The Language of Second Order Arithmetic.
... sequence of real numbers could have many limits, and it will be easiest to pick out a natural one, namely the lim sup. We will define a set T to consist of those pairs (n, k) such that Sn is within 2−k of the lim sup of the sequence. More precisely, (n, k) ∈ T if there are only finitely many m such ...
... sequence of real numbers could have many limits, and it will be easiest to pick out a natural one, namely the lim sup. We will define a set T to consist of those pairs (n, k) such that Sn is within 2−k of the lim sup of the sequence. More precisely, (n, k) ∈ T if there are only finitely many m such ...
Finding the Equation of a Tangent Line
... point on the line, denoted by (x1, y1), and the slope of the line, denoted by m, to calculate the slope-intercept formula for the line. ...
... point on the line, denoted by (x1, y1), and the slope of the line, denoted by m, to calculate the slope-intercept formula for the line. ...
Automata, tableaus and a reduction theorem for fixpoint
... Definition 2.6 We say the decomposability property holds on a fixpoint algebra M over a signature when, for any f 2 with (f ) 6= 0, for any d 2 M , any e1 , . . . , e(f ) 2 M if d fM (e1 ; ; e(f ) ) then there exists d1 , . . . , d(f ) 2 M such that, for any i, di ei and d = fM (d1 ...
... Definition 2.6 We say the decomposability property holds on a fixpoint algebra M over a signature when, for any f 2 with (f ) 6= 0, for any d 2 M , any e1 , . . . , e(f ) 2 M if d fM (e1 ; ; e(f ) ) then there exists d1 , . . . , d(f ) 2 M such that, for any i, di ei and d = fM (d1 ...
HERE
... function is the slope of its graph, and so it must be negative when x is very small and positive when x is large (or vice-versa if the first term of the polynomial is negative). This means that the derivative of an even polynomial function cannot be even. A similar argument shows that the derivative ...
... function is the slope of its graph, and so it must be negative when x is very small and positive when x is large (or vice-versa if the first term of the polynomial is negative). This means that the derivative of an even polynomial function cannot be even. A similar argument shows that the derivative ...
Pre calculus Topics
... Linear approximation. Interpretations and properties of definite integrals. Instantaneous rate of change Area between curves Average rate of change. ...
... Linear approximation. Interpretations and properties of definite integrals. Instantaneous rate of change Area between curves Average rate of change. ...
Full Text (PDF format)
... the invariants [S • (g)]g defines a map of algebras ϕD : [S • (g)]g → [U (g)]g . 1.1.1. M. Kontsevich deduced from his theorem on cup-products on the tangent cohomology [K] the following generalization of the Duflo theorem. Theorem. There exists a canonical map ϕ˜D : H • (g; S • (g)) → H • (g; U (g)) ...
... the invariants [S • (g)]g defines a map of algebras ϕD : [S • (g)]g → [U (g)]g . 1.1.1. M. Kontsevich deduced from his theorem on cup-products on the tangent cohomology [K] the following generalization of the Duflo theorem. Theorem. There exists a canonical map ϕ˜D : H • (g; S • (g)) → H • (g; U (g)) ...
Solution - Math TAMU
... 1. On what interval(s) is the function x2 ex decreasing? Solution. By the product rule, the derivative equals x2 ex + 2xex or x(x + 2)ex . Since ex is always positive, the derivative is negative only on the interval (−2, 0). This is the interval on which the function is decreasing. 2. Suppose f (x) ...
... 1. On what interval(s) is the function x2 ex decreasing? Solution. By the product rule, the derivative equals x2 ex + 2xex or x(x + 2)ex . Since ex is always positive, the derivative is negative only on the interval (−2, 0). This is the interval on which the function is decreasing. 2. Suppose f (x) ...
Math 121. Construction of a regular 17-gon 1
... 1/4). We will use this positivity to identity ι(x1 ) as an explicit quadratic irrationality over Q inside R below. (Recall that [K1 : Q] = [K1 : K0 ] = 2, so the field K1 = Q(x1 ) is real quadratic.) Likewise, since G2 = {1, σ 4 , σ 8 , σ 12 } = {1, −4, −1, 4}, we have x2 = ζ + σ 4 (ζ) + σ 8 (ζ) + σ ...
... 1/4). We will use this positivity to identity ι(x1 ) as an explicit quadratic irrationality over Q inside R below. (Recall that [K1 : Q] = [K1 : K0 ] = 2, so the field K1 = Q(x1 ) is real quadratic.) Likewise, since G2 = {1, σ 4 , σ 8 , σ 12 } = {1, −4, −1, 4}, we have x2 = ζ + σ 4 (ζ) + σ 8 (ζ) + σ ...
Euler`s Formula and the Fundamental Theorem of Algebra
... is allowed to take on some, but not all, values of B. It is legitimate, for example, to write f : R → R with f (x) := x2 , even though f (x) does not ever take on any of the negative values. It would be legitimate (and more informative) to speak of f : R → [0, ∞). If we set the domain to be C instea ...
... is allowed to take on some, but not all, values of B. It is legitimate, for example, to write f : R → R with f (x) := x2 , even though f (x) does not ever take on any of the negative values. It would be legitimate (and more informative) to speak of f : R → [0, ∞). If we set the domain to be C instea ...
HALL-LITTLEWOOD POLYNOMIALS, ALCOVE WALKS, AND
... [10, 23]. On the combinatorial side, we have the Lascoux-Schützenberger formula for the Kostka-Foulkes polynomials in type A [13], but no generalization of this formula to other types is known. Other applications of the type A Hall-Littlewood polynomials that extend to arbitrary type are those rela ...
... [10, 23]. On the combinatorial side, we have the Lascoux-Schützenberger formula for the Kostka-Foulkes polynomials in type A [13], but no generalization of this formula to other types is known. Other applications of the type A Hall-Littlewood polynomials that extend to arbitrary type are those rela ...
Basic concept of differential and integral calculus
... If relation between two variables is expressed via third variable. The third variable is called parameter. More precisely a relation expressed between two variables x and y in the form x=f(t),y=g(t) is said to be parametric form with t is a parameter. ...
... If relation between two variables is expressed via third variable. The third variable is called parameter. More precisely a relation expressed between two variables x and y in the form x=f(t),y=g(t) is said to be parametric form with t is a parameter. ...
Automatic differentiation
In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation or computational differentiation, is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.Automatic differentiation is not: Symbolic differentiation, nor Numerical differentiation (the method of finite differences).These classical methods run into problems: symbolic differentiation leads to inefficient code (unless carefully done) and faces the difficulty of converting a computer program into a single expression, while numerical differentiation can introduce round-off errors in the discretization process and cancellation. Both classical methods have problems with calculating higher derivatives, where the complexity and errors increase. Finally, both classical methods are slow at computing the partial derivatives of a function with respect to many inputs, as is needed for gradient-based optimization algorithms. Automatic differentiation solves all of these problems.