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Definition 16 A context free grammar is called regular if for every production T → w of G, all letters of w, with a possible exception for the last one, are terminals ∗(V ∪{e}) Example V = {S,T};Σ = {a,b} S → abaT T → bbaSaa Theorem 17 A language is regular iffR that assigns a real weight w(e) to each edge e, which may be positive, negative, or zero This chapter describes several algorithms to find the minimum spanning tree of G, that is, the spanning tree T that minimizes the function w(T) = X CBA356 90 H Y R HI T A L R K K yP H V R F L L S V G G D r d v n s e g v A DSD K Y L R L L E q S E H R K S F Q a S v L A E L N N N gF 153 fruit fly

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Aeg nv stock-5 h e a lt h s c r e e n in g; 42 Directed Graphs Digraphs A directed graph (or digraph) is a set of vertices and a collection of directed edges that each connects an ordered pair of vertices We say that a directed edge points from the first vertex in the pair and points to the second vertex in the pair We use the names 0 through V1 for the vertices in a Vvertex graph Glossary

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S E P T E T T E G L A S S E S R E L A P S E I M I T A T E S M E A R E D T A N N E R Y Again, the rows and columns can be transposed to form another valid rectangle For example, a 4×8 rectangle can also be written as an 8×4 rectangle Other formsV I N G S T O N 1 R A N G E L W I S R A N G E R A N G E F L A T H D S W A N NATIONAL A N G E C W H I T E F I S H R A N E BOW CROW PROV INCIAL FOR EST FLA THEAD PROV INCIAL FORES Crandell Mountain Belly River Belly River Goat Haunt Goat Haunt Overlook Kintla Lake Polebridge Bowm an L ke Red Rock Canyon Bison paddock Golf course Park E ntra ceA speci c node s 2V and a tree T = (V;E0), E0 E Give an algorithm that checks whether T is a shortestpath tree for G with starting point s Your algorithm should run in linear time SOLUTION Do a BFS on T to compute the distance from s to all vertices along the

Remark Contextfree grammars are sometimes defined as G =(V N,V T,P,S) The correspondence with our definition is thatΣ= V T and N = V N,sothatV = V N ∪V TThus, in this other definition, it is necessary to assume that V T ∩V N = ∅ Example 1 G 1 =({E,a,b},{a,b},P,E), where P is the set of rules E −→ aEb, E −→ ab7 c on ta c t ,t r a c in g/qu a r an t in e ;S t O M a t t h e w s A C e m e t e r y Beth SJacob Cemetery S t A d a l b e r t C e m e t e r y M a r y h i l l DC e m e t e r y R i d g e w o o d C e m e t e r y

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SCRABBLE® is a registered trademark All intellectual property rights in and to the game are owned in the USA and Canada by Hasbro Inc, and throughout the rest of the world by JW Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc Mattel and Spear are not affiliated with HasbroExact same vertex set, so f may also be thought of as a function from V(G) to V(H) Moreover, f is still a bijection The question that remains is "does f preserve adjacency in the complimentary graphs?" Let u,v ∈ V(G) = V(G) {u,v} is an edge in G iff {u,v} is not an edge in G by the definition of complimentary graphFinal Velocity (t) v f = v i at m/s Final Velocity (d) v f 2 = v i 2 2ad m/s Speed (circular) v = 2pr/T m/s Angular Speed ω = Δθ/Δt rad/s Angular Accel α = Δω/Δt rad/s 2 Acceleration a = Dv/Dt m/s 2 Acceleration (cent) a c = v 2 /r m/s 2 Acceleration (gravity) g = F/m m/s 2 Force F = ma N or kgm/s 2 Weight F

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The Largest Database for the Root Solutions on the InternetGraph G =(V,E) together with a function w E !In a network flow problem, it is useful to work with a cut of the graph, particularly an st cut Definition 6 An st cut of network G is a partition of the vertices V into 2 groups S and S¯ = V \S such that s ∈ S and t ∈ S¯ s t S S Figure 63 An illustration of an st cut There might be both edges from S to S¯ and from S¯ to S

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Subset of vertices U ⊂ V Output The lightest spanning tree in which the nodes of U are leaves (there might be other leaves in this tree as well) Consider the minimum spanning tree T = (V,Eˆ) of G and the leaves of the tree T as L(T) Three possible situations are feasibleInput Undirected graph G = (V,E); STK10/LOK is also called pololike kinase kinase 1 in Xenopus (xPlkk1) It is highly expressed in lymphocytes and is responsible in regulating leukocyte function associated antigen (LFA1)mediated lymphocyte adhesion It plays a role in regulating the CD28 responsive element in T cells, and may also function as a regulator of pololike kinase

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E je2Sg The vector g= f w1 S is a ow of value jgj= jfj w S, and jE (g)jPART 1 MODULE 2 SET INTERSECTION, SET UNION, SET COMPLEMENT SUMMARY The intersection of two sets denotes the elements that the sets have in common, or the "overlap" of the two sets S ∩ T = {xx∈ S and x∈ T} The union of two sets merges the two sets into one "larger" set S ∪ T = {xx ∈ S or x ∈ T} pfam (PSSM ID ) Conserved Protein Domain Family SRCR, These domains are disulphide rich extracellular domains

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P14 5m i n r lake nx2 w a l k oakland alameda emeryville fru i t va le piedmont trestle glen grand lake koreatown uptown old oakland temescal chinatown alameda pointAnd the corresponding parse tree is S S ∗ ( S ) S ∪ S 0 S S S ∗ 1 ( S ) S S 1 0 3 (a) Suppose that language A1 has a contextfree grammar G1 = (V1,Σ,R1,S1), and language has a contextfree grammar G2 = (V2,Σ,R2,S2), where, for i= 1,2, Vi is the set of variables, Ri is the set of rules, and Si is the start variable for CFG GiThe CFGs have the same set of terminals ΣE is a connected graph on V, and T f → e has the same number of nodes as T, implying T f → e is a spanning tree An edge of T An edge of G not in T Proposition 1 Let G = (V, E, W) be a weighted graph, let T = (V, S) be an MST for G, and let e = xy be an edge of G not in T Then i) w (e) ≥ w (f) for any edge f on the path in T from x to

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(1) assignment statement, of the form id = E, where E is any arithmetic expression (generated by the grammar using T and F that we presented in class) (2) conditional statement, eg, "if E < E then statement", or while statement , eg "while E < E do statement" (3) goto statement;, an MST is a subset of the edges T E such that T is a spanning tree whose sum of edge weights is minimized ("spanning" means that the tree encompasses all vertices in G) Cayley's Theorem There are nn2 spanning trees of K n (up to node labeling) 5 23 10 21 14 24 16 6 4 18 9 7 11 8 6 11 G = (V, E) T, e T c e = 50 can't solve by brute forceP O T O M A C U ON N T R I V E R Y OVE O C C O Q U A N Base Fare R I V E R O C C O Q U A N COTY and Metro Park is E O G R I V E R G MER of VE of X of S CRC MS of R O Y ANDC of ERO Y Y FA AS ON O A ST

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This video is Part 1 of the Alphabet ABC Phonics Series, covering letters A, B, C, D, E, F, and GThis series goes through each of the letters, starting withYour current implementation will compute the correct number of paths in a DAG However, by not marking paths it will take exponential time For example, in the illustration below, each stage of the DAG increases the total number of paths by a multiple of 3Links with this icon indicate that you are leaving the CDC website The Centers for Disease Control and Prevention (CDC) cannot attest to the accuracy of a nonfederal website Linking to a nonfederal website does not constitute an endorsement by CDC or any of its employees of the sponsors or the information and products presented on the website

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