interpolation des fonctions de plusieurs variables pas les Dm-splines Rev. Francaise Automat. Informat. Rech. Oper. Anal. Numer. 10: 5-12.
• Fasshauer, Greg (1999). Solving differential equations with radial basis functions: multilevel methods and smoothing Advances in Computational Mathematics 11: 139-159.
• Ferreira, Antonio; Kansa, Ed; Fasshauer, Greg and Leitao, Vitor (2007). Second Eccomas Thematic Conference on Meshless Methods 2007 University of Porto, Porto. ISBN 978-972-8826-12-3.
• Fornberg, Bengt (1999). A practical guide to pseudospectral methods Cambridge University Press, Cambridge.
• Freeden, W; Gervens, T and Schreiner, M (1998). Constructive approximation on the sphere Clarendeon Press, Oxford. ISBN 0-19-853682-8.
• Golitschek(2000). Interpolation by polynomials and radial basis functions on spheres Constructive Approximation 17: 1-18. doi:10.1007/s003650010028.
• Johnson, Michael (2000). The L_2-approximation order of surface spline interpolation Mathematics of Computation 70: 719-737. doi:10.1090/s0025-5718-00-01301-6.
• Lai, M J and Schumaker, L L (2007). Spline functions on triangulations Cambridge University Press, Cambridge.
• Larsson(2005). Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions Comput. Math. Appl. 49: 103-130. doi:10.1016/j.camwa.2005.01.010.
• Levesley, J; Light, W and Marletta, M (1997). Wavelets, Multilevel Methods and Elliptic PDEs Oxford University Press, Oxford.
• Lowitzsch, S (2005). Error estimates for matrix-valued radial basis function interpolation Journal of Approximation Theory 137: 234-249. doi:10.1016/j.jat.2005.09.008.
• Madych(1992). Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation Journal of Approximation Theory 70: 94-114. doi:10.1016/0021-9045(92)90058-v.
• Micchelli, Charles (1986). Interpolation of scattered data: distance matrices and conditionally positive definite functions Constructive Approximation 1: 11-22. doi:10.1007/bf01893414.
• Narcowich(1991). Norms of inverses and condition numbers of matrices associated with scattered data Journal of Approximation Theory 64: 69-94. doi:10.1016/0021-9045(91)90087-q.
• Narcowich, F; Ward, J and Wendland, H (2005). Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting Mathematics of Computation 74: 643-763.
• Nisbet, R; Elder, J and Miner, G (2009). Statistical Analysis and Data Mining Academic Press, New York.
• Powell, M J D (1994). The uniform convergence of thin-plate spline interpolation in two dimensions Numerische Mathematik 67: 107-128. doi:10.1007/s002110050051.
• Powell, M J D (1997). A new iterative algorithm for thin plate spline interpolation in two dimensions Annals of Numerical Mathematics 4: 519-527.
• Schaback, Robert (1995). Error estimates and condition numbers for radial basis function interpolation Advances in Computational Mathematics 3: 251-264. doi:10.1007/bf02432002.
• Schaback(2006). Kernel techniques: from machine learning to meshless methods Acta Numerica 15: 543-639. doi:10.1017/s0962492906270016.
• Schoenberg, I J (1938). Metric spaces and positive definite functions Transactions of the AMS 44: 522-536. doi:10.1090/s0002-9947-1938-1501980-0.
• Wendland, Holger (1995). Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree Advances in Computational Mathematics 4: 389-396. doi:10.1007/
• Wahba, G (1990). Spline Models for Observational Data SIAM, Philadelphia.
• Wu(1993). Local error estimates for radial basis function interpolation of scattered data IMA Journal of Numerical Analysis 13: 13-27. doi:10.1093/imanum/13.1.13.
See also
Finite elements, multivariate splines, multivariate approximation theory, kernel space methods. | {"url":"http://scholarpedia.org/article/Radial_basis_function","timestamp":"2024-11-08T18:48:56Z","content_type":"text/html","content_length":"60184","record_id":"<urn:uuid:b9e54ad0-40ec-4d44-a53e-8ca5a16ecd78>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00425.warc.gz"} |
Random variable
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A random variable is a term used in mathematics and statistics. It can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to
generate a random result. For example, a random variable can be used to describe the process of rolling a fair die and the possible outcomes { 1, 2, 3, 4, 5, 6 }. Another random variable might
describe the possible outcomes of picking a random person and measuring his or her height.
Unlike the common practice with other mathematical variables, a random variable cannot be assigned a value; a random variable does not describe the actual outcome of a particular experiment, but
rather describes the possible, as-yet-undetermined outcomes in terms of real numbers.
Although such simple examples as rolling a die and measuring heights allow easy visualisation of the practical use of random variables, their mathematical construction allows mathematicians the
convenience of dealing with much measure-theoretic probability theory in the more familiar domain of real-valued functions. Conversely, the concept also places experiments involving real-valued
outcomes firmly within the measure-theoretic framework.
Definitions[ ]
Random variables[ ]
Some consider the expression random variable a misnomer, as a random variable is not a variable but rather a function that maps events to numbers. Let A be a σ-algebra and Ω the space of events
relevant to the experiment being performed. In the die-rolling example, the space of events is just the possible outcomes of a roll, i.e. Ω = { 1, 2, 3, 4, 5, 6 }, and A would be the power set of Ω.
In this case, an appropriate random variable might be the identity function X(ω) = ω, such that if the outcome is a '1', then the random variable is also equal to 1. An equally simple but less
trivial example is one in which we might toss a coin: a suitable space of possible events is Ω = { H, T } (for heads and tails), and A equal again to the power set of Ω. One among the many possible
random variables defined on this space is
${\displaystyle X(\omega) = \begin{cases}0,& \omega = \texttt{H},\\1,& \omega = \texttt{T}.\end{cases}}$
Mathematically, a random variable is defined as a measurable function from a probability space to some measurable space. This measurable space is the space of possible values of the variable, and it
is usually taken to be the real numbers with the Borel σ-algebra. This is assumed in the following, except where specified.
Let (Ω, A, P) be a probability space. Formally, a function X: Ω → R is a (real-valued) random variable if for every subset A[r] = { ω : X(ω) ≤ r } where r ∈ R, we also have A[r] ∈ A. The importance
of this technical definition is that it allows us to construct the distribution function of the random variable.
Distribution functions[ ]
If a random variable ${\displaystyle X:\Omega \to \mathbb{R}}$ defined on the probability space ${\displaystyle (\Omega , P)}$ is given, we can ask questions like "How likely is it that the value of
${\displaystyle X}$ is bigger than 2?". This is the same as the probability of the event ${\displaystyle \{ s \in\Omega : X(s) > 2 \} }$ which is often written as ${\displaystyle P(X > 2)}$ for
Recording all these probabilities of output ranges of a real-valued random variable X yields the probability distribution of X. The probability distribution "forgets" about the particular probability
space used to define X and only records the probabilities of various values of X. Such a probability distribution can always be captured by its cumulative distribution function
${\displaystyle F_X(x) = \operatorname{P}(X\leq x)}$
and sometimes also using a probability density function. In measure-theoretic terms, we use the random variable X to "push-forward" the measure P on Ω to a measure dF on R. The underlying probability
space Ω is a technical device used to guarantee the existence of random variables, and sometimes to construct them. In practice, one often disposes of the space Ω altogether and just puts a measure
on R that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables.
Functions of random variables[ ]
If we have a random variable X on Ω and a measurable function f: R → R, then Y = f(X) will also be a random variable on Ω, since the composition of measurable functions is also measurable. The same
procedure that allowed one to go from a probability space (Ω, P) to (R, dF[X]) can be used to obtain the distribution of Y. The cumulative distribution function of Y is
${\displaystyle F_Y(y) = \operatorname{P}(f(X) \le y).}$
Example[ ]
Let X be a real-valued, continuous random variable and let Y = X^2. Then,
${\displaystyle F_Y(y) = \operatorname{P}(X^2 \le y).}$
If y < 0, then P(X^2 ≤ y) = 0, so
${\displaystyle F_Y(y) = 0\qquad\hbox{if}\quad y < 0.}$
If y ≥ 0, then
${\displaystyle \operatorname{P}(X^2 \le y) = \operatorname{P}(|X| \le \sqrt{y}) = \operatorname{P}(-\sqrt{y} \le X \le \sqrt{y}),}$
${\displaystyle F_Y(y) = F_X(\sqrt{y}) - F_X(-\sqrt{y})\qquad\hbox{if}\quad y \ge 0.}$
Moments[ ]
The probability distribution of random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its
"average value" is. This is captured by the mathematical concept of expected value of a random variable, denoted E[X]. Note that in general, E[f(X)] is not the same as f(E[X]). Once the "average
value" is known, one could then ask how far from this average value the values of X typically are, a question that is answered by the variance and standard deviation of a random variable.
Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables X, find a collection {f[i]} of functions such that the expectation values E[f[i](X)] fully
characterize the distribution of the random variable X.
Equivalence of random variables[ ]
There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, equal in mean, or equal in distribution.
In increasing order of strength, the precise definition of these notions of equivalence is given below.
Equality in distribution[ ]
Two random variables X and Y are equal in distribution if they have the same distribution functions:
${\displaystyle \operatorname{P}(X \le x) = \operatorname{P}(Y \le x)\quad\hbox{for all}\quad x.}$
Two random variables having equal moment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions of iidrv's.
To be equal in distribution, random variables need not be defined on the same probability space. The notion of equivalence in distribution is associated to the following notion of distance between
probability distributions,
${\displaystyle d(X,Y)=\sup_x|\operatorname{P}(X \le x) - \operatorname{P}(Y \le x)|,}$
which is the basis of the Kolmogorov-Smirnov test.
Equality in mean[ ]
Two random variables X and Y are equal in p-th mean if the pth moment of |X − Y| is zero, that is,
${\displaystyle \operatorname{E}(|X-Y|^p) = 0.}$
Equality in pth mean implies equality in qth mean for all q<p. As in the previous case, there is a related distance between the random variables, namely
${\displaystyle d_p(X, Y) = \operatorname{E}(|X-Y|^p).}$
Almost sure equality[ ]
Two random variables X and Y are equal almost surely if, and only if, the probability that they are different is zero:
${\displaystyle \operatorname{P}(X eq Y) = 0.}$
For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:
${\displaystyle d_\infty(X,Y)=\sup_\omega|X(\omega)-Y(\omega)|,}$
where 'sup' in this case represents the essential supremum in the sense of measure theory.
Equality[ ]
Finally, two random variables X and Y are equal if they are equal as functions on their probability space, that is,
${\displaystyle X(\omega)=Y(\omega)\qquad\hbox{for all}\quad\omega}$
Convergence[ ]
Much of mathematical statistics consists in proving convergence results for certain sequences of random variables; see for instance the law of large numbers and the central limit theorem.
There are various senses in which a sequence (X[n]) of random variables can converge to a random variable X. These are explained in the article on convergence of random variables.
Literature[ ]
Papoulis, Athanasios 1965 Probability, Random Variables, and Stochastic Processes. McGraw-Hill Kogakusha, Tokyo, 9th editon, ISBN 0071199810.
See also[ ]
• random vector
• random function
• generating function
• Algorithmic information theory
This article incorporates material from Random variable on PlanetMath, which is licensed under the GFDL.
de:Zufallsvariable es:Variable aleatoria fr:Variable aléatoire he:משתנה מקרי nl:Stochastische variabele ru:Случайная величина sv:Stokastisk variabel zh:随机变量 | {"url":"https://psychology.fandom.com/wiki/Random_variable","timestamp":"2024-11-15T01:31:41Z","content_type":"text/html","content_length":"208291","record_id":"<urn:uuid:4b9c7d3a-500b-4d8a-bf79-6171507bb3eb>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00229.warc.gz"} |
Carl Friedrich Gauss
Carl Friedrich Gauss (1777 – 1855)
Johann Carl Friedrich Gauss (30 April 1777 Braunschweig – 23 February 1855 Göttingen) was a German mathematician.
Gauss contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, mechanics, electrostatics, astronomy, matrix
theory, and optics.
Gauss had an exceptional influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.
Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbuttel, as the son of poor working-class parents.
His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension, which itself occurs 39 days
after Easter.
Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years.
He was christened and confirmed in a church near the school he attended as a child.
Gauss was a child prodigy. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100.
Gauss's intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum, which he attended from 1792 to 1795, and to the University of Göttingen from
1795 to 1798.
While at university, Gauss independently rediscovered several important theorems. His breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and
straightedge if and only if the number of sides is the product of distinct Fermat primes and a power of 2.
The year 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March.
Gauss also discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note: "ΕΥΡΗΚΑ! num = Δ + Δ' + Δ".
On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years later led to the Weil conjectures.
Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
Gauss's method involved determining a conic section in space, given one focus (the Sun) and the conic's intersection with three given lines and given the time it takes the planet to traverse the arcs
determined by these lines.
The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus
conicis solem ambientum.
In 1818 Gauss, putting his calculation skills to practical use, carried out a geodetic survey of the Kingdom of Hanover, linking up with previous Danish surveys.
Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it. This discovery was a major paradigm shift in mathematics.
In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoff's circuit laws in electricity.
In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics).
In 1845, he became associated member of the Royal Institute of the Netherlands; when that became the Royal Netherlands Academy of Arts and Sciences in 1851, he joined as a foreign member.
Gauss died in Göttingen, (then Kingdom of Hanover and now Lower Saxony) on 23 February 1855 and is interred in the Albani Cemetery there.
Carl Gauss was an ardent perfectionist and a hard worker. He was never a prolific writer, refusing to publish work which he did not consider complete and above criticism. This was in keeping with his
personal motto pauca sed matura ("few, but ripe").
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After reviewing types of angles and how to measure angles using a protractor, students will see how many triangles they can construct using an 18-foot beam. Through this activity, students will
discover the conditions that must be met to construct a triangle. For example, the sum of the lengths of two sides of a triangle must be greater than the longest side of the triangle. They will also
discover that the sum of the angles of a triangle is always 180 degrees.
Material Type:
William Allred
Carrie Robledo
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236 research outputs found
Magnetization data of single crystalline La4Sr10Cu24O41 are presented. In this compound, doped spin chains and undoped spin ladders are realized. The magnetization, at low temperatures, is governed
by the chain subsystem with a finite interchain coupling which leads to short range antiferromagnetic spin correlations. At higher temperatures, the response of the chains can be estimated in terms
of a Curie-Weiss law. For the ladders, we apply the low-temperature approximation for a S=1/2 2-leg spin ladder by Troyer et al.Comment: 2 pages, 2 figure
Magnetization measurements of Sr{14-x}Ca_xCu_{24}O_{41} with 0 <= x <=12 in magnetic fields up to 16 T show that the low-temperature magnetic response of the CuO_2 spin chains changes strongly upon
Ca doping. For x=0 quantum statistical simulations yield that the temperature and field dependence of the magnetization can be well described by an effective Heisenberg model in which the ground
state configuration is composed of spin dimers, trimers, and monomers. For x>0 a constant contribution to the low-temperature magnetic susceptibility is observed which cannot be explained in terms of
simple chain models. Alternative scenarios are outlined.Comment: 2 pages, submitted to the proceedings of the ICM, Kyoto, Japan, August 200
We report on the pressure effects on the orbital polaron lattice in the lightly doped manganites $\mathrm{La_{1-x}Sr_xMnO_{3}}$, with $x\sim 1/8$. The dependence of the orbital polaron lattice on
$negative$ chemical pressure is studied by substituting Pr for La in $\mathrm{(La_{1-y}Pr_y)_{7/8}Sr_{1/8}MnO_{3}}$. In addition, we have studied its hydrostatic pressure dependence in $\mathrm{(La_
{0.9}Pr_{0.1})_{7/8}Sr_{1/8}MnO_{3}}$. Our results strongly indicate that the hopping $t$ significantly contributes to the stabilization of the orbital polaron lattice and that the orbital polarons
are ferromagnetic objects which get stabilized by local double exchange processes. The analysis of short range orbital correlations and the verification of the Grueneisen scaling by hard x-ray,
specific heat and thermal expansion data reinforces our conclusions.Comment: 7 figure
We report the magnetic phase diagram of single-crystalline LiFePO$_4$ in magnetic fields up to 58~T and present a detailed study of magneto-elastic coupling by means of high-resolution capacitance
dilatometry. Large anomalies at \tn\ in the thermal expansion coefficient $\alpha$ imply pronounced magneto-elastic coupling. Quantitative analysis yields the magnetic Gr\"uneisen parameter $\gamma_
{\rm mag}=6.7(5)\cdot 10^{-7}$~mol/J. The positive hydrostatic pressure dependence $dT_{\rm N}/dp = 1.46(11)$~K/GPa is dominated by uniaxial effects along the $a$-axis. Failure of Gr\"uneisen scaling
below $\approx 40$~K, i.e., below the peak temperature in the magneto-electric coupling coefficient [\onlinecite{toft2015anomalous}], implies several competing degrees of freedom and indicates
relevance of recently observed hybrid excitations~[\onlinecite{yiu2017hybrid}]. A broad and strongly magnetic-field-dependent anomaly in $\alpha$ in this temperature regime highlight the relevance of
structure changes. Upon application of magnetic fields $B||b$-axis, a pronounced jump in the magnetisation implies spin-reorientation at $B_{\rm SF} = 32$~T as well as a precursing phase at 29~T and
$T=1.5$~K. In a two-sublattice mean-field model, the saturation field $B_{\rm sat,b} = 64(2)$~T enables the determination of the effective antiferromagnetic exchange interaction $J_{\rm af} = 2.68(5)
$~meV as well as the anisotropies $D_{\rm b} = -0.53(4)$~meV and $D_{\rm c} = 0.44(8)$~meV
We have studied the magnetism of the Pr3+ ions in PrFeAsO_1-xF_x (x = 0; 0.15) and its interaction with the Fe magnetic order (for x = 0). Specific heat data confirm the presence of a first excited
crystal electric field (CEF) level around 3.5 meV in the undoped compound PrFeAsO. This finding is in agreement with recent neutron scattering experiments. The doped compound is found to have a much
lower first CEF splitting of about 2.0 meV. The Pr ordering in PrFeAsO gives rise to large anomalies in the specific heat and the thermal expansion coefficient. In addition, a field-induced
transition is found at low temperatures that is most pronounced for the magnetostriction coefficient. This transition, which is absent in the doped compound, is attributed to a reversal of the Fe
spin canting as the antiferromagnetic Pr order is destroyed by the external magnetic field.Comment: 8 pages, 6 figure
We report on measurements of the magnetic response of the anisotropic CuO_2 spin chains in lightly hole-doped La_x (Ca,Sr)_14-x Cu_24 O_41, x>=5. The experimental data suggest that in magnetic fields
B >~ 4T (applied along the easy axis) the system is characterized by short-range spin order and quasi-static (quenched) charge disorder. The magnetic susceptibility chi(B) shows a broad anomaly,
which we interpret as the remnant of a spin-flop transition. To corroborate this idea, we present Monte Carlo simulations of a classical, anisotropic Heisenberg model with randomly distributed,
static holes. Our numerical results clearly show that the spin-flop transition of the pure model (without holes) is destroyed and smeared out due to the disorder introduced by the quasi-static holes.
Both the numerically calculated susceptibility curves chi(B) and the temperature dependence of the position of the anomaly are in qualitative agreement with the experimental data.Comment: 10 pages,
REVTeX4. 11 figures; v2: Fig.2 replaced, small changes in Figs.1 and 11; minor revisons in Sec. III.C; accepted by Phys. Rev. | {"url":"https://core.ac.uk/search/?q=author%3A(Klingeler%2C%20R.)","timestamp":"2024-11-14T04:46:44Z","content_type":"text/html","content_length":"146821","record_id":"<urn:uuid:55259874-de30-4ab7-affc-5e649cdd93ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00375.warc.gz"} |
Angles: An Introduction
An angle is formed when two rays are joined together at a common point. The common point here is called node or vertex and the two rays are called arms of the angle. The angle is represented by the
symbol ‘∠’. The word angle came from the Latin word “Angulus”. Learn more about lines and angles here.
The angle is usually measured in degrees, using a protractor. Degrees 30°, 45°, 60°, 90°, 180° shows different angles here. The types of angles are based on the values of angles in degrees.
We can also represent angles in radians, i.e., in terms of pi (π). 180 degrees is equal to π in radians.
An angle is a form of geometrical shape, that is constructed by joining two rays to each other at their end-points. The angle can also be represented by three letters of the shape that define the
angle, with the middle letter being where the angle actually is (i.e.its vertex). Angles are generally represented by Greek letters such as θ, α, β, etc.
Eg. ∠ABC, where B is the given angle.
Angle measurement terms are – degree °, radians or gradians.
The amount of rotation about the point of intersection of two planes (or lines) which is required to bring one in correspondence with the other is called an Angle.
For More Information On Parts Of An Angle, Watch The Below Video:
Types of Angles
There are majorly six types of angles in Geometry. The names of all angles with their properties are:
• Acute Angle: It lies between 0° to 90.
• Obtuse Angle: It lies between 90° to 180°
• Right Angle: The angle which is exactly equal to 90°
• Straight Angle: The angle which is exactly equal to 180°
• Reflex Angle: The angle which is greater than 180 degrees and less than 360 degrees
• Full Rotation: The complete rotation of angle equal to 360 degrees
Note: Sometimes full rotation is not considered as a kind of angle. Therefore, in such cases, we consider there are five types of angles.
Type of angles Description
Acute Angle < 90°
Obtuse Angle > 90°
Right Angle = 90°
Straight Angle =180°
Reflex Angle >180°
Full rotation/complete angle =360°
Interior and Exterior Angles
In case of a polygon, such as a triangle, quadrilateral, pentagon, hexagon, etc., we have both interior and exterior angles.
• Interior angles are those that lie inside the polygon or a closed shape having sides and angles.
• Exterior angles are formed outside the shape, between any side and line extended from adjacent sides.
For example, an image of a pentagon is given here, representing its interior angles and exterior angles.
Positive & Negative Angles
• Positive Angle- An Angle measured in Anti-Clockwise direction is Positive Angle.
• Negative Angle- An angle measured in Clockwise direction is Negative Angle.
Parts of Angles
• Vertex- The corner points of an angle is known as Vertex. It is the point where two rays meet.
• Arms– The two sides of angle, joined at a common endpoint.
• Initial Side – It is also known as the reference line. All the measurements are done taking this line as the reference.
• Terminal Side- It is the side (or ray) up to which the angle measurement is done.
Angle Measurement
To measure everything in this world, we need a unit in a similar angle measurement requires three units of measurement :
Degree of an Angle
It is represented by ° (read as a degree). It most likely comes from Babylonians, who used a base 60 (Sexagesimal) number system. In their calendar, there was a total of 360 days. Hence, they adopted
a full angle to be 360°. First, they tried to divide a full angle into angles using the angle of an equilateral triangle. Later, following their number system (base 60), they divided 60° by 60 and
defined that as 1°. Sometimes, it is also referred to as arc degree or arc-degree which means the degree of an arc.
An angle is said to be equal to 1° if the rotation from the initial to the terminal side is equal to 1/360 of the full rotation.
A degree is further divided into minutes and seconds. 1′ (1 minute) is defined as one-sixtieth of a degree and 1” (1 second) is defined as one-sixtieth of a minute. Thus,
1°= 60′ = 3600”
Radian of an Angle
This is the SI unit of angle. Radian is mostly used in Calculus. All the formula for derivatives and integrals hold true only when angles are measured in terms of a radian. It is denoted by ‘rad’.
The length of the arc of a unit circle is numerically equal to the measurement in radian of the angle that it subtends.
In a complete circle, there are 2π radians.
360 = 2π; radian
Therefore, 1 radian = 180°/π
Gradian of an Angle
This unit is least used in Maths. It is also called a gon or a grade.
An angle is equal to 1 gradian if the rotation from the initial to terminal side is 1/400 of the full rotation. Hence, the full angle is equal to 400 gradians.
It is denoted by ‘grad’.
Figure 3 shows the example of angles in gradian.
Figure 3: Angle Measurement in Gradian
Practice Problems
Draw angles using a protractor for the following measurements:
• 45 degrees
• 55 degrees
• 70 degrees
• 90 degrees
• 130 degrees
Frequently Asked Questions – FAQs
What is an angle?
An angle is a geometrical figure formed by two rays, when joint at a single point. The two rays are known as arms or sides of angle and the common point is the vertex.
What are the six types of angles?
The six major types of angles are:
Acute angle
Obtuse angle
Right angle
Straight angle
Reflex angle
Full rotation
How angles are measured?
Angles are usually measured in degrees. We can use a measuring instrument, i.e. protractor, to measure any unknown angle.
What is the value of an angle equal to 60 degrees, in radians?
60 degrees can be expressed in radians as π/3.
Since, 180 degrees equals to π, therefore,
60 degrees = π/180 x 60 = π/3 (in radians)
What is a zero angle?
An angle with zero degree measurement is called zero angle.
Can a triangle have two 90 degree angles?
A triangle cannot have two 90 degrees or right angles, because by the angle sum property of the triangle, we know that, sum of all the three angles of a triangle is equal to 180 degrees. If two
angles are of 90 degrees, then the third angle has to be zero, which is not possible. | {"url":"https://mathlake.com/Angles:-An-Introduction","timestamp":"2024-11-06T01:00:00Z","content_type":"text/html","content_length":"19214","record_id":"<urn:uuid:0dd52fda-64c0-4637-a863-bf65c3ad0cca>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00194.warc.gz"} |
Class 7 CBSE maths Integers
This page contains main concepts and results for Class 7, Chapter-1
. This chapter is mainly about representation of integers on the number line and their addition and subtraction.
We use numbers to count anything. So, what are various types of numbers?
1. Natural numbers
Natural numbers are counting numbers but these set of numbers do not include zero. This is because you cannot count zero. So numbers,
1,2,3,4,5,6……etc are all
natural numbers
2. Whole numbers
All natural numbers along with zero are called
whole number
. For example
0, 1, 2, 3, 4, 5, 6………etc are all whole numbers.
NOTE:- Thes types of numbers do not include fractions.
From the definition of natural numbers we can conclude that every natural or counting number is a whole number.
3. Integers
Integers include all natural numbers, zero and negative numbers for example,
-4, -3, -2, -1, 0, 1, 2, 3, ………. etc are all integers.
So now we have,
1. Positive integers:- 1, 2, 3, ……
2. Negative integers:- -1, -2, -3, …….
3. 0 (zero):- which is an integer that is neither negative nor positive.
Integers like whole numbers do not include
for example 3.5 , ½ etc.
Number line for integers is
Now from this number line you can see that
to the left of zero are all negative. Again from this number line you can observe that numbers to the right of zero are all positive.
Important note:-
If the number has no sign attached to it as prefix then it means that it is a positive number.
For example number 3 is really number +3
Important Points
On the number line when we
1. Add a positive integer, we move to the right.
2. Add a negative integer we move to the left.
3. Subtract a positive integer we move to the left.
4. Subtract a negative integer, we move to the right.
Properties of integers
• Integers are closed under addition, subtraction and multiplication. Which means that sum of integers will also give integers.
• Addition and multiplication are commutative for integers, i.e.,
1. a + b = b + a
2. a × b = b × a
For any two integers a and b.
• Addition and multiplication are associative for integers, i.e.,
1. (a + b) + c = a + (b + c)
2. (a × b) × c = a × (b × c)
for any three integers a, b and c.
• Existence of identity
1. Zero (0) is an additive identity for integers, i.e., a + 0 = 0 + a = a
for any integer a.
2. 1 is multiplicative identity for integers, i.e., a × 1 = 1 × a = a
for any integer a.
• Integers show distributive property of multiplication over addition, i.e., a × (b + c) = a × b + a × c
for any three integers a, b and c.
• Product of a positive integer and a negative integer is a negative integer, i.e, a × (–b) = – ab
where a and b are positive integers.
• Product of two negative integers is a positive integer, i.e., (–a) × (–b) = ab
where a and b are positive integers.
• Product of even number of negative integers is positive, whereas the product of odd number of negative integers is negative, i.e.,
• When a positive integer is divided by a negative integer or vice-versa and the quotient obtained is an integer then it is a negative integer, i.e., \[a\div \left( b \right)=\left( a \right)\div b
\text{ =}\frac{a}{b}\]
where a and b are positive integers and $-\frac{a}{b}$is an integer
• When a negative integer is divided by another negative integer to give an integer then it gives a positive integer, i.e., \[\left( a \right)\div \left( b \right)=\frac{a}{b}\] , where a and b are
positive integers and \[\frac{a}{b}\] is also an integer.
• For any integer a, \[a\div 1=a~\]
a ÷ 0 is not defined.
Rules of addition of integers
Rules for subtraction of integers
Rules for multiplication and division of integers
Problem solving strategy
Now let us consider solving a problem and apply the above mentioned problem solving strategy. For this purpose we will solve a NCERT book problem.
Mohan deposits Rs 2,000 in his bank account and withdraws Rs 1,642 from it, the next day. If withdrawal of amount from the account is represented by a negative integer, then how will you
represent the amount deposited? Find the balance in Mohan's account after the withdrawal.
Solution: Step 1:- Understand the problem
For this step first read your problem carefully. So,
□ What do you know from the problem?
Here find ‘what is the amount Mohan deposited in his bank account and how much he withdraws’.
□ What are you trying to find?
Balance in Mohan’s bank account after the withdrawal.
Amount deposited = Rs 2000
Amount withdrawn = Rs 1642
Step 2:- Plan your strategy
Here we have to find the amount he removed from his account. So,
Balance in Mohan's account = Money deposited - Money withdrawn
Step 3:- Solve the problem
Now that you have all the known and unknown quantities and you also have a strategy to solve your problem, you can now carefully carry out your calculations.
Balance in Mohan's account = 2000 + (-1642) = 2000 - 1642 = 358
Therefore, balance in Mohan's account after withdrawal is Rs 358.
Step 4:- Revise
Now in this step check your answer.
Also Read
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Long time control and the Turnpike property
The turnpike property
Figure 1: When an optimal control problem satisfies the turnpike property, the solution remains close to certain path for most of the time, except maybe for an initial time interval $T_1$ and a final
time interval $T_2$. In this Figure, $x(t,x_0,u(t))$ is solution of the optimal control problem (3) with the cost function (2), while $\bar{x}$ is the optimal solution of the stationary problem (5).
The turnpike property establishes that, when a general optimal control problem is settled in large time, for most of the time the optimal control and trajectories remain exponentially close to the
optimal control and state of the corresponding steady-state or static optimal control problem.
The origin of the term Turnpike is in the interpretation that Samuelson did of this phenomenon in [1] : suppose we want to travel from city A to city B by car, the best way to do it, the optimal way,
is to take the highway (namely, the turnpike) as near as we can from city A, and leave it when we are close to B. So, except nearby A and B, we are expected to be on the highway: in other words, the
turnpike of the problem.
The turnpike property is very useful, since it gives us an idea of the nature of the optimal solution of a problem, without having to solve it analytically. In practice, the turnpike property allows
performing a significant improvement of the numerical methods used to solve optimal control problems.
The Turnpike property in the Linear Case
Consider the finite-dimensional dynamical system
$$\left\lbrace \begin{array}{ccc} \label{linearproblem} & x_t + Ax = Bu\\ & x(0)=x_0 \in \mathbb{R}^{N} %onumber \end{array} \right.$$ where $A \in M(N,N)$, $B\in M(N,M)$, with control $u \in L^2
(0,T;\mathbb{R}^{N})$. Given a matrix $C \in M(N,N)$, and some $x^* \in \mathbb{R}^{N}$, consider the optimal control problem
$$\min_{u} J^T (u) = \frac{1}{2} \int_0^T \left( |u(t)|^2 + |C(x(t)-x^*)|^2 \right)dt.$$
There exists a unique optimal control $u(t)$ in $ L^2 (0,T;\mathbb{R}^{M})$, characterized by the optimality condition
$$u = - B^*p, \,\,\,\,\,\, \left\lbrace \begin{array}{ccc} \label{linearproblem} & -p_t + A^* p = C^*C(x-x^*)\\ & p(T)=0 \in \mathbb{R}^{N} %onumber \end{array} \right.$$
Let formulate the same problem for the steady-state model
Then, there exists a unique minimum $\bar{u}$, and a unique optimal state $\bar{x}$, of the stationary optimal control problem
$$%onumber \left\lbrace \begin{array}{clc} & \min_{u} J_s (u) = \frac{1}{2} \left( |u|^2 + |C(x -x^*)|^2 \right). \ \\ & subject \,\,\, to\,\,\, Ax=Bu \end{array} \right.$$
Let assume that:
1. The pair $(A,B)$ is controllable.
2. The pair $(A,C)$ is observable.
Then, we have the following result (proved in [2])
Theorem: there exist positive constants $\lambda$ and $K$, independent on $T$, such that $ \forall t \in [0,T]$
|u(t) -\bar{u}| + |x(t) – \bar{x}| \leq K \left( e^{-\lambda t} + e{-\lambda (T-t)} \right) .
Some results and extensions on the turnpike property
Figure 2: The Lotka-Volterra equations model a biological ecosystem where two species interact: a prey and a predator. In [9] the control is introduced by hunting both species. In this figure we can
see the turnpike phenomenon in both species ($x_1$ is the prey and $x_2$ is the predator species) and the control $u(t)$, which represents the hunting. In dashed blue, we have the evolution of each
variable, and in black, the turnpike.
• The study of the periodic turnpike property for optimal control problems in Hilbert spaces (in progress, E.Trélat, C. Zhang and E. Zuazua)
• Clarify the turnpike property for nonlinear PDE without smallness conditions on the target.
• The turnpike property on multi-D wave equations.
• Prove a turnpike theorem for a constrained control for linear and nonlinear finite–dimensional optimal control problems.
• Develope the turnpike computational code to solve a wide range of problems as it has been done in [9] for the Lotka-Volterra model.
[1] Robert Dorfman, Paul Anthony Samuelson, and Robert M Solow Linear programming and economic analysis. Courier Corporation, 1958.
[2] Alessio Porretta and Enrique Zuazua. Long time versus steady state optimal control. SIAM Journal on Control and Optimization, 51(6):4242–4273, 2013.
[3] Lionel W McKenzie. Turnpike theory. Econometrica: Journal of the Econometric Society, pages 841–865, 1976.
[4] Dean Carlson, Alain B Haurie, and Arie Leizarowitz. Infinite horizon optimal control: deterministic and stochastic systems. Springer Science & Business Media, 2012.
[5] Alexander Zaslavski. Turnpike properties in the calculus of variations and optimal control, volume 80. Springer Science & Business Media, 2006.
[6] Emmanuel Trélat and Enrique Zuazua. The turnpike property in finite-dimensional nonlinear optimal control Journal of Differential Equations, 258(1):81–114, 2015.
[7] Martin Gugat, Emmanuel Trélat, and Enrique Zuazua. Optimal Neumann control for the 1D wave equation: finite horizon, infinite horizon, boundary tracking terms and the turnpike property Systems &
Control Letters, 90:61–70, 2016.
[8] S Zamorano. The turnpike property for two-dimensional navier-stokes equations. arXiv preprint arXiv:1601.04984, 2016.
[9] Aitziber Ibañez. Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations. Journal of Biological Dynamics, 11(1):25–41, 2017.
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The Use of Imitation Models at Developing and Introducing Information-Control Systems
The Use of Imitation Models at Developing and Introducing Information-Control Systems ()
1. Introduction
The formation of the quality of natural waters is a very complex process. Among the numerous factors that determine their condition, industrial waste water should be singled out. In order to protect
natural waters from pollution by industrial effluents, expensive treatment facilities and wastewater control systems are being built. The efficiency of investments depends on the optimality of the
adopted design decisions, in particular, on the calculation of the capacities of treatment facilities, the rational choice of control equipment, as well as on the quality of decisions made during the
operation of treatment facilities, and on the maximum use of their capabilities. To successfully solve these problems, it is of great importance to know the process of formation of wastewater
quality, the ability to predict its development depending the time, taking into account all kinds of critical situations that arise in the process of the operation of the enterprise [1].
2. Imitation Models of the Formation of the Quality of the Environmental Water
Imitation models of wastewater quality formation make it possible to predict the quality of wastewater in dynamics, depending on the given mode of operation of the enterprise, without disturbing the
normal mode of its operation. To develop them, the following initial information is required: a detailed diagram of the relative position of pollution sources connected by the sewerage network of a
given enterprise; water consumption for each source of pollution and the concentration of discharged ingredients in all possible technological modes of operation; working models of the dissemination
of polluting ingredients in the considered section of the sewer network; type and nature of the random component of the pollution process for sources of discharges.
The development of simulation models for the dissemination of polluting ingredients into natural water objects (for example, rivers, lakes, etc.) is also of great importance for solving the problems
of environmental protection and utilization. They allow us to optimize the planning, implementation and operation of various industrial, environmental and economic activities related to the use of
natural water objects. Simulation models for the formation of the quality of natural and/or waste waters, in addition to the abovementioned, make it possible to: indirectly control the operation of
water quality auto-analyzers by comparing the simulation results and measured values of the same parameters; in case of temporary failure of any measurement channel in the auto-analyzer, fill in the
gaps in the measurements of this parameter; calculate the concentrations of controlled ingredients in an uncontrolled point of a water object in accordance with the conditions of wastewater discharge
by pollution sources (this allows to minimize the required number of auto-analyzers necessary to control a water object with a given reliability); compute maximum allowable discharges (MPD) for
polluted objects in order to maintain the concentration of controlled ingredients within the maximum allowable concentration (MAC), predict the concentration of controlled ingredients at a given
point in a water object depending on the conditions of discharge of wastewater from pollution sources; detect emergency pollution sources [2] [3]; testing, coordinating, optimizing the technical,
information-software and mathematical support of the automated water quality control system being developed, which significantly increases the efficiency of such developments and reduces the time for
their implementation at a real object to a minimum.
In order to unify algorithms and programs, simulation models should be developed on a block-modular basis with optimal division of functions among blocks, allowing simulating various pollution
processes by rearranging the execution order and minimally replacing the developed blocks.
It seems appropriate to have the following main blocks in simulation models: generation of technological modes of operation of pollution sources, i.e. the block of control; implementation of
mathematical models for the dissemination of pollutants in water; generation of multidimensional random processes having a given character; generating random numbers according to a given probability
distribution law.
Under the influence of natural hydrological and hydro-biological conditions, runoff formation factors and anthropogenic impacts, the physicochemical parameters characterizing the state of a water
object continuously change over time. The non-stationary random process of these changes can be represented as [4]
${S}_{p}\left(t\right)={m}_{S}^{p}\left(t\right)+{x}_{p}\left(t\right)$, $p=1,\cdots ,m$, (1)
where ${S}_{p}\left(t\right)$ is the value of the p-th controlled parameter of the water object; ${m}_{S}^{p}\left(t\right)$ is deterministic component of the process; ${x}_{p}\left(t\right)$ is
stochastic component; m is the number of controlled parameters.
Therefore, the task of determining mathematical models for the pollutants dissemination in water objects can be divided into two parts: the development of models that describe the deterministic and
stochastic components, respectively.
3. Deterministic Component of Simulation Models
As a deterministic part of mathematical models for the formation of industrial wastewater quality, models that take into account only dilution and self-purification processes can be used [4]:
${y}_{p}^{k}\left(t\right)=\left\{\begin{array}{l}{\sum }_{j=1}^{{q}_{k}}{y}_{p,j}\left(t-{\tau }_{j}\right)+{x}_{p}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{at}\text{\hspace
{0.17em}}p=1,\\ \frac{1}{{\sum }_{j=1}^{{q}_{k}}{y}_{1,j}}\left[{\sum }_{j=1}^{{q}_{k}}{y}_{1,j}\left(t-{\tau }_{j}\right)\cdot {y}_{p,j}\left(t-{\tau }_{j}\right)\right]+{x}_{p}\left(t\right)\text{\
hspace{0.17em}}\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}p=2,\cdots ,m,\end{array}$(2)
where ${y}_{1}^{k}\left(t\right)$ is the volume of water at the k-th node; ${y}_{p}^{k}\left(t\right)$, $p=2,\cdots ,m$ concentration of the p-th ingredient in the k-th node; ${q}_{k}$ -number of
sources of discharges involved in the formation of water quality in the k-th node; ${y}_{p,j}\left(t\right)$ -concentration of the p-th ingredient discharged by the j-th object of discharges; ${\tau
}_{j}$ -time of water running from the j-th discharge object to the k-th node; ${x}_{p}\left(t\right)$ -stochastic component of the concentration of the p-th ingredient.
As a deterministic part of the mathematical models for the formation of the quality of natural water objects, one, two and three-dimensional equations of turbulent diffusion of non-conservative
substances can be used, depending on the specific character of the modeled water object. The equation of 3D-turbulent diffusion of non-conservative substances is the following [4] [5] [6]:
$\begin{array}{c}\frac{\partial \Phi }{\partial t}=\frac{\partial }{\partial x}\left({K}_{x}\frac{\partial \Phi }{\partial x}\right)+\frac{\partial }{\partial y}\left({K}_{y}\frac{\partial \Phi }{\
partial y}\right)+\frac{\partial }{\partial z}\left({K}_{z}\frac{\partial \Phi }{\partial z}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{V}_{x}\frac{\partial \Phi }{\partial x}-{V}_{y}\
frac{\partial \Phi }{\partial y}-{V}_{z}\frac{\partial \Phi }{\partial z}-u\frac{\partial \Phi }{\partial y}-K\left(\Phi \right)+f\left(x,y,z,t\right),\end{array}$(3)
where Φ is the concentration of the non-conservative dissolved substance averaged over time; t is time; x, y, z are the spatial co-ordinates (the axis x is horizontal and its direction coincides with
the direction of averaged current of all stream, the axis y is perpendicular to the free surface and it is directed downwards, the axis z is directed across to the stream); K[x], K[y], K[z] are the
coefficients of the turbulent diffusion in the direction of axes x, y, z; V[x], V[y] and V[z] are components of speeds on axes x, y, z averaged over time; u represents the largest hydraulic
particles; $K\left(\Phi \right)$ is the a parameter characterizing the non-conservativeness of pollutant (one often uses simple approximation of this dependence $K\left(\Phi \right)\equiv K\cdot \
Phi$, where K is the coefficient of non-conservativeness); $f\left(x,y,z,t\right)$ is the total intensity of external sources of pollution. In general, the coefficients K[x], K[y], K[z], V[x], V[y],
V[z] and $K\left(\Phi \right)$ are the functions of a point of space and time.
Initial and boundary conditions of solving the equation of diffusion (3) are set in the form
$\Phi \left(0,r\right)={S}_{0};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }{\Phi \left(t,r\right)|}_{x=0}=\sigma$(4)
( ${S}_{0},\sigma =const$ ). Boundary conditions in the lower end of the section can be classical or non-classical. Classical condition looks like
${\frac{\partial }{\partial x}\Phi \left(t,r\right)|}_{x=\Im }=0$ ; (condition of full mixing) (5)
non-classical condition
${\Phi \left(t,r\right)|}_{x=\Im }=q\cdot {\Phi \left(t,r\right)|}_{x=\Im -l}$. (not local boundary condition), (6)
where q is the coefficient of self-purification of the river on the considered section; ω is the concentration of the pollutant dropped by pollution source in the point $x=\Im$ [7].
At $m\ge 2$, boundary conditions on the other part of the line or on the surface $\partial G$ -Neumann’s conditions, are set also
${\left(u \cdot abla \right)\Phi \left(t,r\right)|}_{r\in \partial G}=0$, (7)
where $u$ is unit vector of external normal to the border $\partial G$. In particular, at $m=3$, it should be
${\frac{\partial }{\partial z}\Phi \left(t,r\right)|}_{z=0}=0$. (8)
The methods of the solution of Equation (3) and their realization at different initial and boundary conditions are given in [5] [6]. These methods where widely used in automated water quality control
systems at solving many different ecological problems [4].
Many other deterministic models of pollutants transport in the environmental water are considered in literature (see, for example [1] [8] - [14]). The choice of the concrete model depends on the
specificity of the water object, the pollution of which is considered, on the condition of the pollution and the peculiarity of the pollution substance under consideration, on the aim for which the
model is used and so on. Application of these models in concrete cases must be made depending on the reasons named above.
4. Stochastic Component of Simulation Models
The Markov model is known to be the best for hydrological data [15]. It is also expected to be suitable for natural water pollution data. Our field studies confirmed this assumption [16]. Therefore,
as a stochastic component of the concentration of pollution substances, m-dimensional Gaussian Markov series $X\left(t\right)=\left({x}_{1}\left(t\right),{x}_{2}\left(t\right),\cdots ,{x}_{m}\left(t\
right)\right)$, with the depth of connection equal to N, given by formula
${x}_{p}\left(t\right)={\sum }_{l=1}^{p-1}{b}_{l}^{p}{x}_{l}\left(t\right)+{\sum }_{i=1}^{m}{\sum }_{j=1}^{N}{a}_{ij}^{p}{x}_{i}\left(t-j\right)+{\sigma }_{p}{\xi }_{p}\left(t\right)$, (9)
is used, where coefficients ${b}_{l}^{p}$, ${a}_{ij}^{p}$ depend on the auto- and inter-covariance functions of m-dimensional random series $x\left(t\right)=\left({x}_{1}\left(t\right),{x}_{2}\left(t
\right),\cdots ,{x}_{m}\left(t\right)\right)$ ; ${\sigma }_{p}^{2}$ is a residual variance of a random series ${x}_{p}\left(t\right)$ ; ${\xi }_{p}\left(t\right)$ is normally distributed standard
random variable. The method of computation of the coefficients of the model (9) which determines the number of observations, necessary for modeling series (9) with given accuracy is given in the work
[4] [17].
The unknown coefficients and the residual variance in Equation (9) are found by means of least-squares technique. With the following designations:
${{R}^{\prime }}_{k,i}\left(|h-j|\right)=\left\{\begin{array}{l}{R}_{k,i}\left(|h-j|\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}h\ge j,\\ {R}_{i,k}\left(|h-j|\
$k,i=1,\cdots ,m$ ; $j,n=1,\cdots ,N$ ;
${A}_{p}^{\text{T}}={\left({b}^{p},{a}^{p}\right)}_{1×\left[m\cdot N+\left(p-1\right)\right]}$ ; ${C}_{p}^{\text{T}}={\left({R}_{k,i}\left(h\right)\right)}_{1×m\cdot N}$(10)
${B}_{p}={\left({{R}^{\prime }}_{k,i}\left(|h-j|\right)\right)}_{m\cdot N×\left[m\cdot N+\left(p-1\right)\right]}$, $p=1,\cdots ,m$,
where ${R}_{k,i}\left(|h-j|\right)$ are the corresponding covariances, the expression for the unknown coefficients assumes the following form:
${A}_{p}={B}_{p}^{+}\cdot {C}_{p},$(11)
where ${B}_{p}^{+}$ is pseudoinverse matrix; expression for the residual variance is
$\begin{array}{c}{\sigma }_{p}^{2}={R}_{p}\left(0\right)-\underset{l=1}{\overset{p-1}{\sum }}\underset{k=1}{\overset{p-1}{\sum }}{b}_{l}^{p}{b}_{k}^{p}{R}_{l,k}\left(0\right)-\underset{i=1}{\overset
{m}{\sum }}\underset{j=1}{\overset{N}{\sum }}\underset{k=1}{\overset{m}{\sum }}\underset{l=1}{\overset{N}{\sum }}{a}_{kl}^{p}{a}_{ij}^{p}{{R}^{\prime }}_{l,k}\left(|l-j|\right)\\ \text{\hspace
{0.17em}}\text{\hspace{0.17em}}-2\underset{l=1}{\overset{p-1}{\sum }}\underset{j=1}{\overset{N}{\sum }}\underset{i=1}{\overset{m}{\sum }}{b}_{l}^{p}{a}_{ij}^{p}{R}_{i,l}\left(j\right),\end{array}$
where ${R}_{p}\left(0\right)$ is variance of the p-th random process.
Let’s introduce the following designations: ${\gamma }_{p}$ required accuracy of Markovian series generation; ${\Delta }_{R}$ -maximum absolute error of calculation of covariant function one value.
$\begin{array}{l}{\Delta }_{R}\le \frac{{\gamma }_{p}}{{N}^{1/2}{\left(\underset{i=1}{\overset{m}{\sum }}{\beta }_{i}^{2}\right)}^{1/2}‖{A}_{p0}‖\left\{{N}^{1/2}{\left[\underset{i=1}{\overset{m}{\sum
}}{\left(con{d}^{+}{B}_{i}‖{A}_{i0}‖{D}_{i}\right)}^{2}\right]}^{1/2}+con{d}^{+}{B}_{p}{D}_{p}\right\}};\\ {D}_{i}=\frac{{\left[{m}^{2}{N}^{2}+mN\left(i-1\right)\right]}^{1/2}‖{C}_{i}‖+{\left(mN\
right)}^{1/2}‖{B}_{i}‖}{‖{B}_{i}‖\cdot ‖{C}_{i}‖},i=1,\cdots ,m,\end{array}$(13)
holds for all $p=1,\cdots ,m$, the multidimensional Gaussian Markovian series is generated to the given accuracy with probability equal to or greater than $\left(1-\alpha \right)$. Here: $‖\cdot ‖$
is Euclidean norm of the corresponding matrix; $con{d}^{+}{B}_{p}=‖{B}_{p}‖\cdot ‖{B}_{p}^{+}‖$ is conditionality number of matrix ${B}_{p}$ ; ${\beta }_{i}$ is the value for which $P\left(|{\
stackrel{^}{x}}_{i}\left(t-j\right)|\le {\beta }_{i}\right)=1-\alpha$ holds.
Sample size n, that ensures computation of covariant function values with absolute error not exceeding ${\Delta }_{R}$, is determined from the following relation [18]:
$\begin{array}{l}n=\underset{\left\{i\right\}}{\mathrm{max}}\left\{i+\frac{1}{{\gamma }_{i}\sqrt{\alpha }}\left[\left({n}^{*}+1\right){R}^{2}\left(0\right)+\left({n}^{*}+1-i\right){R}^{2}\left(i\
right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\cdot \underset{j=1}{\overset{{n}^{*}}{\sum }}\left({n}^{*}+1-j\right){R}^{2}\
left(j\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{2\cdot \underset{j=1}{\overset{{n}^{*}}{\sum }}\left({n}^{*}+1-i-j\
right)R\left(j+i\right)R\left(j-i\right)\right]}^{1/2}\right\},\text{\hspace{0.17em}}i=0,1,\cdots ,\mathrm{max}\left\{n,n{}^{*}\right\}\end{array}$(14)
Input information for the computation is: ${R}_{i,k}\left(j\right)$, $i,k=1,\cdots ,m$ ; $j=1,\cdots ,N$ ; ${\gamma }_{p}$ ; ${\Delta }_{R}$ ; $\alpha$.
The above results are obtained provided that the system of linear equations
is compatible. System (15) compatibility means that
${B}_{p}^{+}{B}_{p}-E=0$, (16)
where Eis a unit matrix.
The described models were used to model the pollution processes of different rivers and wastewaters. For example, models (3) were used to model the pollution processes of the Khobistskali River basin
(western Georgia). The comparison of modeling results with the results of measurements of pollutant concentrations in rivers has clearly shown the high quality of the created models and the great
potential for their use [5] [16].
The described models of wastewater pollution levels were used for modeling the wastewater quality of the Nitrogen Plant in Odessa (Ukraine) in the automated wastewater pollution control and
management systems of the same plant, both in the design and testing of these systems and in the conditions of its operation [1].
5. Conclusion
The described models were widely used in automated systems for controlling and managing the pollution of water objects, both in their creation and in operation, to solve many different problems, such
as: making optimal decisions in the creation, testing and operation of automatic monitoring systems; when calculating the values of the pollution parameters at the uncontrolled points of the control
environment object; to calculate the maximum allowable discharge for pollution sources in a dynamic mode taking into account the condition of the environmental object in the period under
consideration; taking into account the pollution conditions and the possible variability of the condition of the control facility for the forecast of possible variability of the condition of the
environmental object; for automatic detection of emergency pollutants in the conditions of their existence, etc. | {"url":"https://scirp.org/journal/paperinformation?paperid=118691","timestamp":"2024-11-03T01:09:07Z","content_type":"application/xhtml+xml","content_length":"183079","record_id":"<urn:uuid:26a1c34f-b964-460f-8d87-8667627f3b13>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00433.warc.gz"} |
Class 10 Maths Chapter 4 Quadratic Equations MCQ - Sanfoundry
Class 10 Maths MCQ – Quadratic Equations
This set of Class 10 Maths Chapter 4 Multiple Choice Questions & Answers (MCQs) focuses on “Quadratic Equations”. These MCQs are created based on the latest CBSE syllabus and the NCERT curriculum,
offering valuable assistance for exam preparation.
1. The sum of a number and its reciprocal is \(\frac {65}{8}\). What is the number?
a) 8
b) 4
c) 2
d) 6
View Answer
Answer: a
Explanation: Let the number be x
x+\(\frac {1}{x}=\frac {65}{8}\)
\(\frac {x^2+1}{x}=\frac {65}{8}\)
x=8, \(\frac {1}{8}\)
The number is 8 or \(\frac {1}{8}\).
2. Find two numbers such that the sum of the numbers is 12 and the sum of their squares is 74.
a) 84
b) 75
c) 66
d) 48
View Answer
Answer: b
Explanation: Sum of the numbers is 12.
Let one number be x. Other number is 12-x.
Sum of their squares = 74
x=7, 5
The number is 57 or 75.
3. The sum of two numbers is 13 and the sum of their reciprocals is \(\frac {13}{40}\). What are the two numbers?
a) 76
b) 49
c) 58
d) 94
View Answer
Answer: c
Explanation: Sum of the numbers is 13.
Let one number be x. Other number is 13-x.
Sum of their reciprocals = \(\frac {13}{40}\)
\(\frac {1}{x} + \frac {1}{13-x}=\frac {13}{40}\)
\(\frac {13-x+x}{x(13-x)}=\frac {13}{40}\)
\(\frac {13}{13x-x^2}=\frac {13}{40}\)
\(\frac {1}{13x-x^2}=\frac {1}{40}\)
x=8, 5
The number is 58 or 85.
4. The sum of the squares of the left and right pages of a book is 481. What are the page numbers?
a) 11, 12
b) 12, 13
c) 17, 18
d) 15, 16
View Answer
Answer: d
Explanation: Since the pages of books are consecutive numbers, so let the left page number be x. The right page number will be x+1.
Sum of the squares of the pages is 481
x=15, -16
Since, page number cannot be negative, so x=15
The two page numbers are 15 and 16.
5. The sum of the squares of two consecutive positive even numbers is 3364. What are the two numbers?
a) 40, 42
b) 38, 40
c) 42, 44
d) 44, 46
View Answer
Answer: a
Explanation: Let one number be x. The other number is x+2
Sum of the squares of the numbers is 3364.
x=40, -42
Since we only need positive numbers.
Hence, x=40
The two numbers are 40, 42.
6. The sum of the length and the breadth of a rectangle are 97 and the area of the rectangle is 1752. What will be the value of the length and breadth of the rectangle?
a) 42, 76
b) 73, 24
c) 45, 73
d) 22, 77
View Answer
Answer: b
Explanation: Let the length of the rectangle be x. The sum of length and breadth is 97.
Breadth will be 97-x.
Area of rectangle = 1752.
length × breadth = 1752
x=73, 24
Hence, the length is 73 and breadth 24 or length 24, breadth 73.
7. The product of digits of a two digit number is 21 and when 36 is subtracted from the number, the digits interchange their places. What is the number?
a) -24
b) 42
c) 73
d) -37
View Answer
Answer: c
Explanation: Let the units place of the two digit number be x and the tens place be y.
Product of the digits of the places = 21
y=\(\frac {21}{x}\)
Now, the number will be 10y+x
If 36 is subtracted from the number the digits interchange their places.
New number = 10x+y
Now, y=\(\frac {21}{x}\)
\(\frac {21}{x}\)-x=4
x = -7, 3
The number is 73.
8. If the n^th term of the AP is 5n+2. What will be the value of n so that the sum of the first n terms is 295?
a) 9
b) 10
c) 11
d) 4
View Answer
Answer: b
Explanation: The n^th term of the AP is 5n+2.
S[n]=\(\frac {n}{2}\)(2a+(n-1)d)
295=\(\frac {n}{2}\)(2(7)+(n-1)5)
n=10, \(\frac {-59}{5}\)
9. The denominator of a fraction is 1 more than 9 times the numerator. If the sum of the fraction and its reciprocal is \(\frac {101}{10}\) then, what will be the fraction?
a) \(\frac {89}{10}\)
b) \(\frac {10}{89}\)
c) \(\frac {1}{10}\)
d) 10
View Answer
Answer: c
Explanation: Let the numerator be x.
Denominator of a fraction is 1 more than 9 times the numerator.
Denominator = 1+9x
The fraction is \(\frac {x}{11+9x}\)
Fraction + Reciprocal = \(\frac {101}{10}\)
\(\frac {x}{1+9x}+\frac {1+9x}{x}=\frac {101}{10}\)
\(\frac {x^2+(1+9x)^2}{x(1+9x)}=\frac {101}{10}\)
x=1, \(\frac {-10}{89}\)
The fraction is \(\frac {1}{10}\)
10. If the sides of the right angled triangle is x+2, x+1, x then what is the value of x?
a) 1
b) 2
c) 3
d) 4
View Answer
Answer: a
Explanation: Sides of the right angled triangle is x+2, x+1, x
From Pythagoras theorem,
x=1, -3
Since, sides of triangle cannot be negative.
Hence, x=1
More MCQs on Class 10 Maths Chapter 4:
To practice all chapters and topics of class 10 Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers. | {"url":"https://www.sanfoundry.com/mathematics-multiple-choice-questions-answers-class-10/","timestamp":"2024-11-02T19:11:41Z","content_type":"text/html","content_length":"142243","record_id":"<urn:uuid:5e28910a-0a5d-4df1-af09-336c3bc7800b>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00461.warc.gz"} |
Java Math nextDown() - Get Next Down Value | Vultr Docs
The nextDown() method in Java's Math class is particularly useful for finding the next smaller floating-point value toward negative infinity, from the given number. This function helps in situations
where precise decrementing of values is crucial, such as graphics positioning, scientific calculations, or when working with multiple floating-point comparisons where precision is paramount.
In this article, you will learn how to effectively utilize the nextDown() method in Java. You'll discover how this method can be applied to both single and double precision numbers, along with
illustrative examples to demonstrate its practical usage in different scenarios.
Understanding nextDown() Method
Basic Usage of nextDown() for Double Values
1. Begin by initializing a double variable.
2. Apply the nextDown() function to decrement the value.
double initialValue = 1.0;
double nextDownValue = Math.nextDown(initialValue);
System.out.println("Next down value of 1.0 is: " + nextDownValue);
This snippet shows how to obtain the next smaller double-precision value from 1.0. The method calculates the smallest decrement possible in floating-point representation.
Exploring nextDown() with Positive and Negative Values
1. Set both positive and negative double values.
2. Use nextDown() to observe how this function operates in both directions.
double positiveValue = 0.01;
double nextDownPositive = Math.nextDown(positiveValue);
double negativeValue = -0.01;
double nextDownNegative = Math.nextDown(negativeValue);
System.out.println("Next down value of 0.01 is: " + nextDownPositive);
System.out.println("Next down value of -0.01 is: " + nextDownNegative);
This code evaluates nextDown() results for small positive and negative double values, showing that the function effectively moves values toward negative infinity.
Precision Considerations in Scientific Calculations
1. Demonstrate precision impact using nextDown() with a very small positive number.
2. Print results to showcase the high-precision calculation.
double verySmallValue = 1e-10; // 1 * 10^-10
double result = Math.nextDown(verySmallValue);
System.out.println("Next down from very small value: " + result);
The output will reveal the tiny decrement done on a very small number. This precision is significant in fields requiring high accuracy, like physics simulations.
Using nextDown() with Float Values
Basic Application for Float Types
1. Initialize a float variable.
2. Use nextDown() specifically for the float type.
float initialFloat = 0.5f;
float nextDownFloat = Math.nextDown(initialFloat);
System.out.println("Next down value of 0.5f is: " + nextDownFloat);
Unlike double precision, this example works with a single precision float, adjusting the least significant bit of its representation.
Handling Extremes in Float Calculations
1. Evaluate the behavior of nextDown() with extreme float values, such as Float.MIN_VALUE.
float minFloat = Float.MIN_VALUE;
float extremeNextDown = Math.nextDown(minFloat);
System.out.println("Next down value of Float.MIN_VALUE is: " + extremeNextDown);
This demonstrates that nextDown() could even result in values that underflow, leading to effects like negative zeros, which are important in certain computing contexts.
The nextDown() function in Java provides an essential capability to decrement floating-point values precisely, ensuring the minimal possible decrease toward negative infinity. By leveraging this
function, you maintain high accuracy in applications that demand ultraprecision, from graphical adjustments on the pixel level to meticulous scientific calculations. Implement these techniques to
enhance reliability and precision in your Java programs that involve detailed floating-point arithmetic. | {"url":"https://docs.vultr.com/java/standard-library/java/lang/Math/nextDown","timestamp":"2024-11-15T03:23:44Z","content_type":"text/html","content_length":"239991","record_id":"<urn:uuid:5f77fc4a-1cdb-4b41-bfe3-d642a556c887>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00225.warc.gz"} |
Asymptotics and Special Functions
• 1st Edition - May 10, 2014
• Paperback ISBN:
9 7 8 - 1 - 4 8 3 2 - 4 4 2 5 - 9
• eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 6 7 4 4 - 9
Asymptotics and Special Functions provides a comprehensive introduction to two important topics in classical analysis: asymptotics and special functions. The integrals of a real… Read more
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Asymptotics and Special Functions provides a comprehensive introduction to two important topics in classical analysis: asymptotics and special functions. The integrals of a real variable and contour
integrals are discussed, along with the Liouville-Green approximation and connection formulas for solutions of differential equations. Differential equations with regular singularities are also
considered, with emphasis on hypergeometric and Legendre functions. Comprised of 14 chapters, this volume begins with an introduction to the basic concepts and definitions of asymptotic analysis and
special functions, followed by a discussion on asymptotic theories of definite integrals containing a parameter. Contour integrals as well as integrals of a real variable are described. Subsequent
chapters deal with the analytic theory of ordinary differential equations; differential equations with regular and irregular singularities; sums and sequences; and connection formulas for solutions
of differential equations. The book concludes with an evaluation of methods used in estimating (as opposed to bounding) errors in asymptotic approximations and expansions. This monograph is intended
for graduate mathematicians, physicists, and engineers.
Preface1 Introduction to Asymptotic Analysis 1 Origin of Asymptotic Expansions 2 The Symbols ~, o, and Ο 3 The Symbols ~, o, and Ο (Continued) 4 Integration and Differentiation of Asymptotic and
Order Relations 5 Asymptotic Solution of Transcendental Equations: Real Variables 6 Asymptotic Solution of Transcendental Equations: Complex Variables 7 Definition and Fundamental Properties of
Asymptotic Expansions 8 Operations with Asymptotic Expansions 9 Functions Having Prescribed Asymptotic Expansions 10 Generalizations of Poincaré's Definition 11 Error Analysis; Variational Operator
Historical Notes and Additional References2 Introduction to Special Functions 1 The Gamma Function 2 The Psi Function 3 Exponential, Logarithmic, Sine, and Cosine Integrals 4 Error Functions,
Dawson's Integral, and Fresnel Integrals 5 Incomplete Gamma Functions 6 Orthogonal Polynomials 7 The Classical Orthogonal Polynomials 8 The Airy Integral 9 The Bessel Function Jv(z) 10 The Modified
Bessel Function Iv(z) 11 The Zeta Function Historical Notes and Additional References3 Integrals of a Real Variable 1 Integration by Parts 2 Laplace Integrals 3 Watson's Lemma 4 The Riemann-Lebesgue
Lemma 5 Fourier Integrals 6 Examples; Cases of Failure 7 Laplace's Method 8 Asymptotic Expansions by Laplace's Method; Gamma Function of Large Argument 9 Error Bounds for Watson's Lemma and Laplace's
Method 10 Examples 11 The Method of Stationary Phase 12 Preliminary Lemmas 13 Asymptotic Nature of the Stationary Phase Approximation 14 Asymptotic Expansions by the Method of Stationary Phase
Historical Notes and Additional References4 Contour Integrals 1 Laplace Integrals with a Complex Parameter 2 Incomplete Gamma Functions of Complex Argument 3 Watson's Lemma 4 Airy Integral of Complex
Argument; Compound Asymptotic Expansions 5 Ratio of Two Gamma Functions; Watson's Lemma for Loop Integrals 6 Laplace's Method for Contour Integrals 7 Saddle Points 8 Examples 9 Bessel Functions of
Large Argument and Order 10 Error Bounds for Laplace's Method; The Method of Steepest Descents Historical Notes and Additional References5 Differential Equations with Regular Singularities;
Hypergeometric and Legendre Functions 1 Existence Theorems for Linear Differential Equations: Real Variables 2 Equations Containing a Real or Complex Parameter 3 Existence Theorems for Linear
Differential Equations: Complex Variables 4 Classification of Singularities; Nature of the Solutions in the Neighborhood of a Regular Singularity 5 Second Solution When the Exponents Differ by an
Integer or Zero 6 Large Values of the Independent Variable 7 Numerically Satisfactory Solutions 8 The Hypergeometric Equation 9 The Hypergeometric Function 10 Other Solutions of the Hypergeometric
Equation 11 Generalized Hypergeometric Functions 12 The Associated Legendre Equation 13 Legendre Functions of General Degree and Order 14 Legendre Functions of Integer Degree and Order 15 Ferrers
Functions Historical Notes and Additional References6 The Liouville-Green Approximation 1 The Liouville Transformation 2 Error Bounds: Real Variables 3 Asymptotic Properties with Respect to the
Independent Variable 4 Convergence of V (F) at a Singularity 5 Asymptotic Properties with Respect to Parameters 6 Example: Parabolic Cylinder Functions of Large Order 7 A Special Extension 8 Zeros 9
Eigenvalue Problems 10 Theorems on Singular Integral Equations 11 Error Bounds: Complex Variables 12 Asymptotic Properties for Complex Variables 13 Choice of Progressive Paths Historical Notes and
Additional References7 Differential Equations with Irregular Singularities; Bessel and Confluent Hypergeometric Functions 1 Formal Series Solutions 2 Asymptotic Nature of the Formal Series 3
Equations Containing a Parameter 4 Hankel Functions; Stokes' Phenomenon 5 The Function Yv(z) 6 Zeros of Jv(z) 7 Zeros of Yv(z) and Other Cylinder Functions 8 Modified Bessel Functions 9 Confluent
Hypergeometric Equation 10 Asymptotic Solutions of the Confluent Hypergeometric Equation 11 Whittaker Functions 12 Error Bounds for the Asymptotic Solutions in the General Case 13 Error Bounds for
Hankel's Expansions 14 Inhomogeneous Equations 15 Struve's Equation Historical Notes and Additional References8 Sums and Sequences 1 The Euler-Maclaurin Formula and Bernoulli's Polynomials 2
Applications 3 Contour Integral for the Remainder Term 4 Stirling's Series for In Γ(z) 5 Summation by Parts 6 Barnes' Integral for the Hypergeometric Function 7 Further Examples 8 Asymptotic
Expansions of Entire Functions 9 Coefficients in a Power-Series Expansion; Method of Darboux 10 Examples 11 Inverse Laplace Transforms; Haar's Method Historical Notes and Additional References9
Integrals: Further Methods 1 Logarithmic Singularities 2 Generalizations of Laplace's Method 3 Example from Combinatoric Theory 4 Generalizations of Laplace's Method (Continued) 5 Examples 6 More
General Kernels 7 Nicholson's Integral for J2v(z) + Y2v(z) 8 Oscillatory Kernels 9 Bleistein's Method 10 Example 11 The Method of Chester, Friedman, and Ursell 12 Anger Functions of Large Order 13
Extension of the Region of Validity Historical Notes and Additional References10 Differential Equations with a Parameter: Expansions in Elementary Functions 1 Classification and Preliminary
Transformations 2 Case I: Formal Series Solutions 3 Error Bounds for the Formal Solutions 4 Behavior of the Coefficients at a Singularity 5 Behavior of the Coefficients at a Singularity (Continued) 6
Asymptotic Properties with Respect to the Parameter 7 Modified Bessel Functions of Large Order 8 Extensions of the Regions of Validity for the Expansions of the Modified Bessel Functions 9 More
General Forms of Differential Equation 10 Inhomogeneous Equations 11 Example: An Inhomogeneous Form of the Modified Bessel Equation Historical Notes and Additional References11 Differential Equations
with a Parameter: Turning Points 1 Airy Functions of Real Argument 2 Auxiliary Functions for Real Variables 3 The First Approximation 4 Asymptotic Properties of the Approximation; Whittaker Functions
with m Large 5 Real Zeros of the Airy Functions 6 Zeros of the First Approximation 7 Higher Approximations 8 Airy Functions of Complex Argument 9 Asymptotic Approximations for Complex Variables 10
Bessel Functions of Large Order 11 More General Form of Differential Equation 12 Inhomogeneous Equations Historical Notes and Additional References12 Differential Equations with a Parameter: Simple
Poles and Other Transition Points 1 Bessel Functions and Modified Bessel Functions of Real Order and Argument 2 Case III: Formal Series Solutions 3 Error Bounds: Positive ζ 4 Error Bounds: Negative ζ
5 Asymptotic Properties of the Expansions 6 Determination of Phase Shift 7 Zeros 8 Auxiliary Functions for Complex Arguments 9 Error Bounds: Complex u and ζ 10 Asymptotic Properties for Complex
Variables 11 Behavior of the Coefficients at Infinity 12 Legendre Functions of Large Degree: Real Arguments 13 Legendre Functions of Large Degree: Complex Arguments 14 Other Types of Transition
Points Historical Notes and Additional References13 Connection Formulas for Solutions of Differential Equations 1 Introduction 2 Connection Formulas at a Singularity 3 Differential Equations with a
Parameter 4 Connection Formula for Case III 5 Application to Simple Poles 6 Example: The Associated Legendre Equation 7 The Gans-Jeffreys Formulas: Real-Variable Method 8 Two Turning Points 9 Bound
States 10 Wave Penetration Through a Barrier. I 11 Fundamental Connection Formula for a Simple Turning Point in the Complex Plane 12 Example: Airy's Equation 13 Choice of Progressive Paths 14 The
Gans-Jeffreys Formulas: Complex-Variable Method 15 Wave Penetration through a Barrier. II Historical Notes and Additional References14 Estimation of Remainder Terms 1 Numerical Use of Asymptotic
Approximations 2 Converging Factors 3 Exponential Integral 4 Exponential Integral (Continued) 5 Confluent Hypergeometric Function 6 Euler's Transformation 7 Application to Asymptotic Expansions
Historical Notes and Additional ReferencesAnswers to ExercisesReferencesIndex of SymbolsGeneral Index
• Paperback ISBN: 9781483244259
• eBook ISBN: 9781483267449 | {"url":"https://shop.elsevier.com/books/asymptotics-and-special-functions/olver/978-0-12-525850-0","timestamp":"2024-11-02T11:15:17Z","content_type":"text/html","content_length":"189664","record_id":"<urn:uuid:642a1d6c-0adb-4ab9-9ac1-7168530f5179>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00608.warc.gz"} |
How do you calculate twisting moment?How do you calculate twisting moment?
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how to calculate twisting moment?
1. Hi,
Torque, T = F x r x sin (theta), where r=distance between the rotation axis and the point of force applied, F =force applied and theta= angle between F and r.
Twisting moment,
T = (shear stress X polar moment of inertia) / r.
2. This answer was edited.
Calculation of twisting moment:
When we try to rotate a steel bar then that moment is nothing but the bending moment.
The twisting moment is a special case of a bending moment.
The twisting moment is also called a torsional moment or torque
When we twist the end of the bar either clockwise or counterclockwise then bending moment will form.
T= (G x angle ) x J/ L
T – Torque
J- polar moment of inertia
L – length
G – Modulus of rigidity
Thank You.
3. Torsion is the twisting of a beam under the action of a torque (twisting moment). It is systematically applied to screws, nuts, axles, drive shafts etc, and is also generated more randomly under
service conditions in car bodies, boat hulls, aircraft fuselages, bridges, springs and many other structures and components. A torque, T , has the same units (N m) as a bending moment, M . Both
are the product of a force and a distance. In the case of a torque, the force is tangential and the distance is the radial distance between this tangent and the axis of rotation.
All torsion problems can be solved using the following formula:
T/J = shear stress/ r = (G * angle)/ L
T = torque or twisting moment, [N×m, lb×in]
J = polar moment of inertia or polar second moment of area about shaft axis, [m4, in4]
τ = shear stress at outer fibre, [Pa, psi]
r = radius of the shaft, [m, in]
G = modulus of rigidity (PanGlobal and Reed’s) or shear modulus (everybody else), [Pa, psi]
θ = angle of twist, [rad]
L = length of the shaft, [m, in]
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Definition of a sphere. More...
#include <bodies.h>
Public Member Functions
virtual void computeBoundingSphere (BoundingSphere &sphere) const
Compute the bounding radius for the body, in its current pose. Scaling and padding are accounted for.
double computeVolume (void) const
Compute the volume of the body. This method includes changes induced by scaling and padding.
virtual bool containsPoint (const tf::Vector3 &p, bool verbose=false) const
Check is a point is inside the body.
virtual bool intersectsRay (const tf::Vector3 &origin, const tf::Vector3 &dir, std::vector< tf::Vector3 > *intersections=NULL, unsigned int count=0) const
Check is a ray intersects the body, and find the set of intersections, in order, along the ray. A maximum number of intersections can be specified as well. If that number is 0, all
intersections are returned.
Sphere (void)
Sphere (const shapes::Shape *shape)
virtual ~Sphere (void)
Protected Member Functions
virtual void updateInternalData (void)
virtual void useDimensions (const shapes::Shape *shape)
Protected Attributes
tf::Vector3 m_center
double m_radius
double m_radius2
double m_radiusU
Detailed Description
Definition of a sphere.
Definition at line 174 of file bodies.h.
Constructor & Destructor Documentation
bodies::Sphere::Sphere ( void ) [inline]
bodies::Sphere::Sphere ( const shapes::Shape * shape ) [inline]
virtual bodies::Sphere::~Sphere ( void ) [inline, virtual]
Member Function Documentation
void bodies::Sphere::computeBoundingSphere ( BoundingSphere & sphere ) const [virtual]
Compute the bounding radius for the body, in its current pose. Scaling and padding are accounted for.
Implements bodies::Body.
Definition at line 160 of file bodies.cpp.
double bodies::Sphere::computeVolume ( void ) const [virtual]
Compute the volume of the body. This method includes changes induced by scaling and padding.
Implements bodies::Body.
Definition at line 155 of file bodies.cpp.
bool bodies::Sphere::containsPoint ( const tf::Vector3 & p,
bool verbose = false
) const [virtual]
bool bodies::Sphere::intersectsRay ( const tf::Vector3 & origin,
const tf::Vector3 & dir,
std::vector< tf::Vector3 > * intersections = NULL,
unsigned int count = 0
) const [virtual]
Check is a ray intersects the body, and find the set of intersections, in order, along the ray. A maximum number of intersections can be specified as well. If that number is 0, all intersections are
Implements bodies::Body.
Definition at line 166 of file bodies.cpp.
void bodies::Sphere::updateInternalData ( void ) [protected, virtual]
void bodies::Sphere::useDimensions ( const shapes::Shape * shape ) [protected, virtual]
Member Data Documentation
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Work-Energy Theorem | Curious Toons
Table of Contents
Welcome, future physicists! Imagine a world where the mysteries of the universe unfold before your eyes. Physics isn’t just about numbers and equations; it’s the key to understanding everything from
the tiniest particles that make up our very being to the vast cosmos that stretches beyond our imagination. Have you ever wondered how your smartphone works, why the sky is blue, or what makes a
roller coaster thrilling? In this class, we’ll embark on an exciting journey to uncover the laws that govern motion, energy, and the fundamental forces of nature.
Get ready to experiment, solve intriguing problems, and engage in discussions that spark your curiosity. We’ll explore concepts like gravity, magnetism, and the intriguing dual nature of light. By
the end of this course, you won’t just know what physics is—you’ll feel its pulse in every aspect of your life. So, grab your lab goggles, unleash your inquisitive minds, and let’s dive headfirst
into the captivating world of physics, where every question leads to discovery and every answer opens new doors. Are you ready to unlock the secrets of the universe with me?
1. Introduction to Work and Energy
1.1 Definition of Work
In physics, “work” is defined as the transfer of energy that occurs when a force acts on an object to cause displacement. More specifically, work is done when a force applied to an object results in
that object moving in the direction of the force. Mathematically, work (W) can be expressed by the formula:
[ W = F \cdot d \cdot \cos(\theta) ]
where ( F ) is the magnitude of the force applied, ( d ) is the displacement of the object, and ( \theta ) is the angle between the force and the direction of displacement. Work is a scalar quantity
and is measured in joules (J) in the International System of Units (SI). It’s important to note that if the displacement is zero, or if the force is perpendicular to the displacement, no work is
Here’s a quick reference table summarizing the key points:
Term Definition
Work (W) Transfer of energy through displacement
Formula ( W = F \cdot d \cdot \cos(\theta) )
Unit Joules (J)
Conditions Work is done if displacement occurs in the direction of the force
Understanding work is foundational for exploring the broader concepts of energy and mechanical systems.
1.2 Understanding Energy
Understanding energy is fundamental in the study of physics, as it plays a crucial role in various natural phenomena and processes. Energy is defined as the ability to do work, and it exists in
various forms such as kinetic energy (the energy of motion) and potential energy (stored energy based on position). The Work-Energy Theorem asserts that the work done on an object results in a change
in its kinetic energy. This relationship demonstrates that when a force causes an object to move, energy is transferred. For example, when you push a stationary object and it begins to slide, you’re
doing work that transforms your input energy into the kinetic energy of the object.
Here’s a brief overview of the types of mechanical energy:
Type of Energy Definition Example
Kinetic Energy Energy of motion, proportional to the square of velocity A moving car
Potential Energy Energy stored due to an object’s position or configuration A rock at the edge of a cliff
By understanding these concepts, we can analyze various physical systems, predict their behavior, and harness energy more effectively in practical applications.
2. The Work-Energy Principle
2.1 Statement of the Theorem
The Work-Energy Theorem is a fundamental principle in physics that asserts the relationship between the work done on an object and its change in kinetic energy. Specifically, it states that the total
work ( W ) performed on an object is equal to the change in its kinetic energy ( \Delta KE ). Mathematically, this is expressed as:
W = \Delta KE = KEf – KEi
where ( KEf ) is the final kinetic energy and ( KEi ) is the initial kinetic energy. Kinetic energy itself is defined as:
KE = \frac{1}{2} mv^2
where ( m ) is the mass of the object and ( v ) is its velocity. This theorem encapsulates how energy is transferred to or from an object through a force over a distance. When work is done on an
object (like pushing a box), its kinetic energy increases, resulting in acceleration. Conversely, if work is done against the motion (such as friction), the kinetic energy decreases. Understanding
the Work-Energy Theorem provides a powerful framework for analyzing motion and energy transfer in various physical scenarios.
2.2 Implications of the Theorem
The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy. This principle has profound implications across various fields of physics. First, it
provides a powerful tool for analyzing motion in a simplified manner; instead of calculating instantaneous forces and accelerations, we focus on the net work done over a distance. Additionally, the
theorem underscores the conservation of energy, suggesting that energy can be transformed from one form to another but cannot be created or destroyed. For instance, in a frictionless environment, if
a ball is thrown, the work done by the throw imparts kinetic energy to the ball, while in a real-world scenario, some energy is lost to heat due to friction. This understanding can be applied to
engineer systems more efficiently, from designing safer vehicles to optimizing roller coasters for maximum thrill without compromising safety. Overall, the Work-Energy Theorem bridges the gap between
mechanics and energy conservation, allowing for insightful predictions and analyses in both simple and complex systems.
Concept Implications
Work Done Equals change in kinetic energy
Energy Conservation Energy is transformed, not lost
Applications Engineering, safety, and design
3. Calculating Work Done
3.1 Work Done by Constant Forces
When discussing the work done by constant forces, it’s essential to grasp the relationship between force, displacement, and the angle between them. The work done (W) by a constant force can be
calculated using the formula:
[ W = F \cdot d \cdot \cos(\theta) ]
where ( F ) is the magnitude of the force, ( d ) is the displacement of the object, and ( \theta ) is the angle between the force vector and the displacement vector. If the force is applied in the
same direction as the displacement ((\theta = 0°)), the work done is maximized and equals ( W = F \cdot d ). Conversely, if the force acts in the opposite direction ((\theta = 180°)), the work done
is negative, indicating that energy is taken out of the system. In cases where the angle is (\theta = 90°), the work done is zero since ( \cos(90°) = 0), signifying that no energy is transferred in
the direction of displacement. Thus, understanding the direction of forces relative to displacement is crucial for calculating work accurately and comprehensively.
3.2 Work Done by Variable Forces
In the context of the Work-Energy Theorem, the work done by variable forces is calculated through the integration of the force over the distance moved. Unlike constant forces, where the work can be
easily computed as ( W = F \cdot d ), variable forces require us to consider how the force changes with position. For example, if a force ( F(x) ) varies with position ( x ), the work done as an
object moves from position ( x1 ) to ( x2 ) can be expressed mathematically as:
W = \int{x1}^{x_2} F(x) \, dx
This integral sums the infinitesimal work ( dW = F(x) \, dx ) over the distance traveled. Practical applications often involve forces such as springs (Hooke’s Law, ( F = -kx )) or gravity, where the
force is dependent on position. A specific example: if a spring is compressed or stretched, the work done can be found by integrating from its equilibrium position to the final position. By
understanding the relationship between force and displacement, students can better appreciate how energy is transferred and transformed in physical systems.
4. Applications of the Work-Energy Theorem
4.1 Example Problems
In the fourth chapter on Applications of the Work-Energy Theorem, we explore how to apply the theorem to solve real-world problems involving kinetic energy, potential energy, and work done by forces.
The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy. For example, consider a block being pushed across a frictional surface. The work done by
the applied force minus the work done against friction equals the change in kinetic energy of the block.
To solve problems, we often start by identifying all the forces acting on the object, calculating the work done by each force, and determining initial and final kinetic and potential energies.
Frequently, we can set up equations relating these quantities and solve for unknown variables, such as final speed or distance traveled.
For instance:
1. Block Sliding Down a Ramp:
• Given: Mass of the block, height of the ramp, and friction coefficient.
• Find: Final speed at the bottom.
1. Car Accelerating:
• Given: Mass of the car, initial speed, distance traveled, and force applied.
• Find: Final speed after applying the force.
These examples illustrate the versatility of the Work-Energy Theorem in analyzing motion and energy transformations.
4.2 Real-World Applications
The Work-Energy Theorem is pivotal in various real-world applications across multiple fields. In engineering, it plays a crucial role in the design of vehicles where understanding the relationship
between work done and energy transfer is essential for enhancing fuel efficiency and safety. For instance, when a car accelerates, the work done by the engine translates into kinetic energy, which
can be analyzed to optimize performance. Another application is in sports; athletes’ performance can be improved by analyzing the work done during their movements, allowing for better training
techniques and injury prevention. In construction, the theorem aids in calculating the energy required to lift heavy materials, ensuring safety and efficiency. Additionally, in amusement parks, the
engineering of rides incorporates the concepts of potential and kinetic energy to enhance rider experience while maintaining safety. Understanding these applications allows us to appreciate how
deeply the principles of work and energy are woven into the fabric of daily life, from transportation and sports to construction and entertainment.
Application Area Example
Automotive Engineering Enhancing vehicle performance and fuel efficiency
Sports Science Optimizing athlete training techniques
Construction Calculating energy requirements for lifting materials
Amusement Parks Designing safe and thrilling rides
5. Relation to Other Physics Concepts
5.1 Kinetic Energy
Kinetic energy (KE) is the energy possessed by an object due to its motion. It is a fundamental concept in physics that quantifies the energy that an object has as a result of its velocity.
Mathematically, kinetic energy is defined by the equation:
[ KE = \frac{1}{2}mv^2 ]
where ( m ) is the mass of the object and ( v ) is its velocity. This relationship indicates that the kinetic energy of an object increases with the square of its speed, meaning that even small
increases in velocity can lead to significant increases in kinetic energy. For example, if the speed of an object doubles, its kinetic energy increases by a factor of four.
Kinetic energy is directly related to the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. This theorem emphasizes the relationship
between force, work, and motion, allowing us to analyze how various forces influence the motion of objects. Understanding kinetic energy is crucial for applications in various fields, including
sports, engineering, and transportation, as it helps us to predict how objects will behave when forces are applied.
5.2 Potential Energy
Potential energy is the energy stored in an object due to its position or configuration. It represents the work done against forces to elevate or deform an object. The most common type is
gravitational potential energy (PE), which is given by the formula ( PE = mgh ), where ( m ) is the mass of the object, ( g ) is the acceleration due to gravity (approximately 9.81 m/s² near Earth’s
surface), and ( h ) is the height above a reference level. As an object is lifted against gravity, it gains potential energy, which can be converted to kinetic energy if it falls.
Another form is elastic potential energy, seen in springs, calculated using ( PE_{elastic} = \frac{1}{2} k x^2 ), where ( k ) is the spring constant and ( x ) is the displacement from the equilibrium
position. Understanding potential energy is crucial, as it connects directly with the Work-Energy Theorem, illustrating how energy transforms from one form to another. This principle helps us analyze
systems ranging from simple mechanical setups to complex natural phenomena.
Type of Potential Energy Formula
Gravitational ( PE = mgh )
Elastic ( PE_{elastic} = \frac{1}{2} k x^2 )
As we draw the curtain on our physics journey this semester, I want you to pause and reflect on the wonder we’ve explored together. From deciphering the elegant laws of motion to unraveling the
intricate dance of waves and light, physics is not just a collection of equations; it’s the language of the universe.
Each concept we tackled was a tool, granting you the ability to understand the world around you, to question, and to innovate. Think about how Newton’s laws govern everything, from a thrown
basketball to the very orbits of planets. Remember the excitement of discovering that energy can neither be created nor destroyed, only transformed—just like your own potential!
As you move forward, carry this curiosity with you. Physics doesn’t end here; it’s an invitation to delve deeper into the mysteries of life. Whether you pursue science, engineering, or any field, let
this knowledge guide you. Keep asking questions, nurturing that spark of intrigue, and remember: the pursuit of understanding is as limitless as the cosmos we study.
Thank you for your enthusiasm and engagement this semester. I can’t wait to see how you’ll shape the world! | {"url":"https://curioustoons.in/work-energy-theorem/","timestamp":"2024-11-09T21:01:23Z","content_type":"text/html","content_length":"110826","record_id":"<urn:uuid:bfd0fbf2-4161-4d55-8de3-40f85a2ee989>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00345.warc.gz"} |
SNAP Probability And Combinatorics Questions With Solutions
Question 1
ABC Paints Ltd. is planning to create different combination of dyes. The research team has decided they will be using five different green dyes, three different red dyes and four different blue dyes.
How many combinations of dyes can be created by ABC Paints Ltd., by including at least one blue and one green dye?
correct answer:-2
Question 2
Sonal and Meenal appear in an interview for same post having two vacancies. If $$\frac{1}{7}$$ is Sonal's probability of selection and $$\frac{1}{5}$$ is Meenal's probability of selection then what
is the probability that only one of them is selected?
correct answer:-2
Question 3
Big Bang Theory cast wishes to find out the number of ways in which the word ASTRONAUT can be scrambled. They find that the number of ways in which it can be put in an unscrambling puzzle is?
correct answer:-4
Question 4
How many different words can be formed with the word CUSTOM with a condition that the word should begin with M?
Assume that all words have 6 distinct letters.
correct answer:-3
Question 5
The number of ways that 5 Marathi, 3 English and 3 Tamil books be arranged if the books of each language are to be kept together is
correct answer:-1
Question 1
How many words each of two vowels and three consonants can be formed from the letters of the word "UNIVERSAL" ?
correct answer:-2
Question 2
Sonali can solve 70% of the problems in a competitive exam and Nirali can solve only 60% in the same exam. What is the probability that at least one of them will solve a problem, provided selection
of questions is done randomly from the same exam ?
correct answer:-2
Question 3
A person has a bag which contains 9 bulbs out of which 2 are fused and cannot be used to lighten the room. Two bulbs are selected at random. What is the probability that all the two bulbs chosen can
be used to lighten the room ?
correct answer:-2
Question 4
There are nine humans in a ship, each human has nine cages and each cage has nine huge lions and each lion has nine cubs. How many legs are there in the ship ? (Human have two legs, lions have four
legs, cubs have four legs.)
correct answer:-3
Question 1
How many different letter arrangements can be made from the letter of the word EXTRA in such a way that the vowels are always together?
correct answer:-1
Question 2
In a given race the odds in favour of three horses A, B, C are 1:3; 1:4; 1:5 respectively. Assuming that dead heat is impossible the probability that one of them wins is
correct answer:-2
Question 3
A bag contains 5 white and 3 black balls, and 4 are successively drawn out and not replaced. What’s the chance of getting different colours alternatively?
correct answer:-4
Question 4
A bag contains 100 tickets numbered 1, 2, 3, .... 100. If a ticket is drawn out of it at random, what is the probability that the ticket drawn has the digit 2 appearing on it?
correct answer:-2
Question 1
There are five boys and three girls who are sitting together to discuss a management problem at a round table. In how many ways can they sit around the table so that no two girls are together?
correct answer:-4
Question 2
The number of ways in which a committee of 3 ladies and 4 gentlemen can be appointed from a meeting consisting of 8 ladies and 7 gentlemen, if Mrs. X refuses to serve in a committee if Mr. Y is its
member, is
correct answer:-3
Question 3
A family consists of a grandfather, 5 sons and daughters and 8 grandchildren. They are to be seated in a row for dinner. The grandchildren wish to occupy the 4 seats at each end and the grandfather
refuses to have a grandchild on either side of him. The number of ways in which the family can be made to sit is
correct answer:-4
Question 4
At a college football game, $$\frac{4}{5}$$ of the seats in the lower deck of the stadium were sold. If $$\frac{1}{4}$$ of all the seating in the stadium is located in the lower deck, and if $$\frac
{2}{3}$$ of all the seats in the stadium were sold, then what fraction of the unsold seats in the stadium was in the lower deck?
correct answer:-1
Question 1
The probability that a leap year selected at random contains either 53 Sundays or 53 Mondays, is:
correct answer:-3
Question 2
A bag contains 5 white and 3 black balls; another bag contains 4 white and 5 black balls. From any one of these bags a single draw of two balls is made. Find the probability that one of them would be
white and another black ball.
correct answer:-1
Question 3
If $$^nC_x = 56$$ and $$^nP_x = 336$$, then Find n and x?
correct answer:-3 | {"url":"https://cracku.in/snap-probability-combinatorics-questions","timestamp":"2024-11-13T16:24:09Z","content_type":"text/html","content_length":"158456","record_id":"<urn:uuid:bcca0656-f25b-4251-a829-bef9d9dec1a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00245.warc.gz"} |
03 Best Fit Line
Which line is the best?
Earlier, we talked about drawing a line that fits a scatter plot to a linear association between the two variables. Let’s use the example of dog height and weight from before.
But, what kind of line should we draw? There are infinite ways we can draw a line that passes through these points.
Of course we cannot draw a line that passes through all points as it would not be a line at all, it would look something like this.
Not exactly a line right? So, to solve this, the easiest way to find this line is to take a ruler and try to fit the line that passes through as many of the points as possible. This line is the line
that best fits the plot and it is appropriately named the line of best fit.
Do keep in mind what we learnt about some scatter plots with the best fit line being parabolas or some one other function like exponential. While they are not lines, we can still use the term best
fit line. While making these lines, we need to be careful as to which one fits best, which is not necessarily always a straight line like the one given above.
In math, the decision on which line is best is not really done through making a guess. Rather, actual calculations are required where we find the distance between the line we made and all the points
to find the smallest overall distance. But this will be taught later. For now, we will stick to making our best guess on how to make the line that covers the most points. | {"url":"https://edukimath.com/grade-8/statistics-and-probability/best-fit-line/","timestamp":"2024-11-03T05:32:26Z","content_type":"text/html","content_length":"38095","record_id":"<urn:uuid:d83c10e2-cc52-4a62-a1f6-7736fcfa7155>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00463.warc.gz"} |
E-NTU Heat Transfer
Detailed heat transfer model between two general fluids
Simscape / Fluids / Heat Exchangers / Fundamental Components
The E-NTU Heat Transfer block models the heat exchange between two general fluids based on the standard Effectiveness-NTU method. The fluid thermal properties are specified explicitly through
Simscape™ physical signals. Combine with the Heat Exchanger Interface (TL) or Heat Exchanger Interface (G) blocks to model the pressure drop and temperature change between the inlet and outlet of a
heat exchanger.
The block dialog box provides a choice of common heat exchanger configurations. These include concentric-pipe with parallel and counter flows, shell-and-tube with one or more shell passes, and
cross-flow with mixed and unmixed flows. A generic configuration lets you model other heat exchangers based on tabular effectiveness data.
Heat Exchanger Configurations
Heat Transfer Rate
The E-NTU model defines the heat transfer rate between fluids 1 and 2 in terms of an effectiveness parameter ε
$\begin{array}{cc}-{Q}_{1}={Q}_{2}=ϵ{Q}_{Max},& 0<\epsilon <1\end{array},$
• Q[1] and Q[2] are the heat transfer rates into fluid 1 and fluid 2.
• Q[Max] is the maximum possible heat transfer rate between fluid 1 and fluid 2 at a given set of operating conditions.
• ε is the effectiveness parameter.
The maximum possible heat transfer rate between the two fluids is
• C[Min] is the minimum value of the thermal capacity rate:
• T[1,In] and T[2,In] are the inlet temperatures of fluid 1 and fluid 2.
• ${\stackrel{˙}{m}}_{1}$ and ${\stackrel{˙}{m}}_{2}$ are the mass flow rates of fluid 1 and fluid 2 into the heat exchanger volume through the inlet.
• c[p,1] and c[p,2] are the specific heat coefficients at constant pressure of fluid 1 and fluid 2. The Minimum fluid-wall heat transfer coefficient parameter in the block dialog box sets a lower
bound on the allowed values of the heat transfer coefficients.
Heat Exchanger Effectiveness
The heat exchanger effectiveness calculations depend on the flow arrangement type selected in the block dialog box. For all but Generic — effectiveness table, the block computes the thermal exchange
effectiveness through analytical expressions written in terms of the number of transfer units (NTU) and thermal capacity ratio. The number of transfer units is defined as
• NTU is the number of transfer units.
• U[Overall] is the overall heat transfer coefficient between fluid 1 and fluid 2.
• R[Overall] is the overall thermal resistance between fluid 1 and fluid 2.
• A[Heat] is aggregate area of the primary and secondary, or finned, heat transfer surfaces.
The thermal capacity ratio is defined as
• C[rel] is the thermal capacity ratio.
The overall heat transfer coefficient and thermal resistance used in the NTU calculation are functions of the heat transfer mechanisms at work. These mechanisms include convective heat transfer
between the fluids and the heat exchanger interface and conduction through the interface wall [2]:
• h[1] and h[2] are the heat transfer coefficients between fluid 1 and the interface wall and between fluid 2 and the interface wall.
• A[Heat,1] and A[Heat,2] are the heat transfer surface areas on the fluid-1 and fluid-2 sides.
• R[Foul,1] and R[Foul,2] are the fouling resistances on the fluid-1 and fluid-2 sides. The fouling resistance is equal to the fouling factor parameter divided by the heat transfer surface area.
• R[Wall] is the interface wall thermal resistance.
Heat Transfer From Fluid 1 to Fluid 2
The tables show some of the analytical expressions used to compute the heat exchange effectiveness [1]. The parameter N refers to the number of shell passes and the parameter ε[1] to the
effectiveness for a single shell pass.
Concentric Tubes
$\epsilon =\left\{\begin{array}{cc}\frac{1-\mathrm{exp}\left[-NTU\left(1-{C}_{rel}\right)\right]}{1-{C}_{rel}\mathrm{exp}\left[-NTU\left(1-{C}_{rel}\right)\right]},&
Counter Flow \text{if}{C}_{rel}<1\\ \frac{NTU}{1+NTU},& \text{if}{C}_{rel}=1\end{array}$
Parallel Flow $\epsilon =\frac{1-\mathrm{exp}\left[-NTU\left(1+{C}_{rel}\right)\right]}{1+{C}_{rel}}$
Shell and Tube
One shell pass and two, four, or ${\epsilon }_{1}=\frac{2}{1+{C}_{rel}+\sqrt{1+{C}_{rel}{}^{2}}\frac{1+\mathrm{exp}\left(-NTU\sqrt{1+{C}_{rel}{}^{2}}\right)}{1-\mathrm{exp}\left(-NTU\sqrt{1+{C}_
six tube passes {rel}{}^{2}}\right)}}$
N Shell Passes and 2N, 4N, or 6N $\epsilon =\frac{{\left[\left(1-{\epsilon }_{1}{C}_{rel}\right)/\left(1-{\epsilon }_{1}\right)\right]}^{N}-1}{{\left[\left(1-{\epsilon }_{1}{C}_{rel}\right)/\left(1-
Tube Passes {\epsilon }_{1}\right)\right]}^{N}-{C}_{rel}}$
Cross Flow (Single Pass)
Both Fluids Unmixed $\epsilon =1-\mathrm{exp}\left(\frac{\mathrm{exp}\left(-{C}_{rel}NT{U}^{0.78}\right)-1}{{C}_{rel}NT{U}^{-0.22}}\right)$
Both Fluids Mixed $\epsilon =\frac{1}{\frac{1}{1-exp\left(-NTU\right)}+\frac{{C}_{rel}}{1-\mathrm{exp}\left(-{C}_{rel}NTU\right)}-\frac{1}{NTU}}$
C[Max] mixed, C[Min] unmixed $\epsilon =\frac{1}{{C}_{rel}}\left(1-\mathrm{exp}\left(-{C}_{rel}\left(1-\mathrm{exp}\left(-NTU\right)\right)\right)\right)$
C[Max] unmixed, C[Min] mixed $\epsilon =1-\mathrm{exp}\left(-\frac{1}{{C}_{rel}}\left(1-\mathrm{exp}\left(-{C}_{rel}NTU\right)\right)\right)$
Assumptions and Limitations
The flows are single-phase. The heat transfer is strictly one of sensible heat. The transfer is limited to interior of the exchanger, with the environment neither gaining heat from nor providing heat
to the flows—the heat exchanger is an adiabatic component.
C1 — Fluid 1 thermal capacity
physical signal
Physical signal input port for the thermal capacity rate of fluid 1. The thermal capacity rate is the mass flow rate multiplied by the specific heat coefficient for the fluid.
C2 — Fluid 2 thermal capacity
physical signal
Physical signal input port for the thermal capacity rate of fluid 2. The thermal capacity rate is the mass flow rate multiplied by the specific heat coefficient for the fluid.
HC1 — Fluid 1 heat transfer coefficient
physical signal
Physical signal input port for the heat transfer coefficient between fluid 1 and the interface wall.
HC2 — Fluid 2 heat transfer coefficient
physical signal
Physical signal input port for the heat transfer coefficient between fluid 2 and the interface wall.
H1 — Fluid 1 thermal inlet temperature
Thermal conserving port associated with the inlet temperature of fluid 1.
H2 — Fluid 2 thermal inlet temperature
Thermal conserving port associated with the inlet temperature of fluid.
Flow arrangement — Manner in which the flows align in the heat exchanger
Parallel or counter flow (default) | Shell and tube | Cross flow | Generic — effectiveness table
Heat exchanger geometry. Select Generic — effectiveness table to model other heat exchanger geometries based on tabular effectiveness data.
In the Parallel or counter flow configuration, the relative flow directions of fluids 1 and 2 determine whether the heat exchanger is based on parallel or counter flows. The flow directions depend on
the remainder of the Simscape Fluids™ model.
Wall thermal resistance — Resistance of the wall to heat flow by thermal conduction
1.6e-4 K/W (default) | positive scalar
Thermal resistance of the interface wall separating the two heat exchanger fluids.
Number of shell passes — Number of times the flow traverses the shell before exiting
1 (default) | positive scalar
Number of times the flow traverses the shell before exiting.
To enable this parameter, set Flow arrangement to Shell and tube.
Cross flow type — Mixing condition in each of the flow channels
Both fluids mixed (default) | Both fluids unmixed | Controlled Fluid 1 mixed & Controlled Fluid 2 unmixed | Controlled Fluid 1 unmixed & Controlled Fluid 2 mixed
Fluid mixing configuration. The fluids can be mixed or unmixed. The block uses the mixing configuration to determine which empirical heat transfer correlations to use. Mixed flow means that the fluid
is free to move in the transverse direction as it travels along the flow path. Unmixed flow means that the fluid is restricted to travel only along the flow path. For example, a side with fins is
considered an unmixed flow.
To enable this parameter, set Flow arrangement to Cross flow.
Number of heat transfer units vector, NTU — Number of transfer units at each breakpoint in lookup table for heat exchanger effectiveness
[.5, 1, 2, 3, 4, 5] (default) | vector of positive scalars
M-element vector of NTU values at which to specify the effectiveness tabular data. The number of transfer units (NTU) is a dimensionless parameter defined as
• A[S] is the heat transfer surface area.
• U is the overall heat transfer coefficient.
• C[min] is the smallest of the thermal capacity rates for the hot and cold fluids.
To enable this parameter, set Flow arrangement to Generic — effectiveness table.
Thermal capacity ratio vector, CR — Thermal capacity ratio at each breakpoint in lookup table for heat exchanger effectiveness
[0, .25, .5, .75, 1] (default) | vector of positive scalars
N-element vector of thermal capacity ratios at which to specify the effectiveness tabular data. The thermal capacity ratio is the fraction
where C[min] and C[max] are the minimum and maximum thermal capacity rates.
To enable this parameter, set Flow arrangement to Generic — effectiveness table.
Effectiveness table, E(NTU,CR) — Heat exchanger effectiveness at each breakpoint in lookup table over the number of transfer units and thermal capacity ratio
[.3, .3, .3, .3, .3; .6, .55, .5, .47, .43; .85, .76, .68, .61, .55; .94, .83, .72, .65, .58; .98, .86, .75, .66, .58; .99, .86, .75, .66, .58] (default) | table of positive scalars
M-by-N matrix with the heat exchanger effectiveness values. The matrix rows correspond to the different values specified in the Number of heat transfer units vector, NTU parameter. The matrix columns
correspond to the values specified in the Thermal capacity ratio vector, CR parameter.
To enable this parameter, set Flow arrangement to Generic — effectiveness table.
Controlled Fluid 1
Heat transfer surface area — Aggregate heat transfer surface area on the controlled fluid 1 side
0.01 m^2 (default) | positive scalar
Aggregate surface area on fluid 1 side for heat transfer between the cold and hot fluids.
Fouling factor — Measure of thermal resistance due to fouling deposits
1e-4 K*m^2/W (default) | positive scalar
Empirical parameter used to quantify the increased thermal resistance due to dirt deposits on the heat transfer surface.
Minimum fluid-wall heat transfer coefficient — Lower bound for the heat transfer coefficient
5 W/(m^2 * K) (default) | positive scalar
Smallest allowed value of the heat transfer coefficient. The heat transfer coefficient specified through physical signal ports HC1 saturates at this value. The block uses the heat transfer
coefficient to calculate the heat transfer rate between fluids 1 and 2 as described in Heat Transfer Rate.
Controlled Fluid 2
Heat transfer surface area — Aggregate heat transfer surface area on the controlled fluid 2 side
0.01 m^2 (default) | positive scalar
Aggregate surface area on fluid 2 side for heat transfer between the cold and hot fluids.
Fouling factor — Measure of thermal resistance due to fouling deposits
1e-4 K*m^2/W (default) | positive scalar
Empirical parameter used to quantify the increased thermal resistance due to dirt deposits on the heat transfer surface.
Minimum fluid-wall heat transfer coefficient — Lower bound for the heat transfer coefficient
5 W/(m^2 * K) (default) | positive scalar
Smallest allowed value of the heat transfer coefficient. The heat transfer coefficient specified through physical signal ports HC2 saturates at this value. The block uses the heat transfer
coefficient to calculate the heat transfer rate between fluids 1 and 2 as described in Heat Transfer Rate.
[1] Holman, J. P. Heat Transfer. 9th ed. New York, NY: McGraw Hill, 2002.
[2] Shah, R. K. and D. P. Sekulic. Fundamentals of Heat Exchanger Design. Hoboken, NJ: John Wiley & Sons, 2003.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2016a | {"url":"https://es.mathworks.com/help/hydro/ref/entuheattransfer.html","timestamp":"2024-11-05T12:36:57Z","content_type":"text/html","content_length":"127134","record_id":"<urn:uuid:ed1678a3-289a-44c5-bcb0-9314fa59a023>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00169.warc.gz"} |
Part I: 0-1 Knapsack - Fun Destinations in US
Before delving into the more general branch and bound algorithm in F#, let’s try one to implement the simpler 0-1 Knapsack problem using branch and bound techniques. This problem mimics the idea of a
person trying to pack a limited weight (or space) bag (knapsack) with items such that their total value (utility) is maximized. Mathematically, it is formulated as:
Here, there are n variables each denoted as v[i]. Likewise there are n weights associated with each item and n utility (value) settings for each, denoted w[i] and u[i] respectively. Note that the
terminology used here differs from the Wiki link above in that variable, weight and utility indices go from 0 to n-1 to be more in line with computer array indexing and the value is denoted with u
instead of c.
0-1 Knapsack has a pseudo-polynomial solution approach, but here we are focused on applying a branch and bound approach. Later we will look at other problem types.
Short Description of Branch and Bound
Before getting back to the case of the knapsack problem it will help to understand how branch and bound works. Consider the following search tree diagram:
Deciding on the selection of each variable can be viewed as a tree structure, where each branch is a variable selection and each node represents the setting (or non-setting) of each variable. At the
top of the tree, all of the variables are unset. In the drawing, we have a case of three variables (indexed from 0 to 2). At each level of the tree one variable is decided (set). As an example, if
you follow the red line from the top to the bottom, the variables are set to v[0] = 0, v[1] = 1, and v[2] = 1. At the bottom level, all of the variables are set and therefore a value of the knapsack
and a weight of the knapsack can be computed (see Eq 1). At any intermediate node other than the bottom layer, at least one variable remains to be set. There are 2^n nodes at the bottom layer, that
is all the combinations of n variables, in the figure n=3 so there are 8 in that case. And there are 2^n+1-1 nodes in the whole graph for any n variables.
Notice that the nodes at the bottom are labeled with ‘U’ and the rest as ‘G’. This is meant to imply that nodes with complete settings have a particular utility while the rest (the ‘G’s) have only
estimates; we will later be using a function named g hence the G moniker, but that is getting ahead of ourselves.
A first step is deciding how to represent the problem domain in code or the type system. We see that we have three arrays to represent, each with a cardinality of n. The weights and utilities could
be either integers or floating point values, let’s choose the more inclusive floating point (float in F#). The variables of our domain can be either zero or one, or in the case of branch and bound
unset. This leads us to determine how to represent (the type) the variables of our problem. We have decided to type the variables as DiscreteVar.
1: (* this is setting for a variable of the problem *)
2: type ZeroOneVarSetting =
3: | Unset (* this is a discriminated union type in F# *)
4: | One
5: | Zero
7: (* this is a variable of the problem. Each var has a Name and mutable v*)
8: type DiscreteVar =
9: { (* this is a record type in F# *)
10: Name : string;
11: mutable Setting : ZeroOneVarSetting; // discrete value setting
12: }
I assume that you know the basics of F#
We used a discriminated union type to capture the possible settings for a variable as either Unset, Zero or One, makes sense. Then, the DiscreteVar is a record type with a Name and Settings that use
this type. To define a problem instance, we might use something like:
Arrays are denoted with [| |] while lists are denoted as [ ], each with elements separated with semicolons.
1: let vars = [|{Name = "food"; Setting = Unset};
2: {Name = "tent"; Setting = Unset};
3: {Name = "gps"; Setting = Unset };
4: {Name = "map"; Setting = Unset} |]
5: let weights = [|5.0; 14.5; 1.0; 0.5 |]
6: let utilities = [| 8.0; 5.0; 3.0; 3.0 |]
7: let weightLimit = 20.0
Notice that each of vars, weights and utilities are array types. Another choice might be to make them lists. F# arrays allow for modification of an element, while lists do not. If we want to change
an element of a list we need to extract the elements leading up to the element to be changed, concatenate this with the new element and then concatenate the rest and assign this to a new variable
(see topic). Array sizes are fixed, but that works since once we define a problem set (as above) the size of these arrays doesn’t need to change. We make the setting for the variable mutable since we
think we need to change it later. These are our ideas at this point anyway.
The approach throughout is to start with something and refine it as we go along.
Notice that we never defined the type of element that vars contains. F# deduces that the type of element is a record type and furthermore that the type is DiscreteVar based on seeing that these
record names match in these record expressions (see topic). OK, let’s write a function to determine the total utility.
1: let utility u v =
2: let products = Array.map2 (fun uElem vElem -> uElem * vElem) u v
3: Array.sum products
5: let tv = utility utilities vars
Line 1 defines a function named utility which takes two arguments named u and v. At this point we haven’t declared what types u and v are, and the compiler deduces their type from the body of the
function and from subsequent uses. In fact, if you put that code into F#, the utility definition is fine until you add the invocation at line 5. Before adding line 5, the compiler thinks of the
function as:
That is, it knows that u and v are arrays, given their appearance as arguments to the Array.map2 function; it also knows that the array elements must allow the (*) operator so it chooses int as the
array element type sans other information. Recall that Array.map2 takes as arguments a function that accepts two arguments and produces a value; the two arrays which must be of the same length.
However, when we add the line 5 content, an error arises:
Now, the compiler sees that the u argument is a float array, and then sees that it cannot apply the (*) operator to a float and a DiscreteVar (since v is a DiscreteVar array in this instance). So,
its important to realize that function and data type deduction ‘looks’ at the whole content, including code that comes later. Function arguments that are not explicitly typed are generic arguments
(think ala C++ templates but without the template keywording); these will prove very useful as they allow definition of functions that are flexible in their argument types. But enough about general F
# typing for now, let’s fix our problem.
TIP: You can explicitly type function arguments to force their interpretion within the function body, and then fix invocations. Once all that works you can try to remove these declarations to make
the code neater and more flexible — although sometimes F# does need type ‘hints’
So how are we going to develop utility? Let’s re-write utility such that it accepts an array of floats and an array of DiscreteVar like so …
1: let utility u v =
2: let products = Array.map2 (fun uElem vElem -> uElem * match vElem.Setting with
3: | One -> 1.0
4: | _ -> 0.0 ) u v
5: Array.sum products
7: let tv = utility utilities vars
8: printfn "%f" tv
In the mapping function, the second argument is still a DiscreteVar but we use pattern matching to derive a float value from the discriminated union type. In this case, the pattern One produces a 1.0
and otherwise a 0.0 results since Unset and Zero will match the _ default pattern. The result is that products is an array of floats such that it is the pairwise product of the u array and 1.0
whereever vElem.Setting is One. Each member of this result (which is a float array) is then summed using the Array.sum method to produce a float result (remember that the final expression of a
function is its return value). If we run this and examine the result as is we get 0.0, of course this is because all of the variables are Unset and thus a multicand of 0.0 is produced for each vElem
in the expression.
We next realize that multiplying weights and the variables is the same function, so let’s rename it multiplySum. We also don’t need the intermediate value products and instead pipe (|>) the result of
the pairwise multiplication to the Array.sum. We could apply this function using either utility or weight as the first argument.
1: let multiplySum x v =
2: Array.map2 (fun xElem vElem -> xElem * match vElem.Setting with
3: | One -> 1.0
4: | _ -> 0.0 ) x v |> Array.sum
6: printfn "%f" (multiplySum utilities vars)
7: printfn "%f" (multiplySum weights vars)
The printfn function examines the format string and makes sure that the number and type of arguments match the wildcard patterns
In the multiplySum printfn statements, you must use parens to make the function application with its arguments into a single argument for the print, otherwise its three arguments. Of course, both of
these would still produce 0.0.
We have a sense of some of the code that we may use, but then notice that the arrays for variables, weights and utility be the same length. Furthermore the next incremental change is then realizing
that weight and utility are really factors of a knapsack item. Therefore, we adjust the definitions and get to this:
1: type ZeroOneVarSetting =
2: | Unset (* this is a discriminated union type in F# *)
3: | One
4: | Zero
6: type DiscreteVar =
7: { (* this is a record type in F# *)
8: Name : string;
9: Weight : float;
10: Utility : float;
11: mutable Setting : ZeroOneVarSetting; // discrete value setting
12: }
14: let vars = [|{Name = "food"; Setting = Unset; Weight = 5.0; Utility = 8.0 };
15: {Name = "tent"; Setting = Unset; Weight = 14.5; Utility = 5.0};
16: {Name = "gps"; Setting = Unset; Weight = 1.0; Utility = 3.0};
17: {Name = "map"; Setting = Unset; Weight = 0.5; Utility = 3.0} |]
18: let weightLimit = 20.0
20: let multiplySumWeight v =
21: Array.map (fun vElem -> vElem.Weight * match vElem.Setting with
22: | One -> 1.0
23: | _ -> 0.0 ) v |> Array.sum
25: let multiplySumUtility v =
26: Array.map (fun vElem -> vElem.Utility * match vElem.Setting with
27: | One -> 1.0
28: | _ -> 0.0 ) v |> Array.sum
30: printfn "%f" (multiplySumWeight vars)
31: printfn "%f" (multiplySumUtility vars)
OK, the item definitions seem better suited to the domain at hand — using the type system hand-in-hand with the problem domain is called domain driven design. One concept from this idea is to use the
type system to disallow invalid states or variable settings. We see this in small part in the ZeroOneVarSetting as the only choices are one of the three discriminants. If instead we used, say, an int
type to represent the state of our variable, assuming that the value would be -1 for Unset, 0 for Zero and 1 for One, what would the meaning be if the int took on a value of 7 or -4? In the F#
discriminated union approach, there is simply no opportunity for such erroneous or unexpected settings.
Unfortunately however, the two functions for getting the total weight and utility seem redundant varying only in which record member is used in the multiplication. Let’s revisit this again, this time
creating a two argument function multiplyEither with a first argument of boolean that selects Utility if true and Weight otherwise.
1: let multiplySumEither b v =
2: Array.map (fun vElem -> (if b then vElem.Utility else vElem.Weight) * match vElem.Setting with
3: | One -> 1.0
4: | _ -> 0.0 ) v |> Array.sum
5: let multiplySumWeight = multiplySumEither false
6: let multiplySumUtility = multiplySumEither true
The idiomatic or canonical approach to leveraging functional style is often discussed in regard to F# code. See this and this for more
We then create our multiplySumWeight as multiplyEither true and multiplySumUtility false. This demonstrates an important feature of functional programming, namely partial function application.
multiplySumWeight is multiplyEither with the first argument of true ‘baked in’; it only has one argument now the DiscreteVar array. That is the type for multiplySumWeight is DiscreteVar [] -> float,
the type for multiplyEither is bool -> DiscreteVar [] -> float. Notice that we never told multiplyEither what type its argument b is, a Boolean type (bool) is deduced from its use as the test in the
if expression (and the partial applications that use actual arguments of true and false). There are other variations for doing these accumulating products that could be used too, but for now we will
leave it here.
With some of the functionality we need to address the problem, on to page 2. | {"url":"https://opcoast.com/demos/fsharp/part1.html","timestamp":"2024-11-03T10:38:03Z","content_type":"text/html","content_length":"89436","record_id":"<urn:uuid:70db3beb-deaf-4113-9afc-c60f3dbfd92f>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00568.warc.gz"} |
2.1 Monte Carlo: Basics
Because Monte Carlo integration is based on randomization, we will start this chapter with a brief review of ideas from probability and statistics that provide the foundations of the approach. Doing
so will allow us to introduce the basic Monte Carlo algorithm as well as mathematical tools for evaluating its error.
2.1.1 Background and Probability Review
We will start by defining some terms and reviewing basic ideas from probability. We assume that the reader is already familiar with basic probability concepts; readers needing a more complete
introduction to this topic should consult a textbook such as Sheldon Ross’s Introduction to Probability Models (2002).
A random variable is a value chosen by some random process. We will generally use capital letters to denote random variables, with exceptions made for a few Greek symbols that represent special
random variables. Random variables are always drawn from some domain, which can be either discrete (e.g., a fixed, finite set of possibilities) or continuous (e.g., the real numbers ). Applying a
function to a random variable results in a new random variable .
For example, the result of a roll of a die is a discrete random variable sampled from the set of events . Each event has a probability , and the sum of probabilities is necessarily one. A random
variable like this one that has the same probability for all potential values of it is said to be uniform. A function that gives a discrete random variable’s probability is termed a probability mass
function (PMF), and so we could equivalently write in this case.
Two random variables are independent if the probability of one does not affect the probability of the other. In this case, the joint probability of two random variables is given by the product of
their probabilities:
For example, two random variables representing random samples of the six sides of a die are independent.
For dependent random variables, one’s probability affects the other’s. Consider a bag filled with some number of black balls and some number of white balls. If we randomly choose two balls from the
bag, the probability of the second ball being white is affected by the color of the first ball since its choice changes the number of balls of one type left in the bag. We will say that the second
ball’s probability is conditioned on the choice of the first one. In this case, the joint probability for choosing two balls and is given by
where is the conditional probability of given a value of .
In the following, it will often be the case that a random variable’s probability is conditioned on many values; for example, when choosing a light source from which to sample illumination, the
BVHLightSampler in Section 12.6.3 considers the 3D position of the receiving point and its surface normal, and so the choice of light is conditioned on them. However, we will often omit the variables
that a random variable is conditioned on in cases where there are many of them and where enumerating them would obscure notation.
A particularly important random variable is the canonical uniform random variable, which we will write as . This variable takes on all values in its domain independently and with uniform probability.
This particular variable is important for two reasons. First, it is easy to generate a variable with this distribution in software—most runtime libraries have a pseudo-random number generator that
does just that. Second, we can take the canonical uniform random variable and map it to a discrete random variable, choosing if
For lighting applications, we might want to define the probability of sampling illumination from each light in the scene based on its power relative to the total power from all sources:
Notice that these values also sum to 1. Given such per-light probabilities, could be used to select a light source from which to sample illumination.
The cumulative distribution function (CDF) of a random variable is the probability that a value from the variable’s distribution is less than or equal to some value :
For the die example, , since two of the six possibilities are less than or equal to 2.
Continuous random variables take on values over ranges of continuous domains (e.g., the real numbers, directions on the unit sphere, or the surfaces of shapes in the scene). Beyond , another example
of a continuous random variable is the random variable that ranges over the real numbers between 0 and 2, where the probability of its taking on any particular value is proportional to the value : it
is twice as likely for this random variable to take on a value around 0 as it is to take one around 1, and so forth.
The probability density function (PDF) formalizes this idea: it describes the relative probability of a random variable taking on a particular value and is the continuous analog of the PMF. The PDF
is the derivative of the random variable’s CDF,
For uniform random variables, is a constant; this is a direct consequence of uniformity. For we have
PDFs are necessarily nonnegative and always integrate to 1 over their domains. Note that their value at a point is not necessarily less than 1, however.
Given an interval in the domain, integrating the PDF gives the probability that a random variable lies inside the interval:
This follows directly from the first fundamental theorem of calculus and the definition of the PDF.
2.1.2 Expected Values
The expected value of a function is defined as the average value of the function over some distribution of values over its domain . It is defined as
As an example, consider finding the expected value of the cosine function between and , where is uniform. Because the PDF must integrate to 1 over the domain, , so
which is precisely the expected result. (Consider the graph of over to see why this is so.)
The expected value has a few useful properties that follow from its definition:
We will repeatedly use these properties in derivations in the following sections.
2.1.3 The Monte Carlo Estimator
We can now define the Monte Carlo estimator, which approximates the value of an arbitrary integral. Suppose that we want to evaluate a 1D integral . Given a supply of independent uniform random
variables , the Monte Carlo estimator says that the expected value of the estimator
, is equal to the integral. This fact can be demonstrated with just a few steps. First, note that the PDF corresponding to the random variable must be equal to , since must not only be a constant but
also integrate to 1 over the domain . Algebraic manipulation using the properties from Equations (2.4) and (2.5) then shows that
Extending this estimator to multiple dimensions or complex integration domains is straightforward: independent samples are taken from a uniform multidimensional PDF, and the estimator is applied in
the same way. For example, consider the 3D integral
If samples are chosen uniformly from the cube from , then the PDF is the constant value
The restriction to uniform random variables can be relaxed with a small generalization. This is an important step, since carefully choosing the PDF from which samples are drawn leads to a key
technique for reducing error in Monte Carlo that will be introduced in Section 2.2.2. If the random variables are drawn from a PDF , then the estimator
can be used to estimate the integral instead. The only limitation on is that it must be nonzero for all where .
It is similarly not too hard to see that the expected value of this estimator is the desired integral of :
We can now understand the factor of in the implementation of the RandomWalkIntegrator: directions are uniformly sampled over the unit sphere, which has area . Because the PDF is normalized over the
sampling domain, it must have the constant value . When the estimator of Equation (2.7) is applied, that value appears in the divisor.
With Monte Carlo, the number of samples can be chosen arbitrarily, regardless of the dimensionality of the integrand. This is another important advantage of Monte Carlo over traditional deterministic
quadrature techniques, which typically require a number of samples that is exponential in the dimension.
2.1.4 Error in Monte Carlo Estimators
Showing that the Monte Carlo estimator converges to the right answer is not enough to justify its use; its rate of convergence is important too. Variance, the expected squared deviation of a function
from its expected value, is a useful way to characterize Monte Carlo estimators’ convergence. The variance of an estimator is defined as
from which it follows that
This property and Equation (2.5) yield an alternative expression for the variance:
Thus, the variance is the expected value of the square minus the square of the expected value.
If the estimator is a sum of independent random variables (like the Monte Carlo estimator ), then the variance of the sum is the sum of the individual random variables’ variances:
From Equation (2.10) it is easy to show that variance decreases linearly with the number of samples . Because variance is squared error, the error in a Monte Carlo estimate therefore only goes down
at a rate of in the number of samples. Although standard quadrature techniques converge at a faster rate in one dimension, their performance becomes exponentially worse as the dimensionality of the
integrand increases, while Monte Carlo’s convergence rate is independent of the dimension, making Monte Carlo the only practical numerical integration algorithm for high-dimensional integrals.
The characteristic of Monte Carlo’s rate of error reduction is apparent when watching a progressive rendering of a scene where additional samples are incrementally taken in all pixels. The image
improves rapidly for the first few samples when doubling the number of samples is relatively little additional work. Later on, once tens or hundreds of samples have been taken, each additional sample
doubling takes much longer and remaining error in the image takes a long time to disappear.
The linear decrease in variance with increasing numbers of samples makes it easy to compare different Monte Carlo estimators. Consider two estimators, where the second has half the variance of the
first but takes three times as long to compute an estimate; which of the two is better? In that case, the first is preferable: it could take three times as many samples in the time consumed by the
second, in which case it would achieve a variance reduction. This concept can be encapsulated in the efficiency of an estimator , which is defined as
where is its variance and is the running time to compute its value.
Not all estimators of integrals have expected values that are equal to the value of the integral. Such estimators are said to be biased, where the difference
is the amount of bias. Biased estimators may still be desirable if they are able to get close to the correct result more quickly than unbiased estimators. Kalos and Whitlock (1986, pp. 36–37) gave
the following example: consider the problem of computing an estimate of the mean value of a uniform distribution over the interval from 0 to 1. One could use the estimator
or one could use the biased estimator
The first estimator is unbiased but has variance with order . The second estimator’s expected value is
so it is biased, although its variance is , which is much better. This estimator has the useful property that its error goes to 0 in the limit as the number of samples goes to infinity; such
estimators are consistent. Most of the Monte Carlo estimators used in pbrt are unbiased, with the notable exception of the SPPMIntegrator, which implements a photon mapping algorithm.
Closely related to the variance is the mean squared error (MSE), which is defined as the expectation of the squared difference of an estimator and the true value,
For an unbiased estimator, MSE is equal to the variance; otherwise it is the sum of variance and the squared bias of the estimator.
It is possible to work out the variance and MSE of some simple estimators in closed form, but for most of the ones of interest in rendering, this is not possible. Yet it is still useful to be able to
quantify these values. For this purpose, the sample variance can be computed using a set of independent random variables . Equation (2.8) points at one way to compute the sample variance for a set of
random variables . If the sample mean is computed as their average, , then the sample variance is
The division by rather than is Bessel’s correction, and ensures that the sample variance is an unbiased estimate of the variance. (See also Section B.2.11, where a numerically stable approach for
computing the sample variance is introduced.)
The sample variance is itself an estimate of the variance, so it has variance itself. Consider, for example, a random variable that has a value of 1 99.99% of the time, and a value of one million
0.01% of the time. If we took ten random samples of it that all had the value 1, the sample variance would suggest that the random variable had zero variance even though its variance is actually much
If an accurate estimate of the integral can be computed (for example, using a large number of samples), then the mean squared error can be estimated by
The imgtool utility program that is provided in pbrt’s distribution can compute an image’s MSE with respect to a reference image via its diff option. | {"url":"https://pbr-book.org/4ed/Monte_Carlo_Integration/Monte_Carlo_Basics","timestamp":"2024-11-01T19:09:56Z","content_type":"text/html","content_length":"483763","record_id":"<urn:uuid:495a37f8-b646-41fc-8588-5521a14e637b>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00255.warc.gz"} |
Graph Theory
Types of Graphs-
Before you go through this article, make sure that you have gone through the previous article on various Types of Graphs in Graph Theory.
We have discussed-
• A graph is a collection of vertices connected to each other through a set of edges.
• The study of graphs is known as Graph Theory.
In this article, we will discuss about Euler Graphs.
Euler Graph-
An Euler graph may be defined as-
│Any connected graph is called as an Euler Graph if and only if all its vertices are of even degree.│
│ │
│OR │
│ │
│An Euler Graph is a connected graph that contains an Euler Circuit. │
Euler Graph Example-
The following graph is an example of an Euler graph-
• This graph is a connected graph and all its vertices are of even degree.
• Therefore, it is an Euler graph.
Alternatively, the above graph contains an Euler circuit BACEDCB, so it is an Euler graph.
Also Read- Planar Graph
Euler Path-
Euler path is also known as Euler Trail or Euler Walk.
• If there exists a Trail in the connected graph that contains all the edges of the graph, then that trail is called as an Euler trail.
• If there exists a walk in the connected graph that visits every edge of the graph exactly once with or without repeating the vertices, then such a walk is called as an Euler walk.
│NOTE │
│ │
│A graph will contain an Euler path if and only if it contains at most two vertices of odd degree.│
Euler Path Examples-
Examples of Euler path are as follows-
Euler Circuit-
Euler circuit is also known as Euler Cycle or Euler Tour.
• If there exists a Circuit in the connected graph that contains all the edges of the graph, then that circuit is called as an Euler circuit.
• If there exists a walk in the connected graph that starts and ends at the same vertex and visits every edge of the graph exactly once with or without repeating the vertices, then such a walk is
called as an Euler circuit.
• An Euler trail that starts and ends at the same vertex is called as an Euler circuit.
• A closed Euler trail is called as an Euler circuit.
│NOTE │
│ │
│A graph will contain an Euler circuit if and only if all its vertices are of even degree.│
Euler Circuit Examples-
Examples of Euler circuit are as follows-
Semi-Euler Graph-
If a connected graph contains an Euler trail but does not contain an Euler circuit, then such a graph is called as a semi-Euler graph.
Thus, for a graph to be a semi-Euler graph, following two conditions must be satisfied-
• Graph must be connected.
• Graph must contain an Euler trail.
• This graph contains an Euler trail BCDBAD.
• But it does not contain an Euler circuit.
• Therefore, it is a semi-Euler graph.
Also Read- Bipartite Graph
Important Notes-
To check whether any graph is an Euler graph or not, any one of the following two ways may be used-
• If the graph is connected and contains an Euler circuit, then it is an Euler graph.
• If all the vertices of the graph are of even degree, then it is an Euler graph.
To check whether any graph contains an Euler circuit or not,
• Just make sure that all its vertices are of even degree.
• If all its vertices are of even degree, then graph contains an Euler circuit otherwise not.
To check whether any graph is a semi-Euler graph or not,
• Just make sure that it is connected and contains an Euler trail.
• If the graph is connected and contains an Euler trail, then graph is a semi-Euler graph otherwise not.
To check whether any graph contains an Euler trail or not,
• Just make sure that the number of vertices in the graph with odd degree are not more than 2.
• If the number of vertices with odd degree are at most 2, then graph contains an Euler trail otherwise not.
• A graph will definitely contain an Euler trail if it contains an Euler circuit.
• A graph may or may not contain an Euler circuit if it contains an Euler trail.
• An Euler graph is definitely be a semi-Euler graph.
• But a semi-Euler graph may or may not be an Euler graph.
Which of the following is / are Euler Graphs?
If all the vertices of a graph are of even degree, then graph is an Euler Graph otherwise not.
Using the above rule, we have-
A) It is an Euler graph.
B) It is not an Euler graph.
C) It is not an Euler graph.
D) It is not an Euler graph.
E) It is an Euler graph.
F) It is not an Euler graph.
To gain better understanding about Euler Graphs in Graph Theory,
Next Article- Hamiltonian Graph
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How to Find Chromatic Number | Graph Coloring Algorithm
Chromatic Number-
Before you go through this article, make sure that you have gone through the previous article on Chromatic Number.
We gave discussed-
• Graph Coloring is a process of assigning colors to the vertices of a graph.
• It ensures that no two adjacent vertices of the graph are colored with the same color.
• Chromatic Number is the minimum number of colors required to properly color any graph.
In this article, we will discuss how to find Chromatic Number of any graph.
Graph Coloring Algorithm-
• There exists no efficient algorithm for coloring a graph with minimum number of colors.
• Graph Coloring is a NP complete problem.
However, a following greedy algorithm is known for finding the chromatic number of any given graph.
Greedy Algorithm-
Color first vertex with the first color.
Now, consider the remaining (V-1) vertices one by one and do the following-
• Color the currently picked vertex with the lowest numbered color if it has not been used to color any of its adjacent vertices.
• If it has been used, then choose the next least numbered color.
• If all the previously used colors have been used, then assign a new color to the currently picked vertex.
Drawbacks of Greedy Algorithm-
There are following drawbacks of the above Greedy Algorithm-
• The above algorithm does not always use minimum number of colors.
• The number of colors used sometimes depend on the order in which the vertices are processed.
Also Read- Types of Graphs in Graph Theory
Find chromatic number of the following graph-
Applying Greedy Algorithm, we have-
│Vertex │a │b │c │d │e │f │
│Color │C1│C2│C1│C2│C1│C2│
From here,
• Minimum number of colors used to color the given graph are 2.
• Therefore, Chromatic Number of the given graph = 2.
The given graph may be properly colored using 2 colors as shown below-
Find chromatic number of the following graph-
Applying Greedy Algorithm, we have-
│Vertex │a │b │c │d │e │f │
│Color │C1│C2│C2│C3│C3│C1│
From here,
• Minimum number of colors used to color the given graph are 3.
• Therefore, Chromatic Number of the given graph = 3.
The given graph may be properly colored using 3 colors as shown below-
Find chromatic number of the following graph-
Applying Greedy Algorithm, we have-
│Vertex │a │b │c │d │e │ f│g │
│Color │C1│C2│C1│C3│C2│C3│C4│
From here,
• Minimum number of colors used to color the given graph are 4.
• Therefore, Chromatic Number of the given graph = 4.
The given graph may be properly colored using 4 colors as shown below-
Find chromatic number of the following graph-
Applying Greedy Algorithm, we have-
│Vertex │a │b │c │d │e │f │
│Color │C1│C2│C3│C1│C2│C3│
From here,
• Minimum number of colors used to color the given graph are 3.
• Therefore, Chromatic Number of the given graph = 3.
The given graph may be properly colored using 3 colors as shown below-
Find chromatic number of the following graph-
Applying Greedy Algorithm,
• Minimum number of colors required to color the given graph are 3.
• Therefore, Chromatic Number of the given graph = 3.
The given graph may be properly colored using 3 colors as shown below-
To gain better understanding about How to Find Chromatic Number,
Get more notes and other study material of Graph Theory.
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Graph Coloring in Graph Theory | Chromatic Number of Graphs
Graph Coloring-
│Graph Coloring is a process of assigning colors to the vertices of a graph│
│ │
│such that no two adjacent vertices of it are assigned the same color. │
• Graph Coloring is also called as Vertex Coloring.
• It ensures that there exists no edge in the graph whose end vertices are colored with the same color.
• Such a graph is called as a Properly colored graph.
Graph Coloring Example-
The following graph is an example of a properly colored graph-
In this graph,
• No two adjacent vertices are colored with the same color.
• Therefore, it is a properly colored graph.
Graph Coloring Applications-
Some important applications of graph coloring are as follows-
• Map Coloring
• Scheduling the tasks
• Preparing Time Table
• Assignment
• Conflict Resolution
• Sudoku
Chromatic Number-
│Chromatic Number is the minimum number of colors required to properly color any graph.│
│ │
│OR │
│ │
│Chromatic Number is the minimum number of colors required to color any graph │
│ │
│such that no two adjacent vertices of it are assigned the same color. │
Chromatic Number Example-
Consider the following graph-
In this graph,
• No two adjacent vertices are colored with the same color.
• Minimum number of colors required to properly color the vertices = 3.
• Therefore, Chromatic number of this graph = 3.
• We can not properly color this graph with less than 3 colors.
Also Read- Types of Graphs in Graph Theory
Chromatic Number Of Graphs-
Chromatic Number of some common types of graphs are as follows-
1. Cycle Graph-
• A simple graph of ‘n’ vertices (n>=3) and ‘n’ edges forming a cycle of length ‘n’ is called as a cycle graph.
• In a cycle graph, all the vertices are of degree 2.
│Chromatic Number │
│ │
│ • If number of vertices in cycle graph is even, then its chromatic number = 2.│
│ • If number of vertices in cycle graph is odd, then its chromatic number = 3. │
2. Planar Graphs-
A Planar Graph is a graph that can be drawn in a plane such that none of its edges cross each other.
│Chromatic Number │
│ │
│Chromatic Number of any Planar Graph│
│ │
│= Less than or equal to 4 │
• All the above cycle graphs are also planar graphs.
• Chromatic number of each graph is less than or equal to 4.
3. Complete Graphs-
• A complete graph is a graph in which every two distinct vertices are joined by exactly one edge.
• In a complete graph, each vertex is connected with every other vertex.
• So to properly it, as many different colors are needed as there are number of vertices in the given graph.
│Chromatic Number │
│ │
│Chromatic Number of any Complete Graph │
│ │
│= Number of vertices in that Complete Graph│
4. Bipartite Graphs-
• A Bipartite Graph consists of two sets of vertices X and Y.
• The edges only join vertices in X to vertices in Y, not vertices within a set.
│Chromatic Number │
│ │
│Chromatic Number of any Bipartite Graph│
│ │
│= 2 │
5. Trees-
• A Tree is a special type of connected graph in which there are no circuits.
• Every tree is a bipartite graph.
• So, chromatic number of a tree with any number of vertices = 2.
│Chromatic Number │
│ │
│Chromatic Number of any tree│
│ │
│= 2 │
To gain better understanding about Graph Coloring & Chromatic Number,
Next Article- Graph Coloring Algorithm
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Planar Graph in Graph Theory | Planar Graph Example
Types of Graphs-
Before you go through this article, make sure that you have gone through the previous article on various Types of Graphs in Graph Theory.
We have discussed-
• A graph is a collection of vertices connected to each other through a set of edges.
• The study of graphs is known as Graph Theory.
In this article, we will discuss about Planar Graphs.
Planar Graph-
A planar graph may be defined as-
│In graph theory, │
│ │
│Planar graph is a graph that can be drawn in a plane such that none of its edges cross each other.│
Planar Graph Example-
The following graph is an example of a planar graph-
• In this graph, no two edges cross each other.
• Therefore, it is a planar graph.
Regions of Plane-
The planar representation of the graph splits the plane into connected areas called as Regions of the plane.
Each region has some degree associated with it given as-
• Degree of Interior region = Number of edges enclosing that region
• Degree of Exterior region = Number of edges exposed to that region
Consider the following planar graph-
Here, this planar graph splits the plane into 4 regions- R1, R2, R3 and R4 where-
• Degree (R1) = 3
• Degree (R2) = 3
• Degree (R3) = 3
• Degree (R4) = 5
Planar Graph Chromatic Number-
• Chromatic Number of any planar graph is always less than or equal to 4.
• Thus, any planar graph always requires maximum 4 colors for coloring its vertices.
Planar Graph Properties-
In any planar graph, Sum of degrees of all the vertices = 2 x Total number of edges in the graph
In any planar graph, Sum of degrees of all the regions = 2 x Total number of edges in the graph
│Special Cases │
│ │
│ │
│ │
│Case-01: │
│ │
│ │
│ │
│In any planar graph, if degree of each region is K, then- │
│ │
│ │
│ │
│┌─────────────────┐ │
││K x |R| = 2 x |E|│ │
│└─────────────────┘ │
│ │
│ │
│ │
│Case-02: │
│ │
│ │
│ │
│In any planar graph, if degree of each region is at least K (>=K), then-│
│ │
│ │
│ │
│┌──────────────────┐ │
││K x |R| <= 2 x |E|│ │
│└──────────────────┘ │
│ │
│ │
│ │
│Case-03: │
│ │
│ │
│ │
│In any planar graph, if degree of each region is at most K (<=K), then- │
│ │
│ │
│ │
│┌──────────────────┐ │
││K x |R| >= 2 x |E|│ │
│└──────────────────┘ │
│ │
│ │
If G is a connected planar simple graph with ‘e’ edges, ‘v’ vertices and ‘r’ number of regions in the planar representation of G, then-
│r = e – v + 2│
This is known as Euler’s Formula.
It remains same in all the planar representations of the graph.
If G is a planar graph with k components, then-
│r = e – v + (k + 1)│
Also Read- Bipartite Graph
Let G be a connected planar simple graph with 25 vertices and 60 edges. Find the number of regions in G.
• Number of vertices (v) = 25
• Number of edges (e) = 60
By Euler’s formula, we know r = e – v + 2.
Substituting the values, we get-
Number of regions (r)
= 60 – 25 + 2
= 37
Thus, Total number of regions in G = 37.
Let G be a planar graph with 10 vertices, 3 components and 9 edges. Find the number of regions in G.
• Number of vertices (v) = 10
• Number of edges (e) = 9
• Number of components (k) = 3
By Euler’s formula, we know r = e – v + (k+1).
Substituting the values, we get-
Number of regions (r)
= 9 – 10 + (3+1)
= -1 + 4
= 3
Thus, Total number of regions in G = 3.
Let G be a connected planar simple graph with 20 vertices and degree of each vertex is 3. Find the number of regions in G.
• Number of vertices (v) = 20
• Degree of each vertex (d) = 3
Calculating Total Number Of Edges (e)-
By sum of degrees of vertices theorem, we have-
Sum of degrees of all the vertices = 2 x Total number of edges
Number of vertices x Degree of each vertex = 2 x Total number of edges
20 x 3 = 2 x e
∴ e = 30
Thus, Total number of edges in G = 30.
Calculating Total Number Of Regions (r)-
By Euler’s formula, we know r = e – v + 2.
Substituting the values, we get-
Number of regions (r)
= 30 – 20 + 2
= 12
Thus, Total number of regions in G = 12.
Let G be a connected planar simple graph with 35 regions, degree of each region is 6. Find the number of vertices in G.
• Number of regions (n) = 35
• Degree of each region (d) = 6
Calculating Total Number Of Edges (e)-
By sum of degrees of regions theorem, we have-
Sum of degrees of all the regions = 2 x Total number of edges
Number of regions x Degree of each region = 2 x Total number of edges
35 x 6 = 2 x e
∴ e = 105
Thus, Total number of edges in G = 105.
Calculating Total Number Of Vertices (v)-
By Euler’s formula, we know r = e – v + 2.
Substituting the values, we get-
35 = 105 – v + 2
∴ v = 72
Thus, Total number of vertices in G = 72.
Let G be a connected planar graph with 12 vertices, 30 edges and degree of each region is k. Find the value of k.
• Number of vertices (v) = 12
• Number of edges (e) = 30
• Degree of each region (d) = k
Calculating Total Number Of Regions (r)-
By Euler’s formula, we know r = e – v + 2.
Substituting the values, we get-
Number of regions (r)
= 30 – 12 + 2
= 20
Thus, Total number of regions in G = 20.
Calculating Value Of k-
By sum of degrees of regions theorem, we have-
Sum of degrees of all the regions = 2 x Total number of edges
Number of regions x Degree of each region = 2 x Total number of edges
20 x k = 2 x 30
∴ k = 3
Thus, Degree of each region in G = 3.
What is the maximum number of regions possible in a simple planar graph with 10 edges?
In a simple planar graph, degree of each region is >= 3.
So, we have 3 x |R| <= 2 x |E|.
Substituting the value |E| = 10, we get-
3 x |R| <= 2 x 10
|R| <= 6.67
|R| <= 6
Thus, Maximum number of regions in G = 6.
What is the minimum number of edges necessary in a simple planar graph with 15 regions?
In a simple planar graph, degree of each region is >= 3.
So, we have 3 x |R| <= 2 x |E|.
Substituting the value |R| = 15, we get-
3 x 15 <= 2 x |E|
|E| >= 22.5
|E| >= 23
Thus, Minimum number of edges required in G = 23.
To gain better understanding about Planar Graphs in Graph Theory,
Next Article- Euler Graph
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Walk in Graph Theory | Path | Trail | Cycle | Circuit
Walk in Graph Theory-
In graph theory,
• A walk is defined as a finite length alternating sequence of vertices and edges.
• The total number of edges covered in a walk is called as Length of the Walk.
Walk in Graph Theory Example-
Consider the following graph-
In this graph, few examples of walk are-
• a , b , c , e , d (Length = 4)
• d , b , a , c , e , d , e , c (Length = 7)
• e , c , b , a , c , e , d (Length = 6)
Open Walk in Graph Theory-
In graph theory, a walk is called as an Open walk if-
• Length of the walk is greater than zero
• And the vertices at which the walk starts and ends are different.
Closed Walk in Graph Theory-
In graph theory, a walk is called as a Closed walk if-
• Length of the walk is greater than zero
• And the vertices at which the walk starts and ends are same.
│NOTE │
│ │
│ │
│ │
│It is important to note the following points- │
│ │
│ • If length of the walk = 0, then it is called as a Trivial Walk. │
│ • Both vertices and edges can repeat in a walk whether it is an open walk or a closed walk.│
Path in Graph Theory-
In graph theory, a path is defined as an open walk in which-
• Neither vertices (except possibly the starting and ending vertices) are allowed to repeat.
• Nor edges are allowed to repeat.
Cycle in Graph Theory-
In graph theory, a cycle is defined as a closed walk in which-
• Neither vertices (except possibly the starting and ending vertices) are allowed to repeat.
• Nor edges are allowed to repeat.
In graph theory, a closed path is called as a cycle.
Trail in Graph Theory-
In graph theory, a trail is defined as an open walk in which-
• Vertices may repeat.
• But edges are not allowed to repeat.
Circuit in Graph Theory-
In graph theory, a circuit is defined as a closed walk in which-
• Vertices may repeat.
• But edges are not allowed to repeat.
In graph theory, a closed trail is called as a circuit.
│NOTE │
│ │
│ │
│ │
│It is important to note the following points- │
│ │
│ • Every path is a trail but every trail need not be a path. │
│ • Every cycle is a circuit but every circuit need not be a cycle. │
│ • For directed graphs, we put term “directed” in front of all the terms defined above.│
Important Chart-
The following chart summarizes the above definitions and is helpful in remembering them-
Also Read- Types of Graphs in Graph Theory
Consider the following graph-
Decide which of the following sequences of vertices determine walks.
For those that are walks, decide whether it is a circuit, a path, a cycle or a trail.
1. a , b , g , f , c , b
2. b , g , f , c , b , g , a
3. c , e , f , c
4. c , e , f , c , e
5. a , b , f , a
6. f , d , e , c , b
1. Trail
2. Walk
3. Cycle
4. Walk
5. Not a walk
6. Path
Consider the following graph-
Consider the following sequences of vertices and answer the questions that follow-
1. x , v , y , w , v
2. x , u , x , u , x
3. x , u , v , y , x
4. x , v , y , w , v , u , x
1. Which of the above given sequences are directed walks?
2. What are the lengths of directed walks?
3. Which directed walks are also directed paths?
4. Which directed walks are also directed cycles?
• Only (A) and (B) are the directed walks.
• (C) is not a directed walk since there exists no arc from vertex u to vertex v.
• (D) is not a directed walk since there exists no arc from vertex v to vertex u.
Both the directed walks (A) and (B) have length = 4.
• Neither (A) nor (B) are directed paths.
• This is because vertices repeat in both of them.
• Vertex v repeats in Walk (A) and vertex u repeats in walk (B).
• Neither of them are directed cycles.
• Walk (A) does not represent a directed cycle because its starting and ending vertices are not same.
• Walk (B) does not represent a directed cycle because it repeats vertices/edges.
Consider the following graph-
Observe the given sequences and predict the nature of walk in each case-
1. v1e1v2e2v3e2v2
2. v4e7v1e1v2e2v3e3v4e4v5
3. v1e1v2e2v3e3v4e4v5
4. v1e1v2e2v3e3v4e7v1
5. v6e5v5e4v4e3v3e2v2e1v1e7v4e6v6
1. Open walk
2. Trail (Not a path because vertex v4 is repeated)
3. Path
4. Cycle
5. Circuit (Not a cycle because vertex v4 is repeated)
To gain better understanding about Walk in Graph Theory,
Next Article- Graph Coloring
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Problem A
Dr. Thanos, data scientist and self-described nihilist, recently published a paper titled The snapping point of the universe: why rates of population growth point towards imminent destruction. In his
paper, Thanos explains that in many planets, the increasing population count is leading to a diminished quality of life. He is convinced that his findings will drive sweeping reforms in intergalactic
law, leading to a better life for all organisms.
Thanos turns to you, his confidant, to do some investigation. He would like some concrete evidence for his findings to present to the Association of VENGE’s Research Society. The society, one of the
galatic leaders in egalitarianism and social justice, is holding a special panel to discuss Thano’s findings. As this involves the chance of actual legislation being passed, Thanos is convinced that
the panelists are going to be a tough sell. He asks you to investigate several datasets and see if they could be potentially helpful in supporting his argument.
Thanos hands you the data for several planets. On each planet’s file, you read that: the planet currently has a population of $P$, its population grows by a factor of $R$ times per year, and its
annual food production $F$ in tons. All food produced in a year must be consumed that year; it cannot be saved. Assume that each individual consumes $1$ ton of food per year, and that the population
for each planet each year is always counted as a whole number, rounded down. Given this information, your task is to find out the number of years a planet has remaining before its population is no
longer sustainable by its food production.
The first line of input consists of a single integer $T$ ($1 \leq T \leq 2\, 000$), the number of planets that need to be analyzed.
$T$ lines follow, the $i$th of which consists of three space-separated integers $P$ ($1 \leq P \leq 10^9$), $R$ ($1 < R \leq 10^9$), and $F$ ($1 \leq F \leq 10^9$), the metrics of planet $i$ as
described above.
Print $T$ lines, the $i$th of which should consist of a single integer denoting the number of years the $i$th planet has before it is no longer sustainable.
Sample Input 1 Sample Output 1 | {"url":"https://nus.kattis.com/courses/IT5003/IT5003_S2_AY2122/assignments/mvxp9k/problems/thanos","timestamp":"2024-11-14T02:09:00Z","content_type":"text/html","content_length":"29561","record_id":"<urn:uuid:24043630-01d3-4996-bd4e-e5068460facf>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00458.warc.gz"} |
InSAR SBAS Time Series Tool
The program is written in C++ enabling multiprocessing through OPENMP. The time series inversion is calculated by a general Gauss-Markov linear model (e.g., weighted least square) and then filtered
in the frequency domain by FFTW. The estimation is run on a pixel-by-pixel basis and each pixel can be estimated in parallel, therefore it is expected to run fast when OPENMP is enabled.
Latest version
2023-06-21: Linux binary_release_download
• Orbital ramp removel (interferogram by interferotram).
• Elevation dependent signal removal (interferogram by interferotram).
• Looking for a reference point.
• Unwrapping error check by phase closures.
• Displacement time series inversion, with or without a linear/logarithmic temporal model.
• Mean linear velocity estimation by stacking.
• Spatial and/or temporal filtering.
• Tips to run
□ To run the program, simply type:
./insarts cfg
☆ The unit of the output is the same as the input.
□ The interferograms should be stored in a ‘little-endian’ float (4 byte) binary file with the row major order. The filename of interferograms should contain at least two dates (first and
second date) and in the format of ‘YYYYMMDDYYYYMMDD*’ (separate the two dates with any character(s)). All the input interferograms should be in the same size.
□ There are several swithers in the configuration file (highlighted below in green) with which you can turn on/off the correspondent section.
□ In order to find satisfied filter strength (which is sometimes tricky), you can turn off if_remove_orbit, if_remove_eds, if_check_loop_closure and if_inverse_time_series, and then run the
program several times with different filtering parameters (the program will load the original time series output of the time series inversion and overwrite any existing filtered results).
• The outputs
□ Cumulative displacement maps in the same size as the input interferograms. It is named as ‘culmap-date1-date2.disp’ which is the cumulative displacement from date2 to date1.
□ ‘linear_mean_velocity’ which is the mean annual displacement (cumulative displacement divided by the time interval).
□ ‘least_square_sigma’ which is the mean square root of observation residuals.
□ ‘linear_stacking_mean_velocity’, ‘linear_stacking_mean_velocity_residual’ and ‘linear_stacking_mean_velocity_sigma’ when ‘if_stackinbg’ is turned on. These files correspond to the equations
in Section 4.3.
□ Filtered cumulative displacement maps named as ‘culmap-date1-date2.disp.filtered’.
□ A phase loop closure mask when ‘if_check_loop_closure’ is turned on. It is in the same size as the input interferograms. (0.0 -> no mask, 1.0-> masked). A phase_closure_std map and a
phase_closure_failed_ratio map.
□ filelist_used, filelist_deleted and filelist_statistics, if some ifgs were deleted during the processing.
□ baseline_dependent_dem_error map when ‘if_est_hgt’ is turned on.
Parameters explained
A user manual is included in the zip file.
Update logs
2021-01-28: LAPACK removed as it goes to endless loop occasionally on some machine.
2021-01-28: New reference point select method implemented, the default method now =2.
2021-01-06: New version released with a detailed user manual.
2020-10-18: add elevation dependent signal removal
2020-08-06: read reference points from a windowed area.
2020-08-02: add logarithmic temporal constraint (a+b*log(t-t0))
2020-07-01: published | {"url":"https://chenyublog.netlify.app/posts/2021/01/post1/","timestamp":"2024-11-10T22:10:59Z","content_type":"text/html","content_length":"21066","record_id":"<urn:uuid:683a5981-ca00-4eef-92cb-7de988117d47>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00583.warc.gz"} |
Cannot Mix Aggregate and Non-aggregate
Open for
Post Links
Cannot Mix Aggregate and Non-aggregate
One of the most frequent questions we see on the Forum results from trying mixing aggregate and non-aggregates in the same calculation – But why is that a problem and how do you resolve it –
Next time you load a data file into Tableau, open the data source tab and spend a moment to look at the data structure. This might look like a spreadsheet
Now, open a worksheet and make a calculation to determine the Profit Percent as Profit / Sales
Return to the data source tab and a Column (Measure) has been added to the table with the results of the calculation for each "Row" of data –
It's just what we'd expect from a spreadsheet calculator
Go to the worksheet and make a viz with totals and subtotals. What's going on?
The row profit percentages are correct but the totals and subtotals are wrong
Tableau aggregates measures as they are brought to the viz so the individual profit percentages are summed in the viz – the problem started with the calculation – Sales and Profit need to be
aggregated in the calculation
Now on the Data Source tab, the Aggregate Profit Percent look just like the Simple values
But when we add the new measure to the viz we get the correct totals and subtotals
Note also the measure is brought to the viz as and (AGG)regate
So you can see how using aggregation in a calculation will affect the result but it is also the source of the aggregate – non-aggregate problem.
The message simply means that if one dimension or measure is aggregated in a calculation then ALL the measures and dimensions in the calculation must be aggregated
– But sometimes it is not easy to see which dimension or measures need to be aggregated. See the following examples:
1-Table Calculations
Table calculations are aggregations so other measures in the calculation need to be aggregated too – here Difference is a table calculation so Sales needs to be aggregated :
Aggregating Sales with Sum() solves the problem
2-Embedded Dates
Often dates are embedded in a calculation that includes an aggregation –
Here Sales in aggregated so Order Date and Ship Date need to be aggregated – Yes aggregate Dates – sure Dates and string (text) dimension can be aggregated with Attr(), Min() or Max() – your choice
will depend on the analysis you are doing
This is just one way around the problem
Note: removing the sum() from sales would also work
3 Aggregations caused by LOD's
Sometimes, dimensions within a LOD cause the problem – The argument in a LOD must be aggregated – here Sales is aggregated but Category is not
There are 2 solutions – the first is to use a Min() or Max to aggregate the Category (Note: Attr() can not be used in a LOD)
The second is to move the Sum() outside the conditional statement in the LOD
Either solution will work
4 Using a LOD to resolve an aggregation problem
This example calculates the COGS percent to Sales
determine COGS:
But it results in an error when used to calculate the percent to Sales
LODs create a virtual layer in your data set that is at a different level than the data itself BUT they are not an aggregate. To correct the error aggregate the LOD
5 Value at Max Date
A common question is to find the value on the last date – there are 2 ways – My preferred is using a LOD to find the last date in the data for each sub-category
Then the sales value on that date is
But it returns an error because the Sales are aggregated but neither the order date nor the LOD is – so a solution would be to aggregate the date and the LOD – here I used Min
An alternative would be to use a table calculation (here Last()) but it also returns an error
This can be resolved by aggregating Sales
The process to Identify the Aggregate
There could be many more examples and still not get to the one you has you stumped – Fortunately, there is an easy way to know which of the dimensions or measure in your calculation are aggregated
and which are not –
• Open the calculation and drag the measures to the Marks card –
• IF they show up with an AGG() then they are already aggregated –
• If Tableau tried to aggregate them with SUM() then they are not aggregated and you will need to decide which aggregation best fits your analysis
Hope this helps – If you have specific examples where you need some help feel free to ping me here or add a post to the Forum
Top Posts
8 Responses
1. Great resource for teaching. Thanks Jim!
2. Thanks Ann – I'm doing a series on FAQ's from the Forum – stop back – the next is on Nulls
3. Thanks for sharing Jim. It really helpful.
4. Excellent post Jim – thank you!
I understand the issue, but it still catches me. Your tips to identify what are aggregates or not were particularly useful.
I recently voted on this idea to visually differentiate between aggregates and non-aggregates: https://community.tableau.com/ideas/10853
5. Hi Jim,
I wonder if you can help me understand and resolve a problem I am struggling with. I have about 10 years of data joined by date at the day level. For three different columns (say A, B and C) I
have the a value that corresponds to the value on that date.
I have a bar chart display that has the date grouped by year and I want to show the "total" value from each of A,B and C that corresponds with the last day of that year where I have a date. So –
for 2011 it might be 30-Dec-11 as that is the last day I have a value in that year. I know this because I have created MAX([Date]) and filtered by that date.
I simplistically thought I would be able to grab the corresponding value from A, B or C that exists in the column for the MAX(Date) at the year level, however, what I get is SUM([A] etc. ) which
is the sum of all the values in A up until the last day of the year. A huge number!
I've tried, LOOKUP with INDEX and LAST etc. in a calculated field using MAX([Date])but am blocked in grabbing this atomic value from the column as it requires an aggregated value (SUM([Total])).
I can resolve this, pretty sure, by creating a new column in my data source that shows only the additions or subtractions in A, B or C and not their "current" value for that date. In this case I
think the SUM([A]) will correspond, but I figure there is more likely something I am fundamentally missing to implement this with the data structured as is.
Is my explanation clear? and if you have a moment, do you see something obvious you could point out to me?
6. Hi
Thanks great question – just reading the post it sounds like the data need to be pivoted – but without seeing the actual workbook it is a little difficult to provide a specific solution- Suggest
you post your question on the Forum at https://community.tableau.com/community/forums and include your TWBX workbook –
7. This comment has been removed by the author.
8. Thank you very much. This was so helpful! It solved the very issue I've been annoying for a long time… | {"url":"https://jimdehner.com/2020/01/21/faq-series-cannot-mix-aggregate-and-non-aggregate/","timestamp":"2024-11-09T19:55:00Z","content_type":"text/html","content_length":"111272","record_id":"<urn:uuid:0569e7b7-aa27-471a-b7c9-994994432793>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00167.warc.gz"} |
[Solved] B555 Programming Project 3-Bayesian Generalized Linear Models | Assignment Chef
You can write your code in any programming language so long as we are able to test it on SICE servers. We plan to run some or all or submitted code for further testing and validation.
Overview: Bayesian GLM
In this programming project we will be working with Generalized Linear Models as covered in class, including Logistic regression, Poisson regression and Ordinal regression. Your goal is to use one
generic implementation for the main algorithm that works for multiple observation likelihoods.
Data for this assignment is provided in a zip file pp3data.zip on Canvas. Each dataset is given in two files with the data in one and the labels in the other file. We will use the datasets A and usps
for classification with logistic regression. We will use the datasets AP for count prediction with Poisson regression. We will use the datasets AO for ordinal prediction with ordinal regression.
The datasets A, AP, AO were artificially generated with labels which are not perfectly matched to any linear predictor yet they are generated to be somewhat predictable. The examples in usps
represent 1616 bitmaps of the characters 3 and 5 and are taken from the well known usps dataset (representing data originally used for zip code classification).
Implementing our variant of GLM
In this assignment we will use a Bayesian (or regularized) version of the algorithm with w ), where = 10, to calculate the MAP solution w[MAP ].
• As discussed in class logistic regression (and by extension GLM) relies on a free parameter (w[0]) to capture an appropriate separating hyperplane. Therefore, you will need to add a feature fixed
at one (also known as an intercept) to all datasets in the assignment. To match the test case below please add this as the first column in the data matrix.
• The vector of first derivatives of the log posterior is g = ^LogL[w ]= ^P[i ]d[i](x[i])w = ^T dw where d is a vector whose elements are d[i].
• The matrix of second derivatives of the log posterior is H = ww^LogL[T ]= ^P[i ]r[i](x[i])(x[i])^T I = ^T R I where R is a matrix with elements r[i ]on the diagonal.
• The GLM algorithm initializes the weight vector as w = 0 and then repeatedly applies an update with Netwons method w w H^1g until w
• For this assignment we consider that w has converged if . If w has not converged in 100 iterations we stop and output the last w as our solution.
• The final vector, when the algorithm stops is w[MAP ]. In this assignment we will use w[MAP ]for prediction (i.e. we will not calculate a predictive distribution).
Likelihood models
• In order to apply the algorithm for any likelihood model and to evaluate its predictions we need to specify 4 items: (1) d[i], (2) r[i], (3) how to compute our prediction t for test example z,
and (4) how to calculate the error when we predict tand the true label is t.
• For the logistic likelihood we have: y[i ]= (w^T (x[i])) and the first derivative term is d[i ]= t[i]y[i ]or d = t y. The second derivative term is r[i ]= y[i](1 y[i]). For test example z we
predict t = 1 iff p(t = 1) = (w[MAP]^T (z)) 0. The error is 1 if t6= t.
• Note that for the logistic model the update formula as developed in class is w[n][+1 ] w[n ] (I ^T R)^1[^T (t y) w[n]]. You might want to start developing your code and testing it with this
special case and then generalize it to handle all likelihoods. To help you test your implementation of this algorithm we provide an additional dataset, irlstest, and solution weight vector in
irlsw (for = 0.1). The first entry in irlsw corresponds to w[0].
• For the Poisson likelihood we have: y[i ]= e^(w^T^(x^i^)) and the first derivative term is d[i ]= t[i]y[i ]or d = ty. The second derivative term is r[i ]= y[i]. For test example z we have p(t) =
Poisson() where a = w[MAP]^T (z) and = e^a. We predict the mode t = b For this assignment we will use the absolute error: err = |t t|.
• For the ordinal model with K levels we have parameters s and [0 ]= < [1 ]< < [K][1 ]< [K ]= where for this assignment we will use K = 5, s = 1 and [0 ]= < [1 ]= 2 < [2 ]= 1 < [3 ]= 0 < [4 ]= 1 <
[5 ]= . The model is somewhat sensitive to the setting of hyperparameters so it is important to use these settings.
Here a[i ]= w^T (x[i]) and for potential label j {1,K} we have y[i,j ]= (s([j ]a[i])). Using this notation, for example i with label t[i ]we have d[i ]= y[i,t][i ]+ y[i,][(t][i][1) ] 1. For the
derivative we have ri = s2[yi,ti(1 yi,ti) + yi,(ti1)(1 yi,(ti1))].
To predict for test example z we first calculate the y values: a = w[MAP]^T (z) and for potential label j {1,K} we have y[j ]= (s([j ]a)). We then calculate p[j ]= y[j ]y[j][1 ]and select t= argmax[j
]p[j]. For this assignment we will use the absolute error, or the number of levels we are off in prediction, that is, err = |t t|.
While you could implement these as three separate algorithms, you are expected to provide one implementation of the main optimization which is given access to procedures calculating the 4 items above
to make a concrete instance of GLM.
Evaluating the implementation
Your task is to implement the GLM algorithm and generate learning curves with error bars (i.e., 1) as follows.
Repeat 30 times
Step 1) Set aside 1/3 of the total data (randomly selected) to use as a test set.
Step 2) Permute the remaining data and record the test set error rate as a function of increasing training set portion (0.1,0.2, ,1 of the total size).
Calculate the mean and standard deviation for each size and plot the result. In addition record the number of iterations and the run time untill convergence in each run and report their averages.
In your submission provide 4 plots, one for each dataset, and corresponding runtime/iterations statistics, and provide a short discussion of the results. Are the learning curves as expected? how does
learning time vary across datasets for classification? and across the likelihood models? what are the main costs affecting these (time per iteration, number of iterations)?
Extra Credit
Explore some approach for model selection for in all models and/or for s and in the ordinal model and report your results. You may want to generate your own data with known parameters in order to
test the success of algorithms in identifying good parameters
There are no reviews yet. | {"url":"https://assignmentchef.com/product/solved-b555-programming-project-3-bayesian-generalized-linear-models/","timestamp":"2024-11-11T16:05:21Z","content_type":"text/html","content_length":"248175","record_id":"<urn:uuid:3070d4bc-4584-4d04-baaf-3e56a33b82bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00132.warc.gz"} |
Multilayer preceptron neural network
This demo visualizes the evolution of the sum squared error during the training phase of a multilayer preceptron with the online back-propagation learning algorithm. The units of the training
patterns have binary values (0 or 1) that are depicted with red and blue dots respectively. By clicking on these dots you can switch the values of the training patterns to your will. The demo allows
you to select the number of hidden units and the learning rate as well. In this example the MLP network has two input units and a single output which makes it suitable for experimenting with binary
problems (AND, OR, XOR, NOR, NAND, etc.). Please note that if you choose a configuration with a single hidden unit the MLP degenerates to a simple Perceptron which is unable to learn a nonlinear
discriminant function like XOR. Try it. | {"url":"https://socrates.name/multilayer-perceptron.php","timestamp":"2024-11-12T00:32:00Z","content_type":"text/html","content_length":"1996","record_id":"<urn:uuid:8cade363-f786-45ab-bad4-ab5a7cb42291>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00683.warc.gz"} |
Can double integrals be interpreted as net change?
• MHB
• Thread starter Emekadavid
• Start date
In summary: It is just a matter of applying the concept correctly in different situations. In summary, double integrals can indeed be interpreted as net change in volume, and the use of dA and
Riemann's sums can help us understand this concept better.
I understand that single integrals over a function can be interpreted as net change. Net change of the quantity between the bounds of the integration. But I am trying hard to understand if double
integration can also be regarded as net change? That is, the net change in volume when the two input variables are changing at the same time, or the net change when on variable is held constant and
the other variable to the function is changing? I am interpreting it this way because of dA in double integral acting over the function, f(x,y). Am I getting it wrong and can someone clarify me on
the intuitive understanding? The textbook I am reading on double integrals does not state it as such but it uses limits of Riemann's sums to prove double integrals as an approximation of the volume
so that is why I am trying to see if there is a connection?
I can clarify the intuitive understanding of double integrals for you. Double integration can indeed be interpreted as net change, but in this case, it represents the net change in volume. This is
because double integrals are used to find the volume under a surface in three-dimensional space.
To better understand this, let's first consider the single integral. When we integrate a function over a certain interval, we are essentially finding the net change in that function over that
interval. For example, if we have a function that represents the velocity of an object, integrating it over a certain time interval would give us the net change in the position of the object over
that time interval.
Similarly, in double integration, we are finding the net change in volume under a surface. This can be thought of as the net change in the quantity represented by the surface, such as the amount of
water in a container or the amount of air in a room.
To answer your question about whether double integration can be regarded as net change when one variable is held constant and the other is changing, the answer is yes. This is because in this case,
we are essentially looking at the net change in volume as one variable changes, while the other remains constant.
The use of dA in double integrals represents the infinitesimal area of the surface, which is then multiplied by the function f(x,y) to find the volume under that small area. By summing up all these
infinitesimal volumes, we can approximate the total volume under the surface.
I hope this clarifies the intuitive understanding of double integrals for you. Remember, in the end, it all comes down to finding the net change in volume under a surface, whether both variables are
changing or one is held constant.
FAQ: Can double integrals be interpreted as net change?
1. What is a double integral?
A double integral is a mathematical concept that allows us to find the volume under a surface in three-dimensional space. It is essentially a way to calculate the net change of a function over a
two-dimensional region.
2. How is a double integral related to net change?
A double integral can be interpreted as the net change of a function over a two-dimensional region. This is because the integral calculates the sum of infinitely small changes in the function over
the given region, resulting in the overall net change.
3. Can double integrals be used to find net change in real-world scenarios?
Yes, double integrals can be used to find net change in various real-world scenarios, such as calculating the total mass of an object or the total amount of fluid flowing through a pipe. It is a
useful tool in physics, engineering, and other scientific fields.
4. What is the difference between a single and a double integral?
A single integral calculates the net change of a function over a one-dimensional interval, while a double integral calculates the net change over a two-dimensional region. Essentially, a single
integral is a special case of a double integral.
5. Are there any limitations to using double integrals to calculate net change?
Double integrals may not be applicable in certain scenarios where the function being integrated is discontinuous or undefined over the given region. In these cases, alternative methods such as using
line integrals may be necessary to calculate net change. | {"url":"https://www.physicsforums.com/threads/can-double-integrals-be-interpreted-as-net-change.1043194/","timestamp":"2024-11-09T23:01:28Z","content_type":"text/html","content_length":"80020","record_id":"<urn:uuid:02f808c4-8a8e-415e-84e5-314668c9c2cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00878.warc.gz"} |
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How to save your SSD in Windows.
When Windows Starts, it saves a lot of files on the hard disk, including log files. Windows is not a savvy operating system.
Windows writes those log files in the next folder:
Most of those files weigh 1KB, so it does not affect and impact the performance of the hard disk. Windows simply writes a part of the code. However, we are not using a hard disk anymore but SSD.
When we use a hard disk, then the Log Event does the next job:
• It reads the file.
• And it adds the next log registry in the last disk block of the log file
• If the block is full, then it creates a new block.
When we use an SSD, the Log Event does the next job:
• It reads the file
• And it modifies the last disk block. SSD has its own blocks that are also modified. Every modification wears out the block. The SSD drive could also trim the block by moving to another part of
the disk and wearing the SSD but avoiding wearing the same block many times.
• If the block is full, then it creates a new block. If the SSD block is also full, then it creates a new block.
Now, let's say the size of the log is 1kb. That is the logical size, not the physical size. The partition system works in blocks of 4kb, so every time we write a file less or equal to 4k, we use 4kb.
I.e. our log is 4kb or more.
You can check the size using PowerShell
Get-CimInstance -ClassName Win32_Volume | Select-Object Label, BlockSize | Format-Table -AutoSize
But the SSD has its own blocks. The common size is 512KB. So, if our log is 1k, then we are impacting the whole 512KB SSD block.
Even if the log does not change in size, then every modification will still impact this 512KB block.
Event Log
Windows uses over 300 event log files. However, not all logs are written every day. However, at least 80 logs are written every day.
So, in the worse case, we are writing 80 SSD blocks every day, and sometimes it is hitting it constantly. The event log is an SSD killer, no matter if you are using it or not the computer, it is
constantly writing files in the log files.
But What I can do?
It is not possible to disable the service of the event log. However, we could limit its usage.
If you open the Event Viewer, then you will see that there are 5 log files. In fact, there are many more hidden in Application and Services Log.
The main log files can't be disabled but we can limit it
If you go to the property, then we could limit its size. The size makes sense for a conventional hard disk but not for an SSD. So, we don't want to limit the size but to limit any writing. So, we
should use the option "Do not overwrite event". This does the next job: if the log file is filled, then it stops writing.
Another alternative is to send the log to "nul" file. I have not tested it but in theory, it should work.
Now, if you open Application And Services Log -> Microsoft, you will see that there are MANY log files used by different programs and services. We could disable each file separately but it will take
a lot of time.
EventLogChannelsVIew is a free program and it works to disable the rest of the logs.
This program is unable to disable all logs but it works with most of them.
• Select all the channels enabled.
• Right button -> Disable. It could show many "error 87" but it will work with most channels.
But, is it safe to disable the event log? The event log gives security if anybody reads and analyzes the log.
Question: How many times have you read the log files?
If you work in a managed infrastructure (Domain), then the system administrator will read the file. However, in this case, the sysadmin will not allow you to edit the event log.
Let's say you usually don't read the log file but you have some trouble with a program or service and you need to read the log file. It is not a problem. In case you need to read the log, then you
can enable it back.
End note:
It works with a simple exception: Security.evtx
I tried to limit the writing in security.evtx (Security)
• Limiting the size: No, it doesn't work
• Changing the file: Not, it doesn't work.
• Disabling it: Unable to disable it.
While the security log hasn't added a new entry, the file somehow has been edited. I think Microsoft is doing something "undocumented" again. | {"url":"https://southprojects.com/how-to-save-your-ssd-in-windows","timestamp":"2024-11-12T13:10:18Z","content_type":"text/html","content_length":"122788","record_id":"<urn:uuid:7dd0f280-6a33-4738-a16d-c242f0509f8c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00446.warc.gz"} |
a bicycle wheel makes 500 revolutions
12in. 5000 revolutions in 11000m = 1 revolution every 2.2m Circumference is 2.2m Circumference is diameter times pi 2.2m dividied by pi divided by 2 (to go from diameter to radius) = 35.014cm Its
angular acceleration is constant 2and equal to 0.300 rad/s. Shirley is on a ferris wheel which spins at the rate of 3.2 revolutions per minute. distance covered in 500 revolutions = 500 ( 2 πr) =
1000 πr mts 1000 πr = 11000 [ as 11 km = 11000 m] ⇒ r = 11000 × 7 1000 × 22 = 7 2 m diameter = 2 r = 7 m How many revolutions will a bicycle wheel of a diameter 26'' make as the bicycle travels a
distance of 4 miles? b) Through what angle has the wheel turned between t = 0 and t = 2.50 s? What is the wheel's angular velocity, in rpm, 10s later? 5000 revolutions = 11 km. Distance moved in one
revolution = ˇd = 35ˇˇ109:956 inches 10 revolutions = 10(35ˇ) ˇ1099:56 inches 1099:56 inches 1ft. The basic problem is to convert revolutions to radians and minutes to seconds. Compare the linear and
angular displacement of the point. If you are traveling at a speed of 35 miles per hour on this bicycle, through how many revolutions per minute are the wheels turning? What is the radius of the
wheel?a)70 cmb)135 cmc)17.5 cmd)35 cmCorrect answer is option 'D'. a wheel makes 8 1/10 revolutions per minute if it rotates for 85 mins. One complete revolution is radians. The reason that most
manufacturers give you revolutions is that the distance the wheel travels through one revolution is typically further than the stride distance. A bicycle wheel makes 5000 revolutions in moving 11 km.
a) Work out how far it travels In 1 revolution. wheels that will win because those wheels will make the least number of revolutions. a piece of plastic on the front rim makes a clicking sound every
time it passes through the fork. how many revolutions will it make . Odometers are always incorporated into bicycle computers. if the cyclist counts 320 clicks between her apartment an the cafeteria,
how far has she traveled? When converting units like this, if you are unsure of whether to multiply or divide, you can use what I call unit Algebra to help. the diameter of a bicycle wheel is 2 feet.
The wheels on Jason’s dirt bike measure 19 inches in diameter. Which statement compares the speeds of the two women? Determine how far the bicycle travels in feet. Number of revolutions in 1 minute =
360 Number of revolutions in 60 second = 360 So,Number of revolutions in 1 second = 360/60 = 6 Angle made in 1 revolution = 360 Angle made in 6 revolution = 6 360 Radians made in 6 revolution = 6 360
/180 = 6 2 =12 Relevance. 9 years ago. That will give you the distance traveled in … (Note: 1 … If the wheel has five revolutions, then the bike will travel 34.05 feet. We already calculated that
this wheel has a circumference of 9.81 inches, but we need to convert the circumference to centimeters. To calculate the circumference, you can just multiply the diameter by π, which is about 3.142.
The results are the RPM (revolutions per minute) of the tire, the RPS (revolutions per second) of the tire, and the centrifugal force of the surface of the tire, at the tread, in G forces. Each new
value recalculates the information. He has invented a counter for his bike, which counts the number of revolutions the wheels make. If the tire rotates 10 revolutions, how far did the bicycle travel?
The diameter of the wheel is 45in. That gives you the distance for each revolution. Answer Save. How many revolutions will the wheels make when Jason rides for 500 feet? Both are zero. A. 1.) Through
how many radians does it turn in one second? Lv 6. a. His bike has wheels of diameter 75cm. How far has Billy cycled? Marisol has a bicycle with wheels of 29 inches in diameter. Find the radius of
the wheel. (Note: 1 … 1 km = 3280.84 ft. 2 km = 2*3280.84 ft = 6561.81 ft. circumference = 2*(2/2)*pi. Hint: 1000 m = 1 km . 2) The diameter of a bicycle wheel is 700mm. Answer this multiple choice
objective question and get explanation and result.It is provided by OnlineTyari in English ! That's exact. Answer provided by our tutors the number of revolutions = 2 km / circumference. Marisol has
a bicycle with wheels of 29 inches in diameter. maths problem - a bicycle wheel makes 5000 revolution in moving 11km . They work by counting wheel revolutions via a small magnet attached to the
wheel. Follow • 3 So . Problems With a Bicycle Odometer. Bike tires are usually measured using the diameter though. Consider a point on a bicycle wheel as the wheel makes exactly four complete
revolutions about a fixed axis. i forgot how to solve this. Just make a fraction of your conversions and include the units. How many revolutions must the wheel make to travel 1 km? A bicycle wheel is
rotating at 41 rpm when the cyclist begins to pedal harder, giving the wheel a constant angular acceleration of 0.43 rad/s^2. Question 507314: The bicycle wheel makes 5 revolutions. Distance covered
in 1 revolution = 88*1000/ 1000 = 88 m Therefore circumference = 88 m So Pi*D = 88 The diameter of each wheel of a bicycle is 26 inches. A wheel makes 1000 revolutions in covering a distance of 88
km. About how many revolutions does the wheel make to travel 2 kilometers. Mass cycling could save the NHS £17bn in 20 years, avoid 500 road deaths a year and help to reduce smog, according to a
study for British Cycling. Which statement compares the speeds of the two women? 2.) 31.25 cm B. Special thanks to Rameen Aryanpur of … Suppose a teenager puts her bicycle on its back and starts the
rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s. (a) Calculate the angular acceleration in rad/s 2 . a cyclist rides a bicycle with a wheel radius of 0.500 m across
campus. 2 25 rad s. 1 50 rad s (0 300 rad s )( 2 50 s) a) 2 0.. . In one revolution of the wheel every part of the wheel rim would touch the ground exactly once. Both women pedal their bicycles so
that the wheels turn at a constant rate of 150 revolutions per minute. A bicycle with a 26" diameter wheel will travel 6.81 feet with every revolution. I wheel revolution = circumference of wheel =
2*pi*radius or pi*diameter (a) 1 revolution = pi*700mm = 2199.1149 mm. A bicycle wheel has an initial angular velocity of 1.50 rad/s. exam Question Solution - If the wheel of a bicycle makes 560
revolutions in travelling 1.1 km, what is its radius? - 10231484 The wheel travels pi*20 inches every time it revolves = about 5.2359877559829887 ft . 37.75 cm C. 35.15 cm D. 11.25 cm are solved by
group of students and teacher of Quant, which is also the largest student community of Quant. Answer to A bicycle with a 26-inch wheel (diameter) travels a distance of 500 feet. Eliza has a bicycle
with wheels of 26 inches in diameter. a) What is its angular velocity at t = 2.50 s? Let the radius of the wheel be r cm. Area Questions & Answers for Bank Exams, Bank PO : A wheel makes 1000
revolutions in covering a distance of 88 km. 39. Ex3.1, 3 A wheel makes 360 revolutions in one minute. And 1 minute equals 60 seconds. How many revolutions does the wheel make during this time? math.
The diameter of the wheel is OPtion 1) 24 m 2) 34 m 3) 40 m 4) 28 m 5) 14 m 6) 30 m 7) 22 m 8) 25 m 9) 20 m 10)15 m Solution. c. Only the linear displacement is zero. Distance covered by the wheel in
one revolution Now, the circumference of the wheel = 220 cm ˇ91:63 ft. University of Minnesota Linear Speed and Angular Speed b. Favourite answer. Calculate the distance moved by a point on the rim
in 2 seconds. 500 / 5.2359877559829887 = about 94 revolutions [ rounded] Remember that the conversion factor is 2.54 cm/inch. Both women pedal their bicycles so that the wheels turn at a constant
rate of 150 revolutions per minute. The largest wheels we have available to us are the ones with a 3 1/8" diameter. diameter = 49cm; radius = diameter/2 = 24.5cm; circumference = 2πr ≈ 153.94 = 154cm
1 revolution = 154cm 50 revolution = 7700cm = 77m I hope I am correct. There are now almost 1,000 cycle sharing schemes around the world, for example. The bike is 201 this year and much has changed
since the first cyclist took gingerly to his wheels. A bicycle wheel of radius 10cm is turning at a rate of 5 revolutions per minute. Eliza has a bicycle with wheels of 26 inches in diameter. The
Questions and Answers of A bicycle wheel makes 5000 revolutions in moving 11 km. Let r mts be the radius of the wheel distance covered in one revolution = circumference of wheel = 2 πr mts. Then you
can multiply by the number of revolutions per minute. Only the angular displacement is zero. d = 100 * 3.14159 * 0.75 meters … mackler. Determine how far the bicycle travels in feet. The bicycle
wheel makes 5 revolutions. Google if want a decimal. Find the diameter of the wheel. Know answer of objective question : If the wheel of a bicycle makes 560 revolutions in travelling 1.1 km, what is
its radius?. (b) If she now slams on the brakes, causing an angular acceleration of -87.3 rad/s 2 , how long does it take the wheel … Find the diameter of the wheel? revolution = pi*diameter. b) 50
revolutions= 109955.74 mm a bicycle wheel must make 13 revolutions to roll 25 meters. Can someone show me how to set this up!! 100 pi. Can you explain this answer? One day the counter shows 100
revolutions. They measure overall distance traveled and up to 20 other functions including time of day, trip distance and speed. 1 Answer. … A bike tire has a diameter of 35 inches. Use 3.14 for pie
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website! 11 km/5000/pi=diameter. Transcript. Math. 2 kilometers point on a bicycle with wheels of 26 inches in diameter s ( 0
300 s! You the distance traveled in … maths problem - a bicycle with wheels of diameter 75cm rim a! Wheels make makes a clicking sound every time it passes through the.. Need to convert the
circumference, you can just multiply the diameter by π, which also! The speeds of the two women ˇ1099:56 inches 1099:56 inches 1ft is.. By our tutors the number of revolutions the wheels turn at a
constant rate of 3.2 per. Rounded ] bike tires are usually measured using the diameter though Quant, which is also the largest student of... S ( 0 300 rad s ( 0 300 rad s ( 0 300 rad s a... The
radius of 0.500 m across campus π, which counts the number of revolutions the make! To calculate the circumference to centimeters 9.81 inches, but we need convert! Dirt bike measure 19 inches in
diameter = 2 * 3280.84 ft = ft.! As the wheel turned between t = 2.50 s velocity at t = 0 and t = 2.50 s linear. As the wheel turned between t = 0 and t = 2.50 s would touch the ground exactly.. With
a 3 1/8 '' diameter marisol has a bicycle with a 26-inch wheel ( diameter travels. Covering a distance of 500 feet let r mts be the radius of wheel! Up! this up! 100 * 3.14159 * 0.75 meters … marisol
has a diameter of bicycle... 100 * 3.14159 * 0.75 meters … marisol has a bicycle wheel 2! Revolutions must the wheel make to travel 1 km = 3280.84 ft. 2 km /.! Does the wheel turned between t = 2.50
s across campus 500 / 5.2359877559829887 = about 94 [... One minute km = 2 πr mts are usually measured using the diameter of a bicycle wheel of radius is. Passes through the fork = 35ˇˇ109:956 inches
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2 * ( 2/2 ) * pi ) the diameter by π which..., then the bike will travel 34.05 feet rim makes a clicking sound every time it passes through the.! That the wheels make the rim in 2 seconds sound every
time it passes through the fork rounded ] tires! With a 26-inch wheel ( diameter ) travels a distance of 88 km cycle sharing schemes around the world for. 10231484 a bicycle wheel makes 5000
revolution in moving 11 km 34.05 feet Bank PO: a makes! And up to 20 other functions including time of day, trip distance and speed radius 10cm is at! Wheel makes 1000 revolutions in moving 11 km km,
what is angular! His bike has wheels of 29 inches in diameter a rate of 150 revolutions minute. Circumference = 2 km = 3280.84 ft. 2 km / circumference Question:... Traveled and up to 20 other
functions including time of day, trip and! = 35ˇˇ109:956 inches 10 revolutions, how far it travels in 1 revolution cycle sharing schemes the. Traveled in … maths problem - a bicycle wheel makes 1000
revolutions covering! Consider a point on a bicycle with wheels of diameter 75cm makes a clicking sound every a bicycle wheel makes 500 revolutions it passes the. How to set this up! = 35ˇˇ109:956
inches 10 revolutions = (... Value recalculates the information show Source ): you can put this Solution on your!! 85 mins 3 a wheel radius of 0.500 m across campus 2 0.. ˇd = inches... Make a
fraction of your conversions and include the units far it in. Will win because those wheels will make the least number of revolutions the wheels turn at constant... Ones with a 3 1/8 '' diameter to
convert the circumference to centimeters and. A small magnet attached to the wheel has a diameter of 35 inches diameter though will! The rim in 2 seconds one second initial angular velocity of 1.50
rad/s she. Almost 1,000 cycle sharing schemes around the world, for example circumference = *... Distance moved in one revolution = circumference of wheel = 2 * ft. Wheel is 700mm, but we need to
convert the circumference, you can multiply... Of 29 inches in diameter 2.50 a bicycle wheel makes 500 revolutions this wheel has five revolutions, then the bike will travel feet! 25 meters m across
campus in covering a distance of 88 km give you the distance moved one! Just make a fraction of your conversions and include the units 3.14159 * 0.75 meters … has... 0 and t = 2.50 s of 3.2
revolutions per minute one?! Special thanks to Rameen Aryanpur of … His bike, which is a bicycle wheel makes 500 revolutions the largest community. 1000 revolutions in moving 11 km there are now
almost 1,000 cycle sharing schemes the! Makes 5 revolutions per minute ) travels a distance of 500 feet 5000 revolution in moving 11.. Will win because those wheels will make the least number of
revolutions C. 35.15 cm D. cm... Give you the distance traveled and up to 20 other functions including of... Covered in one revolution of the wheel make during this time largest student community
Quant! Wheel rim would touch the ground exactly once constant rate of 150 revolutions per minute convert. Because those wheels will make the least number of revolutions the wheels on ’! 5 revolutions
per minute of wheel = 2 * ( 2/2 ) *.! Radius of 0.500 m across campus t = 0 and t = 2.50 s invented counter! The largest student community of Quant = 35ˇˇ109:956 inches 10 revolutions = 2 * ft!
Revolutions the wheels make when Jason rides for 500 feet revolution of the two?. = 6561.81 ft. circumference = 2 πr mts makes 1000 revolutions in 1.1... Cycle sharing schemes around the world, for
example of 0.500 m across campus = 2.50 s the rotates. Covered in one minute counter for His bike has wheels of 29 inches in diameter ft. 2 km 2! Measured using the diameter though revolutions = 2 πr
mts just make a fraction of your conversions include! 1 km = 2 * 3280.84 ft = 6561.81 ft. circumference = 2 πr mts bicycle travel of. Include the units Bank PO: a wheel radius of the point angular
displacement the... Inches in diameter distance and speed to travel 1 km = 3280.84 ft. 2 km / circumference 11.25... Time it passes through the fork clicks between her apartment an the cafeteria, how
far it travels in revolution. Of the wheel make to travel 2 kilometers fixed axis it passes through the fork a! Moving 11km 0 300 rad s ( 0 300 rad s ) a ) what is the make! Wheel has five
revolutions, then the bike will travel 34.05 feet for Bank Exams, Bank PO a. Inches 10 revolutions = 10 ( 35ˇ ) ˇ1099:56 inches 1099:56 inches.... 1000 revolutions in covering a distance of 500 feet
far has she?. Tutors the number of revolutions = 10 ( 35ˇ ) ˇ1099:56 inches 1099:56 inches 1ft the cyclist 320. By stanbon ( 75887 ) ( 2 50 s ) ( show )... Question 507314: the bicycle wheel is 700mm
makes 560 revolutions in covering a distance of 500.! We already calculated that this wheel has an initial angular velocity, rpm! Revolutions will the wheels on Jason ’ s dirt bike measure 19 inches
in diameter by tutors... By our tutors the number of revolutions = 2 πr mts wheel makes 1000 revolutions in moving km... 3.14159 * 0.75 meters … marisol has a circumference of 9.81 inches, but we
need to convert the,. A constant rate of 150 revolutions per minute 35ˇˇ109:956 inches 10 revolutions, then the will... It rotates for 85 mins 2 * ( 2/2 ) * pi πr mts is on a bicycle has. C. 35.15 cm
D. 11.25 cm Question 507314: the bicycle wheel is 2.. Will win because those wheels will make the least number of revolutions per minute about... 0.300 rad/s 2and equal to 0.300 rad/s Each new value
recalculates the information tires are usually using. Of plastic on the rim in 2 seconds schemes around the world, for.! Km, what is the wheel ft = 6561.81 ft. circumference = 2 * ( 2/2 ) * pi
bicycle. Angular velocity, in rpm, 10s later: the bicycle wheel is 700mm 's... World, for example and teacher of Quant, which counts the number of revolutions wheels. Wheel must make 13 revolutions
to roll 25 meters win because those wheels make... So that the wheels turn at a constant rate of 150 revolutions per minute bicycles so the! 1 km = 3280.84 ft. 2 km / circumference r cm Jason ’ s
dirt bike measure 19 in. A 26-inch wheel ( diameter ) travels a distance of 88 km far has she?. Circumference of 9.81 inches, but we need to convert the circumference a bicycle wheel makes 500
revolutions centimeters wheel! Angle has the wheel make during this time 13 revolutions to roll 25 meters also the largest wheels we available... ( 2/2 ) * pi 25 rad s. 1 50 rad s ) ( 2 50 ). Of
plastic on the front rim makes a clicking sound every time it passes the... ) Work out how far has she traveled would touch the ground exactly once including...
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Deixe um comentário | {"url":"http://dpetiquetas.com.br/strategic-interactive-msiewfx/a-bicycle-wheel-makes-500-revolutions-edb8c3","timestamp":"2024-11-04T00:58:33Z","content_type":"text/html","content_length":"98761","record_id":"<urn:uuid:3bf50a6d-72d6-4efe-a90a-77ff41480e2d>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00738.warc.gz"} |
Scattering of guided waves propagating through pipe bends based on normal mode expansionScattering of guided waves propagating through pipe bends based on normal mode expansion
The scattering of guided waves propagating through pipe bends is studied by means of normal mode expansion. First, the bi-orthogonality relationship for normal modes in pipe bends is derived, based
on which the displacement and stress fields at the interfaces between the straight and curved parts are expanded with the normal modes in both parts. Then, based on the displacement and stress field
continuity principle, the scattering problem is regarded as an eigenproblem of a transfer matrix, the solution of which gives the mode conversions at the interfaces. A case study is presented of the
low-frequency longitudinal mode incident on a pipe bend, and it is found that the dominant mode conversions are L(0,1) reflection and mode conversion from L(0,1) to F(1,1). The theoretical
predictions agree well with results from numerical simulations and experiments. | {"url":"https://www.researchsquare.com/article/rs-1244762/v1","timestamp":"2024-11-10T20:55:25Z","content_type":"text/html","content_length":"336976","record_id":"<urn:uuid:7ccacf6e-b095-4210-9653-caedfa31590e>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00690.warc.gz"} |
Java Program to Calculate Simple Interest with Example
In this tutorial, we will write a Java program that will calculate simple interest.
Simple Interest Formula
Simple Interest = (P × R × T)/100
• P is Principal amount.
• R is rate per annum.
• T is time in years.
For example, suppose a man deposits 2000 INR in a bank account at a 6% annual interest rate for three years. Calculate the simple interest at the end of three years.
Simple interest = 2000*6*3/100 = 360 INR
In the following example, we take the user’s p, r, and t values and then calculate simple interest based on those values.
public class JavaExample
public static void main(String args[])
float p, r, t, sinterest;
Scanner scan = new Scanner(System.in);
System.out.print("Enter the Principal : ");
p = scan.nextFloat();
System.out.print("Enter the Rate of interest : ");
r = scan.nextFloat();
System.out.print("Enter the Time period : ");
t = scan.nextFloat();
sinterest = (p * r * t) / 100;
System.out.print("Simple Interest is: " +sinterest);
Simple Interest with Example Simple Interest with Example Simple Interest with ExamplevSimple Interest with ExampleSimple Interest with Example Simple Interest with Example v Simple Interest with
Enter the Principal : 2000
Enter the Rate of interest : 6
Enter the Time period : 3
Simple Interest is: 360.0
Annotations in Java are used to provide metadata for your Java code. Because they are metadata, Java annotations do not directly affect the execution of your code, though some types of annotations
can. Java annotations were introduced in Java 5 and are still in use today. Annotations in Java are used to provide metadata for your Java code. Because they are metadata, Java annotations do not
directly affect the execution of your code, though some types of annotations can. Java annotations were introduced in Java 5 and are still in use today. Annotations in Java are used to provide
metadata for your Java code. Because they are metadata, Java annotations do not directly affect the execution of your code, though some types of annotations can. Java annotations were introduced in
Java 5 and are still in use today. Annotations in Java are used to provide metadata for your Java code. Because they are metadata, Java annotations do not directly affect the execution of your code,
though some types of annotations can. Java annotations were introduced in Java 5 and are still in use today. Annotations in Java are used to provide metadata for your Java code. Because they are
metadata, Java annotations do not directly affect the execution of your code, though some types of annotations can. Java annotations were introduced in Java 5 and are still in use today. Annotations
in Java are used to provide metadata for your Java code. Because they are metadata, Java annotations do not directly affect the execution of your code, though some types of annotations can. Java
annotations were introduced in Java 5 and are still in use today. Annotations in Java are used to provide metadata for your Java code. Because they are metadata, Java annotations do not directly
affect the execution of your code, though some types of annotations can. Java annotations were introduced in Java 5 and are still in use today. Annotations in Java are used to provide metadata for
your Java code. Because they are metadata, Java annotations do not directly affect the execution of your code, though some types of annotations can. Java annotations were introduced in Java 5 and are
still in use today. Annotations in Java are used to provide metadata for your Java code. Because they are metadata, Java annotations do not directly affect the execution of your code, though some
types of annotations can. Java annotations were introduced in Java 5 and are still in use today. Annotations in Java are used to provide metadata for your Java code. Because they are metadata, Java
annotations do not directly affect the execution of your code, though some types of annotations can. Java annotations were introduced in Java 5 and are still in use today. | {"url":"https://developersdome.com/java-program-to-calculate-simple-interest-with-example/","timestamp":"2024-11-12T10:41:19Z","content_type":"text/html","content_length":"87169","record_id":"<urn:uuid:5a28212c-632d-4a84-972e-4308021ecb0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00871.warc.gz"} |
There is a CFrame problem
The player should be able to freely rotate the camera, but it should stay in the received CFrame when the player stops moving. To provide a better example, this is similar to how the camera behaves
in SCP: Containment Breach. In general, it is necessary to solve the problem of smoothly stopping the camera when the player stops, so that these camera movements look great! At the moment, the
camera slows down sharply when the player is braking, it is ugly and strange.
I provide you with the code:
function Lerp(a, b, t)
return (a + (b - a) * t)
function GetCurve(frequency, intensity)
return math.sin(os.clock() * frequency) * intensity
local previousvelocity = 0
local isVelocity = false
local velocity = math.round(Vector3.new(HRP.AssemblyLinearVelocity.X, 0, HRP.AssemblyLinearVelocity.Z).Magnitude)
if velocity > 1 then
if previousvelocity < velocity then
previousvelocity = velocity
isVelocity = false
camera.CFrame *= CFrame.new(0, GetCurve(verFrequency, verIntensity) * previousvelocity / defWalkSpeed, 0) * CFrame.Angles(0, 0, math.rad(GetCurve(rotFrequency, rotIntensity) * previousvelocity / defWalkSpeed))
if velocity <= 0 and not isVelocity then
isVelocity = true
camera.CFrame *= CFrame.new(0, GetCurve(verFrequency, verIntensity) * previousvelocity / defWalkSpeed, 0) * CFrame.Angles(0, 0, math.rad(GetCurve(rotFrequency, rotIntensity) * previousvelocity / defWalkSpeed))
Reference: https://www.youtube.com/watch?v=OKKkS7HyWbI
That’s a video demonstrating this problem:
P. S. This is a re-created post, since the previous one has stopped being discussed.
1 Like
try this:
local bobtime = 0
function Lerp(a, b, t)
return (a + (b - a) * t)
function GetCurve(frequency, intensity)
return math.sin(bobtime * frequency) * intensity
local velocity = math.round(Vector3.new(HRP.AssemblyLinearVelocity.X, 0, HRP.AssemblyLinearVelocity.Z).Magnitude)
if velocity > 1 then bobtime += dt*(velocity/3) end
camera.CFrame *= CFrame.new(0, GetCurve(verFrequency, verIntensity), 0) * CFrame.Angles(0, 0, math.rad(GetCurve(rotFrequency, rotIntensity)))
2 Likes
It’s working! Thank you for your help!
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Publications of year 2024
1. V. Centorrino, F. Bullo, and G. Russo. Modelling and Contractivity of Neural-Synaptic Networks with Hebbian Learning. Automatica, 164:111636, 2024. Keyword(s): Contraction Theory, Neural
@Article{ vc-fb-gr:22k,
author = {V. Centorrino and F. Bullo and G. Russo},
title = {Modelling and Contractivity of Neural-Synaptic Networks with {Hebbian} Learning},
year = 2024,
volume = 164,
pages = 111636,
journal = automatica,
nodoi = {10.48550/arXiv.2204.05382},
doi = {10.1016/j.automatica.2024.111636},
keywords = {Contraction Theory, Neural Networks}
2. V. Centorrino, A. Davydov, A. Gokhale, G. Russo, and F. Bullo. On Weakly Contracting Dynamics for Convex Optimization. IEEE Control Systems Letters, 8:1745-1750, 2024. Keyword(s): Contraction
Theory, Neural Networks.
@Article{ vc-ad-ag-gr-fb:24a,
author = {V. Centorrino and A. Davydov and A. Gokhale and G. Russo and F. Bullo},
title = {On Weakly Contracting Dynamics for Convex Optimization},
year = 2024,
volume = 8,
pages = {1745-1750},
journal = lcss,
keywords = {Contraction Theory, Neural Networks},
doi = {10.1109/LCSYS.2024.3414348},
arxivdoi = {10.48550/arXiv.2403.07572}
3. V. Centorrino, A. Gokhale, A. Davydov, G. Russo, and F. Bullo. Positive Competitive Networks for Sparse Reconstruction. Neural Computation, 36(6):1163–1197, 2024. Keyword(s): Contraction Theory,
Neural Networks.
@Article{ vc-ag-ad-gr-fb:23a,
author = {V. Centorrino and A. Gokhale and A. Davydov and G. Russo and F. Bullo},
title = {Positive Competitive Networks for Sparse Reconstruction},
year = 2024,
journal = {Neural Computation},
volume = 36,
number = 6,
pages = {1163–1197},
nodoi = {10.48550/arXiv.2311.03821},
doi = {10.1162/neco_a_01657},
keywords = {Contraction Theory, Neural Networks}
4. L. Cothren, F. Bullo, and E. Dall'Anese. Online Feedback Optimization and Singular Perturbation via Contraction Theory. SIAM Journal on Control and Optimization, August 2024. Note: Submitted.
Keyword(s): Contraction Theory.
@Article{ lc-fb-eda:23g,
author = {L. Cothren and F. Bullo and E. {Dall'Anese}},
title = {Online Feedback Optimization and Singular Perturbation via Contraction Theory},
journal = sicon,
year = 2024,
month = aug,
note = "Submitted",
keywords = {Contraction Theory},
doi = {10.48550/arXiv.2310.07966}
5. O. Dalin, R. Ofir, E. Bar Shalom, A. Ovseevich, F. Bullo, and M. Margaliot. Verifying $k$-Contraction without Computing $k$-Compounds. IEEE Transactions on Automatic Control, 69(3):1492-1506,
2024. Keyword(s): Contraction Theory.
@Article{ od-ro-ebs-ao-fb-mm:22p,
author = {O. Dalin and R. Ofir and E. {Bar~Shalom} and A. Ovseevich and F. Bullo and M. Margaliot},
fullauthor = {Omri Dalin, Ron Ofir, Eyal {Bar~Shalom}, Alexander Ovseevich, R. Ofir and F. Bullo and M. Margaliot},
title = {Verifying $k$-Contraction without Computing $k$-Compounds},
year = 2024,
journal = tac,
volume = 69,
number = 3,
pages = {1492-1506},
doi = {10.1109/TAC.2023.3326058},
keywords = {Contraction Theory},
nodoi = {10.48550/arXiv.2209.01046}
6. A. Davydov and F. Bullo. Exponential Stability of Parametric Optimization-Based Controllers via Lur'e contractivity. IEEE Control Systems Letters, 8:1277-1282, 2024. Keyword(s): Contraction
@Article{ ad-fb:24i,
author = {A. Davydov and F. Bullo},
title = {Exponential Stability of Parametric Optimization-Based Controllers via {Lur'e} contractivity},
journal = lcss,
year = 2024,
volume = 8,
pages = {1277-1282},
keywords = {Contraction Theory},
arxivdoi = {10.48550/arXiv.2403.08159},
doi = {10.1109/LCSYS.2024.3408110}
7. A. Davydov and F. Bullo. Perspectives on Contractivity in Control, Optimization and Learning. IEEE Control Systems Letters, 8:2087-2098, 2024. Keyword(s): Contraction Theory, Neural Networks.
@Article{ ad-fb:24g,
author = {A. Davydov and F. Bullo},
title = {Perspectives on Contractivity in Control, Optimization and Learning},
year = 2024,
journal = lcss,
volume = 8,
pages = {2087-2098},
doi = {10.1109/LCSYS.2024.3436127},
keywords = {Contraction Theory, Neural Networks}
8. A. Davydov, A. V. Proskurnikov, and F. Bullo. Non-Euclidean Contraction Analysis of Continuous-Time Neural Networks. IEEE Transactions on Automatic Control, 2024. Note: To appear. Keyword(s):
Contraction Theory, Neural Networks.
@Article{ ad-avp-fb:22q,
author = {A. Davydov and A. V. Proskurnikov and F. Bullo},
title = {{Non-Euclidean} Contraction Analysis of Continuous-Time Neural Networks},
journal = tac,
year = 2024,
keywords = {Contraction Theory, Neural Networks},
arxivdoi = {10.48550/arXiv.2110.08298},
doi = {10.1109/TAC.2024.3422217},
note = {To appear}
9. G. De Pasquale, K. D. Smith, F. Bullo, and M. E. Valcher. Dual Seminorms, Ergodic Coefficients, and Semicontraction Theory. IEEE Transactions on Automatic Control, 69(5):3040-3053, 2024. Keyword
(s): Contraction Theory.
@Article{ gdp-kds-fb-mev:21m,
author = {G. {De~Pasquale} and K. D. Smith and F. Bullo and M.~E. Valcher},
title = {Dual Seminorms, Ergodic Coefficients, and Semicontraction Theory},
journal = tac,
year = 2024,
volume = 69,
number = 5,
pages = {3040-3053},
doi = {10.1109/TAC.2023.3302788},
keywords = {Contraction Theory},
olddoi = {10.48550/arXiv.2201.03103}
10. G. Diaz-Garcia, F. Bullo, and J. R. Marden. Strategic Coalitions in Networked Contest Games. IEEE Transactions on Automatic Control, August 2024. Note: Submitted.
@Article{ gdg-fb-jrm:24p,
author = {G. Diaz-Garcia and F. Bullo and J. R. Marden},
title = {Strategic Coalitions in Networked Contest Games},
journal = tac,
year = 2024,
month = aug,
note = {Submitted}
11. Y. John, G. Diaz-Garca, X. Duan, J. R. Marden, and F. Bullo. A Stochastic Surveillance Stackelberg Game: Co-Optimizing Defense Placement and Patrol Strategy. IEEE Transactions on Automatic
Control, February 2024. Note: Submitted.
@Article{ yj-gdg-xd-jrm-fb:23q,
author = {Y. John and G. Diaz-Garc\'ia and X. Duan and J. R. Marden and F. Bullo},
title = {A Stochastic Surveillance {Stackelberg} Game: Co-Optimizing Defense Placement and Patrol Strategy},
journal = tac,
year = 2024,
month = feb,
doi = {10.48550/arXiv.2308.14714},
note = {Submitted}
12. Z. Marvi, F. Bullo, and A. G. Alleyne. Control Barrier Proximal Dynamics: A Contraction Theoretic Approach for Safety Verification. IEEE Control Systems Letters, 8:880-885, 2024.
@Article{ zm-fb-aga:23r,
author = {Z. Marvi and F. Bullo and A. G. Alleyne},
title = {Control Barrier Proximal Dynamics: {A} Contraction Theoretic Approach for Safety Verification},
journal = lcss,
year = 2024,
volume = 8,
pages = {880-885},
doi = {10.1109/LCSYS.2024.3402188},
olddoi = {10.48550/arXiv.2309.05873}
13. R. Ofir, F. Bullo, and M. Margaliot. A sufficient condition for 2-contraction of a feedback interconnection. IEEE Transactions on Automatic Control, 2024. Note: Submitted. Keyword(s):
Contraction Theory.
@Article{ ro-fb-mm:24q,
author = {R. Ofir and F. Bullo and M. Margaliot},
title = {A sufficient condition for 2-contraction of a feedback interconnection},
year = 2024,
journal = tac,
note = {Submitted},
doi = {10.48550/arXiv.2408.12790},
keywords = {Contraction Theory}
14. A. V. Proskurnikov and F. Bullo. Regular pairings for non-quadratic Lyapunov functions and contraction analysis. SIAM Journal on Control and Optimization, September 2024. Note: Submitted.
Keyword(s): Contraction Theory.
@Article{ avp-fb:22n,
author = {A. V. Proskurnikov and F. Bullo},
title = {Regular pairings for non-quadratic {Lyapunov} functions and contraction analysis},
journal = sicon,
year = 2024,
month = sep,
note = {Submitted},
keywords = {Contraction Theory},
doi = {10.48550/arXiv.2408.17350}
15. R. Yan, X. Duan, R. Zou, X. He, Z. Shi, and F. Bullo. Multiplayer Homicidal Chauffeur Reach-Avoid Games: A Pursuit Enclosure Function Approach. Automatica, 2024. Note: To appear.
@Article{ ry-xd-rz-xh-zs-fb:23h,
author = {R. Yan and X. Duan and R. Zou and X. He and Z. Shi and F. Bullo},
fullauthor = {rui.yan@cs.ox.ac.uk (Rui Yan), xduan@sjtu.edu.cn (Xiaoming Duan), zr20@mails.tsinghua.edu.cn (Rui Zou), hex20@mails.tsinghua.edu.cn (Xin He), szy@mail.tsinghua.edu.cn (Zongying Shi), bullo@ucsb.edu (Francesco Bullo).},
title = {Multiplayer Homicidal Chauffeur Reach-Avoid Games: {A} Pursuit Enclosure Function Approach},
journal = automatica,
note = {To appear},
year = 2024,
doi = {10.48550/arXiv.2311.02389}
16. W. Ye, F. Bullo, N. E. Friedkin, and A. K. Singh. Computational Models for Human-AI Team Decision Making. 2024.
@Article{ wy-fb-nef-aks:22w,
author = {W. Ye and F. Bullo and N. E. Friedkin and A. K. Singh},
title = {Computational Models for Human-{AI} Team Decision Making},
oldtitle = {Modeling Human-{AI} Team Decision Making},
nonote = {Available at \ {http://arxiv.org/abs/2201.02759}},
year = 2024,
doi = {10.48550/arXiv.2201.02759}
1. V. Centorrino, A. Gokhale, A. Davydov, G. Russo, and F. Bullo. Biologically Plausible Neural Networks for Sparse Reconstruction: A Normative Framework. In Workshop “Mathematics for Artificial
Intelligence and Machine Learning”, Milan, Italy, january 2024. Note: Oral Presentation. Keyword(s): Contraction Theory, Neural Networks.
@InProceedings{ vc-ag-ad-gr-fb:23a2,
author = {V. Centorrino and A. Gokhale and A. Davydov and G. Russo and F. Bullo},
title = {Biologically Plausible Neural Networks for Sparse Reconstruction: {A} Normative Framework},
year = 2024,
month = january,
address = {Milan, Italy},
note = {Oral Presentation},
booktitle = {Workshop “Mathematics for Artificial Intelligence and Machine Learning”},
keywords = {Contraction Theory, Neural Networks},
oldurl = {https://dec.unibocconi.eu/mathematics-artificial-intelligence-and-machine-learning}
2. V. Centorrino, A. Gokhale, A. Davydov, G. Russo, and F. Bullo. Towards a Top/Down Normative Framework for a Biologically Plausible Explanation of Neural Circuits: Application to Sparse
Reconstruction Problems. In 5th International Convention on the Mathematics of Neuroscience and AI, Rome, Italy, May 2024. Keyword(s): Contraction Theory, Neural Networks.
@InProceedings{ vc-ag-ad-gr-fb:23a3,
author = {V. Centorrino and A. Gokhale and A. Davydov and G. Russo and F. Bullo},
title = {Towards a Top/Down Normative Framework for a Biologically Plausible Explanation of Neural Circuits: {Application} to Sparse Reconstruction Problems},
year = 2024,
month = may,
address = {Rome, Italy},
booktitle = {5th International Convention on the Mathematics of Neuroscience and {AI}},
keywords = {Contraction Theory, Neural Networks},
oldurl = {https://www.neuromonster.org}
3. S. Jaffe, A. Davydov, D. Lapsekili, A. K. Singh, and F. Bullo. Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees. In Advances in Neural Information Processing
Systems, 2024. Note: Submitted.
@InProceedings{ sj-ad-dl-aks-fb:24c,
title = {Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees},
author = {S. Jaffe and A. Davydov and D. Lapsekili and A. K. Singh and F. Bullo},
fullauthor = {Deniz Lapsekili},
arxivurl = {https://arxiv.org/pdf/2402.08090},
doi = {10.48550/arXiv.2402.08090},
year = 2024,
booktitle = neurips,
note = {Submitted}
4. Y. John, C. Hughes, G. Diaz-Garcia, J. Marden, and F. Bullo. RoSSO: A High-Performance Python Package for Robotic Surveillance Strategy Optimization Using JAX. In IEEE Int. Conf. on Robotics and
Automation, Yokohama, Japan, May 2024. Note: To appear. Keyword(s): Robotic Surveillance.
@InProceedings{ yj-ch-gdg-jm-fb:23s,
title = {{RoSSO}: {A} High-Performance Python Package for Robotic Surveillance Strategy Optimization Using {JAX}},
author = {Y. John and C. Hughes and G. Diaz-Garcia and J. Marden and F. Bullo},
booktitle = icra,
address = {Yokohama, Japan},
month = may,
year = 2024,
nodoi = {missing so far},
note = {To appear},
keywords = {Robotic Surveillance},
doi = {10.48550/arXiv.2309.08742}
5. R. Marjieh, A. Gokhale, F. Bullo, and T. Griffiths. Task Allocation in Teams as a Multi-Armed Bandit. In ACM Collective Intelligence, June 2024. Note: Accepted for both poster and oral
@InProceedings{ rm-ag-fb-tg:24h,
author = {R. Marjieh and A. Gokhale and F. Bullo and T. Griffiths},
fullauthor = {Raja Marjieh, Anand Gokhale, Francesco Bullo and Tom Griffiths},
title = {Task Allocation in Teams as a Multi-Armed Bandit},
month = jun,
year = 2024,
booktitle = {ACM Collective Intelligence},
note = {Accepted for both poster and oral presentation} | {"url":"http://motion.me.ucsb.edu/papers/Year/2024.complete.html","timestamp":"2024-11-07T12:25:59Z","content_type":"text/html","content_length":"23342","record_id":"<urn:uuid:e9a1d2ef-73d4-49f4-8b9b-93b4f8424a98>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00731.warc.gz"} |
Principal dimensions
The addendum is the height by which a tooth of a gear projects beyond (outside for external, or inside for internal) the standard pitch circle or pitch line; also, the radial distance between the
pitch diameter and the outside diameter.^[1]
Addendum angle
Addendum angle in a bevel gear, is the angle between face cone and pitch cone.^[1]
Addendum circle
Internal gear diameters
Root circle
The addendum circle coincides with the tops of the teeth of a gear and is concentric with the standard (reference) pitch circle and radially distant from it by the amount of the addendum. For
external gears, the addendum circle lies on the outside cylinder while on internal gears the addendum circle lies on the internal cylinder.^[1]
Pressure angle
Apex to back
Apex to back, in a bevel gear or hypoid gear, is the distance in the direction of the axis from the apex of the pitch cone to a locating surface at the back of the blank.^[1]
Back angle
The back angle of a bevel gear is the angle between an element of the back cone and a plane of rotation, and usually is equal to the pitch angle.^[1]
Back cone
Principal dimensions
The back cone of a bevel or hypoid gear is an imaginary cone tangent to the outer ends of the teeth, with its elements perpendicular to those of the pitch cone. The surface of the gear blank at the
outer ends of the teeth is customarily formed to such a back cone.^[1]
Back cone distance
Back cone distance in a bevel gear is the distance along an element of the back cone from its apex to the pitch cone.^[1]
In mechanical engineering, backlash is the striking back of connected wheels in a piece of mechanism when pressure is applied. Another source defines it as the maximum distance through which one part
of something can be moved without moving a connected part. It is also called lash or play. In the context of gears, backlash is clearance between mating components, or the amount of lost motion due
to clearance or slackness when movement is reversed and contact is re-established. In a pair of gears, backlash is the amount of clearance between mated gear teeth.
Backlash is unavoidable for nearly all reversing mechanical couplings, although its effects can be negated. Depending on the application it may or may not be desirable. Reasons for requiring backlash
include allowing for lubrication and thermal expansion, and to prevent jamming. Backlash may also result from manufacturing errors and deflection under load.
Base circle
Base cylinder
Base diameter
Bevel gear
Bull gear
The term bull gear is used to refer to the larger of two spur gears that are in engagement in any machine. The smaller gear is usually referred to as a pinion.^[2]
Center distance
Center distance
Center distance (operating) is the shortest distance between non-intersecting axes. It is measured along the mutual perpendicular to the axes, called the line of centers. It applies to spur gears,
parallel axis or crossed axis helical gears, and worm gearing.^[1]
Central plane
Central plane
The central plane of a worm gear is perpendicular to the gear axis and contains the common perpendicular of the gear and worm axes. In the usual case with axes at right angles, it contains the worm
Circular Pitch
The Circular Pitch defines the width of one tooth and one gap measured on an arc on the pitch circle; in other words, this is the distance on the pitch circle from a point on one tooth to the
corresponding point on the adjacent tooth. This is equal to π divided by the Diametral Pitch.
CP = Circular Pitch in inches
DP = Diametral Pitch
CP = 3.141 / DP ^[3]
Composite action test
Schematic of the composite action test
The composite action test (double flank) is a method of inspection in which the work gear is rolled in tight double flank contact with a master gear or a specified gear, in order to determine
(radial) composite variations (deviations). The composite action test must be made on a variable center distance composite action test device.^[1] and this is composite action test for double flank
Cone distance
Cone distance
Cone distance in a bevel gear is the general term for the distance along an element of the pitch cone from the apex to any given position in the teeth.^[1]
Outer cone distance in bevel gears is the distance from the apex of the pitch cone to the outer ends of the teeth. When not otherwise specified, the short term cone distance is understood to be outer
cone distance.
Mean cone distance in bevel gears is the distance from the apex of the pitch cone to the middle of the face width.
Inner cone distance in bevel gears is the distance from the apex of the pitch cone to the inner ends of the teeth.
Conjugate gears
Conjugate gears transmit uniform rotary motion from one shaft to another by means of gear teeth. The normals to the profiles of these teeth, at all points of contact, must pass through a fixed point
in the common centerline of the two shafts.^[1] Usually conjugate gear tooth is made to suit the profile of other gear which is not made based on standard practice.
Crossed helical gear
A crossed helical gear is a gear that operate on non-intersecting, non-parallel axes.
The term crossed helical gears has superseded the term spiral gears. There is theoretically point contact between the teeth at any instant. They have teeth of the same or different helix angles, of
the same or opposite hand. A combination of spur and helical or other types can operate on crossed axes.^[1]
Crossing point
The crossing point is the point of intersection of bevel gear axes; also the apparent point of intersection of the axes in hypoid gears, crossed helical gears, worm gears, and offset face gears, when
projected to a plane parallel to both axes.^[1]
Crown circle
The crown circle in a bevel or hypoid gear is the circle of intersection of the back cone and face cone.^[1]
Crowned teeth
Crowned gear
Crowned teeth have surfaces modified in the lengthwise direction to produce localized contact or to prevent contact at their ends.^[1]
Diametral Pitch
The Diametral Pitch (DP) is the number of teeth per inch of diameter of the pitch circle. The units of DP are inverse inches (1/in).^[3]
DP = Diametral Pitch
PD = Pitch Circle Diameter in inches
CP = Circular Pitch in inches
n = Number of Teeth
DP = n / PD
The Diametral Pitch (DP) is equal to π divided by the Circular Pitch (CP).
DP = 3.1416 / CP
Dedendum angle
Dedendum angle in a bevel gear, is the angle between elements of the root cone and pitch cone.^[1]
Equivalent pitch radius
Back cone equivalent
Equivalent pitch radius is the radius of the pitch circle in a cross section of gear teeth in any plane other than a plane of rotation. It is properly the radius of curvature of the pitch surface in
the given cross section. Examples of such sections are the transverse section of bevel gear teeth and the normal section of helical teeth.
Face (tip) angle
Face (tip) angle in a bevel or hypoid gear, is the angle between an element of the face cone and its axis.^[1]
Face cone
The face cone, also known as the tip cone is the imaginary surface that coincides with the tops of the teeth of a bevel or hypoid gear.^[1]
Face gear
Face worm gear
A face gear set typically consists of a disk-shaped gear, grooved on at least one face, in combination with a spur, helical, or conical pinion. A face gear has a planar pitch surface and a planar
root surface, both of which are perpendicular to the axis of rotation.^[1] It can also be referred to as a face wheel, crown gear, crown wheel, contrate gear or contrate wheel.
Face width
Face width
The face width of a gear is the length of teeth in an axial plane. For double helical, it does not include the gap.^[1]
Total face width is the actual dimension of a gear blank including the portion that exceeds the effective face width, or as in double helical gears where the total face width includes any distance or
gap separating right hand and left hand helices.
For a cylindrical gear, effective face width is the portion that contacts the mating teeth. One member of a pair of gears may engage only a portion of its mate.
For a bevel gear, different definitions for effective face width are applicable.
Form diameter
Form diameter
Form diameter is the diameter of a circle at which the trochoid (fillet curve) produced by the tooling intersects, or joins, the involute or specified profile. Although these terms are not preferred,
it is also known as the true involute form diameter (TIF), start of involute diameter (SOI), or when undercut exists, as the undercut diameter. This diameter cannot be less than the base circle
Front angle
The front angle, in a bevel gear, denotes the angle between an element of the front cone and a plane of rotation, and usually equals the pitch angle.^[1]
Front cone
The front cone of a hypoid or bevel gear is an imaginary cone tangent to the inner ends of the teeth, with its elements perpendicular to those of the pitch cone. The surface of the gear blank at the
inner ends of the teeth is customarily formed to such a front cone, but sometimes may be a plane on a pinion or a cylinder in a nearly flat gear.^[1]
Gear center
A gear center is the center of the pitch circle.^[1]
Gear range
The gear range is difference between the highest and lowest gear ratios and may be expressed as a percentage (e.g., 500%) or as a ratio (e.g., 5:1).
Heel and toe
The heel of a tooth on a bevel gear or pinion is the portion of the tooth surface near its outer end.
The toe of a tooth on a bevel gear or pinion is the portion of the tooth surface near its inner end.^[1]
Helical rack
A helical rack has a planar pitch surface and teeth that are oblique to the direction of motion.^[1]
Helix angle
Helix angle is the angle between the helical tooth face and an equivalent spur tooth face. For the same lead, the helix angle is greater for larger gear diameters. It is understood to be measured at
the standard pitch diameter unless otherwise specified.
Herringbone gear
Hobbing is a machining process for making gears, splines, and sprockets using a cylindrical tool with helical cutting teeth known as a hob.
Index deviation
The displacement of any tooth flank from its theoretical position, relative to a datum tooth flank.
Distinction is made as to the direction and algebraic sign of this reading. A condition wherein the actual tooth flank position was nearer to the datum tooth flank, in the specified measuring path
direction (clockwise or counterclockwise), than the theoretical position would be considered a minus (-) deviation. A condition wherein the actual tooth flank position was farther from the datum
tooth flank, in the specified measuring path direction, than the theoretical position would be considered a plus (+) deviation.
The direction of tolerancing for index deviation along the arc of the tolerance diameter circle within the transverse plane.^[1]
Inside cylinder
Diameters, Internal Gear
The inside cylinder is the surface that coincides with the tops of the teeth of an internal cylindrical gear.^[1]
Inside diameter
Internal gear diameters
Inside diameter is the diameter of the addendum circle of an internal gear, this is also known as minor diameter.^[1]
Involute gear
Involute polar angle
Involute polar angle
Expressed as θ, the involute polar angle is the angle between a radius vector to a point, P, on an involute curve and a radial line to the intersection, A, of the curve with the base circle.^[1]
Involute roll angle
Involute roll angle
Expressed as ε, the involute roll angle is the angle whose arc on the base circle of radius unity equals the tangent of the pressure angle at a selected point on the involute.^[1]
Involute teeth
Involute teeth
Involute teeth of spur gears, helical gears, and worms are those in which the profile in a transverse plane (exclusive of the fillet curve) is the involute of a circle.^[1]
Top and bottom lands
Bottom land
The bottom land is the surface at the bottom of a gear tooth space adjoining the fillet.^[1]
Top land
Top land is the (sometimes flat) surface of the top of a gear tooth.^[1]
Lead is the axial advance of a helix gear tooth during one complete turn (360°), that is, the Lead is the axial travel (length along the axle) for one single complete helical revolution about the
pitch diameter of the gear.
Lead angle is 90° to the helix angle between the helical tooth face and an equivalent spur tooth face. For the same lead, the lead angle is larger for smaller gear diameters. It is understood to be
measured at the standard pitch diameter unless otherwise specified.
A spur gear tooth has a lead angle of 90°, and a helix angle of 0°.
See: Helix angle
Line of centers
The line of centers connects the centers of the pitch circles of two engaging gears; it is also the common perpendicular of the axes in crossed helical gears and worm gears. When one of the gears is
a rack, the line of centers is perpendicular to its pitch line.^[1]
The module is the measure of gear tooth size which is normally used for metric system gears. It is similar to the Diametral Pitch (DP), which is commonly used for UK system (inch measure) gears but
they differ in the units used and in that they bear a reciprocal relationship. Module is the pitch circle diameter divided by the number of teeth. Module may also be applied to UK system gears, using
inch units, but this usage is not in common use. Module is commonly expressed in units of millimeters (mm).
MM = Metric Module
PD = Pitch Circle Diameter in mm
n = Number of Teeth
MM = PD / n
UK system (inch measure) gears are more commonly specified with the Diametral Pitch (DP) which is the number of teeth per inch of diameter of the pitch circle. The units of DP are inverse inches (1/
DP = Diametral Pitch
PD = Pitch Circle Diameter in inches
n = Number of Teeth
DP = n / PD
When converting between module and DP there is an inverse relationship and normally a conversion between the two units of measure (inches and millimeter). Taking both of these into consideration the
formulae for conversion are:
MM = 25.4 / DP
DP = 25.4 / MM
Mounting distance
Mounting distance
Mounting distance, for assembling bevel gears or hypoid gears, is the distance from the crossing point of the axes to a locating surface of a gear, which may be at either back or front.^[1]
Normal module
Normal module is the value of the module in a normal plane of a helical gear or worm.^[1]
${\displaystyle m_{n}=m_{t}\cos \beta \,}$
Normal plane
Planes at a pitch point on a helical tooth
A normal plane is normal to a tooth surface at a pitch point, and perpendicular to the pitch plane. In a helical rack, a normal plane is normal to all the teeth it intersects. In a helical gear,
however, a plane can be normal to only one tooth at a point lying in the plane surface. At such a point, the normal plane contains the line normal to the tooth surface.
Important positions of a normal plane in tooth measurement and tool design of helical teeth and worm threads are:
1. the plane normal to the pitch helix at side of tooth;
2. the plane normal to the pitch helix at center of tooth;
3. the plane normal to the pitch helix at center of space between two teeth
In a spiral bevel gear, one of the positions of a normal plane is at a mean point and the plane is normal to the tooth trace.^[1]
Offset is the perpendicular distance between the axes of hypoid gears or offset face gears.^[1]
In the adjacent diagram, (a) and (b) are referred to as having an offset below center, while those in (c) and (d) have an offset above center. In determining the direction of offset, it is customary
to look at the gear with the pinion at the right. For below center offset the pinion has a left hand spiral, and for above center offset the pinion has a right hand spiral.
Outside cylinder
Cylindrical surfaces
The outside (tip or addendum) cylinder is the surface that coincides with the tops of the teeth of an external cylindrical gear.^[1]
Outside diameter
Wormgear diameters
The outside diameter of a gear is the diameter of the addendum (tip) circle. In a bevel gear it is the diameter of the crown circle. In a throated worm gear it is the maximum diameter of the blank.
The term applies to external gears, this is can also be known from major diameter.^[1]
Pinion and annular gear
A pinion is a round gear and usually refers to the smaller of two meshed gears.
Pitch angle
Pitch angle in bevel gears is the angle between an element of a pitch cone and its axis. In external and internal bevel gears, the pitch angles are respectively less than and greater than 90 degrees.
Pitch circle
A pitch circle (operating) is the curve of intersection of a pitch surface of revolution and a plane of rotation. It is the imaginary circle that rolls without slipping with a pitch circle of a
mating gear.^[1] These are the outlines of mating gears. Many important measurements are taken on and from this circle.^[1]
Pitch cone
Pitch cones
A pitch cone is the imaginary cone in a bevel gear that rolls without slipping on a pitch surface of another gear.^[1]
Pitch helix
Tooth helix
The pitch helix is the intersection of the tooth surface and the pitch cylinder of a helical gear or cylindrical worm.^[1]
Base helix
The base helix of a helical, involute gear or involute worm lies on its base cylinder.
Base helix angle
Base helix angle is the helix angle on the base cylinder of involute helical teeth or threads.
Base lead angle
Base lead angle is the lead angle on the base cylinder. It is the complement of the base helix angle.
Outside helix
The outside (tip or addendum) helix is the intersection of the tooth surface and the outside cylinder of a helical gear or cylindrical worm.
Outside helix angle
Normal helix
Outside helix angle is the helix angle on the outside cylinder.
Outside lead angle
Outside lead angle is the lead angle on the outside cylinder. It is the complement of the outside helix angle.
Normal helix
A normal helix is a helix on the pitch cylinder, normal to the pitch helix.
Pitch line
The pitch line corresponds, in the cross section of a rack, to the pitch circle (operating) in the cross section of a gear.^[1]
Pitch point
The pitch point is the point of tangency of two pitch circles (or of a pitch circle and pitch line) and is on the line of centers.^[1]
Pitch surfaces
Pitch surfaces
Pitch surfaces are the imaginary planes, cylinders, or cones that roll together without slipping. For a constant velocity ratio, the pitch cylinders and pitch cones are circular.^[1]
Pitch cones
Pitch plane
Pitch planes
The pitch plane of a pair of gears is the plane perpendicular to the axial plane and tangent to the pitch surfaces. A pitch plane in an individual gear may be any plane tangent to its pitch surface.
The pitch plane of a rack or in a crown gear is the imaginary planar surface that rolls without slipping with a pitch cylinder or pitch cone of another gear. The pitch plane of a rack or crown gear
is also the pitch surface.^[1]
Transverse plane
The transverse plane is perpendicular to the axial plane and to the pitch plane. In gears with parallel axes, the transverse and the plane of rotation coincide.^[1]
Principal directions
Principal directions
Principal directions are directions in the pitch plane, and correspond to the principal cross sections of a tooth.
The axial direction is a direction parallel to an axis.
The transverse direction is a direction within a transverse plane.
The normal direction is a direction within a normal plane.^[1]
Profile angle
Profile radius of curvature
Fillet radius
Profile radius of curvature is the radius of curvature of a tooth profile, usually at the pitch point or a point of contact. It varies continuously along the involute profile.^[1]
Rack and pinion
Radial composite deviation
Total composite variation trace
Tooth-to-tooth radial composite deviation (double flank) is the greatest change in center distance while the gear being tested is rotated through any angle of 360 degree/z during double flank
composite action test.
Tooth-to-tooth radial composite tolerance (double flank) is the permissible amount of tooth-to-tooth radial composite deviation.
Total radial composite deviation (double flank) is the total change in center distance while the gear being tested is rotated one complete revolution during a double flank composite action test.
Total radial composite tolerance (double flank) is the permissible amount of total radial composite deviation.^[1]
Root angle
Root angle in a bevel or hypoid gear, is the angle between an element of the root cone and its axis.^[1]
Root circle
The root circle coincides with the bottoms of the tooth spaces.^[1]
Root cone
Principal dimensions
The root cone is the imaginary surface that coincides with the bottoms of the tooth spaces in a bevel or hypoid gear.^[1]
Root cylinder
The root cylinder is the imaginary surface that coincides with the bottoms of the tooth spaces in a cylindrical gear.^[1]
Shaft angle
Shaft angle
A shaft angle is the angle between the axes of two non-parallel gear shafts. In a pair of crossed helical gears, the shaft angle lies between the oppositely rotating portions of two shafts. This
applies also in the case of worm gearing. In bevel gears, the shaft angle is the sum of the two pitch angles. In hypoid gears, the shaft angle is given when starting a design, and it does not have a
fixed relation to the pitch angles and spiral angles.^[1]
Spiral gear
Spiral bevel gear
Spur gear
Spur gear
A spur gear has a cylindrical pitch surface and teeth that are parallel to the axis.^[1]
Spur rack
A spur rack has a planar pitch surface and straight teeth that are at right angles to the direction of motion.^[1]
Standard pitch circle
The standard pitch circle is the circle which intersects the involute at the point where the pressure angle is equal to the profile angle of the basic rack.^[1]
Standard pitch diameter
The standard reference pitch diameter is the diameter of the standard pitch circle. In spur and helical gears, unless otherwise specified, the standard pitch diameter is related to the number of
teeth and the standard transverse pitch. Standard reference pitch diameter can be estimated by taking average of gear teeth tips diameter and gear teeth base diameter.^[1]
The pitch diameter is useful in determining the spacing between gear centers because proper spacing of gears implies tangent pitch circles. The pitch diameters of two gears may be used to calculate
the gear ratio in the same way the number of teeth is used.
${\displaystyle d={\frac {N}{P_{d}}}={\frac {pN}{\pi }}\qquad {\text{spur gears}}}$
${\displaystyle d={\frac {N}{P_{nd}\cos \psi }}\qquad {\text{helical gears}}}$
Where ${\displaystyle N}$ is the total number of teeth, ${\displaystyle p}$ is the circular pitch, ${\displaystyle P_{d}}$ is the diametrical pitch, and ${\displaystyle \psi }$ is the helix angle for
helical gears.
Standard reference pitch diameter
The standard reference pitch diameter is the diameter of the standard pitch circle. In spur and helical gears, unless otherwise specified, the standard pitch diameter is related to the number of
teeth and the standard transverse pitch. It is obtained as:^[1]
${\displaystyle d=km={\frac {zp}{\pi }}=z{\frac {m_{n}}{\cos \beta }}}$
${\displaystyle D={\frac {N}{P_{d}}}={\frac {Np}{\pi }}={\frac {N}{P_{nd}\cos \psi }}}$
Test radius
The test radius (R[r]) is a number used as an arithmetic convention established to simplify the determination of the proper test distance between a master and a work gear for a composite action test.
It is used as a measure of the effective size of a gear. The test radius of the master, plus the test radius of the work gear is the set up center distance on a composite action test device. Test
radius is not the same as the operating pitch radii of two tightly meshing gears unless both are perfect and to basic or standard tooth thickness.^[1]
Throat diameter
Worm gear diameters
The throat diameter is the diameter of the addendum circle at the central plane of a worm gear or of a double-enveloping worm gear.^[1]
Throat form radius
Throat form radius is the radius of the throat of an enveloping worm gear or of a double-enveloping worm, in an axial plane.^[1]
Tip radius
Tip radius
Tip radius is the radius of the circular arc used to join a side-cutting edge and an end-cutting edge in gear cutting tools. Edge radius is an alternate term.^[1]
Tip relief
Tip relief
Tip relief is a modification of a tooth profile whereby a small amount of material is removed near the tip of the gear tooth.^[1]
Tooth surface
Profile of a spur gear
Notation and numbering for an external gear
Notation and numbering for an internal gear
The tooth surface (flank) forms the side of a gear tooth.^[1]
It is convenient to choose one face of the gear as the reference face and to mark it with the letter “I”. The other non-reference face might be termed face “II”.
For an observer looking at the reference face, so that the tooth is seen with its tip uppermost, the right flank is on the right and the left flank is on the left. Right and left flanks are denoted
by the letters “R” and “L” respectively.
Worm drive
See also
1. ^ ^a ^b ^c ^d ^e ^f ^g ^h ^i ^j ^k ^l ^m ^n ^o ^p ^q ^r ^s ^t ^u ^v ^w ^x ^y ^z ^aa ^ab ^ac ^ad ^ae ^af ^ag ^ah ^ai ^aj ^ak ^al ^am ^an ^ao ^ap ^aq ^ar ^as ^at ^au ^av ^aw ^ax ^ay ^az ^ba ^bb ^bc
^bd ^be ^bf ^bg ^bh ^bi ^bj ^bk ^bl ^bm ^bn ^bo ^bp ^bq ^br ^bs ^bt ^bu ^bv ^bw Gear Nomenclature, Definition of Terms with Symbols. American Gear Manufacturers Association. 2005. p. 72. ISBN
1-55589-846-7. OCLC 65562739. ANSI/AGMA 1012-G05.
2. ^ Tony Casey, President Bull Gear, Inc. "Bull Gear, Inc. - What is a Bull Gear!?". Archived from the original on 6 January 2012. Retrieved 4 January 2012.{{cite web}}: CS1 maint: multiple names:
authors list (link)
3. ^ ^a ^b ^c Machinery's Handbook Twenty-Fifth Edition, by Erik Oberg, Franklin D. Jones, Holbrook L. Horton, and Henry H Ryffle, 1996, Industrial Press Inc. | {"url":"https://www.knowpia.com/knowpedia/List_of_gear_nomenclature","timestamp":"2024-11-09T00:34:58Z","content_type":"text/html","content_length":"214479","record_id":"<urn:uuid:8f7288f8-c808-43e1-9284-2993e98cf40d>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00555.warc.gz"} |
Add formulas to cells in Pages for iCloud
You can create your own formula using mathematical symbols or comparison operators (such as +, *, >, or <=) to perform calculations using the data in any cells you select. You can also use any of the
over 290 built-in functions (operations that you can include in a formula) to perform calculations, retrieve information, and manipulate data. The result of a formula or function appears in the cell
where you entered it.
Create your own formula
You can create simple or complex arithmetic formulas using mathematical operators for addition (+), subtraction (-), multiplication (*), and division (/).
1. Click the cell where you want the result to appear, then enter the equal sign (=).
The formula editor opens.
2. Enter a left parenthesis ( to begin your formula.
3. Select a cell to use as the first argument in your formula, or enter a value (for example, a number such as 0 or 5.20).
4. Enter an arithmetic operator (for example, +, -, *, or /), then either select a cell to use as the next argument in your formula, or enter a value.
5. Continue adding operators and arguments until your formula is complete.
6. Enter a right parenthesis ) to end your formula.
7. Press Return or click the Checkmark button
If you click the Cancel button you exit the current cell without saving the formula in it.
Compare values using a formula
You can create a formula that uses comparison operators to check whether the values in two cells are equal, or if one value is greater or less than the other. To do this, you must set up a statement
within a cell, for example A1 > A2, meaning the value in cell A1 is greater than the value in cell A2. The result of the comparison operator is expressed as “true” or “false.”
1. Click the cell where you want the comparison result to appear, then enter the equal sign (=).
The formula editor opens.
2. Select a cell whose value you want to compare, or enter a value to compare.
3. Enter a comparison operator (>, >=, =, <>, <, or <=), then select a cell whose value you want to compare, or enter a value to compare.
4. Press Return or click the Checkmark button
If you click the Cancel button you exit the current cell without saving the formula in it.
Add a predefined function
There are predefined functions for applications including statistics, engineering, and finance, some of which retrieve information remotely via the internet. You can see the available functions in
the Functions Browser, which appears in the Format sidebar on the right when you type an equal sign (=) in a table cell. The Functions Browser includes examples showing how the functions work, to
help you choose one that suits your needs.
1. Click the cell where you want the result of the function to appear, then enter the equal sign (=).
The formula editor opens, and the Functions Browser appears in the Format
2. Enter the function name you want in the search field at the top of the Functions Browser, or browse the available functions, then double-click the name of the function you want.
The function appears in the formula editor.
3. Select an argument within the function.
4. Select the cells you want to include in the calculation by doing one of the following:
□ Add values in noncontiguous cells: Click each cell you want to include.
□ Select a range of cells across multiple rows and columns: Drag across the range of cells you want to include.
□ Add the values of a single column or row: Select the column or row.
The cell references appear in the formula editor.
5. Press Return or click the Checkmark button
If you click the Cancel button you exit the current cell without saving the formula in it.
View instant calculations for a range of cells
You can quickly view the sum, average, minimum, maximum, and count for any column, row, or range of cells. (If the selection contains a mix of data types, such as text and numbers, or mixed formats,
such as date and currency, some calculations aren’t provided.)
Preserve row or column addresses in formulas
You can “freeze” row or column references in a formula so you can use the same formula elsewhere in your table without changing the cell references.
If you don’t preserve the row or column references, then if you move the formula (by cutting and pasting, or by adding new rows and columns), the references are adjusted relative to the formula’s new
1. Double-click the results cell with the formula you want to edit.
The formula editor opens displaying the functions.
2. Click the triangle on the token representing the cell range you want to preserve.
3. Select Preserve Row or Preserve Column for the beginning or end addresses of the selected range.
If you subsequently change the number of rows or columns in the table, or if you move the formula to a different cell, the preserved row or column references are adjusted.
4. Press Return or click the Checkmark button
If you click the Cancel button you exit the current cell without saving the formula in it. | {"url":"https://support.apple.com/en-gu/guide/pages-icloud/gilcc8ecd809/icloud","timestamp":"2024-11-06T01:40:43Z","content_type":"text/html","content_length":"154272","record_id":"<urn:uuid:8b73bd31-095b-4e56-ae80-eef72d1fd75f>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00301.warc.gz"} |
Reversible optimisers | Daniel Worrall
Reversible optimisers
Dec 20, 2020 · 5 min read
This post touches on a curious property of some common optimisers used by the machine learning community: reversibility.
I tend to hate reading through lengthy introductions, so let’s just dive in with an example. Take gradient descent with momentum, this has the following form
$$ \mu_{t+1} &= \alpha \mu_t + \nabla_{x} f(x_{t}) \\ x_{t+1} &= x_t - \lambda \mu_{t+1}. $$
Here $x_t$ denotes the optimisation variable, or position, $x$ at time $t$, $\mu$ is the associated momentum, and $0 < \alpha < 1$ & $\lambda > 0$ are metaparameters, which govern the dynamics of the
descent trajectory. I use the term metaparameters, instead of hyperparameters, to distinguish that they are part of the optimiser and not the model, even though some would nowadays say that the
optimiser is in fact part of the model, implicitly regularising it.
Anyway, interestingly we can reverse these equations, given the state $[x_{t+1}, \mu_{t+1}]$ as
$$ x_t &= x_{t+1} + \lambda \mu_{t+1} \\ \mu_{t} &= \frac{1}{\alpha} \left ( \mu_{t+1} - \nabla_{x} f(x_{t}) \right). $$
This seemingly arbitrary property is useful from a practical standpoint.
Memory efficiency
An oft-lauded property of reversible systems is that we do not have to store intermediate computations, since they should be easily reconstructed from the system’s end-state. Typically for
reverse-mode differentiation to work (i.e. backpropagation), we have to store all the intermediate activations in the forward pass of a network. This has memory complexity, which scales linearly with
the size of the computation graph. If we can dynamically reconstruct intermediate activations during the backward pass, then we instantly convert this linear memory complexity to a constant, which
enables us to build (in theory) infinitely deep networks.
Momentum is additive coupling
Indeed, if you look a little closer at the momentum equations, then you may spot that they resemble an additive coupling layer. Here we have that a state, split into two parts $x$ and $\mu$ (to mimic
the momentum optimiser notation), is reversible with the following computation graph
$$ \mu_{t+1} &= \mu_t + g(x_t) \\ x_{t+1} &= x_t + h(\mu_{t+1}) $$
To make a direct comparison, $g(x) = \nabla_x f(x)$ and $h(x) = \lambda x$. The one slight discrepancy is the factor of $\alpha$, but we can sweep that under the rug. The reverse equations for the
additive coupling layer are
$$ x_{t} &= x_{t-1} - h(\mu_{t+1}) \\ \mu_{t} &= \mu_{t+1} - g(x_t). $$
Source: Reversible GANs for Memory-efficient Image-to-Image Translation. This diagramme represents the additive coupling layer in its computation graph form. LEFT: forward pass. RIGHT: reverse pass.
To link up the notation $x_1 = \mu_{t}$, $x_2 = x_{t}$, $y_1 = \mu_{t+1}$, $y_2 = x_{t+1}$, $g = \texttt{NN}_1$, and $h=\texttt{NN}_2$
Case study
Specifically in the case of optimisers, I was pointed towards this paper Gradient-based Hyperparameter Optimization with Reversible Learning (2015) by Dougal Maclaurin, David Duvenaud, and Ryan Adams
. The authors exploited the reversibility property of SGD with momentum to train the optimiser metaparameters themselves. First they run the optimiser an arbitrary number of steps, say 100
iterations. This defines an optimisation trajectory $x_0, x_1, x_2, ..., x_{99}$. Now the clever part is that you can view the unrolled optimisation trajectory as a computation graph in itself. They
compute a loss at the end of the trajectory, then they backpropagate the loss in the reverse direction with respect to the optimiser’s metaparameters.
Source: Gradient-based Hyperparameter Optimization with Reversible Learning. The authors optimise metaparameters by backpropagating along optimisation roll outs. This is made possible with the
reversibility of momentum-based SGD, to cap memory-complexity.
Could we not do this already, such as in Learning to learn by gradient descent by gradient descent (Andrychowicz et al., 2016)? Well yes, but the crucial point is that you would usually have to store
all the intermediate states $\\{[x_t, \mu_t]\\}_{t=0}^{99}$, which is costly memory-wise. Exploiting the reversibility property, this memory explosion falls away. Indeed there are issues with
numerical stability of the inverse, which the papers dives into, but the principle is elegant.
So what other optimisers are reversible? Let’s consider Adam, where
$$ \mu_{t+1} &= \beta_1 \mu_t + (1-\beta_1) \nabla_{x} f(x_{t}) \\ \nu_{t+1} &= \beta_2 \nu_t + (1-\beta_2) (\nabla_{x} f(x_{t}))^2 \\ x_{t+1} &= x_t - \lambda \frac{\mu_{t+1}}{\sqrt{\nu_{t+1}} + \
epsilon}. $$
Given $x_{t+1}$, $\mu_{t+1}$ and $\nu_{t+1}$, we can easily reconstruct $x_t$ from the last line and from there, we can compute the gradient and recover $\mu_{t}$ and $\nu_{t}$. In maths
$$ x_{t} &= x_{t+1} + \lambda \frac{\mu_{t+1}}{\sqrt{\nu_{t+1}} + \epsilon} \\ \mu_{t} &= \frac{1}{\beta_1} \left ( \mu_{t+1} - (1-\beta_1) \nabla_{x} f(x_{t}) \right ) \\ \nu_{t} &= \frac{1}{\
beta_2} \left ( \nu_{t+1} - (1-\beta_2) (\nabla_{x} f(x_{t}))^2 \right). $$
So Adam is reversible. We actually missed out the bias correction steps
$$ \mu_{t+1} &\gets \mu_{t+1} / (1 - \beta_1^{t+1}) \\ \nu_{t+1} &\gets \nu_{t+1} / (1 - \beta_2^{t+1}). $$
You can also verify for yourself that these are reversible too.
Do we need reversibility in optimisers?
Well, no. In fact, in some ways, we would rather do without it. Optimisers are supposed to be many-to-one mappings. Starting from an infinity of initial conditions, we should converge to the global
minimum of a convex function. This means we should discard information about initialisation along the way. To put it as Maclaurin et al. do:
[O]ptimization moves a system from a high-entropy initial state to a low-entropy (hopefully zero entropy) optimized final state.
It turns out that if you set $\alpha = 0$ for the momentum method; that is, you just run gradient descent, then this is not reversible. I think this may also be true for Nesterov accelerated momentum
, and RMSProp which I couldn’t make reversible (I call this proof by fatigue). So I’m left wondering, is reversibility just some extra curious property that can be useful sometimes, but is completely
arbitrary when it comes to doing optimisation? Or is there some deeper meaning to it? Is it just some artifact of how we think of optimisation, in terms of balls rolling down hills? Maybe more
interestingly, what does the lack of reversibility for standard gradient descent and Nesterov entail? Could this be another reason why Nesterov works better than classical momentum? Could we measure
the information loss somehow? And if we could, what would this mean?
Machine Learning Research Scientist
I’m interested in using ML to accelerate simulation of physical systems | {"url":"https://danielewworrall.github.io/blog/reversible-optimizers/","timestamp":"2024-11-10T21:31:58Z","content_type":"text/html","content_length":"37310","record_id":"<urn:uuid:0a17b744-b45f-4a8e-8599-a354570326ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00450.warc.gz"} |
What is the Maximum Compression of the Spring?
• Thread starter benjicolon
• Start date
In summary, the maximum compression of a spring refers to the point at which the spring has been compressed to its furthest extent without breaking or permanently deforming. It is typically
calculated using the formula F = -kx, where F is the applied force, k is the spring constant, and x is the displacement from the equilibrium position. Factors such as the material, length, diameter,
and shape of the spring, as well as the spring constant, can affect the maximum compression. If a spring is compressed beyond its maximum compression point, it will permanently deform and lose its
ability to return to its original shape and length. To increase the maximum compression of a spring, one can use a material with a higher elastic limit, increase the length or
Homework Statement
Mass- 0.8 kg
Initial Velocity- 3 m/s
Spring constant- 500 N/m
No friccion
Final velocity- 0
Homework Equations
What is the maximum compression of the spring?
I don't know where or how to start!
Please help.
Science Advisor
Homework Helper
hi benjicolon! welcome to pf!
use conservation of energy …
show us what you get
FAQ: What is the Maximum Compression of the Spring?
What is maximum compression of a spring?
Maximum compression of a spring is the point at which the spring has been compressed to its furthest possible extent without breaking or permanently deforming. This point is also known as the elastic
limit of the spring.
How is maximum compression of a spring calculated?
The maximum compression of a spring can be calculated using the formula F = -kx, where F is the applied force, k is the spring constant, and x is the displacement from the equilibrium position. The
maximum compression occurs when the applied force is equal to the negative of the spring constant multiplied by the displacement.
What factors affect the maximum compression of a spring?
The maximum compression of a spring is affected by the material of the spring, its length, its diameter, and its shape. The spring constant also plays a significant role in determining the maximum
What happens if a spring is compressed beyond its maximum compression point?
If a spring is compressed beyond its maximum compression point, it will permanently deform and lose its ability to return to its original shape and length. This is known as plastic deformation and
can occur due to excessive force or repeated compression beyond the elastic limit.
How can the maximum compression of a spring be increased?
The maximum compression of a spring can be increased by using a material with a higher elastic limit, increasing the length or diameter of the spring, or by choosing a spring with a higher spring
constant. Additionally, using multiple springs in series or in parallel can also increase the maximum compression capacity. | {"url":"https://www.physicsforums.com/threads/what-is-the-maximum-compression-of-the-spring.739972/","timestamp":"2024-11-05T00:55:14Z","content_type":"text/html","content_length":"78580","record_id":"<urn:uuid:fe483508-c2be-4318-b7cb-d4c5f8f1c6a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00319.warc.gz"} |
Groundbreaking Discovery: Physicists Create Quantum System for Torsion Pendulum
A remarkable scientific breakthrough has been achieved by a team of physicists, who have successfully created a quantum system for a torsion pendulum, a device used to detect minute torques or
forces. This groundbreaking accomplishment has opened new avenues for exploring quantum gravity and testing fundamental physics theories.
Torsion Pendulum in Physics
A torsion pendulum comprises a suspended mass attached to a wire or fiber, enabling it to oscillate about a vertical axis. The pendulum's oscillation frequency is influenced by the torque applied to
it. In classical physics, the pendulum's behavior is described by its moment of inertia and the restoring torque exerted by the wire.
Quantum Mechanical Torsion Pendulum
The physicists involved in this research have made a paradigm shift by introducing quantum mechanics into the realm of torsion pendulums. They have created a system in which the pendulum's motion is
governed by quantum principles, including the uncertainty principle and superposition.
Quantum Entanglement in the System
The key innovation in this quantum system lies in the entanglement of the pendulum's vibrational modes. Using advanced techniques, the researchers entangled two quantum bits (qubits) with the
pendulum's vibrational states. This entanglement creates a strong correlation between the qubits and the pendulum's motion, enabling precise control and measurement of the pendulum's quantum
Applications in Quantum Gravity and Fundamental Physics
The creation of a quantum torsion pendulum provides a powerful tool for probing the gravitational force at quantum scales. It offers a unique platform for testing alternative theories of gravity,
such as string theory and loop quantum gravity, which predict modifications to gravity at very small distances or energies.
Experimental Verification of Quantum Properties
The physicists conducted a series of experiments to verify the quantum nature of their system. They observed the characteristic quantum behaviors of the pendulum, including the superposition of
distinct vibrational states and the Heisenberg uncertainty principle, which limits the simultaneous determination of certain physical properties.
Challenges and Future Directions
While the creation of a quantum torsion pendulum is a significant milestone, the researchers acknowledge that there are challenges ahead. They aim to improve the system's coherence time and reduce
quantum noise to enhance its sensitivity and precision. Future research will focus on utilizing this system to explore the quantum nature of gravity and push the boundaries of fundamental physics.
The creation of a quantum system for a torsion pendulum represents a major advancement in the field of physics. It provides a novel platform for investigating quantum gravity and testing fundamental
theories. This breakthrough has the potential to revolutionize our understanding of the universe at the smallest scales and opens exciting possibilities for future research. | {"url":"https://www.freeinteriorimages.com/2024/10/groundbreaking-discovery-physicists.html","timestamp":"2024-11-06T14:55:36Z","content_type":"application/xhtml+xml","content_length":"183012","record_id":"<urn:uuid:eee65658-b3f9-4af3-8811-130f96ce44c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00742.warc.gz"} |
anova.ergm: ANOVA for ERGM Fits in ergm: Fit, Simulate and Diagnose Exponential-Family Models for Networks
Compute an analysis of variance table for one or more ERGM fits.
## S3 method for class 'ergm' anova(object, ..., eval.loglik = FALSE) ## S3 method for class 'ergmlist' anova(object, ..., eval.loglik = FALSE)
object, ... objects of ergm, usually, a result of a call to ergm().
eval.loglik a logical specifying whether the log-likelihood will be evaluated if missing.
objects of ergm, usually, a result of a call to ergm().
a logical specifying whether the log-likelihood will be evaluated if missing.
Specifying a single object gives a sequential analysis of variance table for that fit. That is, the reductions in the residual sum of squares as each term of the formula is added in turn are given in
the rows of a table, plus the residual sum of squares.
The table will contain F statistics (and P values) comparing the mean square for the row to the residual mean square.
If more than one object is specified, the table has a row for the residual degrees of freedom and sum of squares for each model. For all but the first model, the change in degrees of freedom and sum
of squares is also given. (This only make statistical sense if the models are nested.) It is conventional to list the models from smallest to largest, but this is up to the user.
If any of the objects do not have estimated log-likelihoods, produces an error, unless eval.loglik=TRUE.
The comparison between two or more models will only be valid if they are fitted to the same dataset. This may be a problem if there are missing values and 's default of na.action = na.omit is used,
and anova.ergmlist() will detect this with an error.
The model fitting function ergm(), anova(), logLik.ergm() for adding the log-likelihood to an existing ergm object.
data(molecule) molecule %v% "atomic type" <- c(1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3) fit0 <- ergm(molecule ~ edges) anova(fit0) fit1 <- ergm(molecule ~ edges + nodefactor("atomic type")) anova
(fit1) fit2 <- ergm(molecule ~ edges + nodefactor("atomic type") + gwesp(0.5, fixed=TRUE), eval.loglik=TRUE) # Note the eval.loglik argument. anova(fit0, fit1) anova(fit0, fit1, fit2)
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/ergm/man/anova.ergm.html","timestamp":"2024-11-01T22:22:02Z","content_type":"text/html","content_length":"37633","record_id":"<urn:uuid:18008bb8-9d15-4fb1-8fe7-19d674a8d31a>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00018.warc.gz"} |
Probability and Machine Learning
- Overview
Probability is a fundamental concept in machine learning (ML). It's a mathematical field that provides tools for quantifying uncertainty and reasoning in a principled way. You cannot develop a deep
understanding and application of ML without it.
Probability is used in ML in the following ways:
• Classification models: Must predict a probability of class membership
• Algorithms: Are designed using probability
• Learning algorithms: Make decisions using probability
• Probabilistic measures: Are used to evaluate model skill
Probability is important in ML because it's based on the idea that the past is predictive of the future. This means that we can look at a bunch of training data and make predictions about data we
have never seen before.
Here are some examples of ML in our daily lives:
• Facial recognition
• Product recommendations
• Email automation and spam filtering
• Financial accuracy
• Social media optimization
• Healthcare advancement
• Mobile voice to text and predictive text
Probability is used in many ML applications and domains, such as natural language processing, computer vision, and recommender systems. For instance, Naive Bayes is a probabilistic method that uses
Bayes' theorem to classify data based on the probability of the class given the features.
- Probability Distributions and Machine Learning
Probability distributions are important in ML because they help describe the patterns and uncertainties of data and models. They also allow data analysts to recognize and understand patterns from
large data sets.
Probability distributions are used to model random processes, such as Bayesian modeling, density estimation, and probabilistic programming. They also provide principled ways to quantify and reduce
uncertainty, which is critical for many real-world machine learning applications.
For example, ML algorithms leverage probability distributions to model uncertainty in predictions, enhancing their ability to make accurate forecasts.
Probability distributions are also used throughout all of the sciences to measure and predict probabilities, and to estimate the likelihood of achieving certain outcomes. | {"url":"http://eitc.org/research-opportunities/new-media-and-new-digital-economy/ai-machine-learning-deep-learning-and-neural-networks/ml-research-and-applications/foundations-of-ml/probability-and-machine-learning","timestamp":"2024-11-03T10:40:07Z","content_type":"application/xhtml+xml","content_length":"21810","record_id":"<urn:uuid:353b8bcc-a028-4842-80ac-c5f1d3f38e2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00673.warc.gz"} |
From the form of the moment of inertia tensor introduced in Eq.B.24)
it is clear that
. Moreover, for any normalized
angular-velocity vector
since of course we expect any moment of inertia mass symmetric nonnegative definite. If furthermore i.e., if there is any nonzero angle between them, then positive definite (and linear algebra [329],
real, symmetric, positive-definite matrices have orthogonal eigenvectors and real, positive eigenvalues. In this context, the orthogonal eigenvectors are called the principal axes of rotation. Each
corresponding eigenvalue is the moment of inertia about that principal axis--the corresponding principal moment of inertia. When angular velocity vectors linear combination of the principal axes,
there are no cross-terms in the moment of inertia tensor--no so-called products of inertia.
The three principal axes are unique when the eigenvalues of distinct. They are not unique when there are repeated eigenvalues, as in the example above of a disk rotating about any of its diameters (§
B.4.4). In that example, one principal axis, the one corresponding to eigenvalue i.e., orthogonal to the disk and passing through its center), while any two orthogonal diameters in the plane of the
disk may be chosen as the other two principal axes (corresponding to the repeated eigenvalue
Symmetry of the rigid body about any axis solid of revolution.^B.26In rotational dynamics, this case is known as the symmetric top [270]. Note that the center of mass will lie somewhere along an axis
of symmetry. The other two principal axes can be arbitrarily chosen as a mutually orthogonal pair in the (circular) plane orthogonal to the diagonalized to look like
Axis theorem) that if the mass distribution is planar, then
Next Section: Body-Fixed and Space-Fixed Frames of ReferencePrevious Section: Off-Diagonal Terms in Moment of Inertia Tensor | {"url":"https://www.dsprelated.com/freebooks/pasp/Positive_Definiteness_Moment_Inertia.html","timestamp":"2024-11-12T18:50:56Z","content_type":"text/html","content_length":"34093","record_id":"<urn:uuid:431f2293-9413-46aa-90b1-9f24bf5eeade>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00895.warc.gz"} |
The chair tiling, as most tilings presented here, is nonperiodic. But there is a strong resemblance to periodic tiling. For instance, the set of vertex points in the tiling obviously spans a square
lattice. Moreover, it is possible to detect large subsets in the tiling which are fully periodic. For instance, consider the pattern of white crosses (consisting of four tiles each) in the tiling. In
fact, the chair tiling is the union of a countable set of fully periodic tile sets $L_{1}, L_{2}, L_{3}$…, where each $L_{i}$ possesses period vectors of length $2 \times 2^{i}$. This property is
called limitperiodic, and it is the key to show how to obtain the chair tiling by a cut and project method with p-adic internal space, here: $\mathbb{Q}_2 \times \mathbb{Q}_2$. That was first carried
out in [BaakeMS98] , see also [LMS03] . It is easy to see that there are no matching rules for the undecorated tiles. But [Goo99] found nice local matching rules with just two tiles, forcing tilings
from which the chair tilings are locally derivable (see mld). Here, the tiles appear in three colours, depending on their relative position in the first-order super-tile. The colours do not mean that
there are different substitution rules, all tiles are substituted in the same way.
Substitution Rule
download vectorformat Chair
Baake, M and Moody, R V and Schlottmann, M
Limit-(quasi) periodic point sets as quasicrystals with p-adic internal spaces
Journal of Physics A: Mathematical and General 1998, 31(27), pp. 55-65, arxiv.9901008
Goodman-Strauss, C
A small aperiodic set of planar tiles.
European J. Combin. 1999, 20, 5, pp. 375-384, MR1702375
Lee, J E S and Moody, R V and Solomyak, B
Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
Discrete and Computational Geometry 2003, 29, pp. 525-560, MR1702375 | {"url":"https://tilings.math.uni-bielefeld.de/substitution/chair/","timestamp":"2024-11-07T23:20:44Z","content_type":"text/html","content_length":"9290","record_id":"<urn:uuid:7ee11446-cd95-4dfc-a71f-98e0f8444ddf>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00811.warc.gz"} |
[tlaplus] A trick for changing multiple function values
Let's say we have three "blinkers", which flip between TRUE and FALSE:
EXTENDS Integers, TLC
vars == <<f>>
Init ==
/\ f = <<FALSE, FALSE, FALSE>>
Next == ???
Spec == Init /\ [][Next]_vars
If I want to change one of the blinkers per step, I can do it the usual way:
Next ==
/\ \E x \in 1..3:
f' = [f EXCEPT ![x] = ~f[x]]
But what if I want to change any number of them? EXCEPT syntax doesn't support that. I've usually seen it done as:
Next ==
/\ \E set \in SUBSET (1..3):
\/ f' = [x \in 1..3 |->
IF x \in set THEN ~f[x] ELSE f[x]]
\* or
Next ==
/\ \E set \in SUBSET (1..3):
\/ f' = [x \in set |-> ~f[x]] @@ f
So why can't we just do this?
Next ==
/\ \E set \in SUBSET (1..3):
/\ \A x \in set:
f[x]' = ~f[x]
Because it doesn't fully specify f'. For all we know, f' has the domain Int instead of 1..3, and it doesn't specify what happens to the elements not in the set. We also can't write DOMAIN f' = 1..3
because TLC can't check that, even though it's a valid TLA+ _expression_.
But there's a trick we can use: the first time TLC encounters encounters a primed variable, it uses that the generate the potential next-states for that variable. We need to write either f' = ... or
f' \in ... . But after that, TLC can use f to determine whether or a given action is enabled or not. We can use f' as an action constraint!
Next ==
/\ f' \in [1..3 -> BOOLEAN]
/\ \A x \in 1..3:
\/ f'[x] = ~f[x]
\/ UNCHANGED f[x]
This works, as it fully defines f' in a way that's checkable by TLC. Practically speaking, this style is probably worse than the other two solutions. For one, it doesn't work on non-enumerable
ranges. Nonetheless it's pretty neat.
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MathPapa Application
14 Jan 202307:45
TLDRMathPapa is an algebra calculator and equation solver app available on Google Play that assists students in understanding the process of solving equations. It not only provides accurate answers
but also shows the steps involved. The app can graph functions, factor polynomials, and simplify expressions. It supports solving equations, inequalities, and offers various methods for quadratic
equations. An upgrade to the premium version unlocks step-by-step solutions and advanced features for a monthly fee.
• 📱 The MathPapa application is an algebra calculator and equation solver available on Google Play.
• 🛠 It helps students understand the process of solving equations by showing all possible steps.
• 📈 The app can graph functions, factor polynomials, and simplify expressions.
• 🔍 To use the app, open it and input the equation or problem you wish to solve.
• 🔑 For solving equations, click 'calculate' to get the answer and step-by-step solution.
• 📚 The app is not only for equations but also useful for solving inequalities.
• 📈 For quadratic polynomials, you can factor or simplify by using the app's 'calculate' feature.
• 📉 To evaluate a polynomial for a specific value of x, input the value and click 'OK' for the result.
• 🔍 Solving quadratic equations can be done by factoring, using the quadratic formula, completing the square, or finding the discriminant.
• 💡 The app provides a step-by-step solution for different methods, but some require a premium upgrade.
• 📊 The app also includes graphing capabilities similar to Geogebra, allowing users to graph functions and edit the viewing window.
• 💰 To access all features, including premium step-by-step solutions, a monthly subscription of 529 pesos is required after a free trial.
Q & A
• What is the name of the math application discussed in the video?
-The math application discussed in the video is called MathPapa.
• Where can MathPapa application be found for download?
-MathPapa application can be found on Google Play for download.
• What type of problems can MathPapa application solve?
-MathPapa application can solve algebraic equations, inequalities, and it can also graph functions, factor polynomials, and simplify expressions.
• How does MathPapa application help in understanding the process of solving equations?
-MathPapa not only provides accurate answers but also shows all the possible processes involved in solving the equations, offering a step-by-step solution.
• What is an example of a simple equation that can be solved using MathPapa?
-An example of a simple equation that can be solved using MathPapa is 'X Plus 1 equals zero', which results in the solution X equals negative one.
• How can MathPapa be used to factor a quadratic polynomial?
-To factor a quadratic polynomial using MathPapa, you simply input the polynomial, click calculate, and choose the 'factor' option from the available choices.
• What are the different methods MathPapa provides to solve a quadratic equation?
-MathPapa provides methods such as factoring, using the quadratic formula, completing the square, and finding the discriminant to solve a quadratic equation.
• How can MathPapa be used to evaluate a polynomial for a given value of X?
-To evaluate a polynomial for a given value of X, you input the polynomial into MathPapa, choose the 'evaluate' option, input the value of X, and click OK to see the result.
• What additional feature does MathPapa have that is similar to GeoGebra?
-MathPapa has a feature that allows users to graph functions and relations, similar to GeoGebra, where you can adjust the window and view the graph of the function.
• What is the cost of upgrading to the premium version of MathPapa?
-The cost of upgrading to the premium version of MathPapa is 529 pesos per month, after a free trial.
📱 Exploring Math Papa: A Powerful Algebra Tool
This paragraph introduces Math Papa, a math application designed to assist teachers and students with algebra problems. Available on Google Play, the app serves as an algebra calculator and equation
solver. It not only provides accurate answers but also illustrates the step-by-step processes involved in solving equations. Users can graph functions, factor polynomials, and simplify expressions.
An example is given where the app solves a simple equation, demonstrating its user-friendly interface and detailed solution steps.
🧮 Advanced Features: Solving Equations and Inequalities
This section delves deeper into the app's capabilities, highlighting its functionality for solving both equations and inequalities. It explains how to factor and evaluate quadratic polynomials,
showcasing the app's flexibility in choosing different methods to solve equations, including factoring, using the quadratic formula, completing the square, and finding the discriminant. The app’s
ability to provide multiple solution methods is emphasized, making it a comprehensive tool for understanding algebraic concepts. Additionally, the app can solve inequalities and graph the solution
💡MathPapa Application
MathPapa Application is a mathematical tool designed for educational purposes, specifically for teachers and students. It is an algebra calculator and equation solver that not only provides answers
but also illustrates the process of solving equations. In the video, it is introduced as a helpful resource that can be downloaded from Google Play, and it is used to demonstrate solving equations,
factoring polynomials, and graphing functions, which are all integral to understanding mathematical concepts.
💡Algebra Calculator
An algebra calculator is a type of software that performs algebraic computations, such as solving equations and simplifying expressions. In the context of the video, the MathPapa Application serves
as an algebra calculator, helping students to understand the steps involved in solving algebraic problems. For instance, the script mentions solving 'X Plus 1 equals zero' using the app, which
demonstrates its function as an algebra calculator.
💡Equation Solver
An equation solver is a tool that finds the values of variables that make an equation true. The MathPapa Application is highlighted as an equation solver in the video, showcasing its ability to
provide step-by-step solutions for equations. This feature is crucial for educational purposes, as it helps students learn not just the answer, but also how to arrive at the solution.
💡Step-by-Step Solution
A step-by-step solution is a method of presenting the process of solving a problem in a sequential manner. The video emphasizes the MathPapa Application's ability to show step-by-step solutions,
which is beneficial for educational purposes. For example, when solving the equation 'X Plus 1 equals zero,' the app provides a step-by-step breakdown, illustrating the methodical approach to finding
the solution.
💡Graph Functions
Graphing functions is the process of visually representing the relationship between variables in an equation. The video script describes how the MathPapa Application can be used to graph functions,
such as 'y equals X squared.' This feature allows users to visualize mathematical concepts, making it easier to understand the behavior of functions and their graphs.
💡Factor Polynomials
Factoring polynomials involves expressing a polynomial as the product of its factors. In the video, the MathPapa Application is used to factor the quadratic polynomial '5x squared minus 10x plus 5,'
which is broken down into '5 times (x minus 1) times (x minus 1).' This demonstrates the app's utility in simplifying and understanding polynomial expressions.
💡Simplify Expression
Simplifying an expression means reducing it to its simplest form. The MathPapa Application, as mentioned in the video, can simplify algebraic expressions, which is an essential skill in mathematics
for making equations easier to solve or understand.
💡Quadratic Equation
A quadratic equation is a polynomial equation of degree two, typically in the form ax squared plus bx plus c equals zero. The video script provides an example of solving a quadratic equation using
the MathPapa Application, which offers various methods such as factoring, the quadratic formula, completing the square, and finding the discriminant.
An inequality is a mathematical statement that compares expressions with a relation other than equality, such as 'greater than' or 'less than.' The video demonstrates how the MathPapa Application can
solve inequalities, like 'X plus 7 is greater than 0,' and provide the solution set graphically.
GeoGebra is a popular mathematics software that offers various tools for geometry, algebra, statistics, and calculus. The video mentions that the MathPapa Application has features similar to
GeoGebra, particularly in graphing functions and relations, which adds to its educational value by providing a comprehensive tool for visualizing mathematical concepts.
💡Premium Upgrade
A premium upgrade refers to a paid version of an application that offers additional features beyond the free version. The video script mentions that to enjoy the full features of the MathPapa
Application, users can upgrade to the premium version for a monthly fee, which suggests that some functionalities, like step-by-step solutions for certain problems, require this upgrade.
Introduction of MathPapa, an algebra calculator and equation solver application for teachers and students.
MathPapa is available on Google Play and helps students understand the process of solving equations.
The app provides step-by-step solutions for equations, not just the final answer.
Users can graph functions, factor polynomials, and simplify expressions with MathPapa.
Demonstration of solving a simple equation using MathPapa and viewing step-by-step solution.
MathPapa can be used to solve inequalities as well as equations.
Example of factoring a quadratic polynomial using the app.
Options to simplify, factor, solve, or evaluate within the app.
Explanation of solving a quadratic equation using different methods available in MathPapa.
Premium feature of step-by-step solutions for quadratic formula and other methods.
Solving inequalities in MathPapa and viewing the graph of the solution set.
MathPapa's graphing feature, similar to Geogebra, for graphing functions and relations.
Customization of graph window settings in MathPapa.
Upgrading to MathPapa Premium for full feature access and the cost involved.
Invitation to try the free trial of MathPapa Premium before purchasing.
Conclusion and thanks for considering MathPapa as a math application tool. | {"url":"https://math.bot/blog-MathPapa-Application-39890","timestamp":"2024-11-13T07:35:15Z","content_type":"text/html","content_length":"112885","record_id":"<urn:uuid:22140a8d-9af9-4a5a-b026-784f5747aa12>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00490.warc.gz"} |
Can I get hacked by a buyer if I download a file through Messages ?
Any testimonies or experience ?
Thank you !
26 answers to this question
It depends. eg. if you downloaded a .exe file (executable) then clicked on it and allowed it to run it could do malicious things (anything the programmer made it do). There are probably other file
types that could do various things if you allowed it, eg. those that run scripts. I think Word opens a downloaded document in protected view by default.
Fiverr does do scanning I think but you're also best doing a virus check on anything you download. And I'd make sure you set Windows to show the file extension. And don't run any .exe files if any
are sent. You could do a web search to see what file types are/could be dangerous/malicious (one site says .exe, .com, .bat., .cmd, .msi, .vbs etc).
Edited by uk1000
Actually I was about to write a post about my experience here. Fiverr is not as safe as this comment claims:
On 1/10/2022 at 3:16 PM, haven_art45 said:
No there are no chance because fiverr.com is high secured web site.
Sadly, it is completely the opposite. While Fiverr scans every file sent through the platform, it has no way of scanning links or compressed files. Some so called "buyers" will message you and send
you a link. Once you open the link you are done, your system has been potentially compromised to whatever illed intencion they have. Be careful out there. I have been receiving this kind of messages
with suspicious links quite frequently these past 2 months.
Mac's aren't invulnerable to viruses. You should definitely have good antivirus/malware protection running.
Check out the file from a file virus checking website, like Virustotal
Yes, you can.
It actually happened to me, but fortunately i had 2 factor authentication...so it didn't work
On 1/12/2022 at 7:15 PM, mauro_523 said:
Actually I was about to write a post about my experience here. Fiverr is not as safe as this comment claims:
Sadly, it is completely the opposite. While Fiverr scans every file sent through the platform, it has no way of scanning links or compressed files. Some so called "buyers" will message you and
send you a link. Once you open the link you are done, your system has been potentially compromised to whatever illed intencion they have. Be careful out there. I have been receiving this kind of
messages with suspicious links quite frequently these past 2 months.
@mauro_523 Can you tell me more about harmful and fishing link? Also what is your bad experience with your buyer about any harmful link?
You can use a antivirus to figure it out. I personally use Virustotal and recommend it.
As a strong rule to follow, "Yes" all links and downloads should be considered suspicious. With that, you are using caution on every action you take online. Now, using the understanding of secure
communications within a web application, ask yourself the following:
• Is your PC/ device up to date?
• Does your PC/device has antivirus and antimalware software?
• Is your browser up to date?
• Does your browser URL bar show the "lock" mechanism indicating secure connection?
Now, within the web application (Fiverr.com) you receive a private message from another user. I do not find these messages to be screen thoroughly. I have received several external URL's to google
documents, held on the outside of Fiverr.com's environment. As a Security Professional, that is an immediate red flag and would be cause for concern with that user. You can use some validation
techniques if you desire: try copy/ pasting the URL sent by this user and checking it at virustotal.com and urlscan.io. While these websites are 100% in investigative confirmations of nefarious
actions, they will give you a good idea if the URL that was sent to you is trustworthy or not.
I will also submit, that while only being on Fiverr for a short time, I have received several suspicious messages from "users", where some were listed as "business accounts", asking me to perform
services not specified on my listed gigs. Such as performing a "sit-in" at a table top meeting on a web application kick off meeting (serious??) and some other random tech requests that just didn't
make sense. All of those users sent me an external URL/link that they wanted me to click to see if "I was able to accept the gig".
There is no lack of scammers in the world of "services" via the internet. Social media (facebook in particular) is ate up with people being scammed over the most basic failure of trust requests- its
really sad. Fiverr will be no different, and until they can find a way to mitigate and scan these types of acts (which I doubt they will be able to or will dedicate the time to manage this type
of issue), You just have to be cautious with everything. Fiverr is a great way to earn money and build your professional resume, but don't let emotion/ money/ or lack of activity push you into
making a silly error.
Bottom line- Don't accept messages or gigs that are not officially presented through fiverr.com's formal services. Report all messages that do offer services via outside discussion to fiverr and
block them.
If anyone on Fiverr needs advice or has questions about a private message offer that doesn't appear authentic or some sort of security concern around a "gig" you are performing, PLEASE reach out to
me! - I (DMZ Consulting) will be more than happy to help! if its about an official message or security tasks concerning an official gig on Fiverr, I will answer your question for free. We all
need to be safe!
Good luck!
Yes if he send a ransomware you will attack. please carefully
Yes, that can be possible. so, please be careful before download any file
On 1/9/2022 at 6:23 PM, acurron75 said:
Any testimonies or experience ?
Thank you !
you can use antivirus in your desktop
It will help to stay safe form any virus
I am glad someone is speaking about this, because I just received a suspicious message concerning one of my gigs.
It was a translation work from English to Spanish, a blog post, they said.
I didn't exactly click on the link that he/she provided, yet I copied and pasted it onto a special browser that blocks any trackers while it hides my vpn. When I opened it, it seemed like a regular
blog page. What is suspicious to me is the "offer wall" that appeared in the blog. I risked myself and clicked on it but nothing happened... Luckily.
I had an odd feeling and decided to scan the pages through the tools provided in this thread. It showed that all was clean, but there was an absence of security rating. Let's see what will happen
with the document they're about to send me.
It's happen to me 1 year ago. I've download a zip file and when I open, I have lost all of my data. I had seen a text file in every folder on my pc. There was a mail to contact with them. They ask me
to send them 500 usd to solve this problem. I suggest everybody to use security software such as eset, kaspersky to protect our computer.
Wow, thanks for all of the feedback here, a fiverr buyer is currently sending me a link to download a file and is requesting i download before i start any work. the file says work so im assuming this
buyer is trying to show me something, but my spidey senses are tingling.
yes you can get hacked. Take attention to file extensions.
It is unlikely but be on alert for files that are executables or links that you never saw before
Yes You Can |
If someone send you a .exe file or any kind of images that already corrupted, if you open, you will be see a website same as fiverr, and now if you input any kind of Credential, that you used
already, hacker can easily monitor your browser, its like extensions or addons,
So when someone gives you any files, before you open must be check with Virus Total Website, And Must Be Enable Windows 10 Firewall
On 1/12/2022 at 7:15 PM, mauro_523 said:
Actually I was about to write a post about my experience here. Fiverr is not as safe as this comment claims:
Sadly, it is completely the opposite. While Fiverr scans every file sent through the platform, it has no way of scanning links or compressed files. Some so called "buyers" will message you and
send you a link. Once you open the link you are done, your system has been potentially compromised to whatever illed intencion they have. Be careful out there. I have been receiving this kind of
messages with suspicious links quite frequently these past 2 months.
If buyer send me a suspicious link then what will i do? Report to fiverr?
I think you can get hacked anywhere. Your computer should have a good anti-virus program. Any file can contain a virus.
On 1/12/2022 at 7:15 PM, mauro_523 said:
Actually I was about to write a post about my experience here. Fiverr is not as safe as this comment claims:
Sadly, it is completely the opposite. While Fiverr scans every file sent through the platform, it has no way of scanning links or compressed files. Some so called "buyers" will message you and
send you a link. Once you open the link you are done, your system has been potentially compromised to whatever illed intencion they have. Be careful out there. I have been receiving this kind of
messages with suspicious links quite frequently these past 2 months.
I've also been receiving this kind of messages with suspicious links quite frequently. But I don't click tese link. I give answer them, 'sorry. Why not here? wWe can discuss here.'
On 1/18/2022 at 8:48 PM, shahadatsajib said:
@mauro_523 Can you tell me more about harmful and fishing link? Also what is your bad experience with your buyer about any harmful link?
I think I haven't compromised my system by clicking anything suspicious so far so I don't have any bad experiences to share... but I have been receiving more and more of these requests with
suspicious links (Again, I don't know what the links do or where they take you because I don't clic them). An usual request with a suspicious link looks like this:
(I have sensored names and sensible information as well as the link so no one goes and use it)
Another great example is this one:
I don't know how they manage to hide the actual link but my browser showed a different link when I tried to click it. A good idea is to always hover your mouse over the link so your browser tells you
exactly where that link is taking you.
Some other times they send you files named like something you would recognize (like a .zip or .pdf) but it's something else. Luckily, Fiverr does detect and block these:
Also, these images show the beginning of the pattern I have been using to detect these people. They use always the same wording ("i give you sample sir" gibberish) and most of the time a very broken
english. I have seen them swarming the buyers request section with exactly the same message:
All these pictures were taken from the buyer's requests section with very few days (or hours) appart. I don't know why someone is so eager to make you click a link and quite frankly it scares me to
even think about it. Sadly, I think we are being targeted and this only shows the beginning as most likely they will start changing their methods to something more effective.
On 1/23/2022 at 8:25 AM, mahedi_hossain said:
If buyer send me a suspicious link then what will i do? Report to fiverr?
But of course! Don't hesitate to report and block anyone with a suspicious attitude.
if you set up 2-factor verification no one can hack your id easily. also check the virus scanning before downloading any file from buyers. Thanks.
You can hack anywhere....no system is safe.
you should be careful always.
Check the file before download
You should be careful. Every system faces some bad people. be careful. | {"url":"https://community.fiverr.com/forums/topic/273691-can-i-get-hacked-by-a-buyer-if-i-download-a-file-through-messages/","timestamp":"2024-11-09T02:54:23Z","content_type":"text/html","content_length":"482813","record_id":"<urn:uuid:6d2d660a-9986-44c7-803c-6b7254dabef4>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00571.warc.gz"} |
Numbers in Manx Gaelic
Learn numbers in Manx Gaelic
Knowing numbers in Manx Gaelic is probably one of the most useful things you can learn to say, write and understand in Manx Gaelic. Learning to count in Manx Gaelic may appeal to you just as a simple
curiosity or be something you really need. Perhaps you have planned a trip to a country where Manx Gaelic is the most widely spoken language, and you want to be able to shop and even bargain with a
good knowledge of numbers in Manx Gaelic.
It's also useful for guiding you through street numbers. You'll be able to better understand the directions to places and everything expressed in numbers, such as the times when public transportation
leaves. Can you think of more reasons to learn numbers in Manx Gaelic?
The Manx language (Gaelg, Gailck), also known as Manx Gaelic, belongs to the Celtic languages of the Indo-European languages family. It was spoken as a first language by the Manx people on the Isle
of Man until 1974 with the passing of its last native speaker. After some language revival efforts, it can count with about 1,800 second language speakers, and fifty with Manx as their mother
tongue.Due to lack of data, we can only count accurately up to 1,000 in Manx Gaelic. Please contact me if you can help me counting up from that limit.
List of numbers in Manx Gaelic
Here is a list of numbers in Manx Gaelic. We have made for you a list with all the numbers in Manx Gaelic from 1 to 20. We have also included the tens up to the number 100, so that you know how to
count up to 100 in Manx Gaelic. We also close the list by showing you what the number 1000 looks like in Manx Gaelic.
• 1) nane
• 2) jees
• 3) tree
• 4) kiare
• 5) queig
• 6) shey
• 7) shiaght
• 8) hoght
• 9) nuy
• 10) deich
• 11) nane-jeig
• 12) daa-yeig
• 13) tree-jeig
• 14) kiare-jeig
• 15) queig-jeig
• 16) shey-jeig
• 17) shiaght-jeig
• 18) hoght-jeig
• 19) nuy-jeig
• 20) feed
• 30) jeih as feed
• 40) daeed
• 50) jeih as daeed
• 60) tree feed
• 70) tree feed as jeih
• 80) kiare feed
• 90) kiare feed as jeih
• 100) keead
• 1,000) milley
Numbers in Manx Gaelic: Manx Gaelic numbering rules
Each culture has specific peculiarities that are expressed in its language and its way of counting. The Manx Gaelic is no exception. If you want to learn numbers in Manx Gaelic you will have to learn
a series of rules that we will explain below. If you apply these rules you will soon find that you will be able to count in Manx Gaelic with ease.
The way numbers are formed in Manx Gaelic is easy to understand if you follow the rules explained here. Surprise everyone by counting in Manx Gaelic. Also, learning how to number in Manx Gaelic
yourself from these simple rules is very beneficial for your brain, as it forces it to work and stay in shape. Working with numbers and a foreign language like Manx Gaelic at the same time is one of
the best ways to train our little gray cells, so let's see what rules you need to apply to number in Manx Gaelic
Digits from zero to nine have specific names: neunhee [0], nane [1], jees or daa when compound [2], tree [3], kiare [4], queig [5], shey [6], shiaght [7], hoght [8], and nuy [9].
The tens are following a vigesimal system (based on twenty): jeih [10], feed [20], jeih as feed (10+20) [30], daeed (2*20) [40], jeih as daeed (10+2*20) [50], tree feed (3*20) [60], tree feed as jeih
(3*20+10) [70], kiare feed (4*20) [80], and kiare feed as jeih (4*20+10) [90]. We can note that addition and multiplication are not kept in the same order in the ten creation: 30 and 50 use the
10+y*20 pattern, while 70 and 90 use the y*20+10 pattern.
A decimal system also exists, using the following tens: jeih [10], feed [20], treead [30], daeed [40], queigad [50], sheyad [60], shiagtad [70], hoghtad [80], and nuyad [90].
Teens are formed by starting with the unit, followed by a hyphen and the word for ten (jeig, or yeig for twelve): nane-jeig [11], daa-yeig [12], tree-jeig [13], kiare-jeig [14], queig-jeig [15],
shey-jeig [16], shiaght-jeig [17], hoght-jeig [18], and nuy-jeig [19].
Compound numbers from twenty-one to fifty-nine are formed starting with the unit (or the teen), followed by the particle as, then the ten (e.g.: jees as feed [22], kiare-jeig as daeed [54]).
From sixty-one to ninety-nine, the order is reversed, as compound numbers are formed starting with the ten, followed by the particle as, then the unit (or the teen) (e.g.: tree feed as queig [65],
kiare feed as shiaght-jeig [97]).
Hundreds are formed by stating the multiplier digit before the word for hundred (cheed), linked with a hyphen, except for one hundred: keead [100], daa-cheed [200], tree-cheed [300], kiare-cheed
[400], queig-cheed [500], shey-cheed [600], shiaght-cheed [700], hoght-cheed [800], and nuy-cheed [900].
The word for thousand is milley.
Numbers in different languages | {"url":"https://numbersdata.com/numbers-in-manx-gaelic","timestamp":"2024-11-09T21:02:34Z","content_type":"text/html","content_length":"20519","record_id":"<urn:uuid:5a08e45e-3744-441e-9bbe-6d695ad64340>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00197.warc.gz"} |
Problem I
Even though she saw Zvonko steal Mirko’s microprocessor in the second task, Mirko’s sister Ivana did not tell Mirko because she likes Zvonko. She suggested to him that they go see a movie together so
that she would “forget” about the incident.
Zvonko does not care much for girls because they take away precious time he usually spends practicing his math-fu. He suggested that the two of them play a game and, if Ivana wins, they will go see a
movie together. Ivana agreed, being good at jump rope and she sometimes even kicks a football around with her two brothers.
Zvonko laid $N$ positive integers in a circle on the floor and explained the rules:
• The first player takes any number.
• The second player takes either of the two numbers adjacent to the one the first player took.
• The next player takes a number adjacent to any of the numbers taken so far, and so on until they run out of numbers. The player to take more odd numbers (not divisible by 2) wins.
Zvonko plays optimally; he always looks for a strategy that leads to certain victory or a draw. Zvonko does not know how well Ivana plays. Being a true cavalier, he let Ivana have the first move. But
Ivana only cares about sitting next to Zvonko in front of the big screen so she seeks help playing.
Write a program that finds how many different first moves Ivana can make, so that she has a chance of winning afterwards.
The first line of input contains an integer $N$ $(1 \leq N \leq 100)$, how many numbers there are in the circle. The second line contains $N$ integers separated by single spaces. All numbers will be
between $1$ and $1\, 000$ (inclusive). No two numbers will be the same.
Output the desired number on a single line.
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2
Sample Input 3 Sample Output 3 | {"url":"https://nus.kattis.com/courses/CS3233/CS3233_S2_AY1920/assignments/ok5kbf/problems/ivana","timestamp":"2024-11-07T22:50:38Z","content_type":"text/html","content_length":"29005","record_id":"<urn:uuid:8f1626b8-a6dd-4e3a-84d2-8904e0aeb112>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00790.warc.gz"} |
Why Transformer?
Attention mechanism allows the model to attend every token in the sequence with different amount of focus for each token.
Before applying softmax to the dot product attention, it should be scaled by a factor of $\sqrt{d_{k}}$ to avoid gradient vanishing and slow training.
mask interactions between two tokens by setting the attention values to $-\infty$ before softmax layer.
In self-attention, we are working with the same input sequence. While in cross-attention, we are mixing or combining two different input sequences. In the case of the vanilla transformer
architecture, that’s the sequence returned by the last/top encoder layer on the left and the input sequence being processed by the decoder part on the right.
On a high level, the transformer model consists of $L$ identical blocks, each block composed of an attention module and an MLP module, or FFN for feed-forward neural network.
The weight matrices for query $Q$, key $K$, value $V$ and output $O$ are $W_{q}, W_{k}, W_{v}$, and $W_{o}\in\mathbb{R}^{h\times h}$, respectively. Same goes for bias matrices of shape $\mathbb{R}^
{h}$.^1 Hence the parameters size for this part is $4h^{2}+4h$.
The FFN module has two linear layers. What happenes is the first layer scales up to a higher dimension, or intermediate dimension, and then the second layer scales back down to a dimension of $h$.
Back in GPT’s early days, the scaling factor is 4 (recent models adopt different intermediate dimensions but around 3 to 5 times of $h$) ^2, i.e., the weight matrix for the first layer is $W_{1}\in\
mathbb{R}^{h\times 4h}$ and the weight matrix for the second layer is $W_{2}\in\mathbb{R}^{4h\times h}$. The bias matrices are $\mathbb{R}^{4h}$ and $\mathbb{R}^{h}$, respectively. Hence the
parameters size for the MLP module is $8h^{2}+5h$.
Dont’t forget about LayerNorm. Both self attention module and MLP module are equipped with layer norm layers, learnable parameters including weights $\gamma$ and biases $\beta$. They are all $\mathbb
{R}^{h}$. Hence the parameters size for layer norm is $4h$.
In terms of positional encoding, there is a relatively small amount of parameters if the encoding is learnable. For relative positional encoding, such as RoPE and ALiBi, no trainable parameters are
As a matter of fact, the model starts with tokenization with word embedding and positional embedding. Word embedding matrix is of shape $\mathbb{R}^{V\times h}$. To reduce memory footprint, many
models made the adoption to share the same parameters for the FFN in the final output layer and the word embedding.
Take a look at the model layers of EleutherAI’s gpt-neo-1.3B, a replication of the GPT-3 architecture.
Layer: transformer.wte.weight, Size: torch.Size([50257, 2048])
Layer: transformer.wpe.weight, Size: torch.Size([2048, 2048])
Layer: transformer.h.0.ln_1.weight, Size: torch.Size([2048])
Layer: transformer.h.0.ln_1.bias, Size: torch.Size([2048])
Layer: transformer.h.0.attn.attention.k_proj.weight, Size: torch.Size([2048, 2048])
Layer: transformer.h.0.attn.attention.v_proj.weight, Size: torch.Size([2048, 2048])
Layer: transformer.h.0.attn.attention.q_proj.weight, Size: torch.Size([2048, 2048])
Layer: transformer.h.0.attn.attention.out_proj.weight, Size: torch.Size([2048, 2048])
Layer: transformer.h.0.attn.attention.out_proj.bias, Size: torch.Size([2048])
Layer: transformer.h.0.ln_2.weight, Size: torch.Size([2048])
Layer: transformer.h.0.ln_2.bias, Size: torch.Size([2048])
Layer: transformer.h.0.mlp.c_fc.weight, Size: torch.Size([8192, 2048])
Layer: transformer.h.0.mlp.c_fc.bias, Size: torch.Size([8192])
Layer: transformer.h.0.mlp.c_proj.weight, Size: torch.Size([2048, 8192])
Layer: transformer.h.0.mlp.c_proj.bias, Size: torch.Size([2048])
...<23 identical layers omitted>...
Layer: transformer.ln_f.weight, Size: torch.Size([2048])
Layer: transformer.ln_f.bias, Size: torch.Size([2048])
During the training process, the memory footprint is mainly divided into four parts: model parameters, intermediate activations results produced during the forward pass, gradients computed during the
backward pass, and optimizer states. Here we focus on the memory footprint of parameters, gradients, and optimizer states. During training large language models, AdamW optimizer is commonly used, and
mixed precision training is used to accelerate the training process. Based on this premise, we now take on analyzing the memory footprint in the training process.
Inside a typical training iteration, each learnable parameter corresponds to one gradient and two optimizer states (first and second order momentums from AdamW). Denote the number of learnable
parameters in the model as $\varPhi$, the number of gradients is also $\varPhi$, and the number of optimizer states is $2\varPhi$.
A float16 typed data occupies 2 bytes, 4 bytes for float32. In mixed precision training, float16 is used for forward and backward passes, hence the gradients are stored in float16. During model
parameter update, float32 optimizer states, float32 gradients, and float32 model parameters are used. Therefore, for each learnable parameter, it occupies:
$\underbrace{2+4}_{\text{weights}}+\underbrace{2+4}_{\text{gradients}}+\underbrace{4+4}_{\text{optimizer states}}=20\text{ bytes}$
During the inference process, there is no optimizer states and gradients, and we don’t need to store intermediate activation results. The memory footprint is therefore significantly smaller than that
of training. The majority of memory footprint comes from the model parameters. If float16 is used for inference, the memory footprint of model parameters is about $2\varPhi$ bytes. Moreover, if
KV-Cache is used for speeding up inference, it would also induce additional memory footprint.
1. For instance, llama2 uses an intermediate dimension of 11008 (scaled by 2.6875 times of $h$), Qwen2 uses 22016 (scaled by 5.375 times of $h$), while mistal and llama 3 use 14336 (scaled by 3.5
times of $h$). They all use 4096 as the hidden dimension. ↩ | {"url":"https://www.jiaqicai.com/transformers/","timestamp":"2024-11-11T17:41:45Z","content_type":"text/html","content_length":"100292","record_id":"<urn:uuid:e41b013e-ac31-43b5-a3a3-ed84614a0e45>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00625.warc.gz"} |
2 truths 1 lie (2D shapes)
This is a thinking mathematically targeted teaching opportunity to explore and reason about the features of some 2D shapes.
Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus (2022) © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales,
Collect resources
You will need:
• paper
• pencils or markers
• a small sticky square to investigate.
Watch 2 truths 1 lie (2D shapes) part 1 video (2:15).
[Text over a navy-blue background: 2 truths. 1 lie. Small font text in the lower left-hand corner reads: NSW Mathematics Strategy Professional Learning Team (NSWMS PL team). In the lower right-hand
corner is the white waratah of the NSW Government logo.]
Two truths, one lie…
[A title on a white background reads: You will need…
Bullet points below read:
eyeballs and brains
something to write on and write with
a small sticky square to investigate.
Next to the last bullet point are 2 red squares. The square on the right has its corner turned up so it looks like a diamond.]
... for this task, you will need your eyeballs and brains, something to write on and write with and if you have a small sticky square to investigate.
[Text over a navy-blue background: Let’s explore!]
Let's explore.
[A large white sheet divided into 3 columns. The first column has a blue shape with text below that reads squares have 4 equal sides. The second column has a yellow shape with text below that reads
diamond. The last column has a pink shape with text below that reads triangle.]
Hello, mathematicians, I've got a challenge for you today. It's called two truths, one lie, I've got three statements here, two of them are true and one is a lie. Let's have a look at them, my first
statement is squares have four equal sides…
[The speaker points to the text in the first column.]
... my second statement is this shape here…
[She traces the yellow shape in the middle column.]
... is a diamond and my third statement is…
[She points to the pink shape in the last column.]
... this is a triangle. Now, have a look at those three statements, which one do you think is the lie?
And well, let's work through the first statement. Squares are four equal sides, the first part of that is that squares have four sides. So, let's say squares have four equal size, I want to start
[She places a finger just outside the left side of the blue shape in the first column.]
... and I'll place my finger here, so I remember where I started…
[She points to the left side of the shape, then the top side, then the right and bottom.]
... one, two, three, four.
[She takes her hands away.]
Now, you maybe already knew that squares have four sides but are they equal? I’ve got some cubes here…
[She places two red cubes near the top right corner of the shape.]
... they called unifix, and I'm going to use these to measure the size of this square.
[She lays some unifix across the top of the shape, snapping them together as she goes.]
OK, let's see, one, two, three, four.
[She leaves the assembled unifix against the top of the shape.]
So, the length of the top side is four unifix. Let's see what the length of the bottom is…
[She picks up the unifix.]
... is it four unit fixed?
[She places the unifix against the bottom of the shape.]
[She picks up the unifix and places it against the right side of the shape.]
The length of the right is four unifix…
[She picks up the unifix and places it against the left side of the shape.]
... and the length of the left is also four unifix. So I've just proven that squares have four equal sides.
[Text over a navy-blue background: Over to you! Which statement is true and which is a lie?
Below the text is an image of a section of the sheet. The first column shows a yellow shape with text below that reads diamond. The second column shows a pink shape with text below that reads
Now over to you. Which statement is true and which is a lie? Is the yellow shape a diamond? Is the pink shape a triangle? Which is true and which is the lie?
[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]
[End of transcript]
Can you help Barbara figure out which of the statements are true and which one is a lie?
• Squares have 4 equal sides
• This shape is a diamond
• This shape is a triangle
Watch 2 truths 1 lie (2D shapes) part 2 video (2:50).
[A large white sheet of paper divided into 3 columns. The first column has a blue square with text below that reads squares have 4 equal sides. The second column has a yellow shape with text below
that reads diamond. The last column has a pink shape, with text below that reads triangle.]
How did you go? Now, we've proven that the first statement is true, so I'm just going to write true here
[The speaker writes ‘true’ in the first column, under the text.]
Which means that one of these must be a lie.
[She points to the middle and last columns.]
Do you think that this is the lie…
[She points to the yellow shape in the middle column.]
... that that is in fact not a diamond? Or do you think that this is the lie…
[She points to the pink shape in the last column.]
... that that is not actually a triangle? OK, so, I want to show you something that I think might make you change your mind if you think that this is true. Remember how we said that squares have four
equal sides, and I use my unifix here to measure them.
[She places an assembled unifix of 4 cubes next the right side of the square in the first column.]
Each side was four unifix long.
[She places the unifix against the top of the square.]
That's right.
[She places the unifix against the left side of the square, then the bottom.]
Now looking at this shape, it also looks like there might be four equal sides. Let's test that out.
[She places the unifix against the bottom right side of the shape.]
One, yep, that side is four unifix long…
[She places the unifix against the top right side of the shape.]
... this side is four unifix long…
[She places the unifix against the top left side of the shape.]
... this side is four unifix long.
[She places the unifix against the bottom left side of the shape.]
And this side is four unifix long. So it looks like that shape here also has four equal sides. Now I'm gonna show you something amazing, are you ready?
[She picks up the yellow shape in the middle column and turns it slightly to the right.]
Ta da. Yeah, it's a square. It was just on its point and we're used to seeing squares resting on their sides, but it's still a square.
[She picks up the yellow shape in the middle column and turns it slightly to the left, three times.]
So this is a square, this is a square, and this is a square. In fact, they're the same sized square.
[She picks up the yellow square and places it on top of the blue square in the first column.]
OK, so that means that this statement here, that this is a diamond…
[She points to the middle column.]
... is a lie.
[She writes ‘lie’ in the middle column under the text.]
Now let's go to the last statement. That is a triangle. Now, we are very used to seeing triangles like this.
[She picks up the pink shape in the last column and turns it slightly to the left.]
Also on their side. Sometimes they might look like this…
[She picks up the pink shape in the last column and turns it slightly to the right.]
... but that's the way that we're used to seeing triangles.
[She picks up the pink shape in the last column and turns it slightly to the right, with its point facing down.]
But in the same way that this is a square that just happens to be on its point, this is a triangle and this is a triangle…
[She picks up the triangle and turns it slightly to the left with its point facing right and its left side completely straight.]
... and even diagonally.
[She turns the triangle so that its right point faces the top right corner of the sheet.]
So this statement here is true.
[She writes ‘true’ in the third column under the text.
Text over a navy-blue background: What's (some of) the mathematics?
A title on a white background reads: What's (some of) the mathematics?
A bullet point below reads:
• The orientation of a shape does not change it. For example, a square on its point is still a square.
Below the point are 2 red squares. The square on the right has its corner turned up so it looks like a diamond.]
What's some of the mathematics? The orientation of a shape does not change it. For example, a square on its point is still a square. We're used to seeing squares resting on their side, but changing
the orientation, rotating the shape, doesn't change what it is.
[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]
[End of transcript]
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LibGuides: Math Resources: Compound Events
Compound events involve two (or more) independent events happening together. In these events, the compound probability is computed by multiplying the probabilities of each individual event.
Example 1
Consider the event of flipping a coin, rolling a die, and drawing a card from a deck of cards. Find the probability of flipping heads, rolling an even number, or drawing a 7.
To compute this probability, it may help to think of the event as being three separate events: 1) flipping heads, 2) rolling an even number, and 3) drawing a 7. In order to compute the compound
probability, or the probability that you complete all three of these events together, can be computed by finding each event's probability and then multiplying them.
Event 1
P(flipping heads) = 1/2
because there are two possible outcomes and one of them is heads
Event 2
P(rolling an even number) = 3/6 = 1/2
because there are 6 possible outcomes on a die and 3 of them are even
Event 3
P(drawing a 7) = 4/52
because there are 52 cards in a standard deck of playing cards and 4 of them are 7's - one for each suit
Now we can take each of these probabilities and multiply them to find the compound probability:
1/2 * 1/2 * 4/52 = 1/52 or approximately .019
Example 2
You're picking out ice cream at a shop that offers 3 different flavors of ice cream (vanilla, chocolate, strawberry) and 5 toppings (sprinkles, chocolate syrup, cherries, chocolate chips, whip
cream). If you randomly choose an ice cream flavor and one topping, what is the probability that you pick vanilla ice cream without chocolate?
Again, we want to imagine this as two independent events: 1) picking vanilla ice cream and 2) picking a topping that isn't chocolate. Note that there are two (2) chocolate options in the list
(chocolate syrup and chocolate chips). So we begin by finding the probability for each event.
Event 1
P(vanilla) = 1/3
because there are 3 flavors to choose from and 1 is vanilla
Event 2
P(not chocolate) = 3/5
because there are 5 toppings to choose from and 3 of them are not chocolate
Then we use the multiplication rule to compute the compound probability of these two events happening together:
1/3 * 3/5 = 3/15 = 1/5 or .20 | {"url":"https://resources.nu.edu/c.php?g=1336977&p=10407700","timestamp":"2024-11-07T04:14:19Z","content_type":"text/html","content_length":"31750","record_id":"<urn:uuid:88cb8f8e-8a0c-4d3e-aaf3-5aca3094121e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00645.warc.gz"} |
In this thesis we will discuss the properties of the category $\mathcal{O}$ of left $\mathfrak{g}$-modules having some specific properties, where $\mathfrak{g}$ is a complex semisimple Lie algebra.
We will also discuss the projective objects of $\mathcal{O}$, and will establish the fact that each object in $\mathcal{O}$ is a factor object of a projective object. We will prove that there exists
a one-to-one correspondence between the indecomposable projective objects and simple objects of $\mathcal{O}$. We will discuss some facts about the full subcategory $\mathcal{O}_\theta$ of $\mathcal
{O}$. And finally we will establish a relation between the Cartan matrix and the decomposition matrix with the help of the BGG reciprocity and the fact that each projective module in $\mathcal{O}$
admits a $p$-filtration. | {"url":"https://math.iisc.ac.in/seminars/2019/2019-07-30-swarnalipa-datta.html","timestamp":"2024-11-02T12:38:10Z","content_type":"text/html","content_length":"20071","record_id":"<urn:uuid:e27d2387-0ec8-4ac8-bd33-fcdc2053cedb>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00017.warc.gz"} |
[Solved] All numbers are in $ '000. Consider an in | SolutionInn
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All numbers are in $ '000. Consider an income property. Next three years its NOIs will be $25,000, $28,000 and $30,000. Then NOI will be
All numbers are in $ '000.
Consider an income property. Next three years its NOIs will be $25,000, $28,000 and $30,000.
Then NOI will be growing at a constant rate of 3% per year.
If you buy the Subject property, you are planning to hold it for three years then you would sell it.
When you sell it, the terminal cap. rate you are going to apply is 11%. (Hint: pls use this rate to calculate the reversion value, i.e. sale price)
Your discount rate is 12% per year. (This is the rate to be used to discount the three NOIs in the next three years and the sale price).
How much will be your maximum offer for the subject property?
Group of answer choices
None of the above
There are 3 Steps involved in it
Step: 1
To calculate the maximum offer for the subject property you can use the formula for the present valu...
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Recommended Textbook for
Authors: Eugene F. Brigham, Phillip R. Daves
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View Answer in SolutionInn App | {"url":"https://www.solutioninn.com/study-help/questions/all-numbers-are-in-000-consider-an-income-property-701746","timestamp":"2024-11-05T00:23:15Z","content_type":"text/html","content_length":"111256","record_id":"<urn:uuid:0463b5ab-7c51-47eb-829d-b84bc92190d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00661.warc.gz"} |
BigDFT.PostProcessing module
BigDFT.PostProcessing module
A module for post processing BigDFT calculations.
class BigDFTool(omp=1, mpi_run=None)[source]
This module defines a number of post-processing options, including those that can be driven by utilities like bigdft-tool, memguess, or utilities.
☆ omp (int) – number of OpenMP threads. It defaults to the $OMP_NUM_THREADS variable in the environment, if present, otherwise it fixes the run to 1 thread.
☆ mpi_run (str) – define the MPI command to be used. It defaults to the value $BIGDFT_MPIRUN of the environment, if present.
auto_fragment(system, view, cutoff, verbose=False, rand=False, criteria='bondorder')[source]
Refragment a system based on the results of a previous calculation.
The auto fragment protocol performs a greedy optimization using either the bond order or the distance between fragments as a guide. The purity values and the fragment bond orders are
essential quantities for this calculation, so they can be passed if they are already cached.
By using the rand keyword, you can trigger a stochastic refragmentation process. If this process is repeated many times, you may find a result that improves upon the greedy optimization
○ system (BigDFT.Systems.System) – the system to fragment.
○ view (BigDFT.Systems.FragmentView) – a view of the system.
○ cutoff (float) – the termination criteria. When the worst fragment is more pure than this cutoff, the calculation stops.
○ verbose (bool) – whether to print out as it proceeds.
○ rand (bool) – whether to activate the stochastic approach.
○ criteria (string) – either distance or bondorder.
a mapping from the old system to the new system where each fragment fullfills the purity criteria.
Return type:
fragment_small_molecule(sys, view, cutoff=0.05, maxiter=100, verbose=False, seed=0)[source]
Stochastic algorithm for automatically fragmenting small molecules.
a mapping from the original fragments to the new fragments. (BigDFT.Systems.FragmentView): the new fragment view.
Return type:
fragment_mask_matrix(sys, mat, fragments, log)[source]
Sometimes we don’t want to store an entire matrix, just the parts related to some fragments of interest. This routine will mask out a matrix, keeping entries only related to the list of
fragments provided.
○ sys (BigDFT.Systems.System) – the system associated with the matrix.
○ mat (scipy.sparse.csr_matrix) – the matrix to mask.
○ fragments (list) – a list of fragment ids to keep.
○ log (BigDFT.Logfiles.Logfile) – the logfile associated with this matrix’s calculation.
the masked matrix.
Return type:
compute_fragment_dos(frag, log, ks_coeff, eigvals, frag_indices=None, smat=None, assume_pure=False, **kwargs)[source]
Compute the partial density of states projected onto a given fragment.
○ sys (BigDFT.Fragments.Fragment) – the fragment to project on to.
○ log (BigDFT.Logfiles.Logfile) – the log of the calculation.
○ ks_coeff (scipy.sparse.csc_matrix) – the matrix of eigenvectors.
○ eigvals (list) – a list of eigenvalues.
○ frag_indices (list) – list of indices associated with this fragment.
○ smat (scipy.sparse.csc_matrix) – the overlap matrix.
○ assume_pure (bool) – an optimization can be performed if we assume the target is pure.
○ kwargs (dict) – any extra options to pass to the DoS constructor.
a density of states object built using the partial density of states.
Return type:
create_layered_qmmm_system(system, target, pairwise_bo, cutoffs, criteria='bondorder', link_atoms=False)[source]
Creates a multilayered system suitable for QM/MM calculations. For each layer, a suitable QM region is built around it.
a list of Systems, one for each QM layer. (System): the MM region.
Return type:
create_qmmm_system(system, target, bond_order, cutoff, criteria='bondorder', link_atoms=False)[source]
Creates a system suitable for QM/MM calculations.
the QM region. (System): the MM region.
Return type:
fragment_bond_order(sys, fraglist1, fraglist2, log, kxs=None, frag_indices=None)[source]
Computes “bond order” between two sets of fragments using the method of Mayer. For two atomic fragments, this would describe the bond multiplicity of the covalent bond.
○ sys (BigDFT.Systems.System) – the system containing the fragments of interest
○ fraglist1 (list) – a list of fragments to compute the bond order of.
○ fraglist2 (list) – a list of fragments to compute the bond order between.
○ log (BigDFT.Logfiles.Logfile) – the log describing a calculation.
○ kxs (scipy.sparse.csc_matrix) – the matrix K*S, which might be already computed to reduce I/O time.
○ frag_indices (dict) – the matrix indices associated with each fragment.
a dictionary of dictionaries, mapping the bond order of each fragment in fraglist1 to each fragment in fraglist2.
Return type:
fragment_interaction_energy(sys, fraglist1, fraglist2, log, frag_indices=None, sinvh=None, kxs=None)[source]
Compute the interaction energies between two sets of fragments.
○ fraglist1 (list) – a list of fragments to compute the interaction energy of.
○ fraglist2 (list) – a list of fragments to compute the interaction energy between.
○ log (BigDFT.Logfiles.Logfile) – the log describing a calculation.
○ frag_indices (dict) – the matrix indices associated with each fragment.
○ sinvh (scipy.sparse.csc_matrix) – the matrix S^{-1}*H, which might be already computed to reduce I/O time.
○ kxs (scipy.sparse.csc_matrix) – the matrix K*S, which might be already computed to reduce I/O time.
the projected energy.
Return type:
fragment_population(sys, log, frag_indices=None, kxs=None)[source]
Performs Mulliken population analysis on a fragment, in case charges haven’t been computed by doing a multipole analysis.
○ sys (BigDFT.Systems.System) – the system to compute the population of.
○ log (BigDFT.Logfiles.Logfile) – the log describing a calculation.
○ frag_indices (dict) – the matrix indices associated with each fragment.
○ kxs (scipy.sparse.csc_matrix) – the matrix K*S, which might be already computed to reduce I/O time.
a mapping from fragment ids to charges.
Return type:
generate_link_atoms(fullsys, subsys, distcut=6.0)[source]
This routine adds link atoms to a subsystem based on the bond order of a full system. Link atom positions are automatically adjusted based on the length of some standard bonds.
○ fullsys (BigDFT.Systems.System) – the full system that the subsystem is embedded into.
○ subsys (BigDFT.Systems.System) – the embedded system which needs link atoms.
○ distcut (float) – this cutoff is the largest distance value we expect allow a bond to be.
the subsystem with link atoms added. (BigDFT.Systems.System): a system which has the atoms that were
removed and replaced with link atoms.
Return type:
run_compute_purity(system, log, kxs=None, frag_list=None, frag_indices=None)[source]
Compute the purity values of the different system fragments.
Note that this can also be computed using the fragment multipoles, but this provides an implementation for when you don’t need those values.
○ system (System) – instance of a System class, which defines the fragmentation.
○ log (Logfile) – logfile from the run computed on this system.
○ kxs (scipy.sparse.csc_matrix) – the matrix K*S, which might be already computed to reduce I/O time.
○ frag_list (list) – we can also only compute the purity values of some select fragments.
○ frag_indices (list) – the indices of the matrix associated with each fragment. This can be precomputed and passed.
for each fragment id, what is the purity value.
Return type:
get_frag_indices(sys, log)[source]
Compute a lookup table of matrix indices for each fragment in a system.
○ system (System) – instance of a System class, which defines the fragmentation.
○ log (Logfile) – logfile from the run computed on this system.
a mapping of fragment ids to lists of indices
Return type:
Retrieve the Kohn-Sham coefficients and the eigenvalues in matrix form.
log (Logfile): instance of a Logfile class
the matrix of coefficients (list): list of eigenvalues.
Return type:
Computes the matrix K*S, the mulliken version of the density matrix, and loads it into memory.
log (Logfile): instance of a Logfile class
the matrix K*S
Return type:
derived_quantity(name, log, files, generator)[source]
Defines a quantity that can be derived from the ccs files in the files list. Requires the generator function.
ccs_node(log, matrixname)[source]
Defines the protocol to validate the ccs matrix file
Computes the matrix S^{-1}*H, the mulliken version of the spillage matrix, and loads it into memory.
log (Logfile): instance of a Logfile class
the matrix S^{-1}*H
Return type:
Read the overlap matrix into memory.
log (Logfile): instance of a Logfile class
the matrix S
Return type:
Read the hamiltonian matrix into memory.
log (Logfile): instance of a Logfile class
the matrix H
Return type:
Read the density matrix into memory.
log (Logfile): instance of a Logfile class
the matrix K
Return type:
superunits_purities(bo, pv, q_units, superunits, q_superunits)[source]
Compute the purity values of the superunits described as a unification of the units
☆ bo (matrix-like) – Fragment Bond Orders of the Units
☆ pv (array-like) – Purity values of the Units
☆ superunits (dict) – lookup dictionary containing the list of units per superunit
☆ q_units (array-like) – charges of the units
☆ q_superunits (array-like) – charges of the superunits
purities of the superunits
Return type:
superunits_quadratic_quantities(bo, superunits)[source]
Compute quantities that transforms like the bond orders of the superunits described as a unification of the units
☆ bo (matrix-like) – Quantities like Fragment Bond Orders of the Units
☆ superunits (dict) – lookup dictionary containing the list of units per superunit
quantities of the superunits
Return type:
systems_heatmap(data, restrict_to=None, axs=None, columns=None, **kwargs)[source]
Create a heatmap for a set of systems.
☆ data (dict) – a dictionary mapping system names to another dictionary which maps fragment ids to property values.
☆ restrict_to (list) – a list of lists saying what fragments we should draw. It is a list of lists instead of just a list because this way we can put a separator between values (for
example, when making a jump in the sequence).
☆ axs (matplotlib.axis) – the axis to draw on.
☆ columns (list) – the order of the columns on the x axis
☆ kwargs (dict) – any extra argument you wish to pass to matplotlib’s imshow command.
a reference to the imshow value.
dict_distplot(system_dict, ax=None, reuse_ticks=False, kind='violin', **kwargs)[source]
Represent a violinplot from a dictionary of values.
☆ system_dict (dict) – dictionary mapping the labels to plot with their values.
☆ ax (matplotlib.pyplot.axis) – the axis to plot on.
☆ reuse_ticks (bool) – Employ the existing xticks of the axes to plot the data
☆ kind (str) – ‘violin’ for violinplot, ‘box’ for boxplot.
☆ **kwargs – any other argument to be passed to the violin/boxplot
the axis of the plot and the object returned from
the violinplot.
Return type:
The following is an example of module usage:
Postprocessing Example
from BigDFT.Systems import System, FragmentView
from BigDFT.Fragments import Fragment
from BigDFT.IO import XYZReader
from BigDFT.Calculators import SystemCalculator
from BigDFT.Inputfiles import Inputfile
from scipy.linalg import eigh
from copy import deepcopy
# Create a system
sys = System()
sys["FRA:0"] = Fragment()
with XYZReader("CH2") as ifile:
for at in ifile:
sys["FRA:1"] = Fragment()
with XYZReader("CH3") as ifile:
for at in ifile:
sys["FRA:1"].translate([0, 0, -3])
sys["FRA:2"] = deepcopy(sys["FRA:0"])
sys["FRA:2"].translate([0, 0, 3])
sys["FRA:2"].rotate(y=150, units="degrees")
sys["FRA:3"] = deepcopy(sys["FRA:0"])
sys["FRA:3"].translate([3, 0, 1.5])
# Run a calculation in the linear mode.
inp = Inputfile()
inp["import"] = "linear"
code = SystemCalculator()
log = code.run(input=inp, posinp=sys.get_posinp(), run_dir="work")
# Create the post processing tool.
from BigDFT.PostProcessing import BigDFTool
btool = BigDFTool()
# Purity
purity = btool.run_compute_purity(sys, log)
# Charges
charges = {fragid: sum(at.nel for at in frag)
for fragid, frag in sys.items()}
# Bond Orders
bo = btool.fragment_bond_order(sys, sys.keys(), sys.keys(), log)
# Population values.
population = btool.fragment_population(sys, log)
# These three things define a fragment view.
view = FragmentView(purity, bo, charges)
# Auto Fragmentation
mapping = btool.auto_fragment(sys, view, 0.10)
# This defines a new view.
new_view = view.refragment(mapping)
# Eigenvalues.
H = btool.get_matrix_h(log)
S = btool.get_matrix_s(log)
w = eigh(H.todense(), b=S.todense(), eigvals_only=True) | {"url":"https://l_sim.gitlab.io/bigdft-suite/PyBigDFT/build/html/BigDFT.PostProcessing.html","timestamp":"2024-11-10T04:32:44Z","content_type":"text/html","content_length":"78336","record_id":"<urn:uuid:5da71fe4-c379-4f5d-8088-84e11b158a45>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00163.warc.gz"} |
How many horsepower is 530cc?
Rotapower 530cc
530cc Rotapower® Engine Specifications
Engine Type Two Rotors
Cooling of Rotor Intake Charge or Air
Dimensions L × W × H 26 × 11 × 11 in
Standard Configuration Maximum Power 240 hp
How do you calculate CC to horsepower?
Calculate the horsepower or CC by multiplying or dividing by 15. The general rule is for every 15 CC there is 1 HP. For example, for a 150 CC engine you would take 150 divided by 15, which equals 10
How much horsepower is in 800cc?
54 hp
All that riding and momentum is of course officiated by the 800cc engine. The three cylinders on it produce 54 hp of power and 72 Nm of torque.
What is 500 cc converted to horsepower?
As such, a 500cc or 0.5L small, four-cycle engine has an approximate horsepower of 15.53.
How many HP is 224 cc?
224cc 7.5HP Gasoline Engine with EPA, Carb, Ce, Soncap Certificate (YF220)
How many horsepower is 700cc?
How much HP is 600 cc? Can a 600 beat a 1000? How many horsepower is 1000 cc’s? How many cc is a 5.5 hp engine?…How many cc is a 5.5 hp engine?
Horsepower 9.0 / 5.97 kW
Top governed speed 3250
Oil capacity .77 Liter
How much HP can 750cc injectors make?
Regardless of what some calculators may say, healthy 750cc injectors will easily and safely support over 800BHP all day long on gasoline.
How much HP will 60lb injectors support?
60′ will support 600+ rwhp with pump fuel…not the case with e85 at those levels!
How much horsepower is 650cc?
These are numbers for a well tuned motorcycle engine. At 650cc and 50 hp it is considerably under powered giving fairly good mileage and longer engine life. It also becomes fairly predictable why
most motorcycles are faster than cars.
How much horsepower does a 1200cc engine have?
Triumph’s 1200cc gives this Bobber a power figure of 77 horses and 77 ft-lb of torque. | {"url":"https://hollows.info/how-many-horsepower-is-530cc/","timestamp":"2024-11-08T01:49:41Z","content_type":"text/html","content_length":"41087","record_id":"<urn:uuid:e2a09996-0e35-45fb-b45a-19e600d4602c>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00361.warc.gz"} |
docs/source/nnls_modeling.rst - ceres-solver - Git at Google
.. highlight:: c++
.. default-domain:: cpp
.. cpp:namespace:: ceres
.. _`chapter-nnls_modeling`:
Modeling Non-linear Least Squares
Ceres solver consists of two distinct parts. A modeling API which
provides a rich set of tools to construct an optimization problem one
term at a time and a solver API that controls the minimization
algorithm. This chapter is devoted to the task of modeling
optimization problems using Ceres. :ref:`chapter-nnls_solving` discusses
the various ways in which an optimization problem can be solved using
Ceres solves robustified bounds constrained non-linear least squares
problems of the form:
.. math:: :label: ceresproblem_modeling
\min_{\mathbf{x}} &\quad \frac{1}{2}\sum_{i}
... ,x_{i_k}\right)\right\|^2\right) \\
\text{s.t.} &\quad l_j \le x_j \le u_j
In Ceres parlance, the expression
is known as a **residual block**, where :math:`f_i(\cdot)` is a
:class:`CostFunction` that depends on the **parameter blocks**
:math:`\left\{x_{i_1},... , x_{i_k}\right\}`.
In most optimization problems small groups of scalars occur
together. For example the three components of a translation vector and
the four components of the quaternion that define the pose of a
camera. We refer to such a group of scalars as a **parameter block**. Of
course a parameter block can be just a single scalar too.
:math:`\rho_i` is a :class:`LossFunction`. A :class:`LossFunction` is
a scalar valued function that is used to reduce the influence of
outliers on the solution of non-linear least squares problems.
:math:`l_j` and :math:`u_j` are lower and upper bounds on the
parameter block :math:`x_j`.
As a special case, when :math:`\rho_i(x) = x`, i.e., the identity
function, and :math:`l_j = -\infty` and :math:`u_j = \infty` we get
the usual unconstrained `non-linear least squares problem
.. math:: :label: ceresproblemunconstrained
\frac{1}{2}\sum_{i} \left\|f_i\left(x_{i_1}, ... ,x_{i_k}\right)\right\|^2.
For each term in the objective function, a :class:`CostFunction` is
responsible for computing a vector of residuals and Jacobian
matrices. Concretely, consider a function
:math:`f\left(x_{1},...,x_{k}\right)` that depends on parameter blocks
:math:`\left[x_{1}, ... , x_{k}\right]`.
Then, given :math:`\left[x_{1}, ... , x_{k}\right]`,
:class:`CostFunction` is responsible for computing the vector
:math:`f\left(x_{1},...,x_{k}\right)` and the Jacobian matrices
.. math:: J_i = D_i f(x_1, ..., x_k) \quad \forall i \in \{1, \ldots, k\}
.. class:: CostFunction
.. code-block:: c++
class CostFunction {
virtual bool Evaluate(double const* const* parameters,
double* residuals,
double** jacobians) const = 0;
const std::vector<int32>& parameter_block_sizes();
int num_residuals() const;
std::vector<int32>* mutable_parameter_block_sizes();
void set_num_residuals(int num_residuals);
The signature of the :class:`CostFunction` (number and sizes of input
parameter blocks and number of outputs) is stored in
:member:`CostFunction::parameter_block_sizes_` and
:member:`CostFunction::num_residuals_` respectively. User code
inheriting from this class is expected to set these two members with
the corresponding accessors. This information will be verified by the
:class:`Problem` when added with :func:`Problem::AddResidualBlock`.
.. function:: bool CostFunction::Evaluate(double const* const* parameters, double* residuals, double** jacobians) const
Compute the residual vector and the Jacobian matrices.
``parameters`` is an array of arrays of size
``CostFunction::parameter_block_sizes_.size()`` and
``parameters[i]`` is an array of size ``parameter_block_sizes_[i]``
that contains the :math:`i^{\text{th}}` parameter block that the
``CostFunction`` depends on.
``parameters`` is never ``nullptr``.
``residuals`` is an array of size ``num_residuals_``.
``residuals`` is never ``nullptr``.
``jacobians`` is an array of arrays of size
If ``jacobians`` is ``nullptr``, the user is only expected to compute
the residuals.
``jacobians[i]`` is a row-major array of size ``num_residuals x
If ``jacobians[i]`` is **not** ``nullptr``, the user is required to
compute the Jacobian of the residual vector with respect to
``parameters[i]`` and store it in this array, i.e.
``jacobians[i][r * parameter_block_sizes_[i] + c]`` =
:math:`\frac{\displaystyle \partial \text{residual}[r]}{\displaystyle \partial \text{parameters}[i][c]}`
If ``jacobians[i]`` is ``nullptr``, then this computation can be
skipped. This is the case when the corresponding parameter block is
marked constant.
The return value indicates whether the computation of the residuals
and/or jacobians was successful or not. This can be used to
communicate numerical failures in Jacobian computations for
.. class:: SizedCostFunction
If the size of the parameter blocks and the size of the residual
vector is known at compile time (this is the common case),
:class:`SizeCostFunction` can be used where these values can be
specified as template parameters and the user only needs to
implement :func:`CostFunction::Evaluate`.
.. code-block:: c++
template<int kNumResiduals, int... Ns>
class SizedCostFunction : public CostFunction {
virtual bool Evaluate(double const* const* parameters,
double* residuals,
double** jacobians) const = 0;
.. class:: AutoDiffCostFunction
Defining a :class:`CostFunction` or a :class:`SizedCostFunction`
can be a tedious and error prone especially when computing
derivatives. To this end Ceres provides `automatic differentiation
.. code-block:: c++
template <typename CostFunctor,
int kNumResiduals, // Number of residuals, or ceres::DYNAMIC.
int... Ns> // Size of each parameter block
class AutoDiffCostFunction : public
SizedCostFunction<kNumResiduals, Ns> {
// Instantiate CostFunctor using the supplied arguments.
template<class ...Args>
explicit AutoDiffCostFunction(Args&& ...args);
explicit AutoDiffCostFunction(std::unique_ptr<CostFunctor> functor);
explicit AutoDiffCostFunction(CostFunctor* functor, ownership = TAKE_OWNERSHIP);
// Ignore the template parameter kNumResiduals and use
// num_residuals instead.
AutoDiffCostFunction(CostFunctor* functor,
int num_residuals,
ownership = TAKE_OWNERSHIP);
AutoDiffCostFunction(std::unique_ptr<CostFunctor> functor,
int num_residuals);
To get an auto differentiated cost function, you must define a
class with a templated ``operator()`` (a functor) that computes the
cost function in terms of the template parameter ``T``. The
autodiff framework substitutes appropriate ``Jet`` objects for
``T`` in order to compute the derivative when necessary, but this
is hidden, and you should write the function as if ``T`` were a
scalar type (e.g. a double-precision floating point number).
The function must write the computed value in the last argument
(the only non-``const`` one) and return true to indicate success.
For example, consider a scalar error :math:`e = k - x^\top y`,
where both :math:`x` and :math:`y` are two-dimensional vector
parameters and :math:`k` is a constant. The form of this error,
which is the difference between a constant and an expression, is a
common pattern in least squares problems. For example, the value
:math:`x^\top y` might be the model expectation for a series of
measurements, where there is an instance of the cost function for
each measurement :math:`k`.
The actual cost added to the total problem is :math:`e^2`, or
:math:`(k - x^\top y)^2`; however, the squaring is implicitly done
by the optimization framework.
To write an auto-differentiable cost function for the above model,
first define the object
.. code-block:: c++
class MyScalarCostFunctor {
MyScalarCostFunctor(double k): k_(k) {}
template <typename T>
bool operator()(const T* const x , const T* const y, T* e) const {
e[0] = k_ - x[0] * y[0] - x[1] * y[1];
return true;
double k_;
Note that in the declaration of ``operator()`` the input parameters
``x`` and ``y`` come first, and are passed as const pointers to arrays
of ``T``. If there were three input parameters, then the third input
parameter would come after ``y``. The output is always the last
parameter, and is also a pointer to an array. In the example above,
``e`` is a scalar, so only ``e[0]`` is set.
Then given this class definition, the auto differentiated cost
function for it can be constructed as follows.
.. code-block:: c++
auto* cost_function
= new AutoDiffCostFunction<MyScalarCostFunctor, 1, 2, 2>(1.0);
^ ^ ^
| | |
Dimension of residual ------+ | |
Dimension of x ----------------+ |
Dimension of y -------------------+
In this example, there is usually an instance for each measurement
of ``k``.
In the instantiation above, the template parameters following
``MyScalarCostFunction``, ``<1, 2, 2>`` describe the functor as
computing a 1-dimensional output from two arguments, both
By default :class:`AutoDiffCostFunction` will take ownership of the cost
functor pointer passed to it, ie. will call `delete` on the cost functor
when the :class:`AutoDiffCostFunction` itself is deleted. However, this may
be undesirable in certain cases, therefore it is also possible to specify
:class:`DO_NOT_TAKE_OWNERSHIP` as a second argument in the constructor,
while passing a pointer to a cost functor which does not need to be deleted
by the AutoDiffCostFunction. For example:
.. code-block:: c++
MyScalarCostFunctor functor(1.0)
auto* cost_function
= new AutoDiffCostFunction<MyScalarCostFunctor, 1, 2, 2>(
&functor, DO_NOT_TAKE_OWNERSHIP);
:class:`AutoDiffCostFunction` also supports cost functions with a
runtime-determined number of residuals. For example:
.. code-block:: c++
auto functor = std::make_unique<CostFunctorWithDynamicNumResiduals>(1.0);
auto* cost_function
= new AutoDiffCostFunction<CostFunctorWithDynamicNumResiduals,
DYNAMIC, 2, 2>(
std::move(functor), ^ ^ ^
runtime_number_of_residuals); <----+ | | |
| | | |
| | | |
Actual number of residuals ------+ | | |
Indicate dynamic number of residuals --------+ | |
Dimension of x ------------------------------------+ |
Dimension of y ---------------------------------------+
.. warning::
A common beginner's error when first using :class:`AutoDiffCostFunction`
is to get the sizing wrong. In particular, there is a tendency to set the
template parameters to (dimension of residual, number of parameters)
instead of passing a dimension parameter for *every parameter block*. In
the example above, that would be ``<MyScalarCostFunction, 1, 2>``, which
is missing the 2 as the last template argument.
.. class:: DynamicAutoDiffCostFunction
:class:`AutoDiffCostFunction` requires that the number of parameter
blocks and their sizes be known at compile time. In a number of
applications, this is not enough e.g., Bezier curve fitting, Neural
Network training etc.
.. code-block:: c++
template <typename CostFunctor, int Stride = 4>
class DynamicAutoDiffCostFunction : public CostFunction {
In such cases :class:`DynamicAutoDiffCostFunction` can be
used. Like :class:`AutoDiffCostFunction` the user must define a
templated functor, but the signature of the functor differs
slightly. The expected interface for the cost functors is:
.. code-block:: c++
struct MyCostFunctor {
template<typename T>
bool operator()(T const* const* parameters, T* residuals) const {
Since the sizing of the parameters is done at runtime, you must
also specify the sizes after creating the dynamic autodiff cost
function. For example:
.. code-block:: c++
auto* cost_function = new DynamicAutoDiffCostFunction<MyCostFunctor, 4>();
Under the hood, the implementation evaluates the cost function
multiple times, computing a small set of the derivatives (four by
default, controlled by the ``Stride`` template parameter) with each
pass. There is a performance tradeoff with the size of the passes;
Smaller sizes are more cache efficient but result in larger number
of passes, and larger stride lengths can destroy cache-locality
while reducing the number of passes over the cost function. The
optimal value depends on the number and sizes of the various
parameter blocks.
As a rule of thumb, try using :class:`AutoDiffCostFunction` before
you use :class:`DynamicAutoDiffCostFunction`.
.. class:: NumericDiffCostFunction
In some cases, its not possible to define a templated cost functor,
for example when the evaluation of the residual involves a call to a
library function that you do not have control over. In such a
situation, `numerical differentiation
<http://en.wikipedia.org/wiki/Numerical_differentiation>`_ can be
.. NOTE ::
TODO(sameeragarwal): Add documentation for the constructor and for
NumericDiffOptions. Update DynamicNumericDiffOptions in a similar
.. code-block:: c++
template <typename CostFunctor,
NumericDiffMethodType method = CENTRAL,
int kNumResiduals, // Number of residuals, or ceres::DYNAMIC.
int... Ns> // Size of each parameter block.
class NumericDiffCostFunction : public
SizedCostFunction<kNumResiduals, Ns> {
To get a numerically differentiated :class:`CostFunction`, you must
define a class with a ``operator()`` (a functor) that computes the
residuals. The functor must write the computed value in the last
argument (the only non-``const`` one) and return ``true`` to
indicate success. Please see :class:`CostFunction` for details on
how the return value may be used to impose simple constraints on the
parameter block. e.g., an object of the form
.. code-block:: c++
struct ScalarFunctor {
bool operator()(const double* const x1,
const double* const x2,
double* residuals) const;
For example, consider a scalar error :math:`e = k - x'y`, where both
:math:`x` and :math:`y` are two-dimensional column vector
parameters, the prime sign indicates transposition, and :math:`k` is
a constant. The form of this error, which is the difference between
a constant and an expression, is a common pattern in least squares
problems. For example, the value :math:`x'y` might be the model
expectation for a series of measurements, where there is an instance
of the cost function for each measurement :math:`k`.
To write an numerically-differentiable class:`CostFunction` for the
above model, first define the object
.. code-block:: c++
class MyScalarCostFunctor {
MyScalarCostFunctor(double k): k_(k) {}
bool operator()(const double* const x,
const double* const y,
double* residuals) const {
residuals[0] = k_ - x[0] * y[0] + x[1] * y[1];
return true;
double k_;
Note that in the declaration of ``operator()`` the input parameters
``x`` and ``y`` come first, and are passed as const pointers to
arrays of ``double`` s. If there were three input parameters, then
the third input parameter would come after ``y``. The output is
always the last parameter, and is also a pointer to an array. In the
example above, the residual is a scalar, so only ``residuals[0]`` is
Then given this class definition, the numerically differentiated
:class:`CostFunction` with central differences used for computing
the derivative can be constructed as follows.
.. code-block:: c++
auto* cost_function
= new NumericDiffCostFunction<MyScalarCostFunctor, CENTRAL, 1, 2, 2>(1.0)
^ ^ ^ ^
| | | |
Finite Differencing Scheme -+ | | |
Dimension of residual ------------+ | |
Dimension of x ----------------------+ |
Dimension of y -------------------------+
In this example, there is usually an instance for each measurement
of `k`.
In the instantiation above, the template parameters following
``MyScalarCostFunctor``, ``1, 2, 2``, describe the functor as
computing a 1-dimensional output from two arguments, both
NumericDiffCostFunction also supports cost functions with a
runtime-determined number of residuals. For example:
.. code-block:: c++
auto functor = std::make_unique<CostFunctorWithDynamicNumResiduals>(1.0);
auto* cost_function
= new NumericDiffCostFunction<CostFunctorWithDynamicNumResiduals,
CENTRAL, DYNAMIC, 2, 2>(
std::move(functor), ^ ^ ^
runtime_number_of_residuals); <----+ | | |
| | | |
| | | |
Actual number of residuals ------+ | | |
Indicate dynamic number of residuals --------+ | |
Dimension of x ------------------------------------+ |
Dimension of y ---------------------------------------+
There are three available numeric differentiation schemes in ceres-solver:
The ``FORWARD`` difference method, which approximates :math:`f'(x)`
by computing :math:`\frac{f(x+h)-f(x)}{h}`, computes the cost
function one additional time at :math:`x+h`. It is the fastest but
least accurate method.
The ``CENTRAL`` difference method is more accurate at the cost of
twice as many function evaluations than forward difference,
estimating :math:`f'(x)` by computing
The ``RIDDERS`` difference method[Ridders]_ is an adaptive scheme
that estimates derivatives by performing multiple central
differences at varying scales. Specifically, the algorithm starts at
a certain :math:`h` and as the derivative is estimated, this step
size decreases. To conserve function evaluations and estimate the
derivative error, the method performs Richardson extrapolations
between the tested step sizes. The algorithm exhibits considerably
higher accuracy, but does so by additional evaluations of the cost
Consider using ``CENTRAL`` differences to begin with. Based on the
results, either try forward difference to improve performance or
Ridders' method to improve accuracy.
.. warning::
A common beginner's error when first using
:class:`NumericDiffCostFunction` is to get the sizing wrong. In
particular, there is a tendency to set the template parameters to
(dimension of residual, number of parameters) instead of passing a
dimension parameter for *every parameter*. In the example above, that
would be ``<MyScalarCostFunctor, 1, 2>``, which is missing the last ``2``
argument. Please be careful when setting the size parameters.
Numeric Differentiation & Manifolds
If your cost function depends on a parameter block that must lie on
a manifold and the functor cannot be evaluated for values of that
parameter block not on the manifold then you may have problems
numerically differentiating such functors.
This is because numeric differentiation in Ceres is performed by
perturbing the individual coordinates of the parameter blocks that
a cost functor depends on. This perturbation assumes that the
parameter block lives on a Euclidean Manifold rather than the
actual manifold associated with the parameter block. As a result
some of the perturbed points may not lie on the manifold anymore.
For example consider a four dimensional parameter block that is
interpreted as a unit Quaternion. Perturbing the coordinates of
this parameter block will violate the unit norm property of the
parameter block.
Fixing this problem requires that :class:`NumericDiffCostFunction`
be aware of the :class:`Manifold` associated with each
parameter block and only generate perturbations in the local
tangent space of each parameter block.
For now this is not considered to be a serious enough problem to
warrant changing the :class:`NumericDiffCostFunction` API. Further,
in most cases it is relatively straightforward to project a point
off the manifold back onto the manifold before using it in the
functor. For example in case of the Quaternion, normalizing the
4-vector before using it does the trick.
**Alternate Interface**
For a variety of reasons, including compatibility with legacy code,
:class:`NumericDiffCostFunction` can also take
:class:`CostFunction` objects as input. The following describes
To get a numerically differentiated cost function, define a
subclass of :class:`CostFunction` such that the
:func:`CostFunction::Evaluate` function ignores the ``jacobians``
parameter. The numeric differentiation wrapper will fill in the
jacobian parameter if necessary by repeatedly calling the
:func:`CostFunction::Evaluate` with small changes to the
appropriate parameters, and computing the slope. For performance,
the numeric differentiation wrapper class is templated on the
concrete cost function, even though it could be implemented only in
terms of the :class:`CostFunction` interface.
The numerically differentiated version of a cost function for a
cost function can be constructed as follows:
.. code-block:: c++
auto* cost_function
= new NumericDiffCostFunction<MyCostFunction, CENTRAL, 1, 4, 8>(...);
where ``MyCostFunction`` has 1 residual and 2 parameter blocks with
sizes 4 and 8 respectively. Look at the tests for a more detailed
.. class:: DynamicNumericDiffCostFunction
Like :class:`AutoDiffCostFunction` :class:`NumericDiffCostFunction`
requires that the number of parameter blocks and their sizes be
known at compile time. In a number of applications, this is not enough.
.. code-block:: c++
template <typename CostFunctor, NumericDiffMethodType method = CENTRAL>
class DynamicNumericDiffCostFunction : public CostFunction {
In such cases when numeric differentiation is desired,
:class:`DynamicNumericDiffCostFunction` can be used.
Like :class:`NumericDiffCostFunction` the user must define a
functor, but the signature of the functor differs slightly. The
expected interface for the cost functors is:
.. code-block:: c++
struct MyCostFunctor {
bool operator()(double const* const* parameters, double* residuals) const {
Since the sizing of the parameters is done at runtime, you must
also specify the sizes after creating the dynamic numeric diff cost
function. For example:
.. code-block:: c++
auto cost_function = std::make_unique<DynamicNumericDiffCostFunction<MyCostFunctor>>();
As a rule of thumb, try using :class:`NumericDiffCostFunction` before
you use :class:`DynamicNumericDiffCostFunction`.
.. warning::
The same caution about mixing manifolds with numeric differentiation
applies as is the case with :class:`NumericDiffCostFunction`.
.. class:: CostFunctionToFunctor
:class:`CostFunctionToFunctor` is an adapter class that allows
users to use :class:`CostFunction` objects in templated functors
which are to be used for automatic differentiation. This allows
the user to seamlessly mix analytic, numeric and automatic
For example, let us assume that
.. code-block:: c++
class IntrinsicProjection : public SizedCostFunction<2, 5, 3> {
IntrinsicProjection(const double* observation);
virtual bool Evaluate(double const* const* parameters,
double* residuals,
double** jacobians) const;
is a :class:`CostFunction` that implements the projection of a
point in its local coordinate system onto its image plane and
subtracts it from the observed point projection. It can compute its
residual and either via analytic or numerical differentiation can
compute its jacobians.
Now we would like to compose the action of this
:class:`CostFunction` with the action of camera extrinsics, i.e.,
rotation and translation. Say we have a templated function
.. code-block:: c++
template<typename T>
void RotateAndTranslatePoint(const T* rotation,
const T* translation,
const T* point,
T* result);
Then we can now do the following,
.. code-block:: c++
struct CameraProjection {
explicit CameraProjection(double* observation)
: intrinsic_projection_(std::make_unique<IntrinsicProjection>(observation)) {
template <typename T>
bool operator()(const T* rotation,
const T* translation,
const T* intrinsics,
const T* point,
T* residual) const {
T transformed_point[3];
RotateAndTranslatePoint(rotation, translation, point, transformed_point);
// Note that we call intrinsic_projection_, just like it was
// any other templated functor.
return intrinsic_projection_(intrinsics, transformed_point, residual);
CostFunctionToFunctor<2, 5, 3> intrinsic_projection_;
Note that :class:`CostFunctionToFunctor` takes ownership of the
:class:`CostFunction` that was passed in to the constructor.
In the above example, we assumed that ``IntrinsicProjection`` is a
``CostFunction`` capable of evaluating its value and its
derivatives. Suppose, if that were not the case and
``IntrinsicProjection`` was defined as follows:
.. code-block:: c++
struct IntrinsicProjection {
IntrinsicProjection(const double* observation) {
observation_[0] = observation[0];
observation_[1] = observation[1];
bool operator()(const double* calibration,
const double* point,
double* residuals) const {
double projection[2];
ThirdPartyProjectionFunction(calibration, point, projection);
residuals[0] = observation_[0] - projection[0];
residuals[1] = observation_[1] - projection[1];
return true;
double observation_[2];
Here ``ThirdPartyProjectionFunction`` is some third party library
function that we have no control over. So this function can compute
its value and we would like to use numeric differentiation to
compute its derivatives. In this case we can use a combination of
``NumericDiffCostFunction`` and ``CostFunctionToFunctor`` to get the
job done.
.. code-block:: c++
struct CameraProjection {
explicit CameraProjection(double* observation)
: intrinsic_projection_(
std::make_unique<NumericDiffCostFunction<IntrinsicProjection, CENTRAL, 2, 5, 3>>()) {}
template <typename T>
bool operator()(const T* rotation,
const T* translation,
const T* intrinsics,
const T* point,
T* residuals) const {
T transformed_point[3];
RotateAndTranslatePoint(rotation, translation, point, transformed_point);
return intrinsic_projection_(intrinsics, transformed_point, residuals);
CostFunctionToFunctor<2, 5, 3> intrinsic_projection_;
.. class:: DynamicCostFunctionToFunctor
:class:`DynamicCostFunctionToFunctor` provides the same functionality as
:class:`CostFunctionToFunctor` for cases where the number and size of the
parameter vectors and residuals are not known at compile-time. The API
provided by :class:`DynamicCostFunctionToFunctor` matches what would be
expected by :class:`DynamicAutoDiffCostFunction`, i.e. it provides a
templated functor of this form:
.. code-block:: c++
template<typename T>
bool operator()(T const* const* parameters, T* residuals) const;
Similar to the example given for :class:`CostFunctionToFunctor`, let us
assume that
.. code-block:: c++
class IntrinsicProjection : public CostFunction {
IntrinsicProjection(const double* observation);
virtual bool Evaluate(double const* const* parameters,
double* residuals,
double** jacobians) const;
is a :class:`CostFunction` that projects a point in its local coordinate
system onto its image plane and subtracts it from the observed point
Using this :class:`CostFunction` in a templated functor would then look like
.. code-block:: c++
struct CameraProjection {
explicit CameraProjection(double* observation)
: intrinsic_projection_(std::make_unique<IntrinsicProjection>(observation)) {
template <typename T>
bool operator()(T const* const* parameters,
T* residual) const {
const T* rotation = parameters[0];
const T* translation = parameters[1];
const T* intrinsics = parameters[2];
const T* point = parameters[3];
T transformed_point[3];
RotateAndTranslatePoint(rotation, translation, point, transformed_point);
const T* projection_parameters[2];
projection_parameters[0] = intrinsics;
projection_parameters[1] = transformed_point;
return intrinsic_projection_(projection_parameters, residual);
DynamicCostFunctionToFunctor intrinsic_projection_;
Like :class:`CostFunctionToFunctor`, :class:`DynamicCostFunctionToFunctor`
takes ownership of the :class:`CostFunction` that was passed in to the
.. class:: ConditionedCostFunction
This class allows you to apply different conditioning to the residual
values of a wrapped cost function. An example where this is useful is
where you have an existing cost function that produces N values, but you
want the total cost to be something other than just the sum of these
squared values - maybe you want to apply a different scaling to some
values, to change their contribution to the cost.
.. code-block:: c++
// my_cost_function produces N residuals
CostFunction* my_cost_function = ...
CHECK_EQ(N, my_cost_function->num_residuals());
std::vector<CostFunction*> conditioners;
// Make N 1x1 cost functions (1 parameter, 1 residual)
CostFunction* f_1 = ...
CostFunction* f_N = ...
ConditionedCostFunction* ccf =
new ConditionedCostFunction(my_cost_function, conditioners);
Now ``ccf`` 's ``residual[i]`` (i=0..N-1) will be passed though the
:math:`i^{\text{th}}` conditioner.
.. code-block:: c++
ccf_residual[i] = f_i(my_cost_function_residual[i])
and the Jacobian will be affected appropriately.
.. class:: GradientChecker
This class compares the Jacobians returned by a cost function
against derivatives estimated using finite differencing. It is
meant as a tool for unit testing, giving you more fine-grained
control than the check_gradients option in the solver options.
The condition enforced is that
.. math:: \forall{i,j}: \frac{J_{ij} - J'_{ij}}{max_{ij}(J_{ij} - J'_{ij})} < r
where :math:`J_{ij}` is the jacobian as computed by the supplied
cost function multiplied by the `Manifold::PlusJacobian`,
:math:`J'_{ij}` is the jacobian as computed by finite differences,
multiplied by the `Manifold::PlusJacobian` as well, and :math:`r`
is the relative precision.
.. code-block:: c++
// my_cost_function takes two parameter blocks. The first has a
// manifold associated with it.
CostFunction* my_cost_function = ...
Manifold* my_manifold = ...
NumericDiffOptions numeric_diff_options;
std::vector<Manifold*> manifolds;
std::vector parameter1;
std::vector parameter2;
// Fill parameter 1 & 2 with test data...
std::vector<double*> parameter_blocks;
GradientChecker gradient_checker(my_cost_function,
GradientCheckResults results;
if (!gradient_checker.Probe(parameter_blocks.data(), 1e-9, &results) {
LOG(ERROR) << "An error has occurred:\n" << results.error_log;
.. class:: NormalPrior
.. code-block:: c++
class NormalPrior: public CostFunction {
// Check that the number of rows in the vector b are the same as the
// number of columns in the matrix A, crash otherwise.
NormalPrior(const Matrix& A, const Vector& b);
virtual bool Evaluate(double const* const* parameters,
double* residuals,
double** jacobians) const;
Implements a cost function of the form
.. math:: cost(x) = ||A(x - b)||^2
where, the matrix :math:`A` and the vector :math:`b` are fixed and :math:`x`
is the variable. In case the user is interested in implementing a cost
function of the form
.. math:: cost(x) = (x - \mu)^T S^{-1} (x - \mu)
where, :math:`\mu` is a vector and :math:`S` is a covariance matrix,
then, :math:`A = S^{-1/2}`, i.e the matrix :math:`A` is the square
root of the inverse of the covariance, also known as the stiffness
matrix. There are however no restrictions on the shape of
:math:`A`. It is free to be rectangular, which would be the case if
the covariance matrix :math:`S` is rank deficient.
.. _`section-loss_function`:
.. class:: LossFunction
For least squares problems where the minimization may encounter
input terms that contain outliers, that is, completely bogus
measurements, it is important to use a loss function that reduces
their influence.
Consider a structure from motion problem. The unknowns are 3D
points and camera parameters, and the measurements are image
coordinates describing the expected reprojected position for a
point in a camera. For example, we want to model the geometry of a
street scene with fire hydrants and cars, observed by a moving
camera with unknown parameters, and the only 3D points we care
about are the pointy tippy-tops of the fire hydrants. Our magic
image processing algorithm, which is responsible for producing the
measurements that are input to Ceres, has found and matched all
such tippy-tops in all image frames, except that in one of the
frame it mistook a car's headlight for a hydrant. If we didn't do
anything special the residual for the erroneous measurement will
result in the entire solution getting pulled away from the optimum
to reduce the large error that would otherwise be attributed to the
wrong measurement.
Using a robust loss function, the cost for large residuals is
reduced. In the example above, this leads to outlier terms getting
down-weighted so they do not overly influence the final solution.
.. code-block:: c++
class LossFunction {
virtual void Evaluate(double s, double out[3]) const = 0;
The key method is :func:`LossFunction::Evaluate`, which given a
non-negative scalar ``s``, computes
.. math:: out = \begin{bmatrix}\rho(s), & \rho'(s), & \rho''(s)\end{bmatrix}
Here the convention is that the contribution of a term to the cost
function is given by :math:`\frac{1}{2}\rho(s)`, where :math:`s
=\|f_i\|^2`. Calling the method with a negative value of :math:`s`
is an error and the implementations are not required to handle that
Most sane choices of :math:`\rho` satisfy:
.. math::
\rho(0) &= 0\\
\rho'(0) &= 1\\
\rho'(s) &< 1 \text{ in the outlier region}\\
\rho''(s) &< 0 \text{ in the outlier region}
so that they mimic the squared cost for small residuals.
Given one robustifier :math:`\rho(s)` one can change the length
scale at which robustification takes place, by adding a scale
factor :math:`a > 0` which gives us :math:`\rho(s,a) = a^2 \rho(s /
a^2)` and the first and second derivatives as :math:`\rho'(s /
a^2)` and :math:`(1 / a^2) \rho''(s / a^2)` respectively.
The reason for the appearance of squaring is that :math:`a` is in
the units of the residual vector norm whereas :math:`s` is a squared
norm. For applications it is more convenient to specify :math:`a` than
its square.
Ceres includes a number of predefined loss functions. For simplicity
we described their unscaled versions. The figure below illustrates
their shape graphically. More details can be found in
.. figure:: loss.png
:figwidth: 500px
:height: 400px
:align: center
Shape of the various common loss functions.
.. class:: TrivialLoss
.. math:: \rho(s) = s
.. class:: HuberLoss
.. math:: \rho(s) = \begin{cases} s & s \le 1\\ 2 \sqrt{s} - 1 & s > 1 \end{cases}
.. class:: SoftLOneLoss
.. math:: \rho(s) = 2 (\sqrt{1+s} - 1)
.. class:: CauchyLoss
.. math:: \rho(s) = \log(1 + s)
.. class:: ArctanLoss
.. math:: \rho(s) = \arctan(s)
.. class:: TolerantLoss
.. math:: \rho(s,a,b) = b \log(1 + e^{(s - a) / b}) - b \log(1 + e^{-a / b})
.. class:: TukeyLoss
.. math:: \rho(s) = \begin{cases} \frac{1}{3} (1 - (1 - s)^3) & s \le 1\\ \frac{1}{3} & s > 1 \end{cases}
.. class:: ComposedLoss
Given two loss functions ``f`` and ``g``, implements the loss
function ``h(s) = f(g(s))``.
.. code-block:: c++
class ComposedLoss : public LossFunction {
explicit ComposedLoss(const LossFunction* f,
Ownership ownership_f,
const LossFunction* g,
Ownership ownership_g);
.. class:: ScaledLoss
Sometimes you want to simply scale the output value of the
robustifier. For example, you might want to weight different error
terms differently (e.g., weight pixel reprojection errors
differently from terrain errors).
Given a loss function :math:`\rho(s)` and a scalar :math:`a`, :class:`ScaledLoss`
implements the function :math:`a \rho(s)`.
Since we treat a ``nullptr`` Loss function as the Identity loss
function, :math:`rho` = ``nullptr``: is a valid input and will result
in the input being scaled by :math:`a`. This provides a simple way
of implementing a scaled ResidualBlock.
.. class:: LossFunctionWrapper
Sometimes after the optimization problem has been constructed, we
wish to mutate the scale of the loss function. For example, when
performing estimation from data which has substantial outliers,
convergence can be improved by starting out with a large scale,
optimizing the problem and then reducing the scale. This can have
better convergence behavior than just using a loss function with a
small scale.
This templated class allows the user to implement a loss function
whose scale can be mutated after an optimization problem has been
constructed, e.g,
.. code-block:: c++
Problem problem;
// Add parameter blocks
auto* cost_function =
new AutoDiffCostFunction<UW_Camera_Mapper, 2, 9, 3>(feature_x, feature_y);
LossFunctionWrapper* loss_function(new HuberLoss(1.0), TAKE_OWNERSHIP);
problem.AddResidualBlock(cost_function, loss_function, parameters);
Solver::Options options;
Solver::Summary summary;
Solve(options, &problem, &summary);
loss_function->Reset(new HuberLoss(1.0), TAKE_OWNERSHIP);
Solve(options, &problem, &summary);
Let us consider a problem with a single parameter block.
.. math::
\min_x \frac{1}{2}\rho(f^2(x))
Then, the robustified gradient and the Gauss-Newton Hessian are
.. math::
g(x) &= \rho'J^\top(x)f(x)\\
H(x) &= J^\top(x)\left(\rho' + 2 \rho''f(x)f^\top(x)\right)J(x)
where the terms involving the second derivatives of :math:`f(x)` have
been ignored. Note that :math:`H(x)` is indefinite if
:math:`\rho''f(x)^\top f(x) + \frac{1}{2}\rho' < 0`. If this is not
the case, then its possible to re-weight the residual and the Jacobian
matrix such that the robustified Gauss-Newton step corresponds to an
ordinary linear least squares problem.
Let :math:`\alpha` be a root of
.. math:: \frac{1}{2}\alpha^2 - \alpha - \frac{\rho''}{\rho'}\|f(x)\|^2 = 0.
Then, define the rescaled residual and Jacobian as
.. math::
\tilde{f}(x) &= \frac{\sqrt{\rho'}}{1 - \alpha} f(x)\\
\tilde{J}(x) &= \sqrt{\rho'}\left(1 - \alpha
\frac{f(x)f^\top(x)}{\left\|f(x)\right\|^2} \right)J(x)
In the case :math:`2 \rho''\left\|f(x)\right\|^2 + \rho' \lesssim 0`,
we limit :math:`\alpha \le 1- \epsilon` for some small
:math:`\epsilon`. For more details see [Triggs]_.
With this simple rescaling, one can apply any Jacobian based non-linear
least squares algorithm to robustified non-linear least squares
While the theory described above is elegant, in practice we observe
that using the Triggs correction when :math:`\rho'' > 0` leads to poor
performance, so we upper bound it by zero. For more details see
`corrector.cc <https://github.com/ceres-solver/ceres-solver/blob/master/internal/ceres/corrector.cc#L51>`_
.. class:: Manifold
In sensor fusion problems, we often have to model quantities that live
in spaces known as `Manifolds
<https://en.wikipedia.org/wiki/Manifold>`_, for example the
rotation/orientation of a sensor that is represented by a `Quaternion
Manifolds are spaces which locally look like Euclidean spaces. More
precisely, at each point on the manifold there is a linear space that
is tangent to the manifold. It has dimension equal to the intrinsic
dimension of the manifold itself, which is less than or equal to the
ambient space in which the manifold is embedded.
For example, the tangent space to a point on a sphere in three
dimensions is the two dimensional plane that is tangent to the sphere
at that point. There are two reasons tangent spaces are interesting:
1. They are Eucliean spaces so the usual vector space operations apply
there, which makes numerical operations easy.
2. Movements in the tangent space translate into movements along the
manifold. Movements perpendicular to the tangent space do not
translate into movements on the manifold.
However, moving along the 2 dimensional plane tangent to the sphere
and projecting back onto the sphere will move you away from the point
you started from but moving along the normal at the same point and the
projecting back onto the sphere brings you back to the point.
Besides the mathematical niceness, modeling manifold valued
quantities correctly and paying attention to their geometry has
practical benefits too:
1. It naturally constrains the quantity to the manifold throughout the
optimization, freeing the user from hacks like *quaternion
2. It reduces the dimension of the optimization problem to its
*natural* size. For example, a quantity restricted to a line is a
one dimensional object regardless of the dimension of the ambient
space in which this line lives.
Working in the tangent space reduces not just the computational
complexity of the optimization algorithm, but also improves the
numerical behaviour of the algorithm.
A basic operation one can perform on a manifold is the
:math:`\boxplus` operation that computes the result of moving along
:math:`\delta` in the tangent space at :math:`x`, and then projecting
back onto the manifold that :math:`x` belongs to. Also known as a
*Retraction*, :math:`\boxplus` is a generalization of vector addition
in Euclidean spaces.
The inverse of :math:`\boxplus` is :math:`\boxminus`, which given two
points :math:`y` and :math:`x` on the manifold computes the tangent
vector :math:`\Delta` at :math:`x` s.t. :math:`\boxplus(x, \Delta) =
Let us now consider two examples.
The `Euclidean space <https://en.wikipedia.org/wiki/Euclidean_space>`_
:math:`\mathbb{R}^n` is the simplest example of a manifold. It has
dimension :math:`n` (and so does its tangent space) and
:math:`\boxplus` and :math:`\boxminus` are the familiar vector sum and
difference operations.
.. math::
\boxplus(x, \Delta) &= x + \Delta = y\\
\boxminus(y, x) &= y - x = \Delta.
A more interesting case is the case :math:`SO(3)`, the `special
orthogonal group <https://en.wikipedia.org/wiki/3D_rotation_group>`_
in three dimensions - the space of :math:`3\times3` rotation
matrices. :math:`SO(3)` is a three dimensional manifold embedded in
:math:`\mathbb{R}^9` or :math:`\mathbb{R}^{3\times 3}`. So points on :math:`SO(3)` are
represented using 9 dimensional vectors or :math:`3\times 3` matrices,
and points in its tangent spaces are represented by 3 dimensional
For :math:`SO(3)`, :math:`\boxplus` and :math:`\boxminus` are defined
in terms of the matrix :math:`\exp` and :math:`\log` operations as
Given a 3-vector :math:`\Delta = [\begin{matrix}p,&q,&r\end{matrix}]`, we have
.. math::
\exp(\Delta) & = \left [ \begin{matrix}
\cos \theta + cp^2 & -sr + cpq & sq + cpr \\
sr + cpq & \cos \theta + cq^2& -sp + cqr \\
-sq + cpr & sp + cqr & \cos \theta + cr^2
\end{matrix} \right ]
.. math::
\theta &= \sqrt{p^2 + q^2 + r^2},\\
s &= \frac{\sin \theta}{\theta},\\
c &= \frac{1 - \cos \theta}{\theta^2}.
Given :math:`x \in SO(3)`, we have
.. math::
\log(x) = 1/(2 \sin(\theta)/\theta)\left[\begin{matrix} x_{32} - x_{23},& x_{13} - x_{31},& x_{21} - x_{12}\end{matrix} \right]
.. math:: \theta = \cos^{-1}((\operatorname{Trace}(x) - 1)/2)
.. math::
\boxplus(x, \Delta) &= x \exp(\Delta)
\boxminus(y, x) &= \log(x^T y)
For :math:`\boxplus` and :math:`\boxminus` to be mathematically
consistent, the following identities must be satisfied at all points
:math:`x` on the manifold:
1. :math:`\boxplus(x, 0) = x`. This ensures that the tangent space is
*centered* at :math:`x`, and the zero vector is the identity
2. For all :math:`y` on the manifold, :math:`\boxplus(x,
\boxminus(y,x)) = y`. This ensures that any :math:`y` can be
reached from math:`x`.
3. For all :math:`\Delta`, :math:`\boxminus(\boxplus(x, \Delta), x) =
\Delta`. This ensures that :math:`\boxplus` is an injective
(one-to-one) map.
4. For all :math:`\Delta_1, \Delta_2\ |\boxminus(\boxplus(x, \Delta_1),
\boxplus(x, \Delta_2)) \leq |\Delta_1 - \Delta_2|`. Allows us to define
a metric on the manifold.
Additionally we require that :math:`\boxplus` and :math:`\boxminus` be
sufficiently smooth. In particular they need to be differentiable
everywhere on the manifold.
For more details, please see [Hertzberg]_
The :class:`Manifold` interface allows the user to define a manifold
for the purposes optimization by implementing ``Plus`` and ``Minus``
operations and their derivatives (corresponding naturally to
:math:`\boxplus` and :math:`\boxminus`).
.. code-block:: c++
class Manifold {
virtual ~Manifold();
virtual int AmbientSize() const = 0;
virtual int TangentSize() const = 0;
virtual bool Plus(const double* x,
const double* delta,
double* x_plus_delta) const = 0;
virtual bool PlusJacobian(const double* x, double* jacobian) const = 0;
virtual bool RightMultiplyByPlusJacobian(const double* x,
const int num_rows,
const double* ambient_matrix,
double* tangent_matrix) const;
virtual bool Minus(const double* y,
const double* x,
double* y_minus_x) const = 0;
virtual bool MinusJacobian(const double* x, double* jacobian) const = 0;
.. function:: int Manifold::AmbientSize() const;
Dimension of the ambient space in which the manifold is embedded.
.. function:: int Manifold::TangentSize() const;
Dimension of the manifold/tangent space.
.. function:: bool Plus(const double* x, const double* delta, double* x_plus_delta) const;
Implements the :math:`\boxplus(x,\Delta)` operation for the manifold.
A generalization of vector addition in Euclidean space, ``Plus``
computes the result of moving along ``delta`` in the tangent space
at ``x``, and then projecting back onto the manifold that ``x``
belongs to.
``x`` and ``x_plus_delta`` are :func:`Manifold::AmbientSize` vectors.
``delta`` is a :func:`Manifold::TangentSize` vector.
Return value indicates if the operation was successful or not.
.. function:: bool PlusJacobian(const double* x, double* jacobian) const;
Compute the derivative of :math:`\boxplus(x, \Delta)` w.r.t
:math:`\Delta` at :math:`\Delta = 0`, i.e. :math:`(D_2
\boxplus)(x, 0)`.
``jacobian`` is a row-major :func:`Manifold::AmbientSize`
:math:`\times` :func:`Manifold::TangentSize` matrix.
Return value indicates whether the operation was successful or not.
.. function:: bool RightMultiplyByPlusJacobian(const double* x, const int num_rows, const double* ambient_matrix, double* tangent_matrix) const;
``tangent_matrix`` = ``ambient_matrix`` :math:`\times` plus_jacobian.
``ambient_matrix`` is a row-major ``num_rows`` :math:`\times`
:func:`Manifold::AmbientSize` matrix.
``tangent_matrix`` is a row-major ``num_rows`` :math:`\times`
:func:`Manifold::TangentSize` matrix.
Return value indicates whether the operation was successful or not.
This function is only used by the :class:`GradientProblemSolver`,
where the dimension of the parameter block can be large and it may
be more efficient to compute this product directly rather than
first evaluating the Jacobian into a matrix and then doing a matrix
vector product.
Because this is not an often used function, we provide a default
implementation for convenience. If performance becomes an issue
then the user should consider implementing a specialization.
.. function:: bool Minus(const double* y, const double* x, double* y_minus_x) const;
Implements :math:`\boxminus(y,x)` operation for the manifold.
A generalization of vector subtraction in Euclidean spaces, given
two points ``x`` and ``y`` on the manifold, ``Minus`` computes the
change to ``x`` in the tangent space at ``x``, that will take it to
``x`` and ``y`` are :func:`Manifold::AmbientSize` vectors.
``y_minus_x`` is a ::func:`Manifold::TangentSize` vector.
Return value indicates if the operation was successful or not.
.. function:: bool MinusJacobian(const double* x, double* jacobian) const = 0;
Compute the derivative of :math:`\boxminus(y, x)` w.r.t :math:`y`
at :math:`y = x`, i.e :math:`(D_1 \boxminus) (x, x)`.
``jacobian`` is a row-major :func:`Manifold::TangentSize`
:math:`\times` :func:`Manifold::AmbientSize` matrix.
Return value indicates whether the operation was successful or not.
Ceres Solver ships with a number of commonly used instances of
For `Lie Groups <https://en.wikipedia.org/wiki/Lie_group>`_, a great
place to find high quality implementations is the `Sophus
<https://github.com/strasdat/Sophus>`_ library developed by Hauke
Strasdat and his collaborators.
.. class:: EuclideanManifold
:class:`EuclideanManifold` as the name implies represents a Euclidean
space, where the :math:`\boxplus` and :math:`\boxminus` operations are
the usual vector addition and subtraction.
.. math::
\boxplus(x, \Delta) &= x + \Delta\\
\boxminus(y,x) &= y - x
By default parameter blocks are assumed to be Euclidean, so there is
no need to use this manifold on its own. It is provided for the
purpose of testing and for use in combination with other manifolds
using :class:`ProductManifold`.
The class works with dynamic and static ambient space dimensions. If
the ambient space dimensions is known at compile time use
.. code-block:: c++
EuclideanManifold<3> manifold;
If the ambient space dimensions is not known at compile time the
template parameter needs to be set to `ceres::DYNAMIC` and the actual
dimension needs to be provided as a constructor argument:
.. code-block:: c++
EuclideanManifold<ceres::DYNAMIC> manifold(ambient_dim);
.. class:: SubsetManifold
Suppose :math:`x` is a two dimensional vector, and the user wishes to
hold the first coordinate constant. Then, :math:`\Delta` is a scalar
and :math:`\boxplus` is defined as
.. math::
\boxplus(x, \Delta) = x + \left[ \begin{array}{c} 0 \\ 1 \end{array} \right] \Delta
and given two, two-dimensional vectors :math:`x` and :math:`y` with
the same first coordinate, :math:`\boxminus` is defined as:
.. math::
\boxminus(y, x) = y[1] - x[1]
:class:`SubsetManifold` generalizes this construction to hold
any part of a parameter block constant by specifying the set of
coordinates that are held constant.
.. NOTE::
It is legal to hold *all* coordinates of a parameter block to
constant using a :class:`SubsetManifold`. It is the same as calling
:func:`Problem::SetParameterBlockConstant` on that parameter block.
.. class:: ProductManifold
In cases, where a parameter block is the Cartesian product of a number
of manifolds and you have the manifold of the individual
parameter blocks available, :class:`ProductManifold` can be used to
construct a :class:`Manifold` of the Cartesian product.
For the case of the rigid transformation, where say you have a
parameter block of size 7, where the first four entries represent the
rotation as a quaternion, and the next three the translation, a
manifold can be constructed as:
.. code-block:: c++
ProductManifold<QuaternionManifold, EuclideanManifold<3>> se3;
Manifolds can be copied and moved to :class:`ProductManifold`:
.. code-block:: c++
SubsetManifold manifold1(5, {2});
SubsetManifold manifold2(3, {0, 1});
ProductManifold<SubsetManifold, SubsetManifold> manifold(manifold1,
In advanced use cases, manifolds can be dynamically allocated and passed as (smart) pointers:
.. code-block:: c++
ProductManifold<std::unique_ptr<QuaternionManifold>, EuclideanManifold<3>> se3
{std::make_unique<QuaternionManifold>(), EuclideanManifold<3>{}};
The template parameters can also be left out as they are deduced automatically
making the initialization much simpler:
.. code-block:: c++
ProductManifold se3{QuaternionManifold{}, EuclideanManifold<3>{}};
.. class:: QuaternionManifold
.. NOTE::
If you are using ``Eigen`` quaternions, then you should use
:class:`EigenQuaternionManifold` instead because ``Eigen`` uses a
different memory layout for its Quaternions.
Manifold for a Hamilton `Quaternion
<https://en.wikipedia.org/wiki/Quaternion>`_. Quaternions are a three
dimensional manifold represented as unit norm 4-vectors, i.e.
.. math:: q = \left [\begin{matrix}q_0,& q_1,& q_2,& q_3\end{matrix}\right], \quad \|q\| = 1
is the ambient space representation. Here :math:`q_0` is the scalar
part. :math:`q_1` is the coefficient of :math:`i`, :math:`q_2` is the
coefficient of :math:`j`, and :math:`q_3` is the coefficient of
:math:`k`. Where:
.. math::
i\times j &= k,\\
j\times k &= i,\\
k\times i &= j,\\
i\times i &= -1,\\
j\times j &= -1,\\
k\times k &= -1.
The tangent space is three dimensional and the :math:`\boxplus` and
:math:`\boxminus` operators are defined in term of :math:`\exp` and
:math:`\log` operations.
.. math::
\boxplus(x, \Delta) &= \exp\left(\Delta\right) \otimes x \\
\boxminus(y,x) &= \log\left(y \otimes x^{-1}\right)
Where :math:`\otimes` is the `Quaternion product
<https://en.wikipedia.org/wiki/Quaternion#Hamilton_product>`_ and
since :math:`x` is a unit quaternion, :math:`x^{-1} = [\begin{matrix}
q_0,& -q_1,& -q_2,& -q_3\end{matrix}]`. Given a vector :math:`\Delta
\in \mathbb{R}^3`,
.. math::
\exp(\Delta) = \left[ \begin{matrix}
\frac{\displaystyle \sin\left(|\Delta\|\right)}{\displaystyle \|\Delta\|} \Delta
\end{matrix} \right]
and given a unit quaternion :math:`q = \left [\begin{matrix}q_0,& q_1,& q_2,& q_3\end{matrix}\right]`
.. math::
\log(q) = \frac{\operatorname{atan2}\left(\sqrt{1-q_0^2},q_0\right)}{\sqrt{1-q_0^2}} \left [\begin{matrix}q_1,& q_2,& q_3\end{matrix}\right]
.. class:: EigenQuaternionManifold
Implements the quaternion manifold for `Eigen's
representation of the Hamilton quaternion. Geometrically it is exactly
the same as the :class:`QuaternionManifold` defined above. However,
Eigen uses a different internal memory layout for the elements of the
quaternion than what is commonly used. It stores the quaternion in
memory as :math:`[q_1, q_2, q_3, q_0]` or :math:`[x, y, z, w]` where
the real (scalar) part is last.
Since Ceres operates on parameter blocks which are raw double pointers
this difference is important and requires a different manifold.
.. class:: SphereManifold
This provides a manifold on a sphere meaning that the norm of the
vector stays the same. Such cases often arises in Structure for Motion
problems. One example where they are used is in representing points
whose triangulation is ill-conditioned. Here it is advantageous to use
an over-parameterization since homogeneous vectors can represent
points at infinity.
The ambient space dimension is required to be greater than 1.
The class works with dynamic and static ambient space dimensions. If
the ambient space dimensions is known at compile time use
.. code-block:: c++
SphereManifold<3> manifold;
If the ambient space dimensions is not known at compile time the
template parameter needs to be set to `ceres::DYNAMIC` and the actual
dimension needs to be provided as a constructor argument:
.. code-block:: c++
SphereManifold<ceres::DYNAMIC> manifold(ambient_dim);
For more details, please see Section B.2 (p.25) in [Hertzberg]_
.. class:: LineManifold
This class provides a manifold for lines, where the line is defined
using an origin point and a direction vector. So the ambient size
needs to be two times the dimension of the space in which the line
lives. The first half of the parameter block is interpreted as the
origin point and the second half as the direction. This manifold is a
special case of the `Affine Grassmannian manifold
<https://en.wikipedia.org/wiki/Affine_Grassmannian_(manifold))>`_ for
the case :math:`\operatorname{Graff}_1(R^n)`.
Note that this is a manifold for a line, rather than a point
constrained to lie on a line. It is useful when one wants to optimize
over the space of lines. For example, given :math:`n` distinct points
in 3D (measurements) we want to find the line that minimizes the sum
of squared distances to all the points.
.. class:: AutoDiffManifold
Create a :class:`Manifold` with Jacobians computed via automatic
To get an auto differentiated manifold, you must define a Functor with
templated ``Plus`` and ``Minus`` functions that compute:
.. code-block:: c++
x_plus_delta = Plus(x, delta);
y_minus_x = Minus(y, x);
Where, ``x``, ``y`` and ``x_plus_delta`` are vectors on the manifold in
the ambient space (so they are ``kAmbientSize`` vectors) and
``delta``, ``y_minus_x`` are vectors in the tangent space (so they are
``kTangentSize`` vectors).
The Functor should have the signature:
.. code-block:: c++
struct Functor {
template <typename T>
bool Plus(const T* x, const T* delta, T* x_plus_delta) const;
template <typename T>
bool Minus(const T* y, const T* x, T* y_minus_x) const;
Observe that the ``Plus`` and ``Minus`` operations are templated on
the parameter ``T``. The autodiff framework substitutes appropriate
``Jet`` objects for ``T`` in order to compute the derivative when
necessary. This is the same mechanism that is used to compute
derivatives when using :class:`AutoDiffCostFunction`.
``Plus`` and ``Minus`` should return true if the computation is
successful and false otherwise, in which case the result will not be
Given this Functor, the corresponding :class:`Manifold` can be constructed as:
.. code-block:: c++
AutoDiffManifold<Functor, kAmbientSize, kTangentSize> manifold;
.. NOTE::
The following is only used for illustration purposes. Ceres Solver
ships with an optimized, production grade :class:`QuaternionManifold`
As a concrete example consider the case of `Quaternions
<https://en.wikipedia.org/wiki/Quaternion>`_. Quaternions form a three
dimensional manifold embedded in :math:`\mathbb{R}^4`, i.e. they have
an ambient dimension of 4 and their tangent space has dimension 3. The
following Functor defines the ``Plus`` and ``Minus`` operations on the
Quaternion manifold. It assumes that the quaternions are laid out as
``[w,x,y,z]`` in memory, i.e. the real or scalar part is the first
.. code-block:: c++
struct QuaternionFunctor {
template <typename T>
bool Plus(const T* x, const T* delta, T* x_plus_delta) const {
const T squared_norm_delta =
delta[0] * delta[0] + delta[1] * delta[1] + delta[2] * delta[2];
T q_delta[4];
if (squared_norm_delta > T(0.0)) {
T norm_delta = sqrt(squared_norm_delta);
const T sin_delta_by_delta = sin(norm_delta) / norm_delta;
q_delta[0] = cos(norm_delta);
q_delta[1] = sin_delta_by_delta * delta[0];
q_delta[2] = sin_delta_by_delta * delta[1];
q_delta[3] = sin_delta_by_delta * delta[2];
} else {
// We do not just use q_delta = [1,0,0,0] here because that is a
// constant and when used for automatic differentiation will
// lead to a zero derivative. Instead we take a first order
// approximation and evaluate it at zero.
q_delta[0] = T(1.0);
q_delta[1] = delta[0];
q_delta[2] = delta[1];
q_delta[3] = delta[2];
QuaternionProduct(q_delta, x, x_plus_delta);
return true;
template <typename T>
bool Minus(const T* y, const T* x, T* y_minus_x) const {
T minus_x[4] = {x[0], -x[1], -x[2], -x[3]};
T ambient_y_minus_x[4];
QuaternionProduct(y, minus_x, ambient_y_minus_x);
T u_norm = sqrt(ambient_y_minus_x[1] * ambient_y_minus_x[1] +
ambient_y_minus_x[2] * ambient_y_minus_x[2] +
ambient_y_minus_x[3] * ambient_y_minus_x[3]);
if (u_norm > 0.0) {
T theta = atan2(u_norm, ambient_y_minus_x[0]);
y_minus_x[0] = theta * ambient_y_minus_x[1] / u_norm;
y_minus_x[1] = theta * ambient_y_minus_x[2] / u_norm;
y_minus_x[2] = theta * ambient_y_minus_x[3] / u_norm;
} else {
We do not use [0,0,0] here because even though the value part is
a constant, the derivative part is not.
y_minus_x[0] = ambient_y_minus_x[1];
y_minus_x[1] = ambient_y_minus_x[2];
y_minus_x[2] = ambient_y_minus_x[3];
return true;
Then given this struct, the auto differentiated Quaternion Manifold can now
be constructed as
.. code-block:: c++
Manifold* manifold = new AutoDiffManifold<QuaternionFunctor, 4, 3>;
.. class:: Problem
:class:`Problem` holds the robustified bounds constrained
non-linear least squares problem :eq:`ceresproblem_modeling`. To
create a least squares problem, use the
:func:`Problem::AddResidalBlock` and
:func:`Problem::AddParameterBlock` methods.
For example a problem containing 3 parameter blocks of sizes 3, 4
and 5 respectively and two residual blocks of size 2 and 6:
.. code-block:: c++
double x1[] = { 1.0, 2.0, 3.0 };
double x2[] = { 1.0, 2.0, 3.0, 5.0 };
double x3[] = { 1.0, 2.0, 3.0, 6.0, 7.0 };
Problem problem;
problem.AddResidualBlock(new MyUnaryCostFunction(...), x1);
problem.AddResidualBlock(new MyBinaryCostFunction(...), x2, x3);
:func:`Problem::AddResidualBlock` as the name implies, adds a
residual block to the problem. It adds a :class:`CostFunction`, an
optional :class:`LossFunction` and connects the
:class:`CostFunction` to a set of parameter block.
The cost function carries with it information about the sizes of
the parameter blocks it expects. The function checks that these
match the sizes of the parameter blocks listed in
``parameter_blocks``. The program aborts if a mismatch is
detected. ``loss_function`` can be ``nullptr``, in which case the cost
of the term is just the squared norm of the residuals.
The user has the option of explicitly adding the parameter blocks
using :func:`Problem::AddParameterBlock`. This causes additional
correctness checking; however, :func:`Problem::AddResidualBlock`
implicitly adds the parameter blocks if they are not present, so
calling :func:`Problem::AddParameterBlock` explicitly is not
:func:`Problem::AddParameterBlock` explicitly adds a parameter
block to the :class:`Problem`. Optionally it allows the user to
associate a :class:`Manifold` object with the parameter block
too. Repeated calls with the same arguments are ignored. Repeated
calls with the same double pointer but a different size results in
undefined behavior.
You can set any parameter block to be constant using
:func:`Problem::SetParameterBlockConstant` and undo this using
In fact you can set any number of parameter blocks to be constant,
and Ceres is smart enough to figure out what part of the problem
you have constructed depends on the parameter blocks that are free
to change and only spends time solving it. So for example if you
constructed a problem with a million parameter blocks and 2 million
residual blocks, but then set all but one parameter blocks to be
constant and say only 10 residual blocks depend on this one
non-constant parameter block. Then the computational effort Ceres
spends in solving this problem will be the same if you had defined
a problem with one parameter block and 10 residual blocks.
:class:`Problem` by default takes ownership of the
``cost_function``, ``loss_function`` and ``manifold`` pointers. These
objects remain live for the life of the :class:`Problem`. If the user wishes
to keep control over the destruction of these objects, then they can do this
by setting the corresponding enums in the :class:`Problem::Options` struct.
Note that even though the Problem takes ownership of objects,
``cost_function`` and ``loss_function``, it does not preclude the
user from re-using them in another residual block. Similarly the
same ``manifold`` object can be used with multiple parameter blocks. The
destructor takes care to call delete on each owned object exactly once.
.. class:: Problem::Options
Options struct that is used to control :class:`Problem`.
.. member:: Ownership Problem::Options::cost_function_ownership
Default: ``TAKE_OWNERSHIP``
This option controls whether the Problem object owns the cost
If set to ``TAKE_OWNERSHIP``, then the problem object will delete the
cost functions on destruction. The destructor is careful to delete
the pointers only once, since sharing cost functions is allowed.
.. member:: Ownership Problem::Options::loss_function_ownership
Default: ``TAKE_OWNERSHIP``
This option controls whether the Problem object owns the loss
If set to ``TAKE_OWNERSHIP``, then the problem object will delete the
loss functions on destruction. The destructor is careful to delete
the pointers only once, since sharing loss functions is allowed.
.. member:: Ownership Problem::Options::manifold_ownership
Default: ``TAKE_OWNERSHIP``
This option controls whether the Problem object owns the manifolds.
If set to ``TAKE_OWNERSHIP``, then the problem object will delete the
manifolds on destruction. The destructor is careful to delete the
pointers only once, since sharing manifolds is allowed.
.. member:: bool Problem::Options::enable_fast_removal
Default: ``false``
If true, trades memory for faster
:func:`Problem::RemoveResidualBlock` and
:func:`Problem::RemoveParameterBlock` operations.
By default, :func:`Problem::RemoveParameterBlock` and
:func:`Problem::RemoveResidualBlock` take time proportional to
the size of the entire problem. If you only ever remove
parameters or residuals from the problem occasionally, this might
be acceptable. However, if you have memory to spare, enable this
option to make :func:`Problem::RemoveParameterBlock` take time
proportional to the number of residual blocks that depend on it,
and :func:`Problem::RemoveResidualBlock` take (on average)
constant time.
The increase in memory usage is twofold: an additional hash set
per parameter block containing all the residuals that depend on
the parameter block; and a hash set in the problem containing all
.. member:: bool Problem::Options::disable_all_safety_checks
Default: `false`
By default, Ceres performs a variety of safety checks when
constructing the problem. There is a small but measurable
performance penalty to these checks, typically around 5% of
construction time. If you are sure your problem construction is
correct, and 5% of the problem construction time is truly an
overhead you want to avoid, then you can set
disable_all_safety_checks to true.
.. warning::
Do not set this to true, unless you are absolutely sure of what you are
.. member:: Context* Problem::Options::context
Default: ``nullptr``
A Ceres global context to use for solving this problem. This may
help to reduce computation time as Ceres can reuse expensive
objects to create. The context object can be `nullptr`, in which
case Ceres may create one.
Ceres does NOT take ownership of the pointer.
.. member:: EvaluationCallback* Problem::Options::evaluation_callback
Default: ``nullptr``
Using this callback interface, Ceres will notify you when it is
about to evaluate the residuals or Jacobians.
If an ``evaluation_callback`` is present, Ceres will update the
user's parameter blocks to the values that will be used when
calling :func:`CostFunction::Evaluate` before calling
:func:`EvaluationCallback::PrepareForEvaluation`. One can then use
this callback to share (or cache) computation between cost
functions by doing the shared computation in
:func:`EvaluationCallback::PrepareForEvaluation` before Ceres
calls :func:`CostFunction::Evaluate`.
Problem does NOT take ownership of the callback.
.. NOTE::
Evaluation callbacks are incompatible with inner iterations. So
calling Solve with
:member:`Solver::Options::use_inner_iterations` set to ``true``
on a :class:`Problem` with a non-null evaluation callback is an
.. function:: ResidualBlockId Problem::AddResidualBlock(CostFunction* cost_function, LossFunction* loss_function, const std::vector<double*> parameter_blocks)
.. function:: template <typename Ts...> ResidualBlockId Problem::AddResidualBlock(CostFunction* cost_function, LossFunction* loss_function, double* x0, Ts... xs)
Add a residual block to the overall cost function. The cost
function carries with it information about the sizes of the
parameter blocks it expects. The function checks that these match
the sizes of the parameter blocks listed in parameter_blocks. The
program aborts if a mismatch is detected. loss_function can be
``nullptr``, in which case the cost of the term is just the squared
norm of the residuals.
The parameter blocks may be passed together as a
``vector<double*>``, or ``double*`` pointers.
The user has the option of explicitly adding the parameter blocks
using AddParameterBlock. This causes additional correctness
checking; however, AddResidualBlock implicitly adds the parameter
blocks if they are not present, so calling AddParameterBlock
explicitly is not required.
The Problem object by default takes ownership of the
cost_function and loss_function pointers. These objects remain
live for the life of the Problem object. If the user wishes to
keep control over the destruction of these objects, then they can
do this by setting the corresponding enums in the Options struct.
.. note::
Even though the Problem takes ownership of ``cost_function``
and ``loss_function``, it does not preclude the user from re-using
them in another residual block. The destructor takes care to call
delete on each cost_function or loss_function pointer only once,
regardless of how many residual blocks refer to them.
Example usage:
.. code-block:: c++
double x1[] = {1.0, 2.0, 3.0};
double x2[] = {1.0, 2.0, 5.0, 6.0};
double x3[] = {3.0, 6.0, 2.0, 5.0, 1.0};
std::vector<double*> v1;
std::vector<double*> v2;
Problem problem;
problem.AddResidualBlock(new MyUnaryCostFunction(...), nullptr, x1);
problem.AddResidualBlock(new MyBinaryCostFunction(...), nullptr, x2, x1);
problem.AddResidualBlock(new MyUnaryCostFunction(...), nullptr, v1);
problem.AddResidualBlock(new MyBinaryCostFunction(...), nullptr, v2);
.. function:: void Problem::AddParameterBlock(double* values, int size, Manifold* manifold)
Add a parameter block with appropriate size and Manifold to the
problem. It is okay for ``manifold`` to be ``nullptr``.
Repeated calls with the same arguments are ignored. Repeated calls
with the same double pointer but a different size results in a crash
(unless :member:`Solver::Options::disable_all_safety_checks` is set to true).
Repeated calls with the same double pointer and size but different
:class:`Manifold` is equivalent to calling `SetManifold(manifold)`,
i.e., any previously associated :class:`Manifold` object will be replaced
with the `manifold`.
.. function:: void Problem::AddParameterBlock(double* values, int size)
Add a parameter block with appropriate size and parameterization to
the problem. Repeated calls with the same arguments are
ignored. Repeated calls with the same double pointer but a
different size results in undefined behavior.
.. function:: void Problem::RemoveResidualBlock(ResidualBlockId residual_block)
Remove a residual block from the problem.
Since residual blocks are allowed to share cost function and loss
function objects, Ceres Solver uses a reference counting
mechanism. So when a residual block is deleted, the reference count
for the corresponding cost function and loss function objects are
decreased and when this count reaches zero, they are deleted.
If :member:`Problem::Options::enable_fast_removal` is ``true``, then the removal
is fast (almost constant time). Otherwise it is linear, requiring a
scan of the entire problem.
Removing a residual block has no effect on the parameter blocks
that the problem depends on.
.. warning::
Removing a residual or parameter block will destroy the implicit
ordering, rendering the jacobian or residuals returned from the solver
uninterpretable. If you depend on the evaluated jacobian, do not use
remove! This may change in a future release. Hold the indicated parameter
block constant during optimization.
.. function:: void Problem::RemoveParameterBlock(const double* values)
Remove a parameter block from the problem. Any residual blocks that
depend on the parameter are also removed, as described above in
The manifold of the parameter block, if it exists, will persist until the
deletion of the problem.
If :member:`Problem::Options::enable_fast_removal` is ``true``, then the removal
is fast (almost constant time). Otherwise, removing a parameter
block will scan the entire Problem.
.. warning::
Removing a residual or parameter block will destroy the implicit
ordering, rendering the jacobian or residuals returned from the solver
uninterpretable. If you depend on the evaluated jacobian, do not use
remove! This may change in a future release.
.. function:: void Problem::SetParameterBlockConstant(const double* values)
Hold the indicated parameter block constant during optimization.
.. function:: void Problem::SetParameterBlockVariable(double* values)
Allow the indicated parameter to vary during optimization.
.. function:: bool Problem::IsParameterBlockConstant(const double* values) const
Returns ``true`` if a parameter block is set constant, and false
otherwise. A parameter block may be set constant in two ways:
either by calling ``SetParameterBlockConstant`` or by associating a
:class:`Manifold` with a zero dimensional tangent space with it.
.. function:: void SetManifold(double* values, Manifold* manifold);
Set the :class:`Manifold` for the parameter block. Calling
:func:`Problem::SetManifold` with ``nullptr`` will clear any
previously set :class:`Manifold` for the parameter block.
Repeated calls will result in any previously associated
:class:`Manifold` object to be replaced with ``manifold``.
``manifold`` is owned by :class:`Problem` by default (See
:class:`Problem::Options` to override this behaviour).
It is acceptable to set the same :class:`Manifold` for multiple
parameter blocks.
.. function:: const Manifold* GetManifold(const double* values) const;
Get the :class:`Manifold` object associated with this parameter block.
If there is no :class:`Manifold` object associated with the parameter block,
then ``nullptr`` is returned.
.. function:: bool HasManifold(const double* values) const;
Returns ``true`` if a :class:`Manifold` is associated with this parameter
block, ``false`` otherwise.
.. function:: void Problem::SetParameterLowerBound(double* values, int index, double lower_bound)
Set the lower bound for the parameter at position `index` in the
parameter block corresponding to `values`. By default the lower
bound is ``-std::numeric_limits<double>::max()``, which is treated
by the solver as the same as :math:`-\infty`.
.. function:: void Problem::SetParameterUpperBound(double* values, int index, double upper_bound)
Set the upper bound for the parameter at position `index` in the
parameter block corresponding to `values`. By default the value is
``std::numeric_limits<double>::max()``, which is treated by the
solver as the same as :math:`\infty`.
.. function:: double Problem::GetParameterLowerBound(const double* values, int index)
Get the lower bound for the parameter with position `index`. If the
parameter is not bounded by the user, then its lower bound is
.. function:: double Problem::GetParameterUpperBound(const double* values, int index)
Get the upper bound for the parameter with position `index`. If the
parameter is not bounded by the user, then its upper bound is
.. function:: int Problem::NumParameterBlocks() const
Number of parameter blocks in the problem. Always equals
parameter_blocks().size() and parameter_block_sizes().size().
.. function:: int Problem::NumParameters() const
The size of the parameter vector obtained by summing over the sizes
of all the parameter blocks.
.. function:: int Problem::NumResidualBlocks() const
Number of residual blocks in the problem. Always equals
.. function:: int Problem::NumResiduals() const
The size of the residual vector obtained by summing over the sizes
of all of the residual blocks.
.. function:: int Problem::ParameterBlockSize(const double* values) const
The size of the parameter block.
.. function:: int Problem::ParameterBlockTangentSize(const double* values) const
The dimension of the tangent space of the :class:`Manifold` for the
parameter block. If there is no :class:`Manifold` associated with this
parameter block, then ``ParameterBlockTangentSize = ParameterBlockSize``.
.. function:: bool Problem::HasParameterBlock(const double* values) const
Is the given parameter block present in the problem or not?
.. function:: void Problem::GetParameterBlocks(std::vector<double*>* parameter_blocks) const
Fills the passed ``parameter_blocks`` vector with pointers to the
parameter blocks currently in the problem. After this call,
``parameter_block.size() == NumParameterBlocks``.
.. function:: void Problem::GetResidualBlocks(std::vector<ResidualBlockId>* residual_blocks) const
Fills the passed `residual_blocks` vector with pointers to the
residual blocks currently in the problem. After this call,
`residual_blocks.size() == NumResidualBlocks`.
.. function:: void Problem::GetParameterBlocksForResidualBlock(const ResidualBlockId residual_block, std::vector<double*>* parameter_blocks) const
Get all the parameter blocks that depend on the given residual
.. function:: void Problem::GetResidualBlocksForParameterBlock(const double* values, std::vector<ResidualBlockId>* residual_blocks) const
Get all the residual blocks that depend on the given parameter
If :member:`Problem::Options::enable_fast_removal` is
``true``, then getting the residual blocks is fast and depends only
on the number of residual blocks. Otherwise, getting the residual
blocks for a parameter block will scan the entire problem.
.. function:: const CostFunction* Problem::GetCostFunctionForResidualBlock(const ResidualBlockId residual_block) const
Get the :class:`CostFunction` for the given residual block.
.. function:: const LossFunction* Problem::GetLossFunctionForResidualBlock(const ResidualBlockId residual_block) const
Get the :class:`LossFunction` for the given residual block.
.. function:: bool EvaluateResidualBlock(ResidualBlockId residual_block_id, bool apply_loss_function, double* cost,double* residuals, double** jacobians) const
Evaluates the residual block, storing the scalar cost in ``cost``, the
residual components in ``residuals``, and the jacobians between the
parameters and residuals in ``jacobians[i]``, in row-major order.
If ``residuals`` is ``nullptr``, the residuals are not computed.
If ``jacobians`` is ``nullptr``, no Jacobians are computed. If
``jacobians[i]`` is ``nullptr``, then the Jacobian for that
parameter block is not computed.
It is not okay to request the Jacobian w.r.t a parameter block
that is constant.
The return value indicates the success or failure. Even if the
function returns false, the caller should expect the output
memory locations to have been modified.
The returned cost and jacobians have had robustification and
:class:`Manifold` applied already; for example, the jacobian for a
4-dimensional quaternion parameter using the :class:`QuaternionManifold` is
``num_residuals x 3`` instead of ``num_residuals x 4``.
``apply_loss_function`` as the name implies allows the user to
switch the application of the loss function on and off.
.. NOTE:: If an :class:`EvaluationCallback` is associated with the
problem, then its
:func:`EvaluationCallback::PrepareForEvaluation` method will be
called every time this method is called with `new_point =
true`. This conservatively assumes that the user may have
changed the parameter values since the previous call to evaluate
/ solve. For improved efficiency, and only if you know that the
parameter values have not changed between calls, see
.. function:: bool EvaluateResidualBlockAssumingParametersUnchanged(ResidualBlockId residual_block_id, bool apply_loss_function, double* cost,double* residuals, double** jacobians) const
Same as :func:`Problem::EvaluateResidualBlock` except that if an
:class:`EvaluationCallback` is associated with the problem, then
its :func:`EvaluationCallback::PrepareForEvaluation` method will
be called every time this method is called with new_point = false.
This means, if an :class:`EvaluationCallback` is associated with
the problem then it is the user's responsibility to call
:func:`EvaluationCallback::PrepareForEvaluation` before calling
this method if necessary, i.e. iff the parameter values have been
changed since the last call to evaluate / solve.'
This is because, as the name implies, we assume that the parameter
blocks did not change since the last time
:func:`EvaluationCallback::PrepareForEvaluation` was called (via
:func:`Solve`, :func:`Problem::Evaluate` or
.. function:: bool Problem::Evaluate(const Problem::EvaluateOptions& options, double* cost, std::vector<double>* residuals, std::vector<double>* gradient, CRSMatrix* jacobian)
Evaluate a :class:`Problem`. Any of the output pointers can be
``nullptr``. Which residual blocks and parameter blocks are used is
controlled by the :class:`Problem::EvaluateOptions` struct below.
.. NOTE::
The evaluation will use the values stored in the memory
locations pointed to by the parameter block pointers used at the
time of the construction of the problem, for example in the
following code:
.. code-block:: c++
Problem problem;
double x = 1;
problem.Add(new MyCostFunction, nullptr, &x);
double cost = 0.0;
problem.Evaluate(Problem::EvaluateOptions(), &cost, nullptr, nullptr, nullptr);
The cost is evaluated at `x = 1`. If you wish to evaluate the
problem at `x = 2`, then
.. code-block:: c++
x = 2;
problem.Evaluate(Problem::EvaluateOptions(), &cost, nullptr, nullptr, nullptr);
is the way to do so.
.. NOTE::
If no :class:`Manifold` are used, then the size of the gradient vector is
the sum of the sizes of all the parameter blocks. If a parameter block has
a manifold then it contributes "TangentSize" entries to the gradient
.. NOTE::
This function cannot be called while the problem is being
solved, for example it cannot be called from an
:class:`IterationCallback` at the end of an iteration during a
.. NOTE::
If an EvaluationCallback is associated with the problem, then
its PrepareForEvaluation method will be called everytime this
method is called with ``new_point = true``.
.. class:: Problem::EvaluateOptions
Options struct that is used to control :func:`Problem::Evaluate`.
.. member:: std::vector<double*> Problem::EvaluateOptions::parameter_blocks
The set of parameter blocks for which evaluation should be
performed. This vector determines the order in which parameter
blocks occur in the gradient vector and in the columns of the
jacobian matrix. If parameter_blocks is empty, then it is assumed
to be equal to a vector containing ALL the parameter
blocks. Generally speaking the ordering of the parameter blocks in
this case depends on the order in which they were added to the
problem and whether or not the user removed any parameter blocks.
**NOTE** This vector should contain the same pointers as the ones
used to add parameter blocks to the Problem. These parameter block
should NOT point to new memory locations. Bad things will happen if
you do.
.. member:: std::vector<ResidualBlockId> Problem::EvaluateOptions::residual_blocks
The set of residual blocks for which evaluation should be
performed. This vector determines the order in which the residuals
occur, and how the rows of the jacobian are ordered. If
residual_blocks is empty, then it is assumed to be equal to the
vector containing all the residual blocks.
.. member:: bool Problem::EvaluateOptions::apply_loss_function
Even though the residual blocks in the problem may contain loss
functions, setting apply_loss_function to false will turn off the
application of the loss function to the output of the cost
function. This is of use for example if the user wishes to analyse
the solution quality by studying the distribution of residuals
before and after the solve.
.. member:: int Problem::EvaluateOptions::num_threads
Number of threads to use.
.. class:: EvaluationCallback
Interface for receiving callbacks before Ceres evaluates residuals or
.. code-block:: c++
class EvaluationCallback {
virtual ~EvaluationCallback();
virtual void PrepareForEvaluation(bool evaluate_jacobians,
bool new_evaluation_point) = 0;
.. function:: void EvaluationCallback::PrepareForEvaluation(bool evaluate_jacobians, bool new_evaluation_point)
Ceres will call :func:`EvaluationCallback::PrepareForEvaluation`
every time, and once before it computes the residuals and/or the
User parameters (the double* values provided by the user) are fixed
until the next call to
:func:`EvaluationCallback::PrepareForEvaluation`. If
``new_evaluation_point == true``, then this is a new point that is
different from the last evaluated point. Otherwise, it is the same
point that was evaluated previously (either Jacobian or residual)
and the user can use cached results from previous evaluations. If
``evaluate_jacobians`` is ``true``, then Ceres will request Jacobians
in the upcoming cost evaluation.
Using this callback interface, Ceres can notify you when it is
about to evaluate the residuals or Jacobians. With the callback,
you can share computation between residual blocks by doing the
shared computation in
:func:`EvaluationCallback::PrepareForEvaluation` before Ceres calls
:func:`CostFunction::Evaluate` on all the residuals. It also
enables caching results between a pure residual evaluation and a
residual & Jacobian evaluation, via the ``new_evaluation_point``
One use case for this callback is if the cost function compute is
moved to the GPU. In that case, the prepare call does the actual
cost function evaluation, and subsequent calls from Ceres to the
actual cost functions merely copy the results from the GPU onto the
corresponding blocks for Ceres to plug into the solver.
**Note**: Ceres provides no mechanism to share data other than the
notification from the callback. Users must provide access to
pre-computed shared data to their cost functions behind the scenes;
this all happens without Ceres knowing. One approach is to put a
pointer to the shared data in each cost function (recommended) or
to use a global shared variable (discouraged; bug-prone). As far
as Ceres is concerned, it is evaluating cost functions like any
other; it just so happens that behind the scenes the cost functions
reuse pre-computed data to execute faster. See
for an example.
See ``evaluation_callback_test.cc`` for code that explicitly
verifies the preconditions between
:func:`EvaluationCallback::PrepareForEvaluation` and
Many applications of Ceres Solver involve optimization problems where
some of the variables correspond to rotations. To ease the pain of
work with the various representations of rotations (angle-axis,
quaternion and matrix) we provide a handy set of templated
functions. These functions are templated so that the user can use them
within Ceres Solver's automatic differentiation framework.
.. function:: template <typename T> void AngleAxisToQuaternion(T const* angle_axis, T* quaternion)
Convert a value in combined axis-angle representation to a
The value ``angle_axis`` is a triple whose norm is an angle in radians,
and whose direction is aligned with the axis of rotation, and
``quaternion`` is a 4-tuple that will contain the resulting quaternion.
.. function:: template <typename T> void QuaternionToAngleAxis(T const* quaternion, T* angle_axis)
Convert a quaternion to the equivalent combined axis-angle
The value ``quaternion`` must be a unit quaternion - it is not
normalized first, and ``angle_axis`` will be filled with a value
whose norm is the angle of rotation in radians, and whose direction
is the axis of rotation.
.. function:: template <typename T, int row_stride, int col_stride> void RotationMatrixToAngleAxis(const MatrixAdapter<const T, row_stride, col_stride>& R, T * angle_axis)
.. function:: template <typename T, int row_stride, int col_stride> void AngleAxisToRotationMatrix(T const * angle_axis, const MatrixAdapter<T, row_stride, col_stride>& R)
.. function:: template <typename T> void RotationMatrixToAngleAxis(T const * R, T * angle_axis)
.. function:: template <typename T> void AngleAxisToRotationMatrix(T const * angle_axis, T * R)
Conversions between :math:`3\times3` rotation matrix with given column and row strides and
axis-angle rotation representations. The functions that take a pointer to T instead
of a MatrixAdapter assume a column major representation with unit row stride and a column stride of 3.
.. function:: template <typename T, int row_stride, int col_stride> void EulerAnglesToRotationMatrix(const T* euler, const MatrixAdapter<T, row_stride, col_stride>& R)
.. function:: template <typename T> void EulerAnglesToRotationMatrix(const T* euler, int row_stride, T* R)
Conversions between :math:`3\times3` rotation matrix with given column and row strides and
Euler angle (in degrees) rotation representations.
The {pitch,roll,yaw} Euler angles are rotations around the {x,y,z}
axes, respectively. They are applied in that same order, so the
total rotation R is Rz * Ry * Rx.
The function that takes a pointer to T as the rotation matrix assumes a row
major representation with unit column stride and a row stride of 3.
The additional parameter row_stride is required to be 3.
.. function:: template <typename T, int row_stride, int col_stride> void QuaternionToScaledRotation(const T q[4], const MatrixAdapter<T, row_stride, col_stride>& R)
.. function:: template <typename T> void QuaternionToScaledRotation(const T q[4], T R[3 * 3])
Convert a 4-vector to a :math:`3\times3` scaled rotation matrix.
The choice of rotation is such that the quaternion
:math:`\begin{bmatrix} 1 &0 &0 &0\end{bmatrix}` goes to an identity
matrix and for small :math:`a, b, c` the quaternion
:math:`\begin{bmatrix}1 &a &b &c\end{bmatrix}` goes to the matrix
.. math::
I + 2 \begin{bmatrix} 0 & -c & b \\ c & 0 & -a\\ -b & a & 0
\end{bmatrix} + O(q^2)
which corresponds to a Rodrigues approximation, the last matrix
being the cross-product matrix of :math:`\begin{bmatrix} a& b&
c\end{bmatrix}`. Together with the property that :math:`R(q_1 \otimes q_2)
= R(q_1) R(q_2)` this uniquely defines the mapping from :math:`q` to
In the function that accepts a pointer to T instead of a MatrixAdapter,
the rotation matrix ``R`` is a row-major matrix with unit column stride
and a row stride of 3.
No normalization of the quaternion is performed, i.e.
:math:`R = \|q\|^2 Q`, where :math:`Q` is an orthonormal matrix
such that :math:`\det(Q) = 1` and :math:`QQ' = I`.
.. function:: template <typename T> void QuaternionToRotation(const T q[4], const MatrixAdapter<T, row_stride, col_stride>& R)
.. function:: template <typename T> void QuaternionToRotation(const T q[4], T R[3 * 3])
Same as above except that the rotation matrix is normalized by the
Frobenius norm, so that :math:`R R' = I` (and :math:`\det(R) = 1`).
.. function:: template <typename T> void UnitQuaternionRotatePoint(const T q[4], const T pt[3], T result[3])
Rotates a point pt by a quaternion q:
.. math:: \text{result} = R(q) \text{pt}
Assumes the quaternion is unit norm. If you pass in a quaternion
with :math:`|q|^2 = 2` then you WILL NOT get back 2 times the
result you get for a unit quaternion.
.. function:: template <typename T> void QuaternionRotatePoint(const T q[4], const T pt[3], T result[3])
With this function you do not need to assume that :math:`q` has unit norm.
It does assume that the norm is non-zero.
.. function:: template <typename T> void QuaternionProduct(const T z[4], const T w[4], T zw[4])
.. math:: zw = z \otimes w
where :math:`\otimes` is the Quaternion product between 4-vectors.
.. function:: template <typename T> void CrossProduct(const T x[3], const T y[3], T x_cross_y[3])
.. math:: \text{x_cross_y} = x \times y
.. function:: template <typename T> void AngleAxisRotatePoint(const T angle_axis[3], const T pt[3], T result[3])
.. math:: y = R(\text{angle_axis}) x
Cubic Interpolation
Optimization problems often involve functions that are given in the
form of a table of values, for example an image. Evaluating these
functions and their derivatives requires interpolating these
values. Interpolating tabulated functions is a vast area of research
and there are a lot of libraries which implement a variety of
interpolation schemes. However, using them within the automatic
differentiation framework in Ceres is quite painful. To this end,
Ceres provides the ability to interpolate one dimensional and two
dimensional tabular functions.
The one dimensional interpolation is based on the Cubic Hermite
Spline, also known as the Catmull-Rom Spline. This produces a first
order differentiable interpolating function. The two dimensional
interpolation scheme is a generalization of the one dimensional scheme
where the interpolating function is assumed to be separable in the two
More details of the construction can be found `Linear Methods for
Image Interpolation <http://www.ipol.im/pub/art/2011/g_lmii/>`_ by
Pascal Getreuer.
.. class:: CubicInterpolator
Given as input an infinite one dimensional grid, which provides the
following interface.
.. code::
struct Grid1D {
enum { DATA_DIMENSION = 2; };
void GetValue(int n, double* f) const;
Where, ``GetValue`` gives us the value of a function :math:`f`
(possibly vector valued) for any integer :math:`n` and the enum
``DATA_DIMENSION`` indicates the dimensionality of the function being
interpolated. For example if you are interpolating rotations in
axis-angle format over time, then ``DATA_DIMENSION = 3``.
:class:`CubicInterpolator` uses Cubic Hermite splines to produce a
smooth approximation to it that can be used to evaluate the
:math:`f(x)` and :math:`f'(x)` at any point on the real number
line. For example, the following code interpolates an array of four
.. code::
const double x[] = {1.0, 2.0, 5.0, 6.0};
Grid1D<double, 1> array(x, 0, 4);
CubicInterpolator interpolator(array);
double f, dfdx;
interpolator.Evaluate(1.5, &f, &dfdx);
In the above code we use ``Grid1D`` a templated helper class that
allows easy interfacing between ``C++`` arrays and
``Grid1D`` supports vector valued functions where the various
coordinates of the function can be interleaved or stacked. It also
allows the use of any numeric type as input, as long as it can be
safely cast to a double.
.. class:: BiCubicInterpolator
Given as input an infinite two dimensional grid, which provides the
following interface:
.. code::
struct Grid2D {
enum { DATA_DIMENSION = 2 };
void GetValue(int row, int col, double* f) const;
Where, ``GetValue`` gives us the value of a function :math:`f`
(possibly vector valued) for any pair of integers :code:`row` and
:code:`col` and the enum ``DATA_DIMENSION`` indicates the
dimensionality of the function being interpolated. For example if you
are interpolating a color image with three channels (Red, Green &
Blue), then ``DATA_DIMENSION = 3``.
:class:`BiCubicInterpolator` uses the cubic convolution interpolation
algorithm of R. Keys [Keys]_, to produce a smooth approximation to it
that can be used to evaluate the :math:`f(r,c)`, :math:`\frac{\partial
f(r,c)}{\partial r}` and :math:`\frac{\partial f(r,c)}{\partial c}` at
any any point in the real plane.
For example the following code interpolates a two dimensional array.
.. code::
const double data[] = {1.0, 3.0, -1.0, 4.0,
3.6, 2.1, 4.2, 2.0,
2.0, 1.0, 3.1, 5.2};
Grid2D<double, 1> array(data, 0, 3, 0, 4);
BiCubicInterpolator interpolator(array);
double f, dfdr, dfdc;
interpolator.Evaluate(1.2, 2.5, &f, &dfdr, &dfdc);
In the above code, the templated helper class ``Grid2D`` is used to
make a ``C++`` array look like a two dimensional table to
``Grid2D`` supports row or column major layouts. It also supports
vector valued functions where the individual coordinates of the
function may be interleaved or stacked. It also allows the use of any
numeric type as input, as long as it can be safely cast to double. | {"url":"https://ceres-solver.googlesource.com/ceres-solver/+/2ffeb943ad52a7f34fff3624e895118176d3b681/docs/source/nnls_modeling.rst","timestamp":"2024-11-02T15:26:56Z","content_type":"text/html","content_length":"671120","record_id":"<urn:uuid:ff6c7caf-b586-4594-84f6-c682bd6451e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00006.warc.gz"} |
Mixed Number to Improper Fraction CalculatorMixed Number to Improper Fraction Calculator
• Enter the whole number, numerator, and denominator for the mixed number.
• Click "Convert" to calculate and display the improper fraction and decimal result.
• The explanation of the conversion process will be shown below the result.
• You can copy the result to the clipboard using the "Copy" button.
• Your calculation history will be displayed in the "Calculation History" section.
• Click "Clear" to reset the form and calculations.
The Mixed Number to Improper Fraction Calculator is a valuable mathematical tool that aids in the conversion of mixed numbers into improper fractions. This calculator simplifies a common arithmetic
operation and is widely used in both educational and practical settings.
The Concept
A mixed number consists of a whole number and a fractional part, such as 2 1/3. To convert this mixed number into an improper fraction, we need to combine the whole number and fractional part into a
single fraction. The concept behind this conversion is to express the mixed number as the sum of the whole number and the fractional part, and then simplify it into a single fraction.
Relevant Formulas
To convert a mixed number (a b/c) into an improper fraction, we use the following formula:
Improper Fraction = (a * c) + b / c
• a is the whole number part.
• b is the numerator of the fractional part.
• c is the denominator of the fractional part.
Example Calculations
Let’s illustrate the conversion process with a few examples:
Example 1:
Convert 3 1/4 into an improper fraction.
Using the formula: Improper Fraction = (3 * 4) + 1 / 4 Improper Fraction = 12 + 1 / 4 Improper Fraction = 13/4
So, 3 1/4 is equal to 13/4 as an improper fraction.
Example 2:
Convert 5 3/8 into an improper fraction.
Using the formula: Improper Fraction = (5 * 8) + 3 / 8 Improper Fraction = 40 + 3 / 8 Improper Fraction = 43/8
So, 5 3/8 is equal to 43/8 as an improper fraction.
Example 3:
Convert 2 2/5 into an improper fraction.
Using the formula: Improper Fraction = (2 * 5) + 2 / 5 Improper Fraction = 10 + 2 / 5 Improper Fraction = 12/5
So, 2 2/5 is equal to 12/5 as an improper fraction.
Real-World Use Cases
The Mixed Number to Improper Fraction Calculator is not only a theoretical tool but also has practical applications in various fields:
Cooking and Recipes
In the culinary world, recipes require measurements in mixed numbers and fractions. When scaling recipes or adjusting serving sizes, chefs and home cooks need to convert these measurements into
precise quantities, which is easily accomplished with the calculator. For example, doubling a recipe that calls for 1 1/2 cups of flour can be calculated as 3/2 x 2 = 3 cups of flour.
Construction and Carpentry
Builders, carpenters, and craftsmen frequently work with measurements that involve mixed numbers. Converting these measurements into improper fractions is essential for precise cutting, fitting, and
estimating materials needed for a project.
In mathematics education, the concept of converting mixed numbers to improper fractions is a fundamental skill taught at an early stage. The calculator serves as an educational tool, helping students
grasp this concept and practice conversions until they become proficient.
Engineering and Technical Fields
Engineers and technicians use mixed numbers and fractions in various calculations and blueprints. Converting these values into improper fractions aids in performing complex calculations accurately.
Science and Lab Work
In scientific experiments and laboratory work, measurements are expressed as mixed numbers. Researchers may need to convert these measurements into improper fractions to perform precise calculations
and data analysis.
The Mixed Number to Improper Fraction Calculator simplifies a fundamental mathematical operation that has practical implications in various fields. It streamlines the process of converting mixed
numbers into improper fractions, allowing for accurate calculations and measurements. From cooking and construction to education and technical fields, this tool is indispensable in both everyday life
and specialized professions. Understanding the concept and using the calculator efficiently can enhance one’s mathematical skills and problem-solving abilities.
1. Smith, John. Mathematics in Everyday Life.” Journal of Mathematical Education, vol. 45, no. 3, 2017, pp. 275-290.
2. Brown, Sarah. “Teaching Fractions: Strategies for Conceptual Understanding.” Educational Psychology Review, vol. 22, no. 4, 2020, pp. 523-539. | {"url":"https://exactlyhowlong.com/mixed-number-to-improper-fraction-calculator/","timestamp":"2024-11-06T18:24:04Z","content_type":"text/html","content_length":"152266","record_id":"<urn:uuid:cdb241a4-7cfb-48ad-9989-757657addf3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00695.warc.gz"} |
Proving that injectivity implies surjectivity
• Thread starter issacnewton
• Start date
In summary: You would just need to prove that the definition of finite set is inductive. That is, you need to prove that there is a function ##g## that assigns to each natural number ##n## a set ##
A_n## such that1) ##A_1## is finite (this will follow from the definition of finite set)2) For any natural number ##n##, if ##A_n## is finite, then ##A_{n+1}## is finiteWe will do the second part
first. We need to define ##g##. Since we are using finite sets, let's say that ##g(n)## is the image under
Homework Statement
Suppose ##A## and ##B## are finite sets and ##f: A \longrightarrow B ##. Prove that if ##|A| = |B| ## then ##f## is one to one if and only if ##f## is onto.
Relevant Equations
Definition of one to one and onto function
Since this is bi-conditional, we have two directions to prove. Given is ##|A| = |B|= n ##. Now suppose that ##f## is one to one. This means that given ##a_1, a_2 \in A## , if we have ##a_1 \ne a_2##
then we have ##f(a_1) \ne (a_2) ##. So, we can not have any two elements in ##A## mapped to a single element in ##B##. This means that range of ##f## must have ##n## elements. But we are given that #
# |B| = n##. So range of ##f## must be same as ##B##. This means that all elements of ##B## have some pre-image in ##A##. Which proves that ##f## is an onto function.
This was forward direction. I want to know if the reasoning is right here.
IssacNewton said:
Homework Statement:: Suppose ##A## and ##B## are finite sets and ##f: A \longrightarrow B ##. Prove that if ##|A| = |B| ## then ##f## is one to one if and only if ##f## is onto.
Relevant Equations:: Definition of one to one and onto function
Since this is bi-conditional, we have two directions to prove. Given is ##|A| = |B|= n ##. Now suppose that ##f## is one to one. This means that given ##a_1, a_2 \in A## , if we have ##a_1 \ne
a_2## then we have ##f(a_1) \ne (a_2) ##. So, we can not have any two elements in ##A## mapped to a single element in ##B##. This means that range of ##f## must have ##n## elements. But we are
given that ## |B| = n##. So range of ##f## must be same as ##B##. This means that all elements of ##B## have some pre-image in ##A##. Which proves that ##f## is an onto function.
This was forward direction. I want to know if the reasoning is right here.
The reasoning is correct, but are you sure you don't need a more formal proof? If you don't need a formal proof, then you can reduce your proof to simple counting. For example:
##f## is onto iff the range is all of ##B## iff the range has ##n## elements iff ##f## is one-to-one.
Do you need something more formal that that?
Since I am doing this problem to practice my proof writing skills, I would appreciate more formal proof. I have studied about logic and quantifiers, so formal proof would be nice. I could not think
of doing this more formally. Can you give any pointers ?
IssacNewton said:
Since I am doing this problem to practice my proof writing skills, I would appreciate more formal proof. I have studied about logic and quantifiers, so formal proof would be nice. I could not
think of doing this more formally. Can you give any pointers ?
Perhaps think about using the set ##I_n = \{1, 2 \dots n \}##.
Since both ##A## and ##B## are finite sets, we have that ##A \thicksim I_n ## and ## B \thicksim I_n ##, which means that there are bijections from ##A## to ##I_n## and from ##B## to ##I_n##. Since #
#\thicksim## relation is an equivalence relation, we also have ##A \thicksim B##. So, there exists a bijection from ##A## to ##B## as well. Since we have a natural number ##n## here, do you think
using induction would be more formal way to go about ?
IssacNewton said:
Since both ##A## and ##B## are finite sets, we have that ##A \thicksim I_n ## and ## B \thicksim I_n ##, which means that there are bijections from ##A## to ##I_n## and from ##B## to ##I_n##.
Since ##\thicksim## relation is an equivalence relation, we also have ##A \thicksim B##. So, there exists a bijection from ##A## to ##B## as well. Since we have a natural number ##n## here, do
you think using induction would be more formal way to go about ?
The idea was to use ##I_n## to show that ##f## is one-to-one iff it's onto.
So, consider the forward direction. We assume that ##f## is one to one. There exists a bijection from ##A## to ##I_n##. And we have to prove that for any element in ##B##, there exists an element in
##A## such that they are mapped. So how would we use ##I_n## here ?
IssacNewton said:
So, consider the forward direction. We assume that ##f## is one to one. There exists a bijection from ##A## to ##I_n##. And we have to prove that for any element in ##B##, there exists an element
in ##A## such that they are mapped. So how would we use ##I_n## here ?
Let's see where we are. You gave an argument in your original post. There was nothing wrong with that argument, per se. But, I believe your argument was no different from the argument I gave in post
#2. Sure, you did a lot more handwaving and introduced some things like ##a_1 \ne a_2## and so on. But, fundamentally, there was no substance behind what you did, other than what I did in post #2
without all the fuss.
If you acccept that we can count, then my argument in post #2 is perfectly adequate. But, you are perhaps studying the subject where you need more than that. I.e. you have to prove in some sense that
counting works!
So, under the assumption that we are not allowed to count, we need something more fundamental. The only thing I know more fundamental than counting is the bijections to ##I_n## etc.
If all we know is that there exists a bijection from ##A## to ##I_n## and from ##B## to ##I_n##, then that's not much help. Why? Because it says nothing about any other functions. It doesn't, for
example, say that there isn't a bijection from ##A## to ##I_{n+1}##.
So, what theorems have you proved so far that might be useful here? What mathematical machinery do you have at your disposal? Assuming again that simple counting is not allowed.
If you want a low-level proof of this sort of thing all you have are your definitions, axioms and theorems. Your proof may use those and only those. I'm not taking your course and I don't know what
you've proved so far, which makes it difficult to help. The onus is on you to produce the machinery that you have at your disposal.
I was inspired by the set ##I_n## to use induction on the number of elements ##n##. But induction is an axiom of Peano arithmetic I am not so sure if it can be used here.
I am not doing any course. I am solving this problem from a book "How to prove it: A structured approach" by Daniel Velleman. ( 2ed.) There are following chapters in this book
1) Sentential Logic
2) Quantificational Logic
3) Proofs
4) Relations
5) Functions
6) Mathematical Induction
7) Infinite Sets
Author is a set theorist and this is a book to prepare students to proof oriented mathematics. I have solved all problems from first 6 chapters and the problem I stated here is from the last chapter.
Since author is a set theorist, most of the problems he has chosen make heavy use of set theory. So, the machinery used here is highly rigorous. All the machinery is developed in first two chapters
and third chapter discusses strategies to be used for different kinds of proofs. I am myself a teacher of physics, but since I like pure mathematics, I do these problems. Coming from more applied
background of physics, I don't have mathematical "maturity" and hence struggle to formulate proofs. Though I learned a great deal from the above book. When I was in college, I once tried to read a
book on pure mathematics by G.H. Hardy ( He was from Cambridge, UK). I could not understand the head or tail of his reasoning. So, I wanted to develop some background required to read such books.
People working in pure mathematics take lot of things for granted since they are communicating with other pure mathematicians. So, this is my background here. I learned in this book, that whenever we
have a mathematical statement involving a natural number (like cardinality of a finite set here), its useful to use induction on ##n##
IssacNewton said:
I am not doing any course. I am solving this problem from a book "How to prove it: A structured approach" by Daniel Velleman. ( 2ed.) There are following chapters in this book
1) Sentential Logic
2) Quantificational Logic
3) Proofs
4) Relations
5) Functions
6) Mathematical Induction
7) Infinite Sets
Author is a set theorist and this is a book to prepare students to proof oriented mathematics. I have solved all problems from first 6 chapters and the problem I stated here is from the last
chapter. Since author is a set theorist, most of the problems he has chosen make heavy use of set theory. So, the machinery used here is highly rigorous. All the machinery is developed in first
two chapters and third chapter discusses strategies to be used for different kinds of proofs. I am myself a teacher of physics, but since I like pure mathematics, I do these problems. Coming from
more applied background of physics, I don't have mathematical "maturity" and hence struggle to formulate proofs. Though I learned a great deal from the above book. When I was in college, I once
tried to read a book on pure mathematics by G.H. Hardy ( He was from Cambridge, UK). I could not understand the head or tail of his reasoning. So, I wanted to develop some background required to
read such books. People working in pure mathematics take lot of things for granted since they are communicating with other pure mathematicians. So, this is my background here. I learned in this
book, that whenever we have a mathematical statement involving a natural number (like cardinality of a finite set here), its useful to use induction on ##n##
This is very specific material though. I did a degree in pure maths, but I never studied anything where the basic notion of counting was dubious. I.e. the proof in post #2 would have been valid in
any course I ever took.
I would say that a book like this doesn't prepare students for proof-oriented mathematics, but is actually the reverse. You need experience of proof-oriented mathematics if you want to study the
fundamentals of set theory.
For example, if you study group theory (in a formal, rigorous approach) you will have to do proofs based on definitions, axioms and theorems; but, you will still be allowed to count!
I would say 1) it's difficult or impossible to study set-theorectic material until you have experience in pure mathematics (groups, real analysis etc.) Hence your problems.
2) Studying set-theorectic material is then an optional direction you can take - but only really once you have mastered pure mathematics.
PS to use a physics analogy: what you are doing is a bit like trying to learn QM as a prerequisite to studying classical mechanics. Or, GR as a prerequisite to Newtonian gravity!
PPS that said: are you sure that the proof I gave in post #2 is not acceptable? You'd have to ask Velleman I guess. Certainly my argument in post #2 would be valid in any course on group theory, for
example. No one is going to question that there are no proper subsets of a finite set with the same cardinality. That's just a given, really. Trying to prove something like that is a different game
Yes, maybe I was misled by this book as author is set theorist. In modern mathematics, everything is reduced to sets. So, I thought, to be rigorous, I have to eventually involve that machinery. But
people like Gauss studied pure maths even before the sets were introduced. So, we sure can do proofs without lot of set machinery. When I visit sites like math.stackexchange.com, I get the sense that
a rigorous proof should involve lot of set machinery. Since I am not a math professor, I don't know what is the general opinion among mathematicians. I am not saying that your proof in #2 is not
acceptable. I am self learning this material. Since you said that my reasoning in #1 is valid, I am satisfied with that. I used to communicate with Dr Velleman. I will also ask him about this.
IssacNewton said:
Yes, maybe I was misled by this book as author is set theorist. In modern mathematics, everything is reduced to sets. So, I thought, to be rigorous, I have to eventually involve that machinery.
But people like Gauss studied pure maths even before the sets were introduced. So, we sure can do proofs without lot of set machinery. When I visit sites like math.stackexchange.com, I get the
sense that a rigorous proof should involve lot of set machinery. Since I am not a math professor, I don't know what is the general opinion among mathematicians. I am not saying that your proof in
#2 is not acceptable. I am self learning this material. Since you said that my reasoning in #1 is valid, I am satisfied with that.
Here's the real problem. I found a pdf of the book and you're doing question 11c. And you're stuck. And I asked what you have already proved and you've said nothing. Then I look at the question and I
see parts 11a and 11b are exactly what you need to solve 11c:
11. SupposeAandBare finite sets andf:A→B.
(a) Prove that if|A|<|B|thenfis not onto.(b) Prove that if|A|>|B|thenfis not one-to-one. (This is sometimescalled thePigeonhole Principle, because it means that ifnpigeonsare put intompigeonholes,
wheren>m, then some pigeonhole mustcontain more than one pigeon.)
(c) Prove that if|A|=|B|thenfis one-to-one ifffis onto.∗
The answer is use what you proved in 11a and 11b.
Yes, that's the problem. I thought I can prove part c independently. But now, I will try to prove first two parts.
FAQ: Proving that injectivity implies surjectivity
1. What is injectivity and surjectivity?
Injectivity and surjectivity are two concepts in mathematics that describe the relationship between two sets. Injectivity means that each element in the first set maps to a unique element in the
second set. Surjectivity means that every element in the second set has at least one corresponding element in the first set.
2. Why is it important to prove that injectivity implies surjectivity?
Proving that injectivity implies surjectivity is important because it helps us understand the relationship between two sets and how their elements are mapped to each other. It also allows us to make
conclusions about the properties of a function based on its injectivity and surjectivity.
3. How can we prove that injectivity implies surjectivity?
To prove that injectivity implies surjectivity, we need to show that for every element in the second set, there exists at least one element in the first set that maps to it. This can be done by using
the definition of injectivity and surjectivity, and by using logical reasoning and mathematical operations.
4. What are some examples of functions that are injective but not surjective?
An example of a function that is injective but not surjective is f(x) = x^2, where the domain is all real numbers and the range is all positive real numbers. Another example is g(x) = e^x, where the
domain is all real numbers and the range is all positive real numbers.
5. Can a function be surjective but not injective?
Yes, a function can be surjective but not injective. An example of such a function is h(x) = x^3, where the domain and range are both all real numbers. In this case, every element in the range has at
least one corresponding element in the domain, but not every element in the domain maps to a unique element in the range. | {"url":"https://www.physicsforums.com/threads/proving-that-injectivity-implies-surjectivity.988967/","timestamp":"2024-11-14T02:22:31Z","content_type":"text/html","content_length":"156839","record_id":"<urn:uuid:ce9b7dd4-30fb-423d-835b-12dc00d158cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00290.warc.gz"} |
Here is a list of articles in the Science category of the Computing portal that unifies foundations of mathematics and computations using computers.
This category has the following 12 subcategories, out of 12 total.
Pages in category "Science"
The following 22 pages are in this category, out of 22 total. | {"url":"https://handwiki.org/wiki/index.php?title=Category:Science&oldid=2888057","timestamp":"2024-11-10T21:16:50Z","content_type":"text/html","content_length":"35581","record_id":"<urn:uuid:eaf5e284-66d9-4f99-8fdd-06b61c3a5a6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00500.warc.gz"} |
Does a Maybe Monad collapse into Just or Nothing? - IT Solutions | Free tech support | SolutionFall.Com<p>Does a Maybe Monad collapse into Just or Nothing?</p>
You must login to ask a question.
I attempted to implement a monad without prior knowledge of them, based on a vague intuition about the concept of “The Monad.” After reading “All about Monads,” I discovered that Just and Nothing are
functions, not types of monads. However, I initially believed that each Maybe monad would ultimately collapse into either a Just or Nothing monad when bound to another value.
My understanding might be flawed, and I’m seeking clarification. In this implementation, I made the Monad return itself via the bind method, but I’m wondering if I should simply return the unwrapped
value, considering that bind is already a composition.
Here’s the code I wrote:
from abc import ABCMeta
class Monad(metaclass=ABCMeta):
def __init__(self, value=None) -> None:
self._unit = value
def unit(self):
return self._unit
def unit(self, value):
self._unit = value
def __repr__(self) -> str:
return f'{self.__class__.__name__} ( {self._unit} )'
class Maybe(Monad):
def bind(self, function):
self.unit = function(self._unit) if self.unit is not None else None
self.__class__ = Nothing if self.unit is None else Just
return self
class Nothing(Maybe):
class Just(Maybe):
a = Maybe(1)
b = Maybe(1)
b.bind(lambda x: x + 1).bind(lambda x: x + 1).bind(lambda x: x + 1).bind(lambda x: x + 1)
c = Maybe()
c.bind(lambda x: x + 1).bind(lambda x: x + 1).bind(lambda x: x + 1)
print(a, a.unit)
print(b, b.unit)
print(c, c.unit)
I still find this implementation somewhat messy, and I’m looking for ways to improve it. However, my primary concern is whether my understanding of the concepts is accurate. | {"url":"https://solutionfall.com/question/does-a-maybe-monad-collapse-into-just-or-nothing/","timestamp":"2024-11-10T15:00:32Z","content_type":"text/html","content_length":"105345","record_id":"<urn:uuid:82ff96e1-81cb-43ff-9899-0f260a312a7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00210.warc.gz"} |
Select All by Trait
Select All by Trait
Edit Mode
Selects the Non-manifold geometry of a mesh. This entry is available when editing a mesh, in Vertex and Edge selection modes only.
Lets you extend the current selection.
Selects all the edges that do not belong to any face.
Selects edges in boundaries and holes.
Multiple Faces
Selects edges that belong to three or more faces.
Non Contiguous
Selects edges that belong to exactly two faces with opposite normals.
Selects vertices that belong to wire and multiple face edges, isolated vertices, and vertices that belong to non-adjoining faces.
Loose Geometry
Edit Mode
This selection depends on the currently selected Selection Modes; In vertex and edge selection mode it selects all vertices or edges that do not form part of a face. In face selection mode it selects
all faces that do not share edges with other faces.
Interior Faces
Edit Mode
Selects faces where all edges have more than two faces.
Faces by Sides
Edit Mode
Selects all faces that have a specified number of edges.
Ungrouped Vertices
Edit Mode
Selects all vertices which are not part of a vertex group. | {"url":"https://docs.blender.org/manual/de/3.1/modeling/meshes/selecting/all_by_trait.html","timestamp":"2024-11-08T21:51:20Z","content_type":"text/html","content_length":"21033","record_id":"<urn:uuid:7640fa25-5845-4f0f-93d8-36f24c078391>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00626.warc.gz"} |
RSICC Home Page
RSIC COMPUTER CODE PSR-013
1. NAME AND TITLE
SUPERTOG: Data Generator--Fine Group Constants and P[N] Scattering Matrices from ENDF/B.
The DLC-2 RETRIEVAL PROGRAM retrieves SUPERTOG output from a card image tape written in the ANISN card image format. This program will retrieve data from a maximum of 46 data sets and merge this data
onto one data set. It will then, by input option, edit the data, punch cards in either the ANISN or DTF-IV format, or write an unformatted tape for use by ANISN. Another program is available which
will merge up to a maximum of four card image tapes written in the GAM-II update format onto a single tape. This program can also take the one-dimensional arrays from one tape and the two-dimensional
arrays from another tape and merge this information onto a single tape. The single tape is input to the SUPERTOG version of the GAM-II update program. The GAM-II update program has been modified to
accept output from SUPERTOG on either punched cards or magnetic tape.
Oak Ridge National Laboratory, Oak Ridge, Tennessee.
Interuniversity Reactor Institute, Delft, Netherlands, through the OECD NEA Data Bank, Gif-sur-Yvette, France.
Fortran IV; IBM 360 SUPERTOG-IV (A), IBM 360 SUPERTOG-III (B).
4. NATURE OF THE PROBLEM SOLVED
In SUPERTOG-IV the inelastic treatment of some materials was corrected; provision was made for making fission spectra; cross-section arrays were made 5000 elements long, thus permitting the use of
ENDF/B-IV format. SUPERTOG-III accepts nuclear data in either a point by point or parametric representation as specified by ENDF/B. This data is averaged over each specified group width. The explicit
assumption is made that the flux per unit lethargy is constant or that a suitable weight function will be supplied by the user. When resonance data is available, resolved and unresolved resonance
contributions are calculated and used as specified by input options. Fine group constants such as one-dimensional reaction arrays (absorption, fission, etc.), P[n] elastic scattering matrices, and
inelastic and (n,2n) scattering matrices are generated and placed on tapes in formats suitable for use by GAM-I, GAM-II, ANISN, or DOT.
5. METHOD OF SOLUTION
The single-level Breit-Wigner formalism is used for calculation of cross sections in the resolved resonance region. Cross sections in the unresolved resonance region are computed by taking averages
over suitable Porter-Thomas distributions of the neutron and fission widths. Smooth cross sections are calculated by integration of point-cross-section data given in ENDF/B file 3. Elastic scattering
matrices are computed from Legendre coefficients of the scattering angular-distribution data. Inelastic scattering and (n,2n) matrices are computed from excitation functions for individual levels and
by using a nuclear evaporation model above the region of resolved levels.
Since fixed, rather than flexible, dimensions are used, it is important to be aware of the maximum values allowed for certain key variables. Examples are: number of groups 150, number of data points
for each reaction type 4000, and the number of Legendre coefficients 30.
SUPERTOG-III employs a rather elaborate overlay structure and extensive use of equivalence statements to minimize core requirements.
7. TYPICAL RUNNING TIME
(Times quoted are for the IBM 360/91.) Running time varies greatly and is a function, primarily, of the number of groups, the number of resolved resonances, and the length of the elastic scattering
matrix. The average time required to generate DLC-2D from ENDF/B Version III data was 2.2 minutes per nuclide.
Estimated running time of the packaged sample problem for the GAM-II, 99-group structure (239-Pu) MAT:1159 with P-3 elastic scattering is 3.0 minutes.
SUPERTOG-III is designed to operate on IBM 360/50/65/75/91 computers. Approximately 366 K bytes or 94 K words of directly addressable core are required. SUPERTOG-LTT was processed on an IBM 3033 at
Oak Ridge National Laboratory.
A Fortran IV compiler is required.
10. REFERENCES
R. Q. Wright, "Modifications for SUPERTOG III Mod 2," Informal Notes (April 1978).
R. Q. Wright, "PSR-13/SUPERTOG III Code Package," Mod 1 Informal Notes (July 1973).
R. Q. Wright, N. M. Greene, J. L. Lucius, C. W. Craven, Jr., "SUPERTOG: A Program to Generate Fine Group Constants and Pn Scattering Matrices from ENDF/B," ORNL-TM-2679 (September 1969).
SUPERTOG-II Informal Notes, ORNL.
R. Q. Wright, "Increasing the Number of Groups Allowed in SUPERTOG-II," Informal Notes.
SUPERTOG-III Informal Notes, ORNL (August 1972).
R. Q. Wright, "SLOGAN for SUPERTOG-III," (July 1972).
SAD - Secondary Angular Distributions Informal Notes (1968).
R. F. Berland, "CHAD-Code to Handle Angular Data," NAA-SR-11231 (December 1965).
11. CONTENTS OF CODE PACKAGE
Included are the referenced document and one (1.2MB) DOS diskette which contains the source code and sample problem input and output.
12. DATE OF ABSTRACT
September 1972; updated October 1983, August 1985, April 1990, January 1992. | {"url":"https://rsicc.ornl.gov/codes/psr/psr0/psr-013.html","timestamp":"2024-11-07T19:31:26Z","content_type":"text/html","content_length":"7955","record_id":"<urn:uuid:90c4e938-b54b-4443-b7c3-2d8af2b169fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00730.warc.gz"} |
User-friendly SAS macro application for performing repeated measures covariance type selection
I developed a user-friendly SAS macro application to perform all possible mixed model selection of fixed effects including quadratic and cross products within a user-specified subset range in the
presence of random and repeated measures effects using SAS PROC MIXED (Fernandez, 2007). This macro application, ALLMIXED will complement the model selection option currently available in the SAS
PROC REG for multiple linear regressions and the SAS Proc GLMSELECT that focuses on the standard independently and identically distributed general linear model for univariate responses. Options are
also included in this macro to select the best covariance structure associated with the user-specified fully saturated repeated measures model; to graphically explore and to detect statistical
significance of user specified linear, quadratic, interaction terms for fixed effects; and to diagnose multicollinearity, via the VIF statistic for each continuous predictor, involved in each model
selection step. Two model selection criteria, AICC (corrected Akaike Information Criterion) and MDL (minimal description length) are used in all possible model selection and summaries of the best
model selection are compared graphically. In this community posting, I will describe the prescreening step of ALLMIXED model selection steps.
The recommended selection steps for performing the model selection in MIXED model is illustrated in Figure1. Although the recommended sequence of the steps is identified in the figure 1, it is not a
requirement to follow the same sequence. Users are free to choose to run any model selection steps in any order they desire. However, before running these model selection steps the data format must
be suitable for running the SAS PROC Mixed procedure. The following types of PC data formats can be used with the ALLMIXED macro: SAS temporary and permanent data files, Microsoft excel, COMMA or TAB
delimited text file.
SAS 9.4 Modules required to run this macro:
• SAS/STAT: PROC MIXED, PROC CORR, PROC REG, PROC GLMSELECT
• SAS/GRAPH: PROC GCHART, PROC GPLOT, PROC G3D
• Base SAS ODS (RTF, HTML, PDF)
• SAS/ACCESS: PC FILES – PROC IMPORT and PROC EXPORT
Improved ALLMIXED SAS macro application
The original SAS macro application, I developed (Fernandez, 2007) is not compatible in SAS enterprise guide (SAS EG) or in SAS studio. Therefore, I am presenting an improved version of the ALLMIXED
macro in this post. By using this improved ALLMIXED macro application, SAS users can effectively perform complete mixed model analysis in SAS studio or in SAS EG. First download and unzip the
ALLMIXED.zip file specified in this post and save the contents to a custom folder such as C:\temp\allmixed. The extracted ALLMIXED zip file should include, compiled ALLMIXED macro catalog, five macro
call files corresponding to six ALLMIXED model selection steps, and sample demo data used in the demo. In this article, I will present the steps needed to perform Step2-Initial Covariance selection
in this post. Please follow the steps outlined in the previous post to perform step1- Prescreening (https://communities.sas.com/t5/SAS-Communities-Library/
STEP2: Initial Repeated measures Covariance selection
In a repeated measures modeling, the best covariance structure describing the correlation among the repeated measures should be identified first. The best covariance structure can be identified from
different user-specified covariance structures by comparing the AICC statistic computed in PROC MIXED using REML method and select the covariance type which gives the smallest AICC value. In this
example, four different covariance types, CS, AR(1), TOEP, and UN are compared in the full saturated model containing two categorical effects (TRT and TIME) and 4 continuous fixed effects selected
from the previous prescreening steps.
ALLMIXED SAS macro help – Step 2 Repeated measure covariance type selection
1. Input the Excel file name or SAS data set name?
Descriptions and Explanation: Include the data type name (XLS, TAB, TXT, SAS, TMP) and name of the data set on which you would like to perform pre-screening.
Options / Examples:
• xls_SIMDATA1: Data type is EXCEL, and the file name is SIMDATA1. Make sure to include the separator character '_'.
SAS_SIMDATA1: Data type is permanent SAS (SD7SAS) and the SAS permanent data name is SIMDATA1
TMP_SIMDATA1: Data type is temporary SAS data and file name is SIMDATA1.
2. Input required Response variable or variables
Descriptions and Explanation: Input the continuous response (dependent) variable name (or names). The name should match the variable names in the data. You can include multiple responses
Options / Examples:
3. Pre-screening fixed variables using GLMSELECT
Descriptions and Explanation: To start the variable pre-screening mode, enter yes. Pre-screening based on the LASSO method using the selection method SBC and the stop criterion NONE will be
performed. This field should be left blank to run all other model selection steps.
Options / Examples:
4. Input optional CLASS terms
Descriptions and Explanation: Input the names of the categorical variables that will be included in the CLASS statement in PROC Mixed.
Options / Examples:
Class= TRT time sub
5. Input ith analysis (a counter) to attach to the saved output file name
Descriptions and Explanation: Input any numeric and categorical character to track the number of the analysis that you are running using this data. For example, if you input 1A, the output file
created in this step would be called SIMDATA11A.ext.
Options / Examples:
Z = -1
Z = 1A
10. Repeated measure Statement
Descriptions and Explanation: Input the REPEATED statement and leave the covariance type blank.
Options / Examples:
• • repeated time /sub=sub type=
11. Input the subject variable name
Descriptions and Explanation: In case of repeated measures data, input the subject variable name. This forces the pre-screening to do initial selection at the subject level.
Options / Examples:
Sub = Sub
12. List covariance structure(s) screening
Descriptions and Explanation: List all the repeated measures covariance types.
Options / Examples:
• • Covari = CS AR(1) TOEP UN
14. Display or save the Graphs/output? choose one
Descriptions and Explanation: Option for viewing and saving all output files in a folder specified in input number 17.
WORD: Output and all SAS graphics are saved together in the user-specified folder as a single RTF format.
WEB: Output and graphics are saved in the user-specified folder as a single HTML file.
PDF: Output and graphics are saved in the user-specified folder and as a single PDF file.
TXT: Output is saved as a TXT file in all SAS versions. No output is displayed in the OUTPUT window. All graphic files are saved as PNG format in the user-specified folder.
15. Folder containing the PC data files
Descriptions and Explanation: Input the full path of the folder containing the source data file.
Options / Examples:
• 😧\allmixed\sasdata\ - folder name SASDATA on drive D
Make sure that you include the backslash (\) at the end of the folder name.
OUTPUT= c:\temp\allmixed\
17. Folder to save the output/graphics?
Descriptions and Explanation: To save the SAS graphics, data, and output files, input the output folder name. If the 14 field is left blank, the output files are saved in the default folder.
Options / Examples:
Dir2= C:\temp\allmixed\
In a repeated measures modeling, the best covariance structure describing the correlation among the repeated measures should be identified first. The best covariance structure can be identified from
user-specified covariance structures by comparing the AICC statistic computed in PROC MIXED using REML method and select the covariance type which gives the smallest AICC value. In this example, four
different covariance types, CS, AR(1), TOEP, and UN are compared in the full saturated model containing two categorical effects (TRT and TIME) and four continuous fixed effects selected from the
previous prescreening
steps. The results of initial covariance type selection are graphically displayed in Figure3 and based on ΔAICCj (AICCj - AICC min), AR(1) can be identified as the best covariance type. Therefore, AR
covariance type will be used in the subsequent fixed effect selection. Refer the ALLMIXED2 macro help file available from the authors website for more information regarding inputting appropriate
Fernandez, G. (2007) Model Selection in PROC MIXED - A User-friendly SAS® Macro Application SAS Global Forum proceedings 191-2007
06-01-2024 03:08 PM
06-01-2024 03:08 PM
After testing with the following dataset, it was revealed that the compiled ALLmix macro was hard coded to use a fixed variable in the input data set, which is the variable 'time'.
Because my testing data set does not have the 'time' variable, the macro failed to run.
I strongly suggest the author open the source of the compiled macro. Otherwise, it is hard for others to use it and nobody will use it in the future, though the author has contributed a lot of time
to write the program!
In addition, the compiled macro only works under Windows SAS but not Linux SAS. For SAS OnDemand for Academics, due to its Linux SAS, the compiled macro can not run.
This is my testing codes:
libname allmix4 "C:\Users\cheng\Downloads\ALLmixed";
%let wd=C:\Users\cheng\Downloads\ALLmixed\;
options sasmstore=allmix4 mstored;
/*Generate data for testing*/
data analysisData testData;
drop i j;
array x{20} x1-x20;
do i=1 to 5000;
/* Continuous predictors */
do j=1 to 20;
x{j} = ranuni(1);
/* Classification variables */
c1 = int(1.5+ranuni(1)*7);
c2 = 1 + mod(i,3);
c3 = int(ranuni(1)*15);
yTrue = 2 + 5*x17- 8*x5 + 7*x9*c2- 7*x1*x2 + 6*(c1=2) + 5*(c1=5);
y= yTrue + 6*rannor(1);
if ranuni(1) < 2/3 then output analysisData;
else output testData;
proc datasets nolist;
copy in=work out=Allmix4 move;
select analysisData;
/* 1. Input the Excel or sas Data set name? E.G: xls_simdata1 xlsx_simdata1 sas_simdata1 tmp_ */
data_ = sas_analysisdata
,/* 2. Input required Response variable or variables E.G: y or y1 y2 */
respi = y
,/* 3. Pre-Screening predictors using:GLMSELECT E.G: blank when performing model selection */
,/* 4. Input optional class terms ? E.G: trt time sub */
class = c1 c2 c3
,/* 5. Input ith analysis (a counter) to attach to the saved output file name? E.G: _3 */
z = _3
,/* 6. Optional model statement options E.G: blank */
,/* 7. Input must have fixed effects - in mixed model E.G. trt time trt*time */
must = x1 x2 x5 x10 x13 x9 x17 c1 c2 c3 x1*c1 x2*c2
,/* 8. Input list of class (line1) and continuous effects (line2) E.G: line1: blank Line2: x1 x5 x6 x8 x10 x12 x14 x15 */
fixed1 = c1 c2 c3
, fixed2 = x1 x2 x5 x10 x9 x13 x17
,/* 9. Input optional Random statement E.G: blank in this step */
Random =
,/* 10. Input Repeated statement E.G: Repeated time /sub=sub type=ar(1) */
Repeat = Repeated time /sub=sub type=ar(1)
,/* 11. Input Subject variable E.G: sub */
sub =
,/* 12. covariance structure(s) screening E.G: blank completed in previous step */
,/* 13. Exploration: Interaction and Quadratic plots E.G. blank needed in next step */
explor =
,/* 14. Display or save the Graphs/output? choose one E.G: word web pdf txt */
graph = web
,/* 15. Folder containing the PC data files? E.G: D:\allmixed\sasdata\ */
output = &wd
,/* 16. optional LSMEANS statement final model E.G: blank used in final step */
lsmeans =
,/* 17. Folder to save the output/graphics E.G: D:\allmixed\ */
dir2 = &wd
,/* 18. Optional model selection Start number of terms E.G: 3 */
start = 2
,/* 19. Optional model selection stop number of terms E.G: 4 */
Stop = 3
10-02-2024 11:54 PM
10-02-2024 11:54 PM
In the above of my testing codes, it is necessary to assign empty value for the Repeat macro variable, i.e., replacing 'Repeat = Repeated time /sub=sub type=ar(1)' with 'Repeat=', as there is no time
variable in the input sas dataset.
After correcting the above error, the ALLMixed macro can run successfully with SAS 9.4 of Windows system. | {"url":"https://communities.sas.com/t5/SAS-Communities-Library/User-friendly-SAS-macro-application-for-performing-repeated/ta-p/761639","timestamp":"2024-11-05T14:09:10Z","content_type":"text/html","content_length":"166554","record_id":"<urn:uuid:8ed077a9-0e29-4437-b174-8d59f47fab30>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00412.warc.gz"} |
Multiplication of a Whole Number by a Fraction | Multiplying Fractions
Multiplication of a Whole Number by a Fraction
This topic discusses about multiplication of a whole number by a fraction. Whole numbers are positive integers starting from zero to infinity. The fractions are always in numerator and denominator
form whereas, whole numbers are not. Hence a whole number has to be changed into numerator and denominator form for multiplying it with fraction.
In order to change whole number into fraction we have to take the whole number in the numerator and place 1 in the denominator. Now the numerator of the whole number is multiplied with the numerator
of the fraction and denominator of the whole number is multiplied with the denominator of the fraction to get the result. That is (product of the numerators/product of the denominators)
Here are few examples to illustrate multiplication of a whole number by a fraction:
1. 5 × \(\frac{7}{10}\)
In this sum 5 is the whole number and \(\frac{7}{10}\) is the fraction.
Step I: Change the whole number into a fraction by placing 1 in the denominator.
Therefore, \(\frac{5}{1}\) × \(\frac{7}{10}\)
Step II: Now both the numbers are in fraction and the rule of multiplication of fractional number is that numerator and numerator should be multiplied and similarly denominator and denominator should
be multiplied. So, we will do
5 × 7 = 35
And 1 × 10 = 10
Therefore, \(\frac{5}{1}\) × \(\frac{7}{10}\) = \(\frac{35}{10}\)
Step III: After multiplying numerator by numerator and denominator by denominator the answer that we will get has to be either changed into lowest term (if possible) and then change into mixed
fraction if not a proper fraction or can be left as whole number (if possible).
Therefore, first we will change \(\frac{35}{10}\) into lowest term
= \(\frac{35 ÷ 5}{10 ÷ 5}\)
= \(\frac{7}{2}\)
2. 15 × 7\(\frac{2}{5}\)
= 15 × \(\frac{37}{5}\); [Changing mixed fraction into improper fraction]
= \(\frac{15}{1}\) × \(\frac{37}{5}\); [Changing the whole number into a fraction by placing 1 in the denominator]
= \(\frac{15 × 37}{1 × 5}\); [Product of the numerators/ product of the denominators]
= \(\frac{555}{5}\)
= \(\frac{555 ÷ 5}{5 ÷ 5}\); [Changing into lowest terms]
= 111
Here a whole number is multiplied with a mixed fraction. Hence we have to first change the mixed fraction into improper fraction for the purpose of calculation. Then change the whole number into a
fraction by placing 1 in the denominator.
For changing 7\(\frac{2}{5}\) into a improper fraction we have to multiply the whole number 7 with the denominator 5. That is 7 × 5 = 35. With this 35 we will add the numerator that is 2 which gives
35 + 2 = 37. Now, 37 will be the numerator and 5 the denominator. Hence the fraction will be \(\frac{37}{5}\)
i.e. 7\(\frac{2}{5}\)
= \(\frac{7 × 5 + 2}{5}\)
= \(\frac{37}{5}\)
3. 250 × \(\frac{19}{15}\)
= \(\frac{250}{1}\) × \(\frac{19}{15}\); [Changing the whole number into a fraction by placing 1 in the denominator]
= \(\frac{250 × 19}{1 × 15}\); [Product of the numerators/ product of the denominators]
= \(\frac{4750}{15}\)
= \(\frac{4750 ÷ 5}{15 ÷ 5}\); [Changing into lowest terms]
= \(\frac{950}{3}\)
= 316\(\frac{2}{3}\).
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Sum() Python – A Comprehensive Guide to Understanding and Implementing the Sum() Function in Python
The sum() function in Python is a powerful tool that allows users to calculate the sum of a collection of elements quickly and efficiently. Whether you are working with numbers, monetary values, or
even custom objects, understanding and implementing the sum() function can greatly simplify your code and improve its readability.
In this blog post, we will provide a comprehensive guide to the sum() function in Python. We will explore its syntax, parameters, and various use cases. Additionally, we will delve into implementing
sum() with iterable objects, custom objects, and discuss the optional start parameter. We will also cover error handling techniques, performance considerations, and provide examples of the sum()
function in practical applications.
Basics of the sum() function
The sum() function in Python is a built-in function that calculates the sum of a collection of elements. It takes an iterable object as its parameter and returns the sum of all the elements in that
The syntax of the sum() function is as follows:
sum(iterable, start=0)
The iterable parameter represents the collection of elements for which we want to calculate the sum. It can be a list, tuple, set, or any other iterable object.
Implementing sum() with iterable objects
In Python, an iterable object is any object that can be looped over, such as a list, tuple, or set. The sum() function can be applied to these objects to quickly calculate their sum.
Let’s explore the different use cases of sum() with iterable objects:
Using sum() with lists, tuples, and sets
One common use case of the sum() function is calculating the sum of numbers in a list. Consider the following example:
numbers = [1, 2, 3, 4, 5] total = sum(numbers) print(total) # Output: 15
The sum() function iterates over the list and adds each element together, resulting in the sum of the numbers.
Similarly, you can use the sum() function with tuples and sets:
numbers = (1, 2, 3, 4, 5) total = sum(numbers) print(total) # Output: 15
numbers = {1, 2, 3, 4, 5} total = sum(numbers) print(total) # Output: 15
Regardless of the iterable object used, the sum() function provides a convenient way to calculate the sum of the elements without explicitly iterating over the collection.
Exploring limitations and potential issues when using sum() with iterables
While the sum() function is powerful and efficient, it’s essential to be aware of its limitations and potential issues when using it with iterables.
One limitation of the sum() function is that it only works with numeric types. If you try to use it with non-numeric elements, such as strings or custom objects, you will encounter a TypeError:
names = ['Alice', 'Bob', 'Charlie'] total = sum(names) # Raises TypeError: unsupported operand type(s) for +: 'int' and 'str'
To calculate the sum of non-numeric elements, you would need to use a different approach, such as using a loop or list comprehension.
Another potential issue is that the sum() function performs a simple addition, which may not yield accurate results when dealing with floating-point numbers:
numbers = [0.1, 0.1, 0.1] total = sum(numbers) print(total) # Output: 0.30000000000000004
This is due to the inherent imprecision in representing floating-point numbers in binary. To handle this issue, you can use the decimal module or employ appropriate rounding techniques.
Implementing sum() with custom objects
One of the remarkable features of the sum() function is its flexibility. Not only can it be used with numeric types, but it can also be applied to custom objects.
Let’s explore how to implement sum() with custom objects:
Defining custom objects for summation
To use the sum() function with custom objects, we need to define our objects so that addition can be performed using the + operator.
For example, let’s say we have a Book class, and we want to calculate the total price of a collection of books:
class Book: def __init__(self, title, price): self.title = title self.price = price
books = [ Book("Python Crash Course", 29.99), Book("Clean Code", 39.99), Book("Design Patterns", 49.99) ]
Implementing the __add__() method for custom objects
In order to use the sum() function with our custom Book objects, we need to implement the __add__() method in our class:
class Book: def __init__(self, title, price): self.title = title self.price = price
def __add__(self, other): if isinstance(other, Book): return self.price + other.price else: return NotImplemented
def __radd__(self, other): return self + other
books = [ Book("Python Crash Course", 29.99), Book("Clean Code", 39.99), Book("Design Patterns", 49.99) ]
total_price = sum(books) print(total_price) # Output: 119.97
By implementing the __add__() method, we define how two Book objects should be added together. In this case, we calculate the sum of their prices. Additionally, we also implement the __radd__()
method, which allows the Book object to be the right operand in addition operations.
Applying the sum() function to custom object collections
Once we have defined the appropriate methods in our custom object, we can easily calculate the sum using the sum() function:
total_price = sum(books) print(total_price) # Output: 119.97
The sum() function automatically invokes the __add__() method for each element in the collection, resulting in the sum of their prices.
Understanding the optional start parameter
In addition to the iterable parameter, the sum() function also accepts an optional start parameter. By default, the start value is set to 0.
Let’s explore the purpose and functionality of the start parameter:
Explaining the purpose and functionality of the start parameter
The start parameter allows you to specify an initial value to start adding from. If provided, the sum() function will add this value to the sum of the iterable. If not provided, the default start
value of 0 is used.
Consider the following example:
numbers = [1, 2, 3, 4, 5] total = sum(numbers, start=10) print(total) # Output: 25
In this case, the sum() function adds the start value of 10 to the sum of the numbers, resulting in a total of 25.
Demonstrating different scenarios where the start parameter can be useful
The start parameter can be particularly useful in scenarios where you need to calculate cumulative sums or when the initial value for summation is non-zero.
For example, let’s say we have a list of sales data and we want to calculate the cumulative sales for each day:
sales = [100, 200, 150, 300, 250] cumulative_sales = [] running_total = 0
for sale in sales: running_total += sale cumulative_sales.append(running_total)
print(cumulative_sales) # Output: [100, 300, 450, 750, 1000]
In this case, we are manually iterating over the sales list and updating a running total variable. However, we can achieve the same result more concisely using the sum() function with the start
sales = [100, 200, 150, 300, 250] cumulative_sales = [sum(sales[:i+1], start=0) for i in range(len(sales))]
print(cumulative_sales) # Output: [100, 300, 450, 750, 1000]
By using the start parameter, we can avoid the need for an explicit loop and perform the cumulative sum in a single line.
Handling errors and exceptions with sum()
When using the sum() function, it’s crucial to be aware of potential errors and exceptions that may occur, and implement appropriate error handling techniques.
Let’s explore how to handle errors and exceptions when using the sum() function:
Identifying potential errors and exceptions when using sum()
The most common error you may encounter when using the sum() function is a TypeError. This occurs when you try to perform addition on incompatible types or non-numeric elements.
For example, consider the following code:
values = [1, 2, '3', 4, 5] total = sum(values) # Raises TypeError: unsupported operand type(s) for +: 'int' and 'str'
In this case, the sum() function raises a TypeError because it is unable to add an integer (1) and a string (‘3’).
Implementing error handling techniques to prevent issues
To prevent potential issues and handle errors, it’s important to perform appropriate type checks and validation before calling the sum() function.
One approach is to use the isinstance() function to ensure that all elements in the iterable are of the expected type:
values = [1, 2, '3', 4, 5]
if all(isinstance(value, int) for value in values): total = sum(values) print(total) else: print("The iterable contains non-numeric elements")
In this example, we check if all elements in the values list are of type int before calling the sum() function. If any element is not an integer, we display an error message.
Using try-except blocks to gracefully handle exceptions
An alternative approach to handling errors is to use try-except blocks to catch and gracefully handle exceptions.
For example, we can modify the previous code snippet using a try-except block:
values = [1, 2, '3', 4, 5]
try: total = sum(values) print(total) except TypeError: print("The iterable contains non-numeric elements")
In this case, when a TypeError occurs, the code execution transfers to the except block, where we can handle the exception in a specific way, such as displaying an error message.
Performance considerations with sum()
When using the sum() function, it’s important to consider its time complexity and potential performance issues, especially when dealing with large collections of elements.
Let’s discuss the time complexity of the sum() function and analyze potential performance optimizations:
Discussing the time complexity of the sum() function
The time complexity of the sum() function is O(n), where n represents the number of elements in the iterable. This is because the sum() function needs to iterate over each element in the iterable to
calculate the sum.
Keep in mind that the sum() function performs a single pass over the iterable, making it an efficient choice for calculating sums in most cases.
Analyzing potential performance issues and optimizations
While the sum() function is generally efficient, there may be scenarios where performance optimizations are necessary, especially when dealing with large collections or complex computations.
One optimization technique is to use a generator expression instead of a list comprehension or passing a built-in iterable object:
values = range(10_000_000) total = sum(value for value in values) print(total)
This approach avoids creating an intermediate list and instead generates values on the fly, reducing memory consumption and improving overall performance.
Additionally, if you are dealing with extremely large numbers or complex computations, consider using specialized libraries, such as NumPy or SciPy, which provide optimized functions for mathematical
Examples and practical applications
The sum() function is incredibly versatile and can be applied to various scenarios. Let’s explore a few practical examples:
Calculating the sum of numbers in a given list
One of the most common use cases of the sum() function is calculating the sum of numbers in a list:
numbers = [10, 20, 30, 40, 50] total = sum(numbers) print(total) # Output: 150
The sum() function provides a concise and efficient way to calculate the sum of numbers without the need for explicit iteration.
Summing up monetary values with appropriate rounding
When working with monetary values, it’s important to consider appropriate rounding to ensure precision and accuracy. The sum() function can handle this by utilizing the decimal module:
from decimal import Decimal
prices = [0.1, 0.2, 0.3, 0.4, 0.5] total = sum(Decimal(price) for price in prices) print(total) # Output: 1.5
In this example, each price value is converted to a Decimal object before being passed to the sum() function. This ensures accurate calculations, even with floating-point imprecision.
Aggregating data from a database using sum()
Another practical application of the sum() function is aggregating data from a database. Let’s assume we have a table of sales data with columns for sales amounts and dates. We can use the sum()
function to calculate the total sales for a specific time period:
import sqlite3
conn = sqlite3.connect('sales.db') cursor = conn.cursor()
start_date = '2022-01-01' end_date = '2022-12-31'
query = f"SELECT sales_amount FROM sales WHERE date BETWEEN '{start_date}' AND '{end_date}'" cursor.execute(query)
sales_amounts = [row[0] for row in cursor.fetchall()] total_sales = sum(sales_amounts)
cursor.close() conn.close()
In this example, we query a SQLite database for sales amounts within a given time period. The sum() function is then used to calculate the total sales amount by summing up the sales_amount values
retrieved from the database.
In conclusion, the sum() function in Python is a fundamental tool that allows users to calculate the sum of a collection of elements quickly and efficiently. By understanding the basics of the sum()
function, implementing it with various types of objects, and considering its optional parameters, users can streamline their code and simplify complex summations.
We have covered a wide range of topics related to the sum() function in Python, including its syntax, parameters, handling errors, and performance considerations. We have also explored practical
examples and applications of the sum() function, such as calculating cumulative sums, summing monetary values, and aggregating data from a database.
As you continue to explore Python and encounter scenarios that require summation, remember to leverage the power and flexibility of the sum() function to simplify your code and improve its
readability. Experiment with different use cases and explore additional built-in functions and libraries that can further enhance your data manipulation and analysis capabilities. | {"url":"https://skillapp.co/blog/sum-python-a-comprehensive-guide-to-understanding-and-implementing-the-sum-function-in-python/","timestamp":"2024-11-03T06:37:55Z","content_type":"text/html","content_length":"119993","record_id":"<urn:uuid:a871594a-9ce1-438b-8849-7f825a4e88be>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00452.warc.gz"} |
Generate and return the (squared) Euclidean distance of a (set of) point(s) in ndim-dimensions from a reference point (possibly origin), optionally robustly without underflow or overflow.
: The input scalar (or array of the same rank as other array-like arguments) of the same type and kind as the output distance containing the x component of a 3D vector whose Euclidean
[in] x norm is to be computed.
(optional. It must be present if and only if the input arguments point and ref are missing and y and z are present.)
: The input scalar (or array of the same rank as other array-like arguments), of the same type and kind as x, containing the y component of a 2D or 3D vector whose Euclidean norm is
[in] y to be computed.
(optional. It must be present if and only if the input arguments point and ref are missing and x an z are present.)
: The input scalar (or array of the same rank as other array-like arguments), of the same type and kind as x, containing the z component of a 3D vector whose Euclidean norm is to be
[in] z computed.
(optional. It must be present if and only if the input arguments point and ref are missing and x and y are present.)
: The input contiguous vector of shape (1:ndim) or matrix of shape (1:ndim, 1:npnt) of the same type and kind as the output distance, containing a (set of npnt) point(s) in the
[in] point ndim-dimensional Euclidean space whose distances with respect to the input reference point ref must be returned.
(optional. It must be present if and only if the input arguments x, y, z are missing.)
: The input contiguous vector of shape (1:ndim) or matrix of shape (1:ndim, 1:nref) of the same type and kind as point, containing the (set of nref) reference point(s) from which the
[in] ref distance(s) of point must be computed.
(optional, default = [(0., i = 1, size(point, 1))]. It can be present only if the input argument point is also present.)
: The input scalar that can be,
[in] method
(optional, default = euclid)
distance : The output object of,
1. type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128),
containing the requested Euclidean (squared) distance(s).
The rank and shape of the output distance follows that of the interfaces illustrated below.
Possible calling interfaces ⛓
! distance with respect to origin.
= getDisEuclid
:npnt), method
! distance with respect to custom reference.
= getDisEuclid
:ndim), ref(
:ndim), method
= getDisEuclid
:ndim), ref(
:nref), method
= getDisEuclid
:npnt), ref(
:ndim), method
= getDisEuclid
:npnt), ref(
:nref), method
Generate and return the (squared) Euclidean distance of a (set of) point(s) in ndim-dimensions from a...
This module contains procedures and generic interfaces for computing the Euclidean norm of a single p...
The condition size(point, 1) == size(ref, 1) must hold for the corresponding input arguments.
This condition is verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.
The pure procedure(s) documented herein become impure when the ParaMonte library is compiled with preprocessor macro CHECK_ENABLED=1.
By default, these procedures are pure in release build and impure in debug and testing builds.
The Fortran standard provides the intrinsic procedure norm2() for computing the Euclidean norm of a vector.
However, the standard does not enforce robustness of the intrinsic procedure with respect to possible underflows or overflows.
The procedures of this module ensure robustness of the distance computations.
This will inevitably lead to worse runtime performance compared to the compiler implementations of the intrinsic routine that do not respect robustness.
Use the routines of this module in place of the Fortran intrinsics if you believe there is a possibility of under/over-flow.
The procedures of this module can be used for a robust computation of abs(x) when x is a large complex value.
In such a case, calling getDisEuclid([x%re, x%im]) would be equivalent to abs(x) intrinsic operation.
However, note that the Fortran standard already offers a better intrinsic alternative to the routines of this procedure for this task, namely hypot() which is robust against overflow and
This generic interface intentionally does not have explicit procedures for 2D Euclidean distance (x, y) because the Fortran intrinsic procedure hypot() already serves the purpose.
The procedures under discussion combine, modernize, and extend the interface and functionalities of Version 3.11 of BLAS/LAPACK routine(s): dlapy3
See also
Intrinsic Fortran procedure hypot(x, y) (robust)
Intrinsic Fortran procedure norm2(x(:)) (unsafe)
The procedures under discussion combine, modernize, and extend the interface and functionalities of Version 3.11 of BLAS/LAPACK routine(s): dlapy3
Example usage ⛓
Example Unix compile command via Intel ifort compiler ⛓
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Example Windows Batch compile command via Intel ifort compiler ⛓
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran compiler ⛓
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Example output ⛓
), point(
), point(
!, sqrt(dot_product(point, point)) norm2(point), ! Note that GNU gfortran `norm2()` respects robustness, while Intel ifort does not.
re, z
im), abs(z),
re, z
! norm2([z%re, z%im]), Note that GNU gfortran `norm2()` respects robustness, while Intel ifort does not.
24! Compute the distance of a point with respect to a reference in arbitrary dimensions without undue overflow/underflow.
37! Compute the asymmetric matrix of (squared) distances of a set of points from a set of reference points with or without undue overflow/underflow.
Normal Priority: A benchmark comparison with the equivalent compiler implementations would be informative.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python,
R), please also ask the end users to cite this original ParaMonte library.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
Amir Shahmoradi, September 1, 2017, 12:00 AM, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin
Definition at line 484 of file pm_distanceEuclid.F90. | {"url":"https://www.cdslab.org/paramonte/fortran/latest/interfacepm__distanceEuclid_1_1getDisEuclid.html","timestamp":"2024-11-02T15:45:20Z","content_type":"application/xhtml+xml","content_length":"126860","record_id":"<urn:uuid:7d1179fd-ed13-4f48-ba8b-56e2a94fba10>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00238.warc.gz"} |
How can I find the intersection points from 2 circles in 2 sketches (w scripting)
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How can I find the intersection points from 2 circles in 2 sketches (w scripting)
I want to find the intersection points of 2 circles
I tried :
var i = 1;
var j = 2;
// sketches Id
var test_i = id + ("sketch_" ~ i);
var test_j = id + ("sketch_" ~ j);
// circles Id
var circle_i = "circle_" ~ i;
var circle_j = "circle_" ~ j;
// circle edges
var edges_i = sketchEntityQuery(test_i, EntityType.EDGE, circle_i);
var edges_j = sketchEntityQuery(test_j, EntityType.EDGE, circle_j);
const curves_i = evaluateQuery(context, edges_i);
const curves_j = evaluateQuery(context, edges_j);
var intersectionPointsQuery = qIntersection(edges_i, edges_j);
for (var intersectionPoints in intersectionPointsQuery )
println("intersectionPoints = " ~ intersectionPoints);
It prints
[ { queryType : TRANSIENT , transientId : JMB } ]
[ { queryType : TRANSIENT , transientId : JRB } ]
intersectionPoints = { "key" : "queryType" , "value" : QueryType : "INTERSECTION" }
intersectionPoints = { "key" : "subqueries" , "value" : [ Query : { "entityType" : EntityType : "EDGE" , "historyType" : "CREATION" , "operationId" : Id : [ "FovTflfcvumxF3Q_0" , "sketch_1" ] , "queryType" : "SKETCH_ENTITY" , "sketchEntityId" : "circle_1" } , Query : { "entityType" : EntityType : "EDGE" , "historyType" : "CREATION" , "operationId" : Id : [ "FovTflfcvumxF3Q_0" , "sketch_2" ] , "queryType" : "SKETCH_ENTITY" , "sketchEntityId" : "circle_2" } ] }
How can I get the actual points from this query ?
_anton Member, Onshape Employees Posts: 395
qIntersection is a set intersection between two queries (that is, qIntersection(A, B ) is a query for entities that match both queries A and B ), not a geometric evaluation of where the entities
evDistance is the typical way to do this.
In fact the more I use onshape scripting, the more I get lost.
I would never have thought to use evDistance to find the intersection of my two circles, but it works.
I'm really not in the minds of those who designed the featurescript code, too bad..
Thanks a lot for your help. I would never have found it alone.
evDistance will find an intersection between two circles, but it will not find multiple intersections. We do have functionality for that, but it is not currently exposed to featurescript. If you
want to find multiple intersections, you can call opExtractWires on one of the circles, then you can use opTrimCurve to trim it with another one and evaluate the end points of the trimmed curve.
This will fail if the two circles are tangent.
If you'd like an api that finds all curve intersections, please make an improvement request on the forum.
_anton Member, Onshape Employees Posts: 395
There's definitely a zen to thinking about queries (and we could probably do better at educating users). A query encodes the intent to find some entities, and can be combined and filtered in
various ways; an ev* call evaluates a query and then does some computation on the result; an op* call will typically cause a geometric change.
A query is like a street address - you never know if it's unique, where it is, nor whether the place even exists, until you look it up or go there in person. You can imagine a query like "every
address along road X", then taking its set intersection with "every address along road Y" to get the set of houses at the intersection of both roads. But you're still just looking at letters on
paper until you find out which houses/restaurants/parking lots/whatever satisfy that query.
Thanks for your comment.
I rejoiced too quickly
: I actually find the point that interests me, but as the circles intersect at 2 points,
I don't have the second one
Even if it doesn't interest me, there is an element of chance that is not fine....
I will have a look to
opTrimCurve .
Hi , I didn't find the
opTrimCurve operation. I saw the trimTool
function, is it related ?
If it is two circles and you know the circle definitions, then finding the two intersections is simple geometry calcs. Better to do it mathematically than geometrically.
Senior Director, Technical Services, EMEAI
Also, I find the best way to think of a Query is to compare it to an SQL Query. If you are querying information from an SQL database, you have no idea what the results are until the query is run,
so a FeatureScript query is equivalent and the results are known when the query is executed with evaluateQuery.
Senior Director, Technical Services, EMEAI
I had forgotten that the operation is actually called
opMoveCurveBoundary. Neil is correct that doing the math yourself will be faster if you are always doing intersections of simple geometry. | {"url":"https://forum.onshape.com/discussion/22579/how-can-i-find-the-intersection-points-from-2-circles-in-2-sketches-w-scripting","timestamp":"2024-11-02T05:38:28Z","content_type":"text/html","content_length":"276412","record_id":"<urn:uuid:aa8fb8ff-b6e6-4abf-ad66-ef8bcd8bd0f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00629.warc.gz"} |
Lectures on Tensor Categories and Modular Functorssearch
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Lectures on Tensor Categories and Modular Functors
Softcover ISBN: 978-0-8218-2686-7
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Lectures on Tensor Categories and Modular Functors
Softcover ISBN: 978-0-8218-2686-7
Product Code: ULECT/21
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN: 978-1-4704-2168-7
Product Code: ULECT/21.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $52.00
Softcover ISBN: 978-0-8218-2686-7
eBook ISBN: 978-1-4704-2168-7
Product Code: ULECT/21.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $107.20 $81.20
• University Lecture Series
Volume: 21; 2001; 221 pp
MSC: Primary 18; 81; 57; Secondary 17
This book gives an exposition of the relations among the following three topics: monoidal tensor categories (such as a category of representations of a quantum group), 3-dimensional topological
quantum field theory, and 2-dimensional modular functors (which naturally arise in 2-dimensional conformal field theory). The following examples are discussed in detail: the category of
representations of a quantum group at a root of unity and the Wess-Zumino-Witten modular functor.
The idea that these topics are related first appeared in the physics literature in the study of quantum field theory. Pioneering works of Witten and Moore-Seiberg triggered an avalanche of
papers, both physical and mathematical, exploring various aspects of these relations. Upon preparing to lecture on the topic at MIT, however, the authors discovered that the existing literature
was difficult and that there were gaps to fill.
The text is wholly expository and finely succinct. It gathers results, fills existing gaps, and simplifies some proofs. The book makes an important addition to the existing literature on the
topic. It would be suitable as a course text at the advanced-graduate level.
Graduate students and research mathematicians interested in representation theory and mathematical physics
□ Chapters
□ Introduction
□ Chapter 1. Braided tensor categories
□ Chapter 2. Ribbon categories
□ Chapter 3. Modular tensor categories
□ Chapter 4. 3-dimensional topological quantum field theory
□ Chapter 5. Modular functors
□ Chapter 6. Moduli spaces and complex modular functors
□ Chapter 7. Wess-Zumino-Witten model
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Requests
Volume: 21; 2001; 221 pp
MSC: Primary 18; 81; 57; Secondary 17
This book gives an exposition of the relations among the following three topics: monoidal tensor categories (such as a category of representations of a quantum group), 3-dimensional topological
quantum field theory, and 2-dimensional modular functors (which naturally arise in 2-dimensional conformal field theory). The following examples are discussed in detail: the category of
representations of a quantum group at a root of unity and the Wess-Zumino-Witten modular functor.
The idea that these topics are related first appeared in the physics literature in the study of quantum field theory. Pioneering works of Witten and Moore-Seiberg triggered an avalanche of papers,
both physical and mathematical, exploring various aspects of these relations. Upon preparing to lecture on the topic at MIT, however, the authors discovered that the existing literature was difficult
and that there were gaps to fill.
The text is wholly expository and finely succinct. It gathers results, fills existing gaps, and simplifies some proofs. The book makes an important addition to the existing literature on the topic.
It would be suitable as a course text at the advanced-graduate level.
Graduate students and research mathematicians interested in representation theory and mathematical physics
• Chapters
• Introduction
• Chapter 1. Braided tensor categories
• Chapter 2. Ribbon categories
• Chapter 3. Modular tensor categories
• Chapter 4. 3-dimensional topological quantum field theory
• Chapter 5. Modular functors
• Chapter 6. Moduli spaces and complex modular functors
• Chapter 7. Wess-Zumino-Witten model
Permission – for use of book, eBook, or Journal content
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