Linear Network Codes and Systems of Polynomial Equations
|
|
- Kathlyn Leonard
- 6 years ago
- Views:
Transcription
1 1 Liner Network Codes nd Systems of Polynomil Equtions Rndll Dougherty, Chris Freiling, nd Kenneth Zeger (Submitted to ISIT 2008) Abstrct If β nd γ re nonnegtive integers nd F is field, then polynomil collection {p 1,..., p β } Z[α 1,..., α γ] is sid to be solvble over F if there exist ω 1,..., ω γ F such tht for ll i = 1,..., β we hve p i(ω 1,..., ω γ) = 0. We sy tht network nd polynomil collection re solvbly equivlent if for ech field F the network hs sclr-liner solution over F if nd only if the polynomil collection is solvble over F. Koetter nd Médrd s work implies tht for ny directed cyclic network, there exists solvbly equivlent polynomil collection. We provide the converse result, nmely tht for ny polynomil collection there exists solvbly equivlent directed cyclic network. (Hence, the problems of network sclr-liner solvbility nd polynomil collection solvbility hve the sme complexity.) The construction of the network is modeled on mtroid construction using finite projective plnes, due to McLne in A set Ψ of prime numbers is set of chrcteristics of network if for every q Ψ, the network hs sclrliner solution over some finite field with chrcteristic q nd does not hve sclr-liner solution over ny finite field whose chrcteristic lies outside of Ψ. We show tht collection of primes is set of chrcteristics of some network if nd only if the collection is finite or co-finite. Two networks N nd N re ls-equivlent if for ny finite field F, N is sclr-linerly solvble over F if nd only if N is sclr-linerly solvble over F. We further show tht every network is ls-equivlent to multiple-unicst mtroidl network. I. INTRODUCTION We first demonstrte certin equivlence between networks nd collections of polynomils. Specificlly, we show tht ssocited with every finite collection of polynomils with integer coefficients is corresponding network which is sclr-linerly solvble precisely over This work ws supported by the Institute for Defense Anlyses, the Ntionl Science Foundtion, nd the UCSD Center for Wireless Communictions. R. Dougherty is with the Center for Communictions Reserch, 4320 Westerr Court, Sn Diego, CA (rdough@ccrwest.org). C. Freiling is with the Deprtment of Mthemtics, Cliforni Stte University, Sn Bernrdino, 5500 University Prkwy, Sn Bernrdino, CA (cfreilin@csusb.edu). K. Zeger is with the Deprtment of Electricl nd Computer Engineering, University of Cliforni, Sn Diego, L Joll, CA (zeger@ucsd.edu). those finite fields where the polynomils hve common root. A consequence is tht the complexity of determining whether networks re sclr-linerly solvble over prticulr finite fields is equivlent to the complexity of determining whether collections of polynomil hve common roots over the corresponding fields. Secondly, we show tht the collections of prime numbers corresponding to the field chrcteristics of sclr-linerly solvble network lphbets re precisely those which re finite or co-finite. Finlly, we show tht for every network, there exists multiple-unicst network which is mtroidl (i.e. obtined from certin mtroid-tonetwork construction), such tht the two networks re sclr-linerly solvble over the sme finite fields. There hs been interest in determining the solvbility, sclr liner solvbility, nd vector liner solvbility of n rbitrry network with respect to chosen lphbet (e.g. [4] [7], [9] [11], [13], [15], [17], [18]). For given finite lphbet, to determine if network is solvble or sclr-linerly solvble, one cn perform finite exhustive serch of ll possible codes for the network. If vector dimension is lso fixed, finite serch cn lso estblish if network is vector-linerly solvble over tht dimension. There is presently no known lgorithm for determining the generl solvbility or vector-liner solvbility of n rbitrry network. The existence of n lgorithm (which is pprently not computtionlly efficient) to determine sclr-liner solvbility of n rbitrry network follows from work in [12]. Their technique ws to construct finite collection of polynomils from n rbitrry network, such tht for ech finite field, the polynomils hve common root over the field if nd only if the network hs sclrliner solution over the field. Throughout, polynomils will hve integer coefficients nd will use the vribles α 1, α 2,.... For nonnegtive integers β nd γ, ny finite set P = {p 1,..., p β } Z[α 1,..., α γ ] will be clled polynomil collection. If F is field, then polynomil collection is sid to be solvble over F if there exist ω 1,..., ω γ F such tht for ll i = 1,..., β we hve p i (ω 1,..., ω γ ) = 0. We sy tht network nd polynomil collection re solvbly
2 equivlent if for ech field F the network hs sclrliner solution over F if nd only if the polynomil collection is solvble over F. We present n lgorithm in Section II for constructing network from ny polynomil system. Our min results re tht: the network is sclr-linerly solvble over the sme fields s those for which the polynomils hve common roots (Theorem I.2), the constructed network is lwys mtroidl (Theorem I.3), every network is sclr-linerly solvble on the sme set of fields s multiple-unicst mtroidl network (Corollry I.8), nd the collections of prime numbers corresponding to the field chrcteristics of sclr-linerly solvble network lphbets re chrcterized s either finite or co-finite (in Theorem I.9). Let Ψ be n rbitrry collection of integers of the form q i, where q is prime nd i 1. We sy tht Ψ is the solvbility set for network (respectively, polynomil collection) if for every finite field F, the network is sclr-linerly solvble (respectively, polynomil collection is solvble) if nd only if F Ψ. The set of primes q, such tht q i lies in the solvbility set for some i 1, is clled the set of chrcteristics 1 for network (respectively, polynomil collection). The following theorem leds to n lgorithm (vi Gröbner bses [2]) for determining whether network hs sclr-liner solution. No such lgorithm is presently known for determining whether network hs generl nonliner solution. Theorem I.1. (follows from Koetter-Médrd [12]) Every directed cyclic network hs solvbly equivlent polynomil collection. In this pper, we provide the following converse result. Theorem I.2. (Converse to Theorem I.1) Any polynomil collection hs solvbly equivlent directed cyclic network. Furthermore, the solvbly equivlent network in Theorem I.2 is given constructively nd is mtroidl, s stted in the next theorem. Theorem I.3. If polynomil collection P is solvble over some finite field, then ny network constructed, s in Section II, from P is mtroidl. The next definition is tken from [8] ( CSLS stnds for coding solvbility, liner solvbility ). Definition I.4. Two networks N nd N re CSLSequivlent if the following two conditions hold: 1 This terminology is tken from [1]. 1) For ny finite lphbet A, N is solvble over A if nd only if N is solvble over A. 2) For ny finite field F nd ny positive integer k, N is vector-linerly solvble over F in dimension k if nd only if N is vector-linerly solvble over F in dimension k. The following definition gives type of equivlence tht is weker thn CSLS (the cronym ls stnds for (sclr) liner solvbility ). Definition I.5. Two networks N nd N re lsequivlent if for ny finite field F, N is sclr-linerly solvble over F if nd only if N is sclr-linerly solvble over F. Theorem I.6. (see [8, Theorem II.1]) Any network is CSLS-equivlent to multiple-unicst network. The next theorem shows tht if Theorem I.6 is pplied to mtroidl network, then the resulting multipleunicst network cn lso be tken to be mtroidl. Theorem I.7. (see [7, Corollry VII.8]) Any mtroidl network is CSLS-equivlent to multipleunicst mtroidl network. The next corollry follows from our min result in Theorem I.2 together with severl previous results. It demonstrtes tht, when considering which finite fields rbitrry networks re sclr-linerly solvble over, it suffices to consider to the subclss of networks which re simultneously multiple-unicst nd mtroidl. Corollry I.8. Any network is ls-equivlent to multiple-unicst mtroidl network. Theorem I.9. A set of prime numbers is the set of chrcteristics of some network if nd only if the set is finite or co-finite. Theorem I.1 nd our Theorem I.2 together indicte tht determining the sclr-liner solvbility of directed cyclic network over field F is computtionlly equivlent to determining whether collection of polynomils hs common root over F. Given ny lgorithm for determining sclr-liner network solvbility, our result gives n lgorithm for determining polynomil solvbility. This is mny-to-one reduction (i.e., it converts single instnce of the polynomil solvbility problem to single instnce of the network sclr-liner solvbility problem with the sme nswer). The reduction cuses t most liner blowup in input size, in the following sense: the number of nodes nd edges in the resulting network is t most liner function of the number Pge 2 of 5
3 of steps (vrible retrievls nd rithmetic opertions) needed to compute the vlues of the polynomils in the collection. In terms of bit representtions, it is t most n O(n ln n) blowup. This mny-to-one reduction hs the dditionl property tht, given sclr-liner solution to the network, we cn directly reconstruct solution to the polynomil collection. It cn be shown (vi Gröbner bses) tht the sets of chrcteristics of polynomil collections re precisely the sets of primes which re finite or co-finite. In contrst, there hs been no known chrcteriztion of the sets of chrcteristics or the solvbility sets of networks. If Ψ is the solvbility set of network nd n Ψ, then n i Ψ for ll positive integers i While there re n uncountble number of sets of powers of primes closed under exponentition, there re only countbly infinite number of solvbility sets since there re only countble number of networks nd polynomil collections. A fundmentl problem is to determine which sets of integers cn be solvbility sets nd which cn be sets of chrcteristics for networks. Theorem I.1 shows tht every network solvbility set is lso polynomil collection solvbility set. Our Theorem I.2 shows tht every polynomil collection solvbility set is lso network solvbility set. Thus, the network solvbility sets re the sme s the polynomil collection solvbility sets. Our Theorem I.9 shows tht set of primes is the set of chrcteristics of network if nd only if the set of primes is finite or co-finite. II. NETWORK CONSTRUCTION FROM POLYNOMIAL SYSTEM In this section we present n lgorithm for constructing directed cyclic network N from finite polynomil collection P = {p 1,..., p β } Z[α 1,..., α γ ], for i = 1,..., β. The network will be built piece by piece from eight building block components, C 0,..., C 7, which re shown in Figures 1 nd 2 (using Tble I). The messges will be, b, nd c. Certin nodes of the network will be lbeled by x q, y q, u q, or z q, where for ech such node, q is some polynomil in Z[α 1,..., α γ ]. For exmple, the sources for, b, c will be nodes x 0, x 1, y 1, respectively. During the construction, we will lbel vrious nodes with polynomils nd will lter demonstrte connection between these polynomils nd the lphbet symbols crried by these nodes. It will be demonstrted tht this construction lgorithm produces network such tht for ny field F, the network hs sclr-liner solution over F if nd only if the polynomil collection P hs solution over F. The network construction process consists of the steps: Step (1): Strt with component C 0 which cretes nodes x 0, x 1, y 1, z 0, nd z. (See Figure 1) Step (2): If γ > 0, then dd components C 1 (1),..., C 1 (γ), creting nodes x α1,... x αγ. Ech of these components is djoined to the network t the nodes x 0, x 1, z, which hve lredy been creted t Step (1). (See Figure 2 nd Tble I) Step (3): Repetedly dd components C 2,..., C 7 to crete nodes x p1(α 1,...,α γ),..., x pβ (α 1,...,α γ). Steps (3)-(3d) describe the cretion of x p1(α 1,...,α γ),..., x pβ (α 1,...,α γ) s well s mny intermedite nodes. (See Figure 2 nd Tble I) Step (3): For ny positive integer n, to crete node lbeled x n : First, dd component C 4 (1) to crete node u 1. Then, for i = 1,..., n 1, dd component C 2 (i) to crete node z i nd dd component C 6 (i, 1) to crete node x i+1. This is possible since x 1, z 0, z hve lredy been creted. Step (3b): For ny positive integer n, to crete node lbeled x n : First, dd component C 2 (1) to crete node z 1, nd dd component C 5 (1) to crete node u 1. Then, for i = 0,..., n 1, dd component C 6 ( i, 1) to crete node x i 1 nd dd component C 2 ( i 1) to crete node z i 1. Step (3c): For ny positive integer n nd ny α {α 1,..., α γ } to crete node lbeled x α n: First, dd component C 3 (α) to crete node y α. Then, for j = 1,..., n 1, dd component C 2 (α j ) to crete node z α j nd dd component C 7 (α j, α) to crete node x α j+1. Step (3d): To crete nodes lbeled by n rbitrry polynomil in Z[α 1,..., α γ ]: Add vrious instnces of components C 6 nd C 7 to crete nodes lbeled by sums nd products of lbels of existing nodes creted bove. (Some instnces of components C 2, C 3, nd C 4 my lso hve to be dded in order to use C 6 nd C 7.) Step (4): Force ech of the nodes x p1(α 1,...,α γ),..., x pβ (α 1,...,α γ) to demnd messge. To construct nodes lbeled by rbitrry polynomils in Z[α 1,..., α γ ] in Step (3) of the lgorithm, one cn use Step (3) to crete ll positive integer coefficients of the polynomils, use Step (3b) to crete ll negtive integer coefficients of the polynomils, use Step (3c) to crete ll vrible powers occurring in the polynomils, nd finlly use Step (3d) to combine the existing network nodes to crete the desired polynomils. This lgorithm converts single instnce of the poly- Pge 3 of 5
4 nomil solvbility problem to single instnce of the network sclr-liner solvbility problem with the sme nswer. The procedure bove is not the most efficient method to crete the network N from the polynomil collection P. A smller network cn in generl be constructed whose size is liner in the size of the representtion of the polynomil collection. c b y 1 x 0 x 1 z 0 z 3 Fig. 1. Network component C 0. The leftmost three nodes re sources, generting messges c,, nd b from top to bottom, respectively. The rightmost four nodes re receivers nd demnd messges, c, b, nd, respectively. Five of the nodes re lbeled by x 0, x 1, y 1, z 0, or z. Input 1 Input 2 Input 3 New node c b 3 New receiver Fig. 2. A generic network component C i for 1 i 7. Input 1, Input 2, nd Input 3 re existing nodes in the network nd the remining nodes nd edges in component C i re new. The rightmost node, New receiver, demnds one messge. Tble I lists seven different instntitions of this generic network component tht re used in network construction. III. MATROIDALITY OF CONSTRUCTED NETWORKS First we review the concepts of mtroids, mtroidl networks, nd the finite projective plne, ech of which New New Comp. Input 1 Input 2 Input 3 node demnd C 1 (i) x 0 x 1 z x αi C 2 (q) z 0 z x q z q c C 3 (q) x q z 1 z 0 y q C 4 (q) x q z 0 z u q c C 5 (q) x 0 z q z u q c C 6 (q, r) z q u r z x q+r C 7 (q, r) z q y r z x qr TABLE I INSTANTIATIONS OF THE GENERIC NETWORK COMPONENT SHOWN IN FIGURE 2. EACH LINE IN THE TABLE GIVES THE FIVE VALUES THAT ARE USED TO FORM A SPECIFIED COMPONENT. will be used in wht follows. A mtroid M (e.g. see [16]) is n ordered pir (S, I), where S is finite set nd I is set of subsets of S stisfying the following three conditions: (I1) I. (I2) If I I nd J I, then J I. (I3) If I, J I nd J < I, then e I J such tht J {e} I. The set S is clled the ground set, the members of I re clled independent sets, nd ny subset of S not in I is clled dependent set. For ny mtroid M = (S, I) nd ny X S, let I X = {I X : I I}, nd let M X = (X, I X). Then M X is mtroid nd the rnk of X, denoted ρ(x), is the (unique) size of mximl independent set of M X. The rnk of the mtroid M is defined to be ρ(s). Let N be network with messge set µ, node set ν, nd edge set ɛ. Let M = (S, I) be mtroid with rnk function ρ. The network N is mtroidl network (see [7]) ssocited with M if there exists function f : µ ɛ S such tht the following conditions hold: (M1) f is one-to-one on µ. (M2) f(µ) I. (M3) ρ(f(in(x))) = ρ(f(in(x) Out(x))), x ν. It ws shown in [7] tht mny interesting networks re mtroidl, including ll networks tht re sclr-linerly solvble over finite field (e.g. solvble multicst networks). The mtroid used is vector spce over the finite field (with dimension the number of messges); the function f mps the messges to elementry vectors (vectors which re ll 0 except for single 1) nd mps the edges to the corresponding globl coding vectors (see, e.g., [11]) for the given sclr-liner code. In [7], method ws presented for constructing, from given mtroids, (mtroidl) networks which reflect some Pge 4 of 5
5 of the mtroids properties. This construction ws used to obtin networks used to prove vrious results in the literture [5], [6], [8]. For exmple, in [7], network ws constructed from the Vámos mtroid tht demonstrtes the insufficiency of using Shnnon-type informtion inequlities to compute network coding cpcity. In wht follows, we will prove tht if network is constructed from solvble polynomil collection s in Section II, then the network is mtroidl. The network construction lgorithm given in Section II ws inspired by the 1936 work of Sunders McLne in [14]. For ny positive integer n, projective plne (e.g. see [3]) comprises set of points, set of lines, nd n incidence reltion between points nd lines stisfying: (P1) Any two points re incident to exctly one line. (P2) Any two lines re incident to exctly one point. (P3) There exist 4 points, no 3 of which re incident to the sme line. Every finite projective plne induces rnk-three mtroid s follows. Let S be the set of ll points in the projective plne, let I be the collection of subsets of S of crdinlity t most 3 tht do not contin 3 colliner points, nd let M = (S, I). It is esy to see tht M stisfies (I1) nd (I2). Suppose I, J I where I > J. Then J {0, 1, 2}. If J < 2, then for ny v I J, we trivilly hve J {v} I. If J = 2 nd if for ech v I J we hve J {v} I, then the 3 points in I re colliner, contrdicting I I. Thus, M lso stisfies (I3), nd therefore M is rnk-3 mtroid. For ny field F, one cn construct projective plne Π F (of order F if F is finite) s follows. Let Π F = (F F ) F { } where two points (, b) nd (c, d) in F F re sid to hve slope s F if c nd s = (d b)(c ) 1, nd slope s = if = c. A line in Π F consists of n element s of F { } (clled point t infinity) together with mximl set of points in F F such tht every two of them hve slope 1/s (where we mke the convention tht v/0 = nd v/ = 0, for ll nonzero v F ). The set of ll points t infinity is lso considered line nd its point t infinity is. It cn be verified tht xioms (P1) (P3) hold for Π F. McLne [14] (see lso [19, pp ]) used this construction s follows. Let P be polynomil collection nd let K be finite field such tht P hs solution over K. Then McLne constructs mtroid M tht is representble over K nd such tht, for ny finite field F, if M is representble over F, then P hs solution over F. However, it is not necessrily true tht, if P hs solution over F, then M is representble over F. Such n if-nd-only-if result is not ttinble in generl for mtroids; for instnce, it is known tht, if mtroid is representble over the 2-element field nd the 3-element field, then it is representble over ll finite fields [16, Theorem 6.6.3]. The extr flexibility of networks llows us to construct network solvbly equivlent to ny given polynomil collection. REFERENCES [1] R. Bines nd P. Vámos, An lgorithm to compute the set of chrcteristics of system of polynomil equtions over the integers, Journl of Symbolic Computtion, vol. 35, pp , [2] T. Becker, V. Weispfenning, nd H. Kredel, Gröbner Bses: A Computtionl Approch to Commuttive Algebr, Springer- Verlg, New York, [3] L.M. Blumenthl, A Modern View of Geometry, Dover Publictions, Inc., New York, [4] R. Dougherty, C. Freiling, nd K. Zeger, Linerity nd solvbility in multicst networks, IEEE Trnsctions on Informtion Theory vol. 50, no. 10, pp , October [5] R. Dougherty, C. Freiling, nd K. Zeger, Insufficiency of liner coding in network informtion flow, IEEE Trnsctions on Informtion Theory, vol. 51, no. 8, pp , August [6] R. Dougherty, C. Freiling, nd K. Zeger, Unchievbility of network coding cpcity, IEEE Trnsctions on Informtion Theory & IEEE/ACM Trnsctions on Networking (joint issue), vol. 52, no. 6, pp , June [7] R. Dougherty, C. Freiling, nd K. Zeger, Networks, mtroids, nd non-shnnon informtion inequlities, IEEE Trnsctions on Informtion Theory, vol. 53, no. 6, pp , June [8] R. Dougherty nd K. Zeger, Nonreversibility nd equivlent constructions of multiple-unicst networks, IEEE Trns. on Info. Theory, vol. 52, no. 11, pp , November [9] M. Feder, D. Ron, nd A. Tvory, Bounds on liner codes for network multicst, Electronic Colloquium on Computtionl Complexity (ECCC), Report 33, pp. 1-9, [10] T. Ho, D. Krger, M. Médrd nd R. Koetter, Network coding from network flow perspective, IEEE Interntionl Symposium on Informtion Theory, Yokohm, Jpn, p. 441, June [11] S. Jggi, P. Snders, P. A. Chou, M. Effros, S. Egner, K. Jin, nd L. Tolhuizen, Polynomil time lgorithms for multicst network code construction, IEEE Trnsctions on Informtion Theory vol. 51, no. 6, pp , June [12] R. Koetter nd M. Médrd, An lgebric pproch to network coding, IEEE/ACM Trnsctions on Networking, vol. 11, no. 5, pp , October [13] S.-Y. R. Li, R. W. Yeung, nd N. Ci, Liner network coding, IEEE Trnsctions on Informtion Theory, vol. IT-49, no. 2, pp , Februry [14] S. McLne, Some interprettions of bstrct liner dependence in terms of projective geometry, Americn Journl of Mthemtics, vol. 58, no. 1, pp , Jnury [15] M. Médrd, M. Effros, T. Ho, D. Krger, On coding for nonmulticst networks, 41st Annul Allerton Conf. on Communiction Control nd Computing, Monticello, Illinois, October [16] J. G. Oxley, Mtroid Theory, Oxford University Press, New York, [17] A. Rsl Lehmn nd E. Lehmn, Complexity clssifiction of network informtion flow problems, 41st Annul Allerton Conference on Communiction Control nd Computing, Monticello, Illinois, October [18] S. Riis, Liner versus non-liner boolen functions in network flow, 38th Annul Conference on Informtion Sciences nd Systems (CISS), Princeton, NJ, Mrch [19] N. L. White (editor), Combintoril Geometries, Encyclopedi of Mthemtics nd its Applictions, Cmbridge Univ. Press, [20] R. W. Yeung, A First Course in Informtion Theory, Kluwer, Pge 5 of 5
Generalized Fano and non-fano networks
Generlized Fno nd non-fno networks Nildri Ds nd Brijesh Kumr Ri Deprtment of Electronics nd Electricl Engineering Indin Institute of Technology Guwhti, Guwhti, Assm, Indi Emil: {d.nildri, bkri}@iitg.ernet.in
More informationIN GAUSSIAN INTEGERS X 3 + Y 3 = Z 3 HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A2 IN GAUSSIAN INTEGERS X + Y = Z HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH Elis Lmpkis Lmpropoulou (Term), Kiprissi, T.K: 24500,
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationarxiv: v1 [math.ra] 1 Nov 2014
CLASSIFICATION OF COMPLEX CYCLIC LEIBNIZ ALGEBRAS DANIEL SCOFIELD AND S MCKAY SULLIVAN rxiv:14110170v1 [mthra] 1 Nov 2014 Abstrct Since Leibniz lgebrs were introduced by Lody s generliztion of Lie lgebrs,
More informationZero-Sum Magic Graphs and Their Null Sets
Zero-Sum Mgic Grphs nd Their Null Sets Ebrhim Slehi Deprtment of Mthemticl Sciences University of Nevd Ls Vegs Ls Vegs, NV 89154-4020. ebrhim.slehi@unlv.edu Abstrct For ny h N, grph G = (V, E) is sid to
More informationLinearly Similar Polynomials
Linerly Similr Polynomils rthur Holshouser 3600 Bullrd St. Chrlotte, NC, US Hrold Reiter Deprtment of Mthemticl Sciences University of North Crolin Chrlotte, Chrlotte, NC 28223, US hbreiter@uncc.edu stndrd
More informationElementary Linear Algebra
Elementry Liner Algebr Anton & Rorres, 1 th Edition Lecture Set 5 Chpter 4: Prt II Generl Vector Spces 163 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 163 คณตศาสตรวศวกรรม 3 สาขาวชาวศวกรรมคอมพวเตอร
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationResearch Article Moment Inequalities and Complete Moment Convergence
Hindwi Publishing Corportion Journl of Inequlities nd Applictions Volume 2009, Article ID 271265, 14 pges doi:10.1155/2009/271265 Reserch Article Moment Inequlities nd Complete Moment Convergence Soo Hk
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationIntuitionistic Fuzzy Lattices and Intuitionistic Fuzzy Boolean Algebras
Intuitionistic Fuzzy Lttices nd Intuitionistic Fuzzy oolen Algebrs.K. Tripthy #1, M.K. Stpthy *2 nd P.K.Choudhury ##3 # School of Computing Science nd Engineering VIT University Vellore-632014, TN, Indi
More informationN 0 completions on partial matrices
N 0 completions on prtil mtrices C. Jordán C. Mendes Arújo Jun R. Torregros Instituto de Mtemátic Multidisciplinr / Centro de Mtemátic Universidd Politécnic de Vlenci / Universidde do Minho Cmino de Ver
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationNumerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More informationBernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
More informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationHere we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
More informationElements of Matrix Algebra
Elements of Mtrix Algebr Klus Neusser Kurt Schmidheiny September 30, 2015 Contents 1 Definitions 2 2 Mtrix opertions 3 3 Rnk of Mtrix 5 4 Specil Functions of Qudrtic Mtrices 6 4.1 Trce of Mtrix.........................
More informationOn the degree of regularity of generalized van der Waerden triples
On the degree of regulrity of generlized vn der Werden triples Jcob Fox Msschusetts Institute of Technology, Cmbridge, MA 02139, USA Rdoš Rdoičić Deprtment of Mthemtics, Rutgers, The Stte University of
More informationa a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationTheory of Computation Regular Languages. (NTU EE) Regular Languages Fall / 38
Theory of Computtion Regulr Lnguges (NTU EE) Regulr Lnguges Fll 2017 1 / 38 Schemtic of Finite Automt control 0 0 1 0 1 1 1 0 Figure: Schemtic of Finite Automt A finite utomton hs finite set of control
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner
More informationPre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs
Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationCHAPTER 9. Rational Numbers, Real Numbers, and Algebra
CHAPTER 9 Rtionl Numbers, Rel Numbers, nd Algebr Problem. A mn s boyhood lsted 1 6 of his life, he then plyed soccer for 1 12 of his life, nd he mrried fter 1 8 more of his life. A dughter ws born 9 yers
More informationFrobenius numbers of generalized Fibonacci semigroups
Frobenius numbers of generlized Fiboncci semigroups Gretchen L. Mtthews 1 Deprtment of Mthemticl Sciences, Clemson University, Clemson, SC 29634-0975, USA gmtthe@clemson.edu Received:, Accepted:, Published:
More informationArithmetic & Algebra. NCTM National Conference, 2017
NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationCSCI FOUNDATIONS OF COMPUTER SCIENCE
1 CSCI- 2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 My 7, 2015 2 Announcements Homework 9 is due now. Some finl exm review problems will be posted on the web site tody. These re prcqce problems not
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationTheory of Computation Regular Languages
Theory of Computtion Regulr Lnguges Bow-Yw Wng Acdemi Sinic Spring 2012 Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring 2012 1 / 38 Schemtic of Finite Automt control 0 0 1 0 1 1 1 0 Figure: Schemtic of
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationBisimulation. R.J. van Glabbeek
Bisimultion R.J. vn Glbbeek NICTA, Sydney, Austrli. School of Computer Science nd Engineering, The University of New South Wles, Sydney, Austrli. Computer Science Deprtment, Stnford University, CA 94305-9045,
More informationTorsion in Groups of Integral Triangles
Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More information1. Extend QR downwards to meet the x-axis at U(6, 0). y
In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions
More informationWe will see what is meant by standard form very shortly
THEOREM: For fesible liner progrm in its stndrd form, the optimum vlue of the objective over its nonempty fesible region is () either unbounded or (b) is chievble t lest t one extreme point of the fesible
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationKRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION
Fixed Point Theory, 13(2012), No. 1, 285-291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd
More informationDecomposition of terms in Lucas sequences
Journl of Logic & Anlysis 1:4 009 1 3 ISSN 1759-9008 1 Decomposition of terms in Lucs sequences ABDELMADJID BOUDAOUD Let P, Q be non-zero integers such tht D = P 4Q is different from zero. The sequences
More informationSemigroup of generalized inverses of matrices
Semigroup of generlized inverses of mtrices Hnif Zekroui nd Sid Guedjib Abstrct. The pper is divided into two principl prts. In the first one, we give the set of generlized inverses of mtrix A structure
More informationMath 61CM - Solutions to homework 9
Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationReversals of Signal-Posterior Monotonicity for Any Bounded Prior
Reversls of Signl-Posterior Monotonicity for Any Bounded Prior Christopher P. Chmbers Pul J. Hely Abstrct Pul Milgrom (The Bell Journl of Economics, 12(2): 380 391) showed tht if the strict monotone likelihood
More informationA Criterion on Existence and Uniqueness of Behavior in Electric Circuit
Institute Institute of of Advnced Advnced Engineering Engineering nd nd Science Science Interntionl Journl of Electricl nd Computer Engineering (IJECE) Vol 6, No 4, August 2016, pp 1529 1533 ISSN: 2088-8708,
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationLECTURE. INTEGRATION AND ANTIDERIVATIVE.
ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationSOME relationships among events happening in
Possible nd Impossible Vector Clock Sets Estebn Meneses nd Frncisco J. Torres-Rojs Abstrct It is well known tht vector clocks cpture perfectly the cuslity reltionship mong events in distributed system.
More informationCS 275 Automata and Formal Language Theory
CS 275 Automt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Problem (II) Chpter II.5.: Properties of Context Free Grmmrs (14) Anton Setzer (Bsed on book drft by J. V. Tucker nd K. Stephenson)
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationIST 4 Information and Logic
IST 4 Informtion nd Logic T = tody x= hw#x out x= hw#x due mon tue wed thr fri 31 M1 1 7 oh M1 14 oh 1 oh 2M2 21 oh oh 2 oh Mx= MQx out Mx= MQx due 28 oh M2 oh oh = office hours 5 3 12 oh 3 T 4 oh oh 19
More informationRudin s Principles of Mathematical Analysis: Solutions to Selected Exercises. Sam Blinstein UCLA Department of Mathematics
Rudin s Principles of Mthemticl Anlysis: Solutions to Selected Exercises Sm Blinstein UCLA Deprtment of Mthemtics Mrch 29, 2008 Contents Chpter : The Rel nd Complex Number Systems 2 Chpter 2: Bsic Topology
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationCS375: Logic and Theory of Computing
CS375: Logic nd Theory of Computing Fuhu (Frnk) Cheng Deprtment of Computer Science University of Kentucky 1 Tble of Contents: Week 1: Preliminries (set lgebr, reltions, functions) (red Chpters 1-4) Weeks
More informationMAC-solutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationSelf-similarity and symmetries of Pascal s triangles and simplices mod p
Sn Jose Stte University SJSU ScholrWorks Fculty Publictions Mthemtics nd Sttistics Februry 2004 Self-similrity nd symmetries of Pscl s tringles nd simplices mod p Richrd P. Kubelk Sn Jose Stte University,
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationBoolean Algebra. Boolean Algebras
Boolen Algebr Boolen Algebrs A Boolen lgebr is set B of vlues together with: - two binry opertions, commonly denoted by + nd, - unry opertion, usully denoted by or ~ or, - two elements usully clled zero
More informationThe Henstock-Kurzweil integral
fculteit Wiskunde en Ntuurwetenschppen The Henstock-Kurzweil integrl Bchelorthesis Mthemtics June 2014 Student: E. vn Dijk First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. A. vn der Schft
More informationA General Dynamic Inequality of Opial Type
Appl Mth Inf Sci No 3-5 (26) Applied Mthemtics & Informtion Sciences An Interntionl Journl http://dxdoiorg/2785/mis/bos7-mis A Generl Dynmic Inequlity of Opil Type Rvi Agrwl Mrtin Bohner 2 Donl O Regn
More informationSolution to Fredholm Fuzzy Integral Equations with Degenerate Kernel
Int. J. Contemp. Mth. Sciences, Vol. 6, 2011, no. 11, 535-543 Solution to Fredholm Fuzzy Integrl Equtions with Degenerte Kernel M. M. Shmivnd, A. Shhsvrn nd S. M. Tri Fculty of Science, Islmic Azd University
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationVariational Techniques for Sturm-Liouville Eigenvalue Problems
Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More informationMath Lecture 23
Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of
More informationHilbert Spaces. Chapter Inner product spaces
Chpter 4 Hilbert Spces 4.1 Inner product spces In the following we will discuss both complex nd rel vector spces. With L denoting either R or C we recll tht vector spce over L is set E equipped with ddition,
More informationA Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions
Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, 2451-2460 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2014.49294 A Convergence Theorem for the Improper Riemnn Integrl of Bnch
More informationNatural examples of rings are the ring of integers, a ring of polynomials in one variable, the ring
More generlly, we define ring to be non-empty set R hving two binry opertions (we ll think of these s ddition nd multipliction) which is n Abelin group under + (we ll denote the dditive identity by 0),
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationIntegral points on the rational curve
Integrl points on the rtionl curve y x bx c x ;, b, c integers. Konstntine Zeltor Mthemtics University of Wisconsin - Mrinette 750 W. Byshore Street Mrinette, WI 5443-453 Also: Konstntine Zeltor P.O. Box
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationAutomata and Languages
Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering Lb. The University of Aizu Jpn Grmmr Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Regulr Lnguges Context Free Lnguges Context Sensitive
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updted /) Lecture 8- - Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More informationUniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More information1.4 Nonregular Languages
74 1.4 Nonregulr Lnguges The number of forml lnguges over ny lphbet (= decision/recognition problems) is uncountble On the other hnd, the number of regulr expressions (= strings) is countble Hence, ll
More informationPositive Solutions of Operator Equations on Half-Line
Int. Journl of Mth. Anlysis, Vol. 3, 29, no. 5, 211-22 Positive Solutions of Opertor Equtions on Hlf-Line Bohe Wng 1 School of Mthemtics Shndong Administrtion Institute Jinn, 2514, P.R. Chin sdusuh@163.com
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationFundamental Theorem of Calculus and Computations on Some Special Henstock-Kurzweil Integrals
Fundmentl Theorem of Clculus nd Computtions on Some Specil Henstock-Kurzweil Integrls Wei-Chi YANG wyng@rdford.edu Deprtment of Mthemtics nd Sttistics Rdford University Rdford, VA 24142 USA DING, Xiofeng
More information