FunctionalRepairbyTransfer Regenerating Codes


 Garey Cameron
 2 years ago
 Views:
Transcription
1 FunctionalRepairbyTransfer Regenerating Codes Kenneth W Shum and Yuchong Hu Abstract In a distributed storage system a data file is distributed to several storage nodes such that the original file can be decoded from any subset of the storage nodes of size larger than or equao a certain threshold Upon the failure of a storage node we would like to regenerate it with minimal amount of data transmissions from the surviving nodes to the new node This performance metric is called the repairbandwidth Another performance metric is the disk input/output (I/O) cost which measures the number of bits a storage node needs to read out from its memory in order to repair the failed node In this paper we give examples of linear regenerating codes with minimal disk I/O cost and repairbandwidth without any linear mixing in the helping storage nodes Index Terms Cloud storage distributed storage system regenerating codes network coding I INTRODUCTION Regenerating codes as introduced by Dimakis et al in [2] is a class of erasure codes for distributed storage systems A source data file is encoded across the storage nodes such that a data collector can decode the source data file by connecting to a fraction of the storage nodes Should a storage node fail the failed node can be repaired by downloading some data packets from the surviving nodes The aim of the design of regenerating codes is to minimize the the totaraffic required in the repair process Besides the bandwidth requirement another metric that arises in practice is the disk I/O cost: we want to minimize the number of bits that a surviving node must read out from its memory during the repair of the failed node In the extreme case where disk I/O cost is minimal the number of bits read out from the memory is exactly equao the number of bits to be sent out Data combining is only required in the receiving end Regenerating code with minimal disk I/O cost is called a repairbytransfer regenerating code Some constructions of repairbytransfer regenerating codes attaining minimum repairbandwidth can be found in [3] [4] In this paper we will consider minimumstorage regenerating codes This class of regenerating codes find applications in multiplecloud storages [5] We give an example of repairbytransfer regenerating code which has the same parameters as in the example in [2 Fig 2] but linear combining is not required in the surviving storage nodes The encoding of packets are shown in Fig 1 The first Part of this work was presented in Information Theory and Applications Workshop San Diego Feb 2012 [1] The work of K W Shum and Y Hu was partially supported by a grant from the University Grants Committee (Project No AoE/E02/08) of the Hong Kong Special Administrative Region China K W Shum and Y Hu are with Institute of Network Coding The Chinese University of Hong Kong Shatin Hong Kong Fig 1 1st packet 2nd packet Node 1 A C + D Node 2 B D + A Node 3 C A + B Node 4 D B + C An example of exact regenerating codes with repairbytransfer packet in each storage node is an uncoded packet the second one is the sum of two packets with addition performed in F 2 the finite field of size 2 Let x y denote the unique integer z { } such that x + y i mod 4 The paritycheck packets in nodes i 2 is the sum of the uncoded packets in node i and node i 1 One can verify that the four information packets can be decoded from any two nodes For example from nodes 1 and 3 packets A and C can be obtained directly and packet B (resp D) can be decoded by adding packet A (resp C) to A+B (resp C +D) If node 1 fails we can repair it by downloading packets D +A from node 2 packet C from node 3 and packet D from node 4 Packet A can be recovered by adding packets D + A and D and packet C + D can be recovered by adding packets C and D By the cyclic structure of the code nodes 2 3 and 4 can be repaired similarly We also note that data update can be performed efficiently; should an information packet be updated we only need to modify one uncoded packet and two paritycheck packets in the system The above example falls under the category of exact repair In this paper we consider functional repair Regenerating codes for functional repair in general is given in Section II and the repairbytransfer subclass is discussed in Section III A family of repairbytransfer regenerating codes for functional repair is detailed in Section IV II NETWORK CODES FOR DISTRIBUTED STORAGE Let n be the number of storage nodes A data file of size M is encoded and distributed to the n storage nodes The amount of data stored in each node is denoted by α We divide the time into stages Initially the system start at stage 0 After a node failure in stage t the failed node is repaired and the time is advanced to the (t + 1)st stage The design objectives of regenerating code for distributed storage network (DSN) are: (Node repair) Upon the failure of a storage node in stage t we recover it by setting up a newcomer who contacts any d surviving nodes called the helpers and downloads β units of data from each of them The total amount of data transmitted from the helpers is called the repairbandwidth and is denoted by γ = dβ (File retrieval) In any stage a data collector can decode the
2 original data file by downloading from any k storage nodes We calhis property the (n k) recovery property If α is equal to M/k which is the minimum possible value for α then the system is said to be maximaldistance separable (MDS) Any code which realizes the (n k) recovery property and repairs a failed node by connecting to d surviving nodes is called an (n d k) regenerating code or simply a regenerating code if the parameters are understood from the context There are two modes of repair The first one is called functional repair and the second one is exact repair In functional repair the content of the newcomer is not necessarily the same as the content in the failed storage nodes Only the (n k) recovery property is preserved In exact repair the content of the newcomer should be exactly the same as in the failed node For a given α there is a fundamental limit for the repair cost measured in terms of the repairbandwidth For functional repair a lower bound of repairbandwidth can be derived via the maxflow bound for singlesource multicasting [2] In the MDS case the lower bound on repairbandwidth is given by γ M k(d + 1 k) (1) It is proved in [6] that for functional repair the lower bound of repairbandwidth in (1) can be achieved by linear network code over a fixed finite field In the following we describe the realization of regenerating codes by linear network codes Let q be a prime power and let the finite field of size q be denoted by F q The parameters B α and β of the DSN are all integers In the following we will refer to an element in F q as a packet as well The file is divided into many stripes of data each consisting of B packets Each stripe of data will be encoded in the same way In the remaining of this paper we will focus on one stripe of data and suppose that the data file consists of B packets m 1 m 2 m B These B symbols are called the message symbols Each of the n storage nodes stores α packets obtained by taking some linear combinations of the message symbols The vector formed by the B coefficients of the linear combination associated to a packet is called the global encoding vector of the packet In the tth stage we denote the global encoding vector of the jth packet stored in the ith node by Γ t (i j) for i = 1 2 n and j = 1 2 α Thus the jth packet in the ith node is equao the dot product of Γ t (i j) and [m 1 m 2 m B ] We call (i j) the index of the vector Γ t (i j) We will view Γ t (i j) as a function mapping an index (i j) to a global encoding vector in stage t The definition of Γ t is extended as a function with sets of indices as domain by defining Γ t (X ) := {Γ t (x) : x X } for any set of indices X Let i be the α B encoding matrix whose rows are precisely the global encoding vectors Γ t (i 1) Γ t (i α) and m be the column vector [m 1 m 2 m B ] T (We will use superscript (t) to indicate that a variable associated with stage t and superscript T for the transpose operator) The packets stored within the ith node in the tth stage are the components in i m The DSN is initialized such that at stage 0 the global encoding vectors in any k storage nodes form a fullrank matrix Suppose that node i fails in stage t and the helpers are nodes h 2 For j = 1 2 d helper h j sends β packets to the newcomer Let be the β α local encoding matrix of helper at stage t The packets sent from helper to the newcomer are the components in m The newcomer takes some linear combinations of the received packets and compute α packets by H (t) i m where H (t) i is an α (dβ) matrix over F q After the repair process in stage t the global encoding matrices can be updated by j if j i j = H (t) i if j = i If we want to repair the DSN exactly the encoding matrix j should equal j for all j For functional repair we just want to maintain the (n k) recovery property namely at any stage and in any set of k storage nodes the rank of the kα global encoding vectors is equao B III REPAIRBYTRANSFER FOR FUNCTIONAL REPAIR In a repairbytransfer regenerating code we choose the local encoding matrix h j such that there is exactly one 1 in each row while the other entries are all zero In the remaining of this paper we will focus on (n n 1 2) repairbytransfer regenerating codes for functional repair with parameters d = n 1 k = 2 B = k(d + 1 k) = 2(n 2) α = B/k = n 2 and β = 1 ie any two storage nodes are sufficient in rebuilding the original file and each node stores B/2 packets (the MDS case) Any regenerating code with the above parameters is optimal with respect to the bound in (1) For repairbytransfer regenerating code with the above parameters the local encoding matrix j is a zeroone row vector If node fails in stage t for j we let χ(t j) be the index of the unique 1 in ie node j sends j
3 packet χ(t j) to node The function χ(t j) is called the choice function and takes value between 1 and α Let R t ( ) := {(j χ(t j) : j {1 2 n} \ { }} be the set of indices of the global encoding vectors used in the repair of node For example in Fig 1 if node 1 fails we can repair node 1 using packets C A + C and B + C from nodes 2 3 and 4 respectively The choice function is χ(t 1 2) = 1 χ(t 1 3) = 1 and χ(t 1 4) = 2 and we have R t (4) = {(2 1) (3 1) (4 2)} In case node fails in stage t the repairbytransfer process consists of the following steps: (i) For j node j sends the χ(t j)th packet in its memory to the newcomer (ii) Put the packets received by the newcomer together as a column vector and multiply it by an α (n 1) local encoding matrix H (t) Update the global encoding matrix of the newly reconstructed node to = H (t) Γ t (1 χ(t 1)) Γ t ( 1 χ(t 1)) Γ t ( + 1 χ(t + 1)) (2) Γ t (n χ(t n)) (iii) Determine the choice function for the repair in the next stage χ(t + 1 i j ) for i and j {1 2 α} \ {i } We introduce some notations Let and C i := {(i 1) (i 2) (i α)} C ij := C i C j The (n 2) property requires that for any t the global encoding vectors indexed by C ij with i j in stage t are linearly independent Example 1 Consider a (5 4 2) regenerating code In stage t the global encoding vectors can be tabulated as Node 1 Γ t (1 1) Γ t (1 2) Γ t (1 3) Node 2 Γ t (2 1) Γ t (2 2) Γ t (2 3) Node 3 Γ t (3 1) Γ t (3 2) Γ t (3 3) Node 4 Γ t (4 1) Γ t (4 2) Γ t (4 3) Node 5 Γ t (5 1) Γ t (5 2) Γ t (5 3) Suppose that node 1 fails in stage t and we want to repair node 1 by the first packets Γ t (2 1) Γ t (3 1) Γ t (4 1) and Γ t (5 1) in node 2 to node 5 respectively In order to maintain the (n 2) recovery property in stage t + 1 we need to guarantee that after the repair each of the five sets of global encoding vectors Γ t+1 (C 12 ) Γ t+1 (C 13 ) Γ t+1 (C 14 ) and Γ t+1 (C 15 ) are linearly independent For integer j between 1 and n let S t ( j) := C j R t ( ) The set S t ( j) is interpreted as the index set of the packets of node j and the packets used in repairing node in stage t Continued from Example 1 we have S t (1 2) = {(2 1) (2 2) (2 3) (3 1) (4 1) (5 1)} S t (1 3) = {(3 1) (3 2) (3 3) (2 1) (4 1) (5 1)} S t (1 4) = {(4 1) (4 2) (4 3) (2 1) (3 1) (5 1)} S t (1 5) = {(5 1) (5 2) (5 3) (2 1) (3 1) (4 1)} Graphically these four sets of indices can be illustrated by the following arrays: We note that for j the set S t ( j) consists of 2(n 2) elements because C j = n 2 R t ( ) = n 1 and C j R t ( ) = 1 When = j S t ( j) = 2n 3 Suppose that the (n 2) recovery property holds in stage t In order to maintain the (n 2) recovery property after the repair of node we need to check that each of the following sets of global encoding vectors Γ t+1 (C ltj) for j {1 2 n} \ { } are linearly independent (We do not need to check Γ t+1 (C jj ) for j j because Γ t (C jj ) = Γ t+1 (C jj )) We make the following observations for j : (i) Each vector in Γ t+1 (C lt j) can be obtained as a linear combination of the vectors in Γ t (S t ( j)) Therefore if the vectors in Γ t (S t ( j)) are linearly dependent then the vectors in Γ t+1 (C ltj) are also linearly dependent (ii) If the vectors in Γ t (S t ( j)) are linearly independent then we can choose a local encoding matrix H (t) in stage t such that the vectors in Γ t+1 (C lt j) are linearly independent In fact we can use a (n 2) (n 1) matrix H whose (r s)entry is { 1 s = r or s = r + 1 H(r s) = 0 otherwise as the linearlycombining matrix H (t) at the newcomer in all stage t and for all node [ being ] repaired For example if n = we have H (t) i = 0110 This matrix works because after 0011 eliminating any column matrix the resulting square matrix has full rank This proves that for each j we can choose H (t) such that a newcomer in the next stage can decode the original data file from the packets in nodes j and A standard application of the SchwartzZippel s lemma in network coding theory [7] shows that there is a choice of H (t) such that all data collectors in the next stage are satisfied if the finite field size is large enough This gives the following lemma
4 Lemma 1 Suppose that node fails in stage t and the choice function χ is given If for each j the set of vectors in Γ t (S t ( j)) are linearly independent then we can choose H (t) such that the vectors in Γ t+1 (C ltj) are linearly independent for all j provided that the underlying finite field is sufficiently large The intuition from Lemma 1 is as follows The (n 2) recovery property per se requires that the global encoding vectors in any two storage nodes are linearly independent However in order to sustain the (n 2) recovery property through repairbytransfer we need to ensure the linear independence of many other sets of global encoding vectors The next lemma serves as a tool for this purpose Recalhat for a set X of indices the notation Γ t (X ) stands for the set of global encoding vector corresponding to the indices in X in stage t Lemma 2 Suppose that node fails in stage t and packets with indices R t ( ) are used to repaired node Let X and Y be two sets of indices of global encoding vectors in stage t and t + 1 respectively such that Y \ X C lt and X \ Y R t ( ) If the set of vectors in Γ t (X ) are linearly independent then we can choose the local encoding matrix in stage t such that the vectors in Γ t+1 (Y) are linearly independent The proof of Lemma 2 is omitted IV MAIN THEOREM The choice function χ has to be carefully chosen otherwise the (n 2) recovery property cannot be preserved for unbounded number of stages As a counterexample suppose that in Example 1 we always pick the first packets in the memory of the surviving nodes in the repair process ie the function χ(t j) is identically equao 1 in all stages Then the dimension of the span of totality of encoding vectors in all nodes will eventually drop to four or less As the dimension of the original data file is six the data recovery will doom to failure The main theorem in this paper is that there is an appropriate choice function which preserve the (n 2) recovery property through repairbytransfer Theorem 3 For all n 4 there exists an (n n 1 2) MDS repairbytransfer regenerating code for functional repair meeting the lower bound on repairbandwidth in (1) Moreover the finite field size can be chosen to be any prime power larger than ( n(n 2) 2(n 2)) We will only sketch the proof of Theorem 3 in this section and go through an example for illustration The idea of proof is to come up with a choice function χ(t i j) and local encoding matrix H (t) i such that there are sufficiently many sets of linearly independent global encoding vectors in every stage We let B = k(d+1 k) = 2(n 2) α = n 2 and β = 1 by normalizing the unit if necessary In the proof of Theorem 3 we assume that no node will fail in two consecutive stages because say if node i fails in stage t and t + 1 we can repeat the repair procedure taken in stage t It is wellknown that under the condition q + 1 n(n 2) we can construct n(n 2) vectors of length B with components drawn from F q such that every subset of B vectors are linearly independent over F q [8] (These vectors form a generating matrix of a maximaldistance separable (MDS) code of dimension B over F q ) In stage 0 we take these n(n 2) vectors as the initial global encoding vectors Since any set of B of them are linear independent by construction the global encoding vectors in any two storage nodes are linear independent Thus the (n 2) recover property holds in the initial stage We will display a choice function by a sequence of n (n 2) arrays For each t 0 we create an n (n 2) array and write i in the rth row and the χ(t i r)th column The array associated with stage t is interpreted as follows If node i fails in stage t for r i node r sends the jth packet to node i if the jth column in the array in stage t is labeled by i An example for n = 5 is shown in Fig 2 In stage 0 we initialize χ(0 i j) = 1 for all i and j ie no matter which node fails the surviving nodes send their first packets to the newcomer In the rth row we write the element in {1 2 5} \ {r} in the column labeled by Packet 1 Suppose that node 1 fails in stage 0 ie l 0 = 1 The newcomer receives packets with global encoding vector Γ 0 (2 1) Γ 0 (3 1) Γ 0 (4 1) and Γ 0 (5 1) The indices of these four packets are R 0 (1) = {(2 1) (3 1) (4 1) (5 1)} The packets in the repaired node 1 are linear combinations of these four packets Then we specify the choice function for stage t = 1 From the hypothesis that no node fails in two consecutive stages node 1 will not fail in stage 1 The 5 3 array associated with stage 1 does not contain the label 1 We design the choice function by evenly spreading the labels 2 to 5 in the array In rows 2 to 5 each entry in the array contains exactly one label Each packet is potentially used in the repair process after stage 1 depending on which node fails in stage 1 This evenly spread property is expressed mathematically as Property (a): For j 1 χ(t i j) = χ(t i j) only if i = i From the discussion at the beginning of this section we try to avoid using the same packet in the repair process of two consecutive stages This motivates the second heuristics in the design of the choice function Since the first packets in nodes 2 to 5 have been used in the repair process before stage 1 we want to impose the requirement that at most one of these four packets are used again in the repair process after stage 1 As there are four possible node failures in stage 1 this implies that exactly one of these four packets is used in the repair of the node failure in stage 1 Indeed the choice function at stage 1 satisfies this requirement The four entries under Packet 1 corresponding to nodes 2 to 5 are distinct Suppose node 2 fails in stage 1 Node 1 and 5 send the first packets in their memory to the newcomer while nodes 3 and 4 send the last packets in their memory to the newcomer The indices of these four packets are R 1 (2) = {(1 1) (3 3) (4 3) (5 1)} We note that only the first packet in node 5 is used twice in the repair after stage 0 and stage 1 In
5 Stage 0 Node Node Node Node Node Stage Stage Stage Fig 2 An example of choice functions satisfying property (a) and (b) Node 1 fails in stage 0 node 2 fails in stage 1 and node 4 fails in stage 2 The integers 1 2 and 4 are highlighted (l 0 = 1 l 1 = 2 and l 2 = 4) settheoreticaerms we can write R 0 (1) R 1 (2) = {(5 1)} The choice function for t = 2 is illustrated in the array under Stage 2 in Fig 2 We note that the choice function in stage 2 satisfies Property (a) ie for r { } each entry in row r contains exactly one label Also the label 2 does not appear because it is assumed that node 2 will not fail in stage 2 We can check that R 1 (2) R 2 (1) = {(3 3)} R 1 (2) R 2 (3) = {(1 1)} R 1 (2) R 2 (4) = {(5 1)} R 1 (2) R 2 (5) = {(4 3)} We formulate this heuristic as follows Property (b) R t ( ) R t+1 (i) 1 for t 0 and i We write X t Y if Y and X satisfy the condition in Lemma 2 Under the choice function as in Fig 2 and l 0 = 1 l 1 = 2 we illustrate the proof idea using the following figure: Stage 0 01 Stage 1 12 Stage 2 The asterisks in each array indicate a set of six global encoding vectors In stage 2 it is required that the global encoding vectors in nodes 1 and 2 are linearly independent By Lemma 1 this can be guaranteed if the global encoding vectors S 1 (2 1) = {(1 1) (1 2) (1 3) (3 3) (4 3) (5 1)} in stage 1 are linearly independent and this in turns can be done as in Lemma 2 if the global encoding vectors with indices {(2 1) (3 1) (3 3) (4 1) (4 3) (5 1)} in stage 0 are linearly independent We initialize the global encoding vectors in stage 0 in such a way that any six global encoding vectors are linearly independent Hence in stage 2 with suitable choice of local encoding matrices the six global encoding vectors in nodes 1 and 2 are linearly independent The main idea of proof is that the above example can be generalized to all C ij in all stages provided that the choice function satisfies Properties (a) and (b) and some other technical conditions The details of the proof is omitted due to space limitation V CONCLUDING REMARKS We give an existence proof of a family of MDS regenerating codes with the property that failed node is repaired by transferring packets received in earlier time This can be regarded as a linear code with coefficients restricted to {0 1} In a recent work by Wang et al in [9] a construction of MDS exactrepair regenerating code with optimal disk I/O cost for any k is given Their construction differs from the construction in this paper as being a vector linear regenerating code; the data sent from a surviving node to the newcomer is divided into many fragments and each fragment is a linear combination of the source data (β > 1) The regenerating code considered in this paper is scalar linear regenerating codes (β = 1) Another vector linear regenerating code for repairing the systematic nodes only can be found in [10] ACKNOWLEDGEMENT The authors would like to thank Raymond Yeung for the useful discussions REFERENCES [1] K W Shum and Y Hu Repairbytransfer in distributed storage system in Information Theory and Applications Workshop San Diego Feb 2012 [2] A G Dimakis P B Godfrey Y Wu M J Wainwright and K Ramchandran Network coding for distributed storage systems IEEE Trans Inf Theory vol 56 no 9 pp Sep 2010 [3] S El Rouayheb and K Ramchandran Fractional repetition codes for repair in distributed storage systems in Allerton conference on commun control and computing Monticello Sep 2010 [4] N B Shah K V Rashmi P V Kumar and K Ramchandran Distributed storage codes with repairbytransfer and nonachievability of interior points on the storagebandwidth tradeoff IEEE Trans Inf Theory vol 58 no 3 pp Mar 2012 [5] Y Hu H C H Chen P P C Lee and Y Tang NCCloud: Applying network coding for the storage repair in a cloudofclouds in Proc of the 10th USENIX Conf on File and Storage Tech (FAST 12) San Jose Feb 2012 [6] Y Wu Existence and construction of capacityachieving network codes for distributed storage IEEE J on Selected Areas in Commun vol 28 no 2 pp Feb 2010 [7] R Kötter and M Médard An algebraic approach to network coding IEEE/ACM Trans on Networking vol 11 no 5 pp Oct 2003 [8] R M Roth Introduction to Coding Theory Cambrdige: Cambridge University Press 2006 [9] Z Wang I Tamo and J Bruck On codes for optimal rebuilding access in Proc of the Allerton Conference Monticello Sep 2011 pp [10] V R Cadambe C Huang J Li and S Mehrotra Polynomial length MDS codes with optimal repair in distributed storage in the 45th Asilomar Conf on Signals Systems and Computers Pacific Grove Nov 2011 pp
Reliability Comparison of Various Regenerating Codes for Cloud Services
Reliability Comparison of Various Regenerating Codes for Cloud Services Yonsei Univ. Seoul, KORE JungHyun Kim, Jin Soo Park, KiHyeon Park, Inseon Kim, MiYoung Nam, and HongYeop Song ICTC 13, Oct. 1416,
More informationNetwork Coding for Distributed Storage
Network Coding for Distributed Storage Alex Dimakis USC Overview Motivation Data centers Mobile distributed storage for D2D Specific storage problems Fundamental tradeoff between repair communication and
More informationA Piggybacking Design Framework for Readand Downloadefficient Distributed Storage Codes
A Piggybacing Design Framewor for Readand Downloadefficient Distributed Storage Codes K V Rashmi, Nihar B Shah, Kannan Ramchandran, Fellow, IEEE Department of Electrical Engineering and Computer Sciences
More informationErasure Coding for Cloud Storage Systems: A Survey
TSINGHUA SCIENCE AND TECHNOLOGY ISSNll10070214ll06/11llpp259272 Volume 18, Number 3, June 2013 Erasure Coding for Cloud Storage Systems: A Survey Jun Li and Baochun Li Abstract: In the current era of
More informationSecurity for Wiretap Networks via RankMetric Codes
Security for Wiretap Networks via RankMetric Codes Danilo Silva and Frank R. Kschischang Department of Electrical and Computer Engineering, University of Toronto Toronto, Ontario M5S 3G4, Canada, {danilo,
More informationWeakly Secure Network Coding
Weakly Secure Network Coding Kapil Bhattad, Student Member, IEEE and Krishna R. Narayanan, Member, IEEE Department of Electrical Engineering, Texas A&M University, College Station, USA Abstract In this
More informationLinear Codes. Chapter 3. 3.1 Basics
Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length
More informationOverview. How distributed file systems work Part 1: The code repair problem Regenerating Codes. Part 2: Locally Repairable Codes
Overview How distributed file systems work Part 1: The code repair problem Regenerating Codes Part 2: Locally Repairable Codes Part 3: Availability of Codes Open problems 1 current hadoop architecture
More informationLinear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.
Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a subvector space of V[n,q]. If the subspace of V[n,q]
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationSECRET sharing schemes were introduced by Blakley [5]
206 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 1, JANUARY 2006 Secret Sharing Schemes From Three Classes of Linear Codes Jin Yuan Cunsheng Ding, Senior Member, IEEE Abstract Secret sharing has
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationBeyond the MDS Bound in Distributed Cloud Storage
Beyond the MDS Bound in Distributed Cloud Storage Jian Li Tongtong Li Jian Ren Department of ECE, Michigan State University, East Lansing, MI 488241226 Email: {lijian6, tongli, renjian}@msuedu Abstract
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationTHE problems of characterizing the fundamental limits
Beamforming and Aligned Interference Neutralization Achieve the Degrees of Freedom Region of the 2 2 2 MIMO Interference Network (Invited Paper) Chinmay S. Vaze and Mahesh K. Varanasi Abstract We study
More informationPractical Data Integrity Protection in NetworkCoded Cloud Storage
Practical Data Integrity Protection in NetworkCoded Cloud Storage Henry C. H. Chen Department of Computer Science and Engineering The Chinese University of Hong Kong Outline Introduction FMSR in NCCloud
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationA Solution to the Network Challenges of Data Recovery in Erasurecoded Distributed Storage Systems: A Study on the Facebook Warehouse Cluster
A Solution to the Network Challenges of Data Recovery in Erasurecoded Distributed Storage Systems: A Study on the Facebook Warehouse Cluster K. V. Rashmi 1, Nihar B. Shah 1, Dikang Gu 2, Hairong Kuang
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationLecture 11. Shuanglin Shao. October 2nd and 7th, 2013
Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous
More informationLecture Notes: Matrix Inverse. 1 Inverse Definition
Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationCodes for Network Switches
Codes for Network Switches Zhiying Wang, Omer Shaked, Yuval Cassuto, and Jehoshua Bruck Electrical Engineering Department, California Institute of Technology, Pasadena, CA 91125, USA Electrical Engineering
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationSecure Network Coding on a Wiretap Network
IEEE TRANSACTIONS ON INFORMATION THEORY 1 Secure Network Coding on a Wiretap Network Ning Cai, Senior Member, IEEE, and Raymond W. Yeung, Fellow, IEEE Abstract In the paradigm of network coding, the nodes
More informationTowards High Security and Fault Tolerant Dispersed Storage System with Optimized Information Dispersal Algorithm
Towards High Security and Fault Tolerant Dispersed Storage System with Optimized Information Dispersal Algorithm I Hrishikesh Lahkar, II Manjunath C R I,II Jain University, School of Engineering and Technology,
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationA Network Flow Approach in Cloud Computing
1 A Network Flow Approach in Cloud Computing Soheil Feizi, Amy Zhang, Muriel Médard RLE at MIT Abstract In this paper, by using network flow principles, we propose algorithms to address various challenges
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More informationOn the Locality of Codeword Symbols
On the Locality of Codeword Symbols Parikshit Gopalan Microsoft Research parik@microsoft.com Cheng Huang Microsoft Research chengh@microsoft.com Sergey Yekhanin Microsoft Research yekhanin@microsoft.com
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationInteger Factorization using the Quadratic Sieve
Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationFactoring Algorithms
Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationMOST errorcorrecting codes are designed for the equal
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 3, MARCH 2007 387 Transactions Letters Unequal Error Protection Using Partially Regular LDPC Codes Nazanin Rahnavard, Member, IEEE, Hossein PishroNik,
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationSolution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationMath 315: Linear Algebra Solutions to Midterm Exam I
Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More informationChapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6
Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a
More informationImplementation and Performance Evaluation of Distributed Cloud Storage Solutions using Random Linear Network Coding
Implementation and erformance Evaluation of Distributed Cloud Storage Solutions using Random Linear Network Coding Frank H.. Fitzek 1,2, Tamas Toth 2,3, Aron Szabados 2,3, Morten V. edersen 1, Daniel E.
More informationSecure Network Coding for Wiretap Networks of Type II
1 Secure Network Coding for Wiretap Networks of Type II Salim El Rouayheb, Emina Soljanin, Alex Sprintson Abstract We consider the problem of securing a multicast network against a wiretapper that can
More informationOptimal Index Codes for a Class of Multicast Networks with Receiver Side Information
Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information Lawrence Ong School of Electrical Engineering and Computer Science, The University of Newcastle, Australia Email: lawrence.ong@cantab.net
More informationGENERATING SETS KEITH CONRAD
GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationBy choosing to view this document, you agree to all provisions of the copyright laws protecting it.
This material is posted here with permission of the IEEE Such permission of the IEEE does not in any way imply IEEE endorsement of any of Helsinki University of Technology's products or services Internal
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all ndimensional column
More informationMathematics of Cryptography
CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationI. INTRODUCTION. of the biometric measurements is stored in the database
122 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL 6, NO 1, MARCH 2011 Privacy Security TradeOffs in Biometric Security Systems Part I: Single Use Case Lifeng Lai, Member, IEEE, SiuWai
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationCofactor Expansion: Cramer s Rule
Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating
More informationImage Compression through DCT and Huffman Coding Technique
International Journal of Current Engineering and Technology EISSN 2277 4106, PISSN 2347 5161 2015 INPRESSCO, All Rights Reserved Available at http://inpressco.com/category/ijcet Research Article Rahul
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationFacts About Eigenvalues
Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v
More informationAN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC
AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l4] on graphs and matroids I used definitions equivalent to the following.
More information1 Determinants. Definition 1
Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described
More informationNetwork Monitoring in Multicast Networks Using Network Coding
Network Monitoring in Multicast Networks Using Network Coding Tracey Ho Coordinated Science Laboratory University of Illinois Urbana, IL 6181 Email: trace@mit.edu Ben Leong, YuHan Chang, Yonggang Wen
More informationMatrix Inverse and Determinants
DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationTwo classes of ternary codes and their weight distributions
Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationThe Cayley Hamilton Theorem
The Cayley Hamilton Theorem Attila Máté Brooklyn College of the City University of New York March 23, 2016 Contents 1 Introduction 1 1.1 A multivariate polynomial zero on all integers is identically zero............
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationSec 4.1 Vector Spaces and Subspaces
Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationFace Recognition using Principle Component Analysis
Face Recognition using Principle Component Analysis Kyungnam Kim Department of Computer Science University of Maryland, College Park MD 20742, USA Summary This is the summary of the basic idea about PCA
More information= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are
This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In
More informationEfficient LDPC Code Based Secret Sharing Schemes and Private Data Storage in Cloud without Encryption
Efficient LDPC Code Based Secret Sharing Schemes and Private Data Storage in Cloud without Encryption Yongge Wang Department of SIS, UNC Charlotte, USA yonwang@uncc.edu Abstract. LDPC codes, LT codes,
More informationDistributed Data Storage Systems with Opportunistic Repair
Distributed Data Storage Systems with Opportunistic Repair Vaneet Aggarwal, Chao Tian, Vinay A. Vaishampayan, and YihFarn R. Chen AT&T LabsResearch, Florham Park, NJ 07932 email: {vaneet, tian, vinay,
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationWeek 5 Integral Polyhedra
Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory
More informationNotes 11: List Decoding Folded ReedSolomon Codes
Introduction to Coding Theory CMU: Spring 2010 Notes 11: List Decoding Folded ReedSolomon Codes April 2010 Lecturer: Venkatesan Guruswami Scribe: Venkatesan Guruswami At the end of the previous notes,
More informationEnergy Benefit of Network Coding for Multiple Unicast in Wireless Networks
Energy Benefit of Network Coding for Multiple Unicast in Wireless Networks Jasper Goseling IRCTR/CWPC, WMC Group Delft University of Technology The Netherlands j.goseling@tudelft.nl Abstract Jos H. Weber
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationSECTION 102 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More informationAdaptive Linear Programming Decoding
Adaptive Linear Programming Decoding Mohammad H. Taghavi and Paul H. Siegel ECE Department, University of California, San Diego Email: (mtaghavi, psiegel)@ucsd.edu ISIT 2006, Seattle, USA, July 9 14, 2006
More informationOn Secure Communication over Wireless Erasure Networks
On Secure Communication over Wireless Erasure Networks Andrew Mills Department of CS amills@cs.utexas.edu Brian Smith bsmith@ece.utexas.edu T. Charles Clancy University of Maryland College Park, MD 20472
More information