on 14-Jan-2017 (Sat)

As you can imagine, the smaller you make the area segments, the closer this gets to the true mass, since your approximation of constant sigma is more accurate for smaller segments. If you let the segment area dA approach zero and N approach infinity, the summation becomes integration, and you have Mass = \(\int_{S}\sigma(x, y)dA\) : This is the area integral of the scalar function sigma(x, y) over the surface S.

Functions that manipulate functions are called higher-order functions.

With higher-order functions, we begin to see a more powerful kind of abstraction: some functions express general methods of computation, independent of the particular functions they call.

naming and functions allow us to abstract away a vast amount of complexity

it is only by virtue of the fact that we have an extremely general evaluation procedure for the Python language that small components can be composed into complex processes.

Like local assignment, local def statements only affect the current local frame. These functions are only in scope while sqrt is being evaluated. Consistent with our evaluation procedure, these local def statements don't even get evaluated until sqrt is called.

This discipline of sharing names among nested definitions is called lexical scoping. Critically, the inner functions have access to the names in the environment where they are defined (not where they are called).

We require two extensions to our environment model to enable lexical scoping.

1. Each user-defined function has a parent environment: the environment in which it was defined.
2. When a user-defined function is called, its local frame extends its parent environment.

The sqrt_update function carries with it some data: the value for a referenced in the environment in which it was defined. Because they "enclose" information in this way, locally defined functions are often called closures.

An important feature of lexically scoped programming languages is that locally defined functions maintain their parent environment when they are returned.

We can use higher-order functions to convert a function that takes multiple arguments into a chain of functions that each take a single argument. More specifically, given a function f(x, y) , we can define a function g such that g(x)(y) is equivalent to f(x, y) . Here, g is a higher-order function that takes in a single argument x and returns another function that takes in a single argument y . This transformation is called currying.

we can compute a*b + c*d without having to name the subexpressions a*b or c*d , or the full expression. In Python, we can create function values on the fly using lambda expressions, which evaluate to unnamed functions. A lambda expression evaluates to a function that has a single return expression as its body. Assignment and control statements are not allowed.

 >>> def compose1 ( f , g ): return lambda x : f ( g ( x )) 

We can understand the structure of a lambda expression by constructing a corresponding English sentence:

 lambda x : f(g(x))
"A function that takes x and returns f(g(x))"

The result of a lambda expression is called a lambda function. It has no intrinsic name (and so Python prints <lambda> for the name), but otherwise it behaves like any other function.

compound lambda expressions are notoriously illegible, despite their brevity.

In general, Python style prefers explicit def statements to lambda expressions, but allows them in cases where a simple function is needed as an argument or return value.

Elements with the fewest restrictions are said to have first-class status. Some of the "rights and privileges" of first-class elements are:

1. They may be bound to names.
2. They may be passed as arguments to functions.
3. They may be returned as the results of functions.
4. They may be included in data structures.

Python awards functions full first-class status, and the resulting gain in expressive power is enormous.

A function is called recursive if the body of the function calls the function itself, either directly or indirectly

it is often clearer to think about recursive calls as functional abstractions.

Treating a recursive call as a functional abstraction has been called a recursive leap of faith. We define a function in terms of itself, but simply trust that the simpler cases will work correctly when verifying the correctness of the function.

Recursive functions leverage the rules of evaluating call expressions to bind names to values, often avoiding the nuisance of correctly assigning local names during iteration. For this reason, recursive functions can be easier to define correctly. However, learning to recognize the computational processes evolved by recursive functions certainly requires practice.

When a recursive procedure is divided among two functions that call each other, the functions are said to be mutually recursive.

A function with multiple recursive calls is said to be tree recursive because each call branches into multiple smaller calls, each of which branches into yet smaller calls, just as the branches of a tree become smaller but more numerous as they extend from the trunk.

We can think of a tree-recursive function as exploring different possibilities.

Sander¹⁴ developed a DSSP algorithm to classify SS into 8 fine-grained states. In particular, DSSP assigns 3 types for helix (G for 3₁₀ helix, H for alpha-helix, and I for pi-helix), 2 types for strand (E for beta-strand and B for beta-bridge), and 3 types for coil (T for beta-turn, S for high curvature loop, and L for irregular).

Reading 22 is organized as follows:

Section 2 discusses the scope of financial statement analysis.

Section 3 describes the sources of information used in financial statement analysis, including the primary financial statements.

Section 4 provides a framework for guiding the financial statement analysis process.

Prąd elektryczny w przewodnikach płynie od potencjału wyższego do potencjału niższego. By było to możliwe, w obwodzie zamkniętym musi znajdować się element, który zapewni dostarczenie nośników ładunku z punktów o niższym potencjale do punktów o wyższym potencjale, czyli w kierunku przeciwnym do działającego na nie pola elektrycznego. Wymaga to dostarczenia energii i dzieje się w elementach nazywanych źródłami prądu. Rolę chwilowego źródła energii w obwodzie może pełnić również element inercyjny (mający zdolność gromadzenia energii) – uprzednio naładowany kondensator, albo cewka indukcyjna z energią zgromadzoną w jej polu magnetycznym.

Źródło prądu – urządzenie, które dostarcza energię elektryczną do zasilania innych urządzeń elektrycznych. Źródło prądu może wytwarzać energię elektryczną kosztem innych form energii, np.: • chemicznej (ogniwo chemiczne) • cieplnej (zjawisko Seebecka) • mechanicznej (prądnica) • świetlnej (fotoogniwo) Źródłem prądu nazywa się również elektryczną sieć energetyczną, a także zasilacze pełniące często rolę przetworników prądu sieciowego. Rozróżnia się zasilacze prądu przemiennego (AC – alternating current) i prądu stałego (DC – direct current).

Prąd stały jest to prąd, którego wartość natężenia jest stała w funkcji czasu. W obwodzie elektrony poruszają się w sposób ciągły, w jednym kierunku

Prąd stały płynie zawsze w tym samym kierunku – nie zmienia się biegunowość prądu => stałe natężenie i napięcie prądu. Może być magazynowany, np. w akumulatorze (ogniwo galwaniczne), który po rozładowaniu można ponownie naładować.

Prąd zmienny jest to prąd, którego wartość natężenia jest zmienna w funkcji czasu.

Prąd okresowo zmienny: jest to prąd zmienny, którego zmiany powtarzają się w czasie.

Prąd przemienny to charakterystyczny przypadek prądu elektrycznego okresowo zmiennego, w którym wartości chwilowe podlegają zmianom w powtarzalny, okresowy sposób, z określoną częstotliwością. Wartości chwilowe natężenia prądu przemiennego przyjmują naprzemiennie wartości dodatnie i ujemne. Stosunkowo największe znaczenie praktyczne mają prąd i napięcie o przebiegu sinusoidalnym.

Prąd okresowo przemienny: jest to prąd przemienny, którego zmiany powtarzają się w czasie.

Prąd sinusoidalny: jest to prąd przemienny, którego wartość i kierunek natężenia, zmieniają się jak funkcja sinus (cosinus).

The quantity or quantity demanded variable is the amount of the product that consumers are willing and able to buy at each price level. The quantity sold can be affected by the business through such activities as sales promotion, advertising, and competitive positioning of the product that would take place under the market model of imperfect competition. Under perfect competition, however, total quantity in the market is influenced strictly by price, while non-price factors are not important. Once consumer preferences are established in the market, price determines the quantity demanded by buyers. Together, price and quantity constitute the firm’s demand curve, which becomes the basis for calculating the total, average, and marginal revenue.

The quantity or quantity demanded variable is the amount of the product that consumers are willing and able to buy at each price level.

Total revenue increases with a greater quantity, but the rate of increase in TR (as measured by marginal revenue) declines as quantity increases.

Average revenue and marginal revenue decrease when output increases, with MR falling faster than price and AR. Average revenue is equal to price at each quantity level.

Total revenue increases with a greater quantity, but the rate of increase in TR (as measured by marginal revenue) declines as quantity increases.

Average revenue and marginal revenue decrease when output increases, with MR falling faster than price and AR.

Average revenue is equal to price at each quantity level.

This shows the relationships among the revenue variables.

3.1.2. Factors of Production

Revenue generation occurs when output is sold in the market. However, costs are incurred before revenue generation takes place as the firm purchases resources, or what are commonly known as the factors of production, in order to produce a product or service that will be offered for sale to consumers. Factors of production, the inputs to the production of goods and services, include:

• land, as in the site location of the business;

• labor, which consists of the inputs of skilled and unskilled workers as well as the inputs of firms’ managers;

• capital, which in this context refers to physical capital—such tangible goods as equipment, tools, and buildings. Capital goods are distinguished as inputs to production that are themselves produced goods; and

• materials, which in this context refers to any goods the business buys as inputs to its production process.1

For example, a business that produces solid wood office desks needs to acquire lumber and hardware accessories as raw materials and hire workers to construct and assemble the desks using power tools and equipment. The factors of production are the inputs to the firm’s process of producing and selling a product or service where the goal of the firm is to maximize profit by satisfying the demand of consumers. The types and quantities of resources or factors used in production, their respective prices, and how efficiently they are employed in the production process determine the cost component of the profit equation.

Clearly, in order to produce output, the firm needs to employ factors of production. While firms may use many different types of labor, capital, raw materials, and land, an analyst may find it more convenient to limit attention to a more simplified process in which only the two factors, capital and labor, are employed. The relationship between the flow of output and the two factors of production is called the production function , and it is represented generally as:

Equation (5) 

Q = f (K, L)

where Q is the quantity of output, K is capital, and L is labor. The inputs are subject to the constraint that K ≥ 0 and L ≥ 0. A more general production function is stated as:

Equation (6) 

Q = f (x₁, x₂, … x_n)

where x_i represents the quantity of the ith input subject to x_i ≥ 0 for n number of different inputs. Exhibit 9illustrates the shape of a typical input–output relationship using labor (L) as the only variable input (all other input factors are held constant). The production function has three distinct regions where both the direction of change and the rate of change in total product (TP or Q, quantity of output) vary as production changes. Regions 1 and 2 have positive changes in TP as labor is added, but the change turns negative in Region 3. Moreover, in Region 1 (L₀ – L₁), TP is increasing at an increasing rate, typically because specialization allows laborers to become increasingly productive. In Region 2, however, (L₁ – L₂), TP is in

...

Revenue generation occurs when output is sold in the market. However, costs are incurred before revenue generation takes place as the firm purchases the factors of production, in order to produce a product that will be offered for sale to consumers.

Factors of production, the inputs to the production of goods and services, include:

• land, as in the site location of the business;

• labor, which consists of the inputs of skilled and unskilled workers as well as the inputs of firms’ managers;

• capital, which in this context refers to physical capital—such tangible goods as equipment, tools, and buildings. Capital goods are distinguished as inputs to production that are themselves produced goods; and

• materials, which in this context refers to any goods the business buys as inputs to its production process.

The types and quantities of resources used in production, their prices, and how efficiently they are employed in the production process determine the cost component of the profit equation.

in order to produce output, the firm needs to employ factors of production.

While firms may use many different types of labor, capital, raw materials, and land, an analyst may find it more convenient to limit attention to a more simplified process in which only the two factors, capital and labor, are employed.

The relationship between the flow of output and the two factors of production is called the production function , and it is represented generally as:

Equation (5) 

Q = f (K, L)

where Q is the quantity of output, K is capital, and L is labor.

The inputs in the production function are subject to the constraint that K ≥ 0 and L ≥ 0.

A more general production function is stated as:

Equation (6) 

Q = f (x₁, x₂, … x_n)

where x_i represents the quantity of the ith input subject to x_i ≥ 0 for n number of different inputs.

Capital goods are distinguished as inputs to production that are themselves produced goods; and

This image illustrates the shape of a typical input–output relationship using labor (L) as the only variable input (all other input factors are held constant). The production function has three distinct regions where both the direction of change and the rate of change in total product (TP or Q, quantity of output) vary as production changes. Regions 1 and 2 have positive changes in TP as labor is added, but the change turns negative in Region 3. Moreover, in Region 1 (L₀ – L₁), TP is increasing at an increasing rate, typically because specialization allows laborers to become increasingly productive. In Region 2, however, (L₁ – L₂), TP is increasing at a decreasing rate because capital is fixed, and labor experiences diminishing marginal returns. The firm would want to avoid Region 3 if at all possible because total product or quantity would be declining rather than increasing with additional input: There is so little capital per unit of labor that additional laborers would possibly “get in each other’s way”. Point A is where TP is maximized.

The production function has three distinct regions where both the direction of change and the rate of change in total product (TP or Q, quantity of output) vary as production changes. Regions 1 and 2 have positive changes in TP as labor is added, but the change turns negative in Region 3.

The firm would want to avoid Region 3 because total product or quantity would be declining rather than increasing with additional input:

There is so little capital per unit of labor that additional laborers would possibly “get in each other’s way”.

Point A is where TP is maximized.