... – NUMBERSAND OPERATIONS REVIEW – Practice Question The number –16 belongs in which of the following sets of numbers? a rational numbers only b whole numbersand integers c whole numbers, ... factor shared by 28 and 21 Practice Question What are the common factors of 48 and 36? a 1, 2, and b 1, 2, 3, and c 1, 2, 3, 6, and 12 d 1, 2, 3, 6, 8, and 12 e 1, 2, 3, 4, 6, 8, and 12 Answer c ... Median, and Mode To find the average, or mean, of a set of numbers, add all of the numbers together and divide by the quantity of numbers in the set mean ϭ sum of numbers in set ᎏᎏᎏ quantity of numbers...
... another command prompt Step Start a fourth Telnet session to router by opening another command prompt Step 10 Check the number of sessions on the host a Open another command prompt on the host and type ... port numbers 3-5 CCNA 2: Routers and Routing Basics v 3.0 - Lab 10.2.5 Copyright 2003, Cisco Systems, Inc Erasing and reloading the router Enter into the ... with the proper IP address, subnet mask and default gateway Step Allow HTTP access to the router a Allow HTTP access by issuing the ip http server command in global configuration mode Step Use...
... number of sessions on the host a Open a command prompt on the host and type netstat /? at the DOS prompt b What options are available for the netstat command? ... access by issuing the ip http server command in global configuration mode Step Use the workstation browser to access the router a Open a browser on Host and type http://ip-address of Router GAD ... information from the privileged exec command mode GAD# copy running-config startup-config Step Configure the host with the proper IP address, subnet mask and default gateway Step Allow HTTP access...
... Ph and y j = Pth, and, since I a ;" + cq*n I < 2, Take now for m the smallest integer such that h being a positive integer such that ~ l + ~ ! + ' ' + y k - < i l + Since a = 7-I andand the numbers ... the definitions and the results (though elementary) borrowed from algebra and from number theory I wish to express my thanks to Dr Abram L Sachar, President of Brandeis University, and to the Department ... THEOREM set o all numbers having the preceding property is denumerable PROOF We again write A& = 4, + en where a, is an integer and c, I = (1 XOn 11 We have and, since we have and an easy calculation...
... CONGRUENCES AND MODULAR EQUATIONS Proposition 1.4 The set Z/n with the operations + and × is a commutative ring and the n n function πn : Z −→ Z/n is a ring homomorphism which is surjective (onto) and ... first summand is {±1} and the second can be taken to be Now for a general n we have n = pr1 pr2 · · · prs s where for each i, pi is a prime with and ri p1 < p < · · · < p s Then the numbers pi ... (a) ≡ r p2r−1 p Then there exists a ∈ Z such that f (a ) ≡ p2r+1 and a ≡ a r p and a ∈ Z, CHAPTER The p-adic norm and the p-adic numbers Let R be a ring with unity = 1R Definition 2.1 A function...
... IN NUMBERS statement (1) let's just PICK A WHOLE BUNCH OF NUMBERS WHOSE GCF IS and watch what happens let's try to make the numbers diverse say, and 6 and 8 and 10 10 and 12 and 10 and 14 and ... 3, 2, andand share only as a factor and share only as a factor and share only as a factor and share only as a factor There are four positive integers, therefore, that are both less than and share ... between m and r, imagine m and r both starting out at 12, and 'sliding' equally in opposite directions, with r moving to the right and m moving to the left (you can't slide r to the left and m to...
... q-Bernoulli numbersand polynomials Duke Math J 15, 987–1000 (1948) [2] Carlitz, L: q-Bernoulli and Eulerian numbers Trans Am Math Soc 76, 332–350 (1954) [3] Kamano, K: p-adic q-Bernoulli numbersand ... q-Bernoulli and q-Euler numbersand polynomials and a class of generalized q-Hurwitz zeta functions Appl Math Comput 215(3), 1185–1208 (2009) [12] Kim, T: On the analogs of Euler numbersand polynomials ... eB(q)t k! and Gq (t) = Ek (q) k=0 tk = eE(q)t , k! where the symbol Bk (q) and Ek (q) are interpreted to mean that (B(q))k and (E(q))k must be replaced by Bk (q) and Ek (q) when we expanded the...
... (white bars) and wild type mice (grey bars) standardised to injury area Neutrophil numbers (A) were determined by NIMP-14 immunoreactivity Microglial and monocyte derived macrophages numbers (B) ... fluorescence threshold (CT) GAPDH mRNA was used as the house keeping gene and MT-I and MT-II mRNA copy numbers were standardized to the copy number of the house-keeping gene, GAPDH Plasma cytokine ... Neutrophil numbers were greatly diminished at DPI and mostly absent from the injury site at DPI No significant differences were found between neutrophil numbers in the injury site of wild-type and MT-I/II...
... q-Bernoulli numbersand polynomials Duke Math J 15, 987–1000 (1948) [2] Carlitz, L: q-Bernoulli and Eulerian numbers Trans Am Math Soc 76, 332–350 (1954) [3] Kamano, K: p-adic q-Bernoulli numbersand ... q-Bernoulli and q-Euler numbersand polynomials and a class of generalized q-Hurwitz zeta functions Appl Math Comput 215(3), 1185–1208 (2009) [12] Kim, T: On the analogs of Euler numbersand polynomials ... eB(q)t k! and Gq (t) = Ek (q) k=0 tk = eE(q)t , k! where the symbol Bk (q) and Ek (q) are interpreted to mean that (B(q))k and (E(q))k must be replaced by Bk (q) and Ek (q) when we expanded the...
... integers can be represented by the q-Bernoulli, q-Euler numbers, and polynomials q-Bernoulli, q-Euler numbersand polynomials related to the Bosonic and the Fermionic p-adic integral on ℤp In this section, ... q-Bernoulli, q-Euler numbers, and polynomials In the complex case, we shall explicitly determine the generating function Fq(t) of qBernoulli numbersand the generating function Gq(t) of q-Euler numbers: ... (1991) doi:10.1016/0022-314X(91)90048-G 11 Choi, J, Anderson, PJ, Srivastava, HM: Carlitz’s q-Bernoulli and q-Euler numbersand polynomials and a class of generalized q-Hurwitz zeta functions...
... Bn,ξ = and n = 1, n > 1, (5) with the usual convention about replacing Bn by Bn,ξ (see [17-19]) Recently, several ξ authors have studied the twisted Bernoulli numbersand q-Bernoulli numbers ... the twisted q-Bernoulli numbersand polynomials related to q-Bernstein polynomials From these properties, we derive some new identities for the twisted q-Bernoulli numbersand polynomials Final ... the twisted Carlitz’s q-Bernoulli numbersand q-Bernstein polynomials On the twisted Carlitz ‘s q-Bernoulli numbers In this section, we assume that n Î ℤ+, ξ Î Tp and q ∈ Cp with |1 - q|p < Let...
... q-Euler numbersand polynomials Proc Jangjeon Math Soc 14, 7–14 (2011) 14 Kim, T, Choi, J, Kim, YH: q-Bernstein polynomials associated with q-Stirling numbersand Carlitz’s q-Bernoulli numbers, ... T: The modified q-Euler numbersand polynomials Adv Stud Contemp Math 16, 161–170 (2008) Kim, T: Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic ... on ℤp and investigate some new identities on the weighted q-Euler numbers related to the weighted q-Bernstein polynomials q-Euler numbers with weight a In this section we assume that a Î N and...
... http://www.advancesindifferenceequations.com/content/2011/1/33 Page of q-Bernoulli numbersand q-Bernoulli polynomials revisited In this section, we perform a further investigation on the q-Bernoulli numbersand qBernoulli polynomials given ... Acikgöz, Erdal and Araci derived some results by using Theorems 1-3 Hence, the other results are incorrect Now, we redefine the generating function of q-Bernoulli numbersand polynomials and correct ... Bernoulli numbers from Definition for any q In particular, by (1) and (2), we get qDq (t, 1) − Dq (t) = t (7) Thus, by (7), we have qBn,q (1) − Bn,q = 1, if 0, if n = 1, n > 1, (8) and n n l−1...
... the Euler and the Genocchi numbersand polynomials,” Advanced Studies in Contemporary Mathematics (Kyungshang), vol 20, no 1, pp 23–28, 2010 T Kim, “A note on the q-Genocchi numbersand polynomials,” ... polynomials attached to χ by using the generating function of those numbersand polynomials Generalized q-Genocchi Numbersand Polynomials For r ∈ N, let us consider the q-extension of the generalized ... generalized Genocchi numbersand polynomials attached to χ The purpose of this paper is to present a systemic study of some families of higher-order generalized q-Genocchi numbersand polynomials...
... related to the q-Bernoulli numbersand q-Bernoulli polynomials and give a new construction of these numbersand polynomials related to the second kind Stirling numbersand q-Bernstein polynomials ... polynomials at negative integers and are associated with q-Bernstein polynomials New Approach to q-Bernoulli Numbersand Polynomials Let N be the set of natural numbersand N∗ q-Bernoulli polynomials ... “q-Bernstein polynomials associated with q-Stirling numbersand Carlitz’s q-Bernoulli numbers, ” Abstract and Applied Analysis in press 15 T Kim, L.-C Jang, and H Yi, “A note on the modified q-Bernstein...
... the higher-order Carlitz’s type q-Bernoulli numbersand polynomials in the p-adic number field On the Generalized Higher-Order q-Bernoulli Numbersand Polynomials In this section, we assume that ... 1798–1804, 2009 L Carlitz, “q-Bernoulli numbersand polynomials,” Duke Mathematical Journal, vol 15, pp 987–1000, 1948 L Carlitz, “q-Bernoulli and Eulerian numbers, ” Transactions of the American ... 2008 Journal of Inequalities and Applications 17 17 E.-J Moon, S.-H Rim, J.-H Jin, and S.-J Lee, “On the symmetric properties of higher-order twisted q-Euler numbersand polynomials,” Advances...
... functions and related integrals,” Journal of Number Theory, vol 76, no 2, pp 320–329, 1999 T Kim, J Choi, and Y.-H Kim, “Some identities on the q-Bernstein polynomials, q-Stirling numbersand q-Bernoulli ... authors cf 8, and the f x n We have references given there For n ∈ N, write fn x n−1 I1 fn f l I1 f 2.3 l This identity is to derives interesting relationships involving Bernoulli numbersand polynomials ... Mathematics-Modelling and Simulation In press M Acikgoz and S Araci, “On the generating function of the Bernstein polynomials,” in Proceedings of the 8th International Conference of Numerical Analysis and Applied...
... T Kim, “On the multiple q-Genocchi and Euler numbers, ” Russian Journal of Mathematical Physics, vol 15, no 4, pp 481–486, 2008 T Kim, “q-Bernoulli numbersand polynomials associated with Gaussian ... t n! By Laurent series and Cauchy residue theorem in 3.1 and 3.3 , we obtain the following theorem Theorem 3.2 Let χ be Dirichlet’s character with odd conductor d ∈ N, and let ζ h, s ∈ C, x / ... gamma functions and related integrals,” Journal of Number Theory, vol 76, no 2, pp 320–329, 1999 T Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the...
... studied by Cenkci and Can , Kim 4–12 , and Ozden et al 16–18 This research for q-Euler numbers seems to be motivated by Carlitz who had constructed the q-Bernoulli numbersand polynomials for ... physics, and so on see 3–18 The purpose of this paper is to give a new construction of the q-extensions of Euler numbersand polynomials It is expected that new constructed q-Euler numbersand polynomials ... made a wider and deeper study of the q-number up to recently see 1–18 In the field of number theory and mathematical physics, zeta functions and l-functions interpolating these numbers in negative...