The Epsilon Calculus
... 3. If f is a function symbol with ar( f ) = 0, then f is a semiterm. 4. If f is a function symbol with ar( f ) = n > 0, and t1, . . . , tn are semi-terms, then f (t1, . . . ,tn) is a semi-term. 5. If P is a predicate symbol with ar(P) = n > 0, and t1, . . . , tn are terms, then P(t1, . . . ,tn) is a ...
... 3. If f is a function symbol with ar( f ) = 0, then f is a semiterm. 4. If f is a function symbol with ar( f ) = n > 0, and t1, . . . , tn are semi-terms, then f (t1, . . . ,tn) is a semi-term. 5. If P is a predicate symbol with ar(P) = n > 0, and t1, . . . , tn are terms, then P(t1, . . . ,tn) is a ...
THE COMPLEX EXPONENTIAL FUNCTION
... Expanding on the left and equating real and imaginary parts, leads to trigonometric identities which can be used to express cos nθ and sin nθ as a sum of terms of the form (cos θ)j (sin θ)k . For example with n = 2 one gets: (cos θ + i sin θ)2 = cos2 θ − sin2 θ + i 2 sin θ cos θ = cos 2θ + i sin 2θ. ...
... Expanding on the left and equating real and imaginary parts, leads to trigonometric identities which can be used to express cos nθ and sin nθ as a sum of terms of the form (cos θ)j (sin θ)k . For example with n = 2 one gets: (cos θ + i sin θ)2 = cos2 θ − sin2 θ + i 2 sin θ cos θ = cos 2θ + i sin 2θ. ...
A(x) - Teaching-WIKI
... Propositional logic is not expressive enough – Suppose we want to capture the knowledge that Anyone standing in the rain will get wet. and then use this knowledge. For example, suppose we also learn that Jan is standing in the rain. – We'd like to conclude that Jan will get wet. But each of these se ...
... Propositional logic is not expressive enough – Suppose we want to capture the knowledge that Anyone standing in the rain will get wet. and then use this knowledge. For example, suppose we also learn that Jan is standing in the rain. – We'd like to conclude that Jan will get wet. But each of these se ...
Differentiation - Keele Astrophysics Group
... instantaneous velocity (speed plus direction, which is given by the sign of the slope) of the object at that point and time. Suppose we want to work out how fast the object is travelling at any moment in time. Speed (or velocity, if we care about the sign) is defined as the rate of change of distanc ...
... instantaneous velocity (speed plus direction, which is given by the sign of the slope) of the object at that point and time. Suppose we want to work out how fast the object is travelling at any moment in time. Speed (or velocity, if we care about the sign) is defined as the rate of change of distanc ...
Stable Models and Circumscription
... CIRC[F ] whenever F is a conjunction such that every conjunctive term • does not contain implication, or • is the universal closure of a formula G → H such that G and H do not contain implication. Remark 2 The equivalence of SMpq [F ] to (6) in Example 1 can be established also in another way, witho ...
... CIRC[F ] whenever F is a conjunction such that every conjunctive term • does not contain implication, or • is the universal closure of a formula G → H such that G and H do not contain implication. Remark 2 The equivalence of SMpq [F ] to (6) in Example 1 can be established also in another way, witho ...
Some Systems of Second Order Arithmetic and Their Use Harvey
... and relative recursivity. In RCA, we can define and prove the basic facts about coding. These include codes for finite sequences of natural numbers, for functions as sets, for finite and infinite sequences of sets and functions, for if-structures, and for partial recursive functions and recursively ...
... and relative recursivity. In RCA, we can define and prove the basic facts about coding. These include codes for finite sequences of natural numbers, for functions as sets, for finite and infinite sequences of sets and functions, for if-structures, and for partial recursive functions and recursively ...
The inverse of secant and trigonometric substitutions
... Doing the substitution this way, there is no difference between using the substitution t = arcsec2 x and t = sgn x arcsec x until the time the integration is done and one is ready to write the result in terms of x. There are other methods for calculating integrals of rational expressions of x and of ...
... Doing the substitution this way, there is no difference between using the substitution t = arcsec2 x and t = sgn x arcsec x until the time the integration is done and one is ready to write the result in terms of x. There are other methods for calculating integrals of rational expressions of x and of ...
U.S. NAVAL ACADEMY COMPUTER SCIENCE DEPARTMENT TECHNICAL REPORT Algorithmic Reformulation of Polynomial Problems
... equations and a host of other problems. In fact, humans aren’t very good at these problems, while computer algebra programs have proven to be remarkably effective. Why approach a mathematical problem like a person? Why not do it right? This paper considers a very fundamental problem — quantifier eli ...
... equations and a host of other problems. In fact, humans aren’t very good at these problems, while computer algebra programs have proven to be remarkably effective. Why approach a mathematical problem like a person? Why not do it right? This paper considers a very fundamental problem — quantifier eli ...
MTH4100 Calculus I - School of Mathematical Sciences
... Geometrical interpretation of this theorem: Any horizontal line crossing the y-axis between f (a) and f (b) will cross the curve y = f (x) at least once over the interval [a, b]. Continuity is essential: if f is discontinuous at any point of the interval, then the function may “jump” and miss some v ...
... Geometrical interpretation of this theorem: Any horizontal line crossing the y-axis between f (a) and f (b) will cross the curve y = f (x) at least once over the interval [a, b]. Continuity is essential: if f is discontinuous at any point of the interval, then the function may “jump” and miss some v ...
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.