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Flashcard 1732481518860

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#differential-equations
Question
[...] contain unknown multivariable functions and their partial derivatives.
Answer
partial differential equation


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In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (A special case is ordinary differential equations (ODEs), which deal with functi

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Partial differential equation - Wikipedia
Wikipedia, the free encyclopedia Jump to: navigation, search [imagelink] A visualisation of a solution to the two-dimensional heat equation with temperature represented by the third dimension <span>In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. PDEs can be used to describe







Flashcard 1736023608588

Tags
#stochastics
Question
The homogeneous Poisson process defined on the real line is called [...].
Answer
the stationary Poisson process


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The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.

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Stochastic process - Wikipedia
constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] <span>The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} ,







Flashcard 1738586590476

Tags
#measure-theory #stochastics
Question
Radon, Fréchet, and others generalized the new real analysis to [...of...on...]
Answer
integrals of real-valued functions on arbitrary spaces


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What Kolmogorov did was to say that the new real analysis that had started with the PhD thesis of Henri Lebesgue (1902) and had been rapidly generalized to integrals of real-valued functions on arbitrary spaces by Radon, Fr´echet, and others (called Lebesgue integration or abstract integration) should also be used in probability theory

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Flashcard 1759103814924

Tags
#hilbert-space
Question
Hilbert spaces arise naturally and frequently in mathematics and physics, typically as [...spaces...]
Answer
infinite-dimensional function spaces.

Think Fourier analysis


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Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces.

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Hilbert space - Wikipedia
n abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. <span>Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tool







opening and closing tags, which are <?php ?>

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Flashcard 1760858606860

Tags
#calculus
Question
The total derivative is the [...description...]
Answer
limiting ratio of ∆ƒ/∆t


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The total derivative of a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value, taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.

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Total derivative - Wikipedia
integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the







Flashcard 1760860179724

Tags
#calculus
Question
The total derivative taking into account the exogenous argument's [...] to the function
Answer
direct and indirect effects


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a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value, taking into account the exogenous argument's <span>direct effect as well as indirect effects via the other arguments of the function. <span><body><html>

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Total derivative - Wikipedia
integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the







Flashcard 1760861752588

Tags
#calculus
Question
The [...] of a function of several variables, e.g., , , , with respect to an exogenous argument is defined as
Answer
total derivative


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The total derivative of a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous ar

Original toplevel document

Total derivative - Wikipedia
integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the







Flashcard 1760880102668

Tags
#calculus
Question
Limits describe the value of a function at a certain input in terms of [...].
Answer
its values at nearby inputs


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In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a nearby input. They capture small-scale behavior in the context of the real number system. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals ge

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Calculus - Wikipedia
ion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals. <span>In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a nearby input. They capture small-scale behavior in the context of the real number system. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits were the first way to provide rigorous foundations for calculus, and for this reason they are the standard approach. Differential calculus[edit source] Main article: Differential calculus [imagelink] Tangent line at (x, f(x)). The derivative f′(x) of a curve at a point is the slope (rise ov







Flashcard 1760886656268

Tags
#calculus
Question

The derivative is defined as [...formula...]

Answer


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The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:

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Calculus - Wikipedia
e behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. <span>The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero: lim h → 0 f ( a + h ) − f ( a ) h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference







Flashcard 1760891374860

Tags
#calculus
Question
derivative defined by limit considers [...] and extracts a consistent value for the exact point
Answer
the behavior of f at nearby inputs


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derivative defined by limit considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero

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Calculus - Wikipedia
e behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. <span>The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero: lim h → 0 f ( a + h ) − f ( a ) h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference







the cap rate equation is: Cap Rate = Net Operating Income (NOI) /Value

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To determine the value of the property, we just switch around the formula like this: Value = NOI Cap /Rate

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The type of property you are buying (A, B, C, or D) will determine what cap rate you use.

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A simple definition of cap rate is the rate of return you expect to get on your investment.

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If you were buying a property that was newer and in a good area, this would be a less risky property to own, so you would expect to get a lower return.

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If you’re interested in a property in a tough area and it had a lot of deferred maintenance, you should expect to get a higher return.

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Properties are categorized with letter grades. A’s are good and D’s are tough.

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A properties were usually built within the last 10 years.

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Your biggest competition in A properties is the single-family home market.

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B properties were built within the last 20 years. Tenants are a mix of white-collar and blue-collar workers. This is where you’ll start to see a little deferred maintenance on the property if it has not been taken care of properly.

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Many white- collar workers live here.

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C properties were built within the last 30 years. Units are filled mostly with blue-collar workers and tenants with Section 8 (subsidized housing) or other housing assistance.

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one of my criteria for Section 8 tenants is that they have a job. This shows me they are responsible. Having a job also means they will make their part of the rent payment, and the government will pay the rest.

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D properties were built more than 30 years ago. They are usually in bad areas, filled with very bad tenants. You may findaDproperty inaCorBarea;ifthis is the case, you can probably reposition that property toaCoraB.

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If the D property is in a lousy area, avoid it.

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you can make just as much money buying better quality properties in better areas.

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The rule of thumb for cap rates for each of these property types is as follows: A: 6–7 B: 8–9 C: 10–11 D: 12+

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As the cap rate lowers, the property becomes more expensive.

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Analyzing Your First Deal

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It’s a 75-unit building.

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the income is $772,000 a year and expenses run $370,000 a year.

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Another important rule of thumb to determine value. It deals with expenses. Expenses on a 2-unit to 20-unit property will run about 35 to 40 percent of income. Expenses on larger properties will run about

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50 percent of income. These numbers assume that tenants are paying the utilities.

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do a quick scan of the expenses: Are they at least 50 percent of the income? = 48% Expense Ratio $370,000 in Expenses $772,000 in Income This one’s a little low at 48 percent. If you want to be a conservative investor, then assume expenses will be a little higher, as in 50 percent.

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You’ll often see the expense figure come in much lower than the rule of thumb. When this happens, do not just assume it’s a great deal and go with the lower numbers. Instead, increase the figure to 50 percent and then do your calculations.

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when the figures come in lower than the rule of thumb, it means one of three things—all of them bad: 1. The seller is not doing regular repairs. 2. The seller doesn’t know the true expenses. 3. The seller is lying.

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How to Analyze a Property for Profit 117 50 percent of income. These numbers assume that tenants are paying the utilities. Simply do a quick scan of the expenses: Are they at least 50 percent of the income? = 48% Expense Ratio $370,000 in Expenses $772,000 in Income This one’s a little low at 48 percent. If you want to be a conservative investor, then assume expenses will be a little higher, as in 50 percent. You’ll often see the expense figure come in much lower than the rule of thumb. When this happens, do not just assume it’s a great deal and go with the lower numbers. Instead, increase the figure to 50 percent and then do your calculations. Why? Because when the figures come in lower than the rule of thumb, it means one of three things—all of them bad: 1. The seller is not doing regular repairs. 2. The seller doesn’t know the true expenses. 3. The seller is lying. It’s usually explanation number one, that the seller has been deferring the repair and maintenance.

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If you’re going to be successful at repositioning such a property, you’ll have to spend more to make those repairs the first year and catch up.

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It may not be one major repair, but a lot of little ones. Still, they can really add up and deplete your cash flow.

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It could be that the seller is a lousy recordkeeper, and actually does not know the true expense picture. Believe it or not, many investors don’t even know whether they’re making a profit.

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Then there are the liars. They know the property value is deter- mined by the income and expenses. If they can show lower expenses, some of these desperate owners will do it

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The seller may even be adamant that the property is running at the lower expense level. In this situation, ask for copies of all previous invoices. If he says he doesn’t have them, ask for permission to contact the vendors to get copies of invoices for the last two years.

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You could even ask to see last year’s bank reconciliation to verify who was paid and when. The bottom line is you know the property can’t be run that cheaply. Do your numbers based on the higher expenses, and make any offers based on those more conservative numbers.

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#calculus

In a formulation of the calculus based on limits, the notation is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, dx , is not a number, and is not being multiplied by f(x) , although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential dx indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator.

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Calculus - Wikipedia
t;the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. <span>In a formulation of the calculus based on limits, the notation ∫ a b ⋯ d x {\displaystyle \int _{a}^{b}\cdots \,dx} is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, dx, is not a number, and is not being multiplied by f(x), although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. The indefinite integral, or antiderivative, is written: ∫ f ( x ) d x . {\di




#calculus

The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration.

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Calculus - Wikipedia
∫ 2 x d x = x 2 + C . {\displaystyle \int 2x\,dx=x^{2}+C.} <span>The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration. Fundamental theorem[edit source] Main article: Fundamental theorem of calculus The fundamental theorem of calculus states that differentiation and integration are inverse operatio




#calculus
The definite integral is an operator that inputs a function and outputs a number

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Calculus - Wikipedia
n as the antiderivative, is the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.) <span>The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis. The technical definition of the definite integral involves the limit of a sum of areas of rectangl




#calculus
the antiderivative of a given function is actually a family of functions differing only by a constant.

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Calculus - Wikipedia
∫ f ( x ) d x . {\displaystyle \int f(x)\,dx.} Functions differing by only a constant have the same derivative, and it can be shown that <span>the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function y = x 2 + C, where C is any constant, is y′ = 2x, the antiderivative of the latter given by: ∫ 2 x




#calculus
The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives.

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Calculus - Wikipedia
tyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).} This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, was key to the proliferation of analytic results after their work became known. <span>The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences. Applications[edit




Flashcard 1763254603020

Tags
#calculus
Question
The [...] integral is an operator that inputs a function and outputs a number
Answer
definite


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The definite integral is an operator that inputs a function and outputs a number

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Calculus - Wikipedia
n as the antiderivative, is the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.) <span>The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis. The technical definition of the definite integral involves the limit of a sum of areas of rectangl







#calculus
Formally, the differential dx indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator.

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the area, as an output. The terminating differential, dx , is not a number, and is not being multiplied by f(x) , although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. <span>Formally, the differential dx indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. <span><body><html>

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Calculus - Wikipedia
t;the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. <span>In a formulation of the calculus based on limits, the notation ∫ a b ⋯ d x {\displaystyle \int _{a}^{b}\cdots \,dx} is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, dx, is not a number, and is not being multiplied by f(x), although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. The indefinite integral, or antiderivative, is written: ∫ f ( x ) d x . {\di




Flashcard 1763259321612

Tags
#calculus
Question
Formally, [...] indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator.
Answer
the differential dx


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Formally, the differential dx indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator.

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Calculus - Wikipedia
t;the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. <span>In a formulation of the calculus based on limits, the notation ∫ a b ⋯ d x {\displaystyle \int _{a}^{b}\cdots \,dx} is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, dx, is not a number, and is not being multiplied by f(x), although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. The indefinite integral, or antiderivative, is written: ∫ f ( x ) d x . {\di







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#calculus
Question
Formally, the differential dx indicates [...] and serves as [...]
Answer
the variable over which the function is integrated, a closing bracket for the integration operator.


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Formally, the differential dx indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator.

Original toplevel document

Calculus - Wikipedia
t;the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. <span>In a formulation of the calculus based on limits, the notation ∫ a b ⋯ d x {\displaystyle \int _{a}^{b}\cdots \,dx} is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, dx, is not a number, and is not being multiplied by f(x), although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. The indefinite integral, or antiderivative, is written: ∫ f ( x ) d x . {\di







Flashcard 1763263253772

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#calculus
Question
the antiderivative of a given function is actually [...]
Answer
a family of functions differing only by a constant.


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the antiderivative of a given function is actually a family of functions differing only by a constant.

Original toplevel document

Calculus - Wikipedia
∫ f ( x ) d x . {\displaystyle \int f(x)\,dx.} Functions differing by only a constant have the same derivative, and it can be shown that <span>the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function y = x 2 + C, where C is any constant, is y′ = 2x, the antiderivative of the latter given by: ∫ 2 x







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Question

The unspecified constant C present in the indefinite integral or antiderivative is known as [...].


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The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration.

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Calculus - Wikipedia
∫ 2 x d x = x 2 + C . {\displaystyle \int 2x\,dx=x^{2}+C.} <span>The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration. Fundamental theorem[edit source] Main article: Fundamental theorem of calculus The fundamental theorem of calculus states that differentiation and integration are inverse operatio







Flashcard 1763266399500

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#calculus
Question
The fundamental theorem provides an algebraic method of computing many definite integrals—without [...]—by finding formulas for antiderivatives.
Answer
performing limit processes


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The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives.

Original toplevel document

Calculus - Wikipedia
tyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).} This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, was key to the proliferation of analytic results after their work became known. <span>The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences. Applications[edit