Some finite difference inequalities of the Volterra type.Abstract In this paper we establish some finite difference A finite difference is a mathematical expression of the form f(x + b) − f(x + a). If a finite difference is divided by b − a, one gets a difference quotient. inequalities This page lists Wikipedia articles about named mathematical inequalities. Pure mathematics
Keywords Keywords are the words that are used to reveal the internal structure of an author's reasoning. While they are used primarily for rhetoric, they are also used in a strictly grammatical sense for structural composition, reasoning, and comprehension. and Phrases: Explicit bounds, Volterra type, Finite difference, Inequalities, Differential equations differential equation Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. . Tamsui Oxford Journal of Mathematical Sciences 21(2) (2005) 201-216 Aletheia University Aletheia University (after Greek αλήθεια, truth) is a university in Tamsui, Taipei County, Taiwan founded by George Leslie Mackay as the Oxford (University) College. 1. Introduction The inequalities which provide explicit bounds on unknown functions play a fundamental role in the development of the theory of differential, integral and finite difference equations, see [1, 3, 4] and the references given therein. In [5, p.6] B. G. Pachpatte proved the following interesting theorems This is a list of theorems, by Wikipedia page. See also
Theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. A. Let U(n) : [N.sub.0] [right arrow] [R.sub.+], K(n, [sigma]), [[DELTA delta [from triangular shape of the Nile delta, like the Greek letter delta], a deposit of clay, silt, and sand formed at the mouth of a river where the stream loses velocity and drops part of its sediment load. ].sub.1]K(n, [sigma]) : [bar.D] [right arrow] [R.sub.+] for (n, [sigma]) [member of] [bar.D] = {(n, [sigma]) [member of] [N.sup.2.sub.0] 0 : 0 [less than or equal to] [sigma] [less than or equal to] n < [infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ]} and C [greater than or equal to] 0 is a constant. ([a.sub.1]) If [U.sup.2](n) [less than or equal to] C + [n-1.summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over ([sigma] = 0)] K (n, [sigma]U([sigma])), for n [member of] [N.sub.0], then U(n) [less than or equal to] [square root of C] + 1/2 [n-1.summation over ([sigma] = 0)] L([sigma]) for n [member of] [N.sub.0], where L(n) = K(n + 1, n) + [n-1.summation over ([sigma] = 0)] [[DELTA].sub.1]K(n, [sigma]), for n [member of] [N.sub.0]. ([a.sub.2]) Let g(U) be a continuous, nondecreasing function defined on [R.sub.+] and g(U) > 0 on (0, [infinity]). If [U.sup.2](n) [less than or equal to] C + [n-1.summation over ([sigma] = 0)] K (n, [sigma])U([sigma])g(U([sigma])), for n [member of] [N.sub.0], then for 0 [less than or equal to] n [less than or equal to] [n.sub.1], n, [n.sub.1] [member of] [N.sub.0], U(n) [less than or equal to] [G.sup.-1][G([square root of C]) + 1/2 [n-1.summation over ([sigma] = 0)]L([sigma])], where L(n) is as defined in ([a.sub.1]), [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ], r > 0, [r.sub.0] > 0, [G.sup.-1] is the inverse (mathematics) inverse - Given a function, f : D -> C, a function g : C -> D is called a left inverse for f if for all d in D, g (f d) = d and a right inverse if, for all c in C, f (g c) = c and an inverse if both conditions hold. of G and n1 2 N0 is chosen so that G([square]) + [1/2] [n-1.summation over ([sigma] = 0)] L([sigma]) [member of] [member of] Dom Dom (dōm), peak, 14,942 ft (4,554 m) high, Valais canton, S Switzerland, in the Mischabelhörner group. It is the highest peak entirely in Switzerland. ([G.sup.-1]) for all n [member of] [N.sub.0] such that 0 [less than or equal to] n [less than or equal to] [n.sub.1]. Theorem B. Let U(m, n) : [N.sup.2.sub.0] [right arrow] [R.sub.+], P(m, n, [sigma], w), [[DELTA].sub.1]P(m, n, [sigma], w), [[DELTA].sub.2]P(m, n, [sigma], w), [[DELTA].sub.2][[DELTA].sub.1]P(m, n, [sigma], w) : [bar.E] [right arrow] [R.sub.+] for (m, n, [sigma], w) [member of] [bar.E] = {(m, n, [sigma], w) [member of] [N.sup.4.sub.0]: 0 [less than or equal to] [sigma] [less than or equal to] m < m < [infinity], 0 [less than or equal to] w [less than or equal to] n < [infinity]} and C [greater than or equal to] 0 is a constant. ([b.sub.1]) If [U.sup.2](m, n) [less than or equal to] C + [m-1.summation of ([sigma]=0)] [n-1.summation of (w=0)] P(m, n, [sigma], w)U([sigma], w), for m, n [member of] [N.sub.0], then U(m, n) [less than or equal to] [square root of C] + 1/2 [m-1.summation of (s=0)] [n-1.summation of (t=0)] H(s,t), for m, n [member of] [N.sub.0], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for m, n [member of] [N.sub.0]. ([b.sub.2]) Let g(U) be as in ([a.sub.2]). If [U.sup.2](m, n) [less than or equal to] C + [m-1.summation of ([sigma]=0)] [n-1.summation of (w=0)] P(m, n, [sigma], w)U([sigma], w)g(U([sigma], w)), For m, n [member of] [N.sub.0], then for 0 [less than or equal to] m [less than or equal to] [m.sub.1], 0 [less than or equal to] n [less than or equal to] [n.sub.1], m [m.sub.1], n [n.sub.1] [member of] [N.sub.0], U(m, n) [less than or equal to] [G-.sup.1] [G([square root of C]) + 1/2 [m-1.summation of (s=0)][n-1.summation of (t=0)] H(s,t)], where H(m, n) is as defined in ([b.sub.1]), G, [G.sup.-1] are as defined in ([a.sub.2]) and [m1.sub.], [n.sub.1] are chosen so that G([square root of C]) + [1/2][m-1.summation of (s=0)][n-1.summation of (t=0)] H(s,t) [member of] Dom([G.sup.-1]) for all m, n [member of] [N.sub.0] lying in 0 [less than or equal to] m [less than or equal to] [m.sub.1], 0 [less than or equal to] n [less than or equal to] [n.sub.1]. The main purpose of the present paper is to offer some results dealing with variants of the above inequalities which can be used to study the qualitative qualitative /qual·i·ta·tive/ (kwahl´i-ta?tiv) pertaining to quality. Cf. quantitative. qualitative pertaining to observations of a categorical nature, e.g. breed, sex. behavior of solutions of certain sum-difference equations, and an application is also given to illustrate the usefulness. 2. Main Results In what follows, R denotes the set of real numbers and [R.sub.+] = [0,[infinity]), [N.sub.0] = {0, 1, 2, ...}, are the given subsets of R. We denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. by, [bar.D] = {(n, [sigma]) [member of] [N.sup.2.sub.0] : 0 [less than or equal to] [sigma] n [less than or equal to] < [infinity]}, [bar.E] = {(m, n, [sigma], w) [member of] [N.sup.4.sub.0] : 0 [less than or equal to] [sigma] [less than or equal to] m < [infinity], 0 [less than or equal to] w [less than or equal to] n < [infinity]. For a function Z(m, n), m, n [member of] [N.sub.0], denote [[DELTA].sub.1]Z(m, n) = Z(m+1, n)-Z(m, n), [[DELTA].sub.2]Z(m, n) = Z(m, n+1)-Z(m, n), [[DELTA].sub.2][[DELTA].sub.1] (m, n) = [[DELTA].sub.2]([[DELTA].sub.1]Z(m, n)). We use the usual conventions that the empty sums and products are taken to be 0 and 1 respectively and assume that all the sums and products involved throughout the discussion exist on the respective domains of their definitions. Theorem 1. Let U(n), f(n), [DELTA]f(n): [N.sub.0] [right arrow] [R.sub.+], K(n, [sigma]), [[DELTA].sub.1]K(n, [sigma]) : [bar.D] [right arrow] [R.sup.+]. ([c.sub.1]) If [U.sup.2](n) [less than or equal to] f(n) + [n-1.summation over ([sigma]=0)] K(n,[sigma])U([sigma]), (1.1) for n [member of] [N.sub.0], then U(n) [less than or equal to] [square root of f(n)] + [1/2] [n-1.summation over ([sigma]=0)] L([sigma]), (1.2) for n [member of] [N.sub.0], where L(n) = K(n+1, n) + [n-1.summation over ([sigma]=0)] [[DELTA].sub.1]K(n, [sigma]), (1.3) for n [member of] [N.sub.0]. ([c.sub.2]) Let g(U) be as in ([a.sub.2]). If [U.sup.2](n) [less than or equal to] f(n) + [n-1.summation over ([sigma]=0)] K(n, [sigma])U([sigma])g(U([sigma])), (1.4) for n [member of] [N.sub.0], then for 0 [less than or equal to] n [less than or equal to] [n.sub.1], n, [n.sub.1] [member of] [N.sub.0], U(n) [less than or equal to] [G.sup.-1] (G([square root of f(n))] + [1/2] [n-1.summation over ([sigma]=0)] L([sigma])], (1.5) where L(n) is defined by (1.3), G, [G.sup.-1] are as defined in ([a.sub.2]) and [n.sub.1] [member of] [N.sub.0] is chosen so that G([square root of f(n)]) + [1/2] [n-1.summation over ([sigma]=0)] L([sigma]) [member of] Dom([G.sup.-1]), for all n [member of] [N.sub.0] such that 0 [less than or equal to] n [less than or equal to] [n.sub.1]. Proof. ([c.sub.1]) Let f(n) > 0 and define a function Z(n) by the right hand side of (1.1). Then Z(0) = f(0), Z(n) is positive and nondecreasing, U(n) [less than or equal to] [square root of Z(n)] for n [member of] [N.sub.0], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.6) By using the facts that [square root of Z(n)] > 0, [square root of Z(n)] [less than or equal to] [square root of Z(n+1)], [square root of f(n)] [less than or equal to] [square root of Z(n)] for n [member of] [N.sub.0] and (1.6), we observe TO OBSERVE, civil law. To perform that which has been prescribed by some law or usage. Dig., 1, 3, 32. that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.7) Now by setting n = [sigma] in (1.7) and then taking the sum over [sigma] from 0 to n - 1 we obtain [square root of Z(n)] - [square root of Z(0)] [less than or equal to] [square root of f(n)] - [square root of f(0)] + 1/2 [n-1.summation over ([sigma]=0)] L([sigma]) (1.8) Using (1.8) in U(n) [less than or equal to] [square root of Z(n)] and the fact that [square root of Z(0)] = [square root of f(0)] we get the desired inequality inequality, in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equation, but it does contain information about the expressions involved. in (1.2). If f(n) is nonnegative non·neg·a·tive adj. Of, relating to, or being a quantity that is either positive or zero. Adj. 1. nonnegative - either positive or zero , we carry out the above procedure with f(n) + [epsilon] instead of f(n), where [epsilon] > 0 is an arbitrary Irrational; capricious. The term arbitrary describes a course of action or a decision that is not based on reason or judgment but on personal will or discretion without regard to rules or standards. small constant, and subsequently pass to the limit as [epsilon] [right arrow] 0 to obtain (1.2). Proof. ([c.sub.2]) First we assume that f(n) > 0. Let T [member of] [N.sub.0] be an arbitrary number. From (1.4) we observe that [U.sup.2](n) [less than or equal to] f(T) + [n-1.summation over ([sigma]=0)] K(n, [sigma])U([sigma])g(U([sigma])) (1.9) for 0 [less than or equal to] n [less than or equal to] T. Define a function W(n) by the right side of (1.9). Then W(0) = f(T), W(n) is positive and nondecreasing, U(n) [less than or equal to] [square root of W(n)] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.10) By using the facts that [square root of W(n)] > 0, [square root of W(n)] [less than or equal to] [square root of W(n + 1)] for n [member of] [N.sub.0] and (1.10), we observe that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.11) From (1.11) and using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we observe that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.12) By taking n = [sigma] in (1.12) and summing up over [sigma] from 0 to n - 1, we have G([square root of W(n)]) [is less than or equal to] G([square root of F(T)]) + [1/2] [n-1.summation over ([sigma]=0)] L([sigma]). (1.13) From (1.13) we have [square root of W(n)]) [is less than or equal to] [G.sup.-1][G([square root of f(T))] + [1/2] [n-1.summation over ([sigma]=0)] L([sigma]) (1.14) for 0 [less than or equal to] n [less than or equal to] T [less than or equal to] [n.sub.1]. Now by taking n = T in (1.14) and using the fact that U(n) [less than or equal to] [square root of W(n)] is true for n = T we obtain U(T) [less than or equal to] [G.sup.-1][G([square root of f(T))] + 1/2 [T-1.summation over ([sigma]=0)] L([sigma])]. (1.15) Since T is arbitrary, the conclusion (1.5) is clear from (1.15). The proof of the case when f(n) [greater than or equal to] 0 can be completed as mentioned in the proof of part ([c.sub.1]). Remark 1. We note that Theorem A is a special case of Theorem 1. In the following theorem we establish a two independent variable version of Theorem 1 which can be used in certain applications. Theorem 2. Let U(m, n), f(m, n), [[DELTA].sub.1]f(m, n), [[DELTA].sub.2]f(m, n) : [N.sup.2.sub.0] [right arrow] [R.sub.+], P(m, n, [sigma], w), [[DELTA].sub.1]P(m, n, [sigma], w), [[DELTA].sub.2]P(m, n, [sigma], w), [[DELTA].sub.2][[DELTA].sub.1]P(m, n, [sigma], w) : [bar.E] [right arrow] [R.sub.+]. ([d.sub.1]) If [U.sup.2](m, n) [less than or equal to] f(m, n) + [m-1.summation over ([sigma]=0)] [n-1.summation over (w=0)] P(m, n, [sigma], w)U([sigma], w), (2.1) for m, n [member of] [N.sub.0], then U(m, n) [less than or equal to] [square of f(m, n)] + 1/2 [m-1.summation over (s=0)] [n-1.summation over (t=0)]H(s, t) (2.2) for m, n [member of] [N.sub.0], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3) for m, n [member of] [N.sub.0]. ([d.sub.2]) Let g(U) be as in ([a.sub.2]). If [U.sup.2](m, n) [less than or equal to] f(m, n) + [m-1.summation over ([sigma]=0)][n-1.summation over (w=0)] P(m, n, [sigma], w)U([sigma], w) g(U([sigma], w)), (2.4) for m, n [member of] [N.sub.0], then for 0 [less than or equal to] m [less than or equal to] [m.sub.1], 0 [less than or equal to] n [less than or equal to] [n.sub.1], m, [m.sub.1], n, [n.sub.1] [member of] [N.sub.0], U(m, n) [less than or equal to] [G.sup.-1] [G([square root of f(m, n)]) + 1/2 [m-1.summation over (s=0)][n-1.summation over (t=0)] H(s,t)], (2.5) where H(m, n) is defined by (2.3), G, [G.sup.-1] are as defined in ([a.sub.2]) and [m.sub.1], [n.sub.1] are chosen so that G ([square root of f(m, n)) + [1/2] [m-1.summation over (s=0)][n-1.summation over (t=0)] H(s,t)] [member of] Dom([G.sup.-1]),) for all m, n [member of] [N.sub.0] lying in 0 [less than or equal to] m [less than or equal to] [m.sub.1], 0 [less than or equal to] n [less than or equal to] [n.sub.1]. Proof. ([d.sub.1]) We first assume that f(m, n) > 0 for m, n [member of] [N.sub.0], and define a function Z(m, n) by the right side of (2.1). Then Z(0, n) = f(0, n), Z(m, n) positive and nondecreasing for m, n [member of] [N.sub.0], U(m, n) [less than or equal to] [square root of Z(m, n)] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.(2.7) From (2.7) and using the fact that [square root of Z(s, t)] [less than or equal to] [square root of Z(s, t+1)] [less than or equal to] [square root of Z(s+1, t+1)], Z(s, t) > 0 for s, t [member of] [N.sub.0], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.8) Now an application of the formula [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.9) From (2.8) and (2.9) we see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.10) By using the facts that [[DELTA].sub.1]f(s,t) [less than or equal to] [[DELTA].sub.1]Z(s,t), [[DELTA].sub.2][square root of Z(s,t)] [greater than or equal to] 0 for s, t [member of] [N.sub.0], [m-1.summation over (s=0)] [[DELTA].sub.1] [square root of f(s, n)] = [square root of f(m, n)] - [square root of f(0, n)], [square root of Z(0, n)] = [square root of f(0, n)], and from (2.10), we observe that [square root of Z(m, n)] [less than or equal to] [square root of f(m, n)] + 1/2 [m-1.summation over (s=0)][n-1.summation over (t=0)] H(s,t) (2.11) The desired inequality in (2.2) now follows by using (2.11) in U(m, n) [less than or equal to] [square root of Z(m, n)]. If f(m, n) is nonnegative we can carry out the above proof with f(m, n) + [epsilon] instead of f(m, n) where [epsilon] > 0 is arbitrary small constant, and subsequently pass to the limit as [epsilon] [right arrow] 0 to obtain (2.2). The proof is complete. Proof. ([d.sub.2]) First we assume that f(m, n) > 0. Let S, T [member of] [N.sub.0] be arbitrary numbers. From (2.4) we observe that [U.sup.2](m, n) [less than or equal to] f(S,T) + [m-1.summation over ([sigma]=0)][n-1.summation over (w=0)] P(m, n, [sigma], w)U([sigma], w) g(U([sigma], w)) (2.12) for 0 [less than or equal to] m [less than or equal to] S, 0 [less than or equal to] n [less than or equal to] T. Define a function W(m, n) by the right side of (2.12). Then W(m, 0) = W(0, n) = f(S, T), U(m, n) [less than or equal to] [square root of W(m, n)] and [[DELTA].sub.2][[DELTA].sub.1]W(m, n) [less than or equal to] H (m, n) [square root of W (m, n)]g([square root of W (m, n)]) (2.13) By using the facts that [square root of W(m, n)] > 0, [[DELTA].sub.1]W(m, n) [greater than or equal to] 0, [square root of W(m + 1, n)], [less than or equal to] [square root of W(m + 1, n + 1)] [square root of W(m, n)] [less than or equal to] [square root of W(m, n+1)], [square root of W(m, n)] [less than or equal to] [square root of W(m, n + 1)], [square root of W(m, n)] [less than or equal to] [square root of W(m + 1, n)] for m, n [member of] [N.sub.0] we observe that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.14) From (2.13) and (2.14) we have [[DELTA].sub.2][[DELTA].sub.1]([square root of W(m,n)]) [less than or equal to] 1/2 H(m,n)g([square root of W(m,n)]). (2.15) From (2.15) and using the facts that [[DELTA].sub.1][square root of W(m,n)] [greater than or equal to] 0, [square root of W(m,n)] [less than or equal to] [square root of W(m,n+1)] for m, n [member of] [N.sub.0], we observe that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.16) Now keeping m fixed in (2.16), set n = t and sum over t from 0 to n - 1 and use the fact that [[DELTA].sub.1]([square root of W(m,0)]) = 0, to obtain the estimate [[DELTA].sup.1]([square root of W(m,0)]) / g([square root of W(m,n)]) [less than or equal to] 1/2 [n-1.summation over (t=0)] H(m,t). (2.17) From (2.17) and using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we observe that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.18) Now keeping n fixed in (2.18), set m = s and sum over s from 0 to m-1 and use the fact that W(0, n) = f(S, T), to obtain the estimate G([square root of W(m,n)]) [less than or equal to] G ([square root of f(S, T)]) + [1/2] [m-1.summation over (s=0)] [n-1.summation over (t=0)] H (s,t) (2.19) for 0 [less than or equal to] m [less than or equal to] S, 0 [less than or equal to] n [less than or equal to] T. From (2.19) we have [square root of W(m,n)] [less than or equal to] for 0 [less than or equal to] m [less than or equal to] S [less than or equal to] [m.sub.1], 0 [less than or equal to] n [less than or equal to] T [less than or equal to] [n.sub.1]. Now by taking m = S, n = and using the fact that U(m, n) [less than or equal to] [square root of W(m, n)] is true for m = S, n = T, we obtain U(S, T) [less than or equal to] [G.sup.-1] [G([square root of f(S, T)] + 1/2 [S-1.summation over (s=0)] [T-1.summation over (t=0)] H(s,t)]. (2.21) Since S, T are arbirary, the conclusion (2.5) is clear from (2.21). The proof of the case when f(m, n) [greater than or equal to] 0 can be completed as mentioned in the proof of part ([d.sub.1]). Remark 2. We note that Theorem B is a special case of Theorem 2. 3. Applications Consider the sum-difference equation of the form [y.sup.2](n) = g(n) + [n-1.summation over ([sigma]=0)] h(n,[sigma], y([sigma])), (3.1) where g : [N.sub.0] [right arrow] R, h : [bar.D] x R [right arrow] R We assume that the functions g, h in (3.1) satisfy the conditions [absolute value of g(n)] [less than or equal to] f(n), [absolute value of h(n, [sigma], y([sigma]))] [less than or equal to] K(n, [sigma]) (3.2) Q(n) = [square root of f(n)] + [1/2] [n-1.summation over ([sigma]=0)] L([sigma]) < [infinity] (3.3) where f(n), K(n, [sigma]), and L([sigma]) are as in Theorem 1. Let y(n), n [member of] [N.sub.0] be a solution of (3.1). From (3.1) and (3.2) we have [[absolute value of y(n)].sup.2] [is less than or equal to] f(n) + [n- 1.summation over ([sigma]=0)] K(n, [sigma])[absolute value of y([sigma])]. (3.4) An application of the inequality in ([c.sub.1]) given in Theorem 1 to (3.4) yields [absolute value of y(n)] [less than or equal to] Q(n) (3.5) for n [member of] [N.sub.0]. From the hypothesis An assumption or theory. During a criminal trial, a hypothesis is a theory set forth by either the prosecution or the defense for the purpose of explaining the facts in evidence. (3.3), the estimation estimation In mathematics, use of a function or formula to derive a solution or make a prediction. Unlike approximation, it has precise connotations. In statistics, for example, it connotes the careful selection and testing of a function called an estimator. in (3.5) implies (logic) implies - (=> or a thin right arrow) A binary Boolean function and logical connective. A => B is true unless A is true and B is false. The truth table is A B | A => B ----+------- F F | T F T | T T F | F T T | T It is surprising at first that A => the boundedness n. 1. (Math.) the quality of being finite. Noun 1. boundedness - the quality of being finite finiteness, finitude quality - an essential and distinguishing attribute of something or someone; "the quality of mercy is not of the solution of (3.1) on [N.sub.0]. Received April 19, 2004, Accepted August 9, 2004. References [1] D. Bainov and P. Simenov, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht Dordrecht (dôr`drĕkht) or Dort (dôrt), city (1994 pop. 113,394), South Holland prov., SW Netherlands, at the point where the Lower Merwede divides to form the Noord and Oude Maas (Old Meuse) rivers. , 1992. [2] W. Okrasinski, On a nonlinear A system in which the output is not a uniform relationship to the input. nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input. convolution convolution /con·vo·lu·tion/ (-loo´shun) a tortuous irregularity or elevation caused by the infolding of a structure upon itself. equation occurring in the theory of water precloation, Ann ANN, Scotch law. Half a year's stipend over and above what is owing for the incumbency due to a minister's relict, or child, or next of kin, after his decease. Wishaw. Also, an abbreviation of annus, year; also of annates. In the old law French writers, ann or rather an, signifies a year. . Polon. Math. 37(1980), 223-229. [3] B. G. Pachpatte, Inequalities for Differential and Integral Equatons, Academic Press, New York New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of , 1998. [4] B. G. Pachpatte, Inequalities for Finite Difference Equations, Marcel Dekker Marcel Dekker is a well-known encyclopedia publishing company with editorial boards found in New York, New York. They are part of the Taylor and Francis publishing group. Initially a textbook publisher, they went to encyclopedia publishing in the late 1990's. Inc., New York, 2001. [5] B. G. Pachatte, Explicit bounds on Volterra integral inequalities, Tamsui Oxford J. Math. Sci. 19 (2003), 13-25. Chyng-Nan Chou Chou (jō), dynasty of China, which ruled from c.1027 B.C. to 256 B.C. The pastoral Chou people migrated from the Wei valley NW of the Huang He c.1027 B.C. and overthrew the Shang dynasty. The Chou built their capital near modern Xi'an in 1027 B.C. Department of General Education, Kuang Wu Institute of Technology, Peito, Taipei Taipei (tībā`), city (1995 est. pop. 2,632,863), N Taiwan, capital of Taiwan and provisional capital of the Republic of China. Taiwan's largest city, it is the administrative, cultural, and industrial center of the island. , Taiwan Taiwan (tī`wän`), Portuguese Formosa, officially Republic of China, island nation (2005 est. pop. 22,894,000), 13,885 sq mi (35,961 sq km), in the Pacific Ocean, separated from the mainland of S China by the 100-mi-wide (161-km) Taiwan , 11271. and Gou-Sheng Yang yang (yang) [Chinese] in Chinese philosophy, the active, positive, masculine principle that is complementary to yin; see yin, under principle. Department of Mathematics, Tamkang University Tamkang University (Traditional Chinese: 淡江大學; Simplified Chinese: 淡江大学 , Tamsui, Taiwan 25137. |
|
||||||||||||||||||
Printer friendly
Cite/link
Email
Feedback
Reader Opinion