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Sets and Propositions
Compound propositions are unary relations (negation), dyadic relations (see the following table) or polyadic relations consisting of the operators. Basis of the induction: The validity of the statement A(n) is shown for some initial value (usually= 0 or n= 1).
Number Systems and their Arithmetic
Determining one or all of the values of the variables for which a given equation is true is called solving the equation. Intersection of the unit circle In the figure, the unit circle|z|= 1 is divided into 6 segments by the solutions of the equation.
Combinatorial Analysis
If k has been chosen from different elements, where 1 ≤k ≤ n and the order is not taken into account, then one speaks of a combination (without repetition). If some of the n different elements occur several times, then one speaks of combination with repetition.
Sequences and Series
The limits of the sequence{sn} of partial sums is called the sum of the series (provided that it exists): lim. The series of functions whose terms are of the form fn(x) = an(x−x0)n, n∈IN0, are called power series centered on x0.
Mathematics of Finance
For C = 0 the investment corresponds to the given conventional interest rate p, for C > 0 its maturity return is higher. The internal rate of return (return-to-maturity) is the amount for which the net present value of the investment equals zero.
Functions of one Independent Variable
According to the Fundamental Theorem of Algebra, any polynomial of degreen can be represented in the form x−xn−1)(x−xn) product representation The numbers xi are the real or complex zeros of the polynomial. The increase in the function is proportional to the product of the impulse and braking factors.
Functions of one Variable: Differential Calculus
A fractional rational function has poles at the zeros of its denominator, provided that the numerator is nonzero at these points ( Fractions of a Rational Function, p. 50). The differential quotient is the slope of the tangent to the graph of f at the point (x, f(x)). Differentiation by inverse function and logarithmic differentiation is used when the inverse function or the function lnf(x) can be differentiated in an “easier” way than the original functions.
If the functional is twice differentiable on the interval (a, b) and has the point of inflection at xw (the point between the intervals of convexity and concavity), then. If is three times continuously differentiable at (a, b), then forxw sufficient with f(xw) = 0 to be a reflection point is the validity of the relation. The maximum profit per unit is reached at the point where the slopes of the price-response function and the average cost function are equal.
Functions of one Variable: Integral Calculus
The area A located between the interval [a, b] of the x-axis and the graph of the bounded function f can be approximately calculated by summations of the form. Let the functionf have a pole at the pointx=b, and letf be bounded and integrable on any interval [a, b−ε] such that 0< ε < b−a. If ϕand ψ are two differentiable functions then force ≤t≤d and if f(x, t) is partially differentiable with respect to having a continuous partial derivative in the domain defined by ϕ(t) < x < ψ(t ), c ≤t ≤ d , then the parameter integral with limitsϕ(t) andψ(t) is differentiable with respect to forc≤t≤d, where.
This is (from the consumer's point of view) a measure of the profitability of the purchase at the equilibrium point (but not before). It is (from the producer's point of view) a measure of the profitability of sales at the equilibrium point (but not before).
Differential Equations
A fails to be diagonalizable; let V be the matrix describing the similarity transformation from the matrix A to the Jordan normal form
A particular solution can be obtained by varying the constants or a trial solution (table p. 95), where in all components all parts of r(x) must be considered.
Difference Equations
The general solution of a linear inhomogeneous differential equation (∗) is the sum of the general solution of the corresponding homogeneous differential equation ∆2y+a∆y+by= 0 and any particular solution. To present the general solution of the homogeneous differential equation associated with (∗), it is necessary to distinguish between three cases where C1,C2 are arbitrary real constants. The general solution of the inhomogeneous equation is the sum of the general solution of the homogeneous equation and the particular solution of the inhomogeneous equation.
The nth order difference equation (1) has exactly one solution k =f(k) if the initial values for the successive values are given. Ifyk, is a particular solution of the inhomogeneous linear difference equation (1) and yk,h is the general solution of the homogeneous linear difference equation (2), then for the general solution of the inhomogeneous linear difference equation (1) the representation yk =yk,h+yk,s is valid. To find a particular solution of the inhomogeneous difference equation (1), in many cases the ansatz method is successful, where the ansatz function is chosen in such a way that it corresponds to the right-hand side with respect to the structure (the second-order difference equation , p. 100).
Differential Calculus for Functions of Several Variables
A point x is called an interior point of the set M ⊂IRn if there exists a neighborhood Uε(x) in M. A number a∈IR is called the limit of the function f at the point x0 if for any point series{xk} converging tox0 such that xk =x0 andxk ∈Df the relation lim. For linearly homogeneous functions, a proportional increase of variables causes a proportional increase of the function value.
If the partial drainages fxixj and fxjxi are continuous around the point x, then the relations apply: fxixj(x) =fxjxi(x). With the transformation xi = xi−N1[xi], the system of normal equations can be simplified, since in this case [xi] = 0. The change in the asking price under the change ∆xi of the i-th input (while keeping the remaining inputs unchanged) can be estimated by means of partial difference ∂Pcall.
Linear Algebra
The sign of the determinant changes if two rows or two columns of the corresponding matrix are exchanged. For later calculation of the value of the extracted variables, the extracted row is "marked". If the considered system of equations is consistent, then by renumbering the rows and columns we can achieve that first a11 and after k steps the element ˜a1,k+1 (i.e. the diagonal elements) can be chosen as the pivot. elements.
In this case, after the process of Gaussian elimination, the system of equations has the form. The multiple of the zeroλ of the characteristic polynomial is denoted as the algebraic multiple of the eigenvalueλ. A is the matrix of buyers' fluctuation and s a non-trivial solution of the linear homogeneous system (A −I)s=0mets1+.
Linear Programming. Transportation Problem
If one assumes a simplex table using the following algorithm, one either obtains an optimal simplex table or recognizes that the programming problem is unsolvable. Starting from a dual simplex table using the following algorithm, one either obtains an optimal simplex table or recognizes that the underlying programming problem is unsolvable. By dividing rowp by bpq and generating zeros in column xNq (except for position p), a new dual simplex table is obtained by Gaussian elimination.
By exchanging the variables xBp ⇐⇒ xNq using the exchange method, a new dual simplex table is obtained. Starting from the normal form of a linear programming problem with the property a ≥ 0, the following algorithm either leads to a simplex table or shows the unsolvability of the linear programming problem. The iterations of the transport algorithm can be represented in the following table form by placing only the variables nexij ∈ X (boxed) with (i, j)∈JS(X) and only the variableswijwith (i, j)∈/JS(X) .
Descriptive Statistics
Wi=pi·qi – the value of goodi n. piqi – the total value of the goods basketW. piτ,pit – price of goodii in the base period and in the given period, respectively basic and actual price, respectively) qiτ bzw.qit – the amount of goodiin the basic and in the given. base and actual quantity, respectively). Paasche's index describes the average relative change of a component (price or quantity) using weights (quantities or prices) during the given period. Laspeyres' index describes the average relative change of a component (price or quantity) using weights (quantities or prices) during the base period.
This method is used to estimate the linear trendT(t) =a+and the quadratic trendT(t) =a+bt+ct2 respectively (see p. 112). These methods serve to estimate the trend component using observation valuesy1. Impact of the smoothing factorα αlarge αsmall take into account “older” values little strong take into account “newer” values very little.
Calculus of Probability
Ai=Ω and Ai∩Aj =∅ (i=j) (i.e. in the result of a trial exactly one of the events Ai always occurs). Axiom 3': The probability of the event that exactly one of the pairwise disjoint events A1, A2 occurs. A random variable X is called discrete (discretely distributed; see below) if its distribution function FX is a step function (i.e. piecewise constant);.
If X1,X2 are independent continuous random variables with densities fX1 and fX2, then Y =X1+X2 is a continuous random variable with density. If X1 and X2 are χ2-distributed with m+n degrees of freedom, respectively, then sumX1+X2 is χ2-distributed with m+n degrees of freedom. If X1,X2 are independent continuous random variables with densities fX1 and fX2, respectively, then Y =X1·X2 is a continuous random variable with density.
Inductive Statistics
Task: In order to assess the accuracy of the estimate of the unknown parameter θ of the distribution, it is necessary to construct intervals, so-called confidence intervals, that cover θ with a high probability. Task: Statistical tests serve to test these statistical hypotheses about (completely or partially) unknown distributions F with the help of appropriate samples. In this case, the distribution of T must be known if H0 is true.) 3. Selection of the critical region K* (part of the range of the test statistic T must be as large as possible, so that the probability p* for the case that T takes the values from K ∗ is not greater than the significance level α(0<.
Index
