LIMITS IN CATEGORY THEORY

Size: px
Start display at page:

Download "LIMITS IN CATEGORY THEORY"

Transcription

1 LIMITS IN CATEGORY THEORY SCOTT MESSICK Abstract. I will start assming no knowledge o category theory and introdce all concepts necessary to embark on a discssion o limits. I will conclde with two big theorems: that a category with prodcts and eqalizers is complete, and that limits in any category can be redced to limits o Hom-sets by means o a natral transormation. Contents 1. Categories 1 2. Fnctors and Natral Transormations 3 3. Limits 4 4. Pllbacks 6 5. Complete Categories 8 6. Another Limit Theorem 9 Reerences Categories Category theory is a scheme or dealing with mathematical strctres in a highly abstract and general way. The basic element o category theory is a category. Deinition 1.1. A category C consists o three components: (1) A collection 1 o objects Ob(C ). Instead o C Ob(C ), we may write simply C C. (2) A collection o morphisms Ar(C ), and with each morphism, two associated objects, called the domain dom and the codomain cod. The set o morphisms with domain A and codomain B is written Hom C (A, B) or simply C (A, B) and called a Hom-set. A morphism can be thoght o as an arrow going rom its domain to its codomain. Indeed, I will se the words morphism and arrow interchangeably. Instead o C (A, B), : A B may be written, where A and B are already nderstood to be objects in C. (3) A composition law, i.e., or every pair o Hom-sets C (A, B) and C (B, C), a binary operation : C (A, B) C (B, C) C (A, C). Instead o (, g) we write g or g. Composition mst satisy the ollowing two axioms. Date: Agst 17, For the scope o this paper, I will not attempt to make the word collection precise. Note, however, that it is oten too big to be a set. 1

2 2 SCOTT MESSICK Category Objects Morphisms Set sets nctions Top topological spaces continos maps Grp grops homomorphisms o grops Any poset elements o the set exactly one arrow or every Table 1. Examples o categories (a) Associativity. I C (A, B), g C (B, C), and h C (C, D), then h (g ) = (h g) = h g = hg. (b) Identities. For every object C C there exists an identity arrow 1 C C (C, C) sch that or every morphism g C (A, C) and h C (C, B), 1 C g = g and h 1 C = h hold. The qintessential example o a category is the category o sets, Set. The objects are all sets, and the morphisms all nctions between sets (with the sal composition o nctions). The categories o topological spaces and grops are similar; in act, there is a category like this or almost every branch o mathematics; the objects are the strctres being stdied, and the morphisms are the strctrepreserving maps. The two axioms or composition o morphisms can be restated diagramatically as ollows. For every object C, there exists an identity arrow 1 C sch that the ollowing diagram commtes or every g, h: A g C h 1 C 1 C 1 C g C h B Given objects A, B, C, D and morphisms between them, the ollowing diagram always commtes: A B h g g g C h D A commtative diagram is one where, between any two given objects, composition along every (directed) path o arrows yields the same morphism between those objects. These diagrams also exempliy the sal way o illstrating concepts o category theory, representing objects as nodes and morphisms as arrows between them. The ollowing concept is important. Deinition 1.2. A morphism : A B is an isomorphism i there exists a morphism g : B A sch that g = 1 B and g = 1 A. Then the objects A, B are said to be isomorphic. Loosely speaking, isomorphic objects look the same in a category, becase arrows to or rom one can be niqely mapped throgh the isomorphism to arrows

3 LIMITS IN CATEGORY THEORY 3 to or rom the other. In terms o categorical strctre, thereore, they are indistingishable. For example, two sets o the same cardinality are isomorphic in Set, and are indeed the same thing i nctions between sets are all that is being considered. There are also generalizations o srjective and injective arrows, bt they are not needed here. 2. Fnctors and Natral Transormations Categories, in part, embody the idea that any notion o a mathematical object shold come with a notion o maps between two sch objects. Sets come with nctions, grops with homomorphisms, topological spaces with continos maps, and so on. Similarly, categories come with nctors. Deinition 2.1. A nctor F : C D is a map which associates with every object C C and object F (C) D, and with every morphism C (C 1, C 2 ) a morphism F () DF (C 1 ), F (C 2 ), and which preserves composition and identities, as in: F (1 C ) = 1 F (C) holds or every object C C. Whenever h, g, are arrows in C sch that h = g, it also holds that F (h) = F (g) F (). Parentheses may be omitted, as in C F C and F. There are many simple examples o nctors, orgetl nctors rom Grp or Top to Set which take objects to their nderlying sets, ree nctors going the other way (e.g. ptting the trivial topology on every set), nctors rom little categories that pick ot diagrams in their codomain (which will be important later), and so on. One important kind o nctor is given by Hom-sets. Observe that i C is any category, with any object C, then Hom C (C, ) gives a nctor H : C Set. This nctor maps arrows by let-composition, i.e., given : A B, H : Hom C (C, A) Hom C (C, B) is deined by (H)(g) = g. A natral transormation is, in trn, a morphism o nctors. Given nctors F, G : C D, one may imagine the image o F as a bnch o objects sitting in D, with some arrows between them highlighted. Similarly one may imagine the image o G as another bnch o objects. Loosely speaking, a natral transormation will be a way o getting rom the irst pictre to the second pictre, sing the arrows o D, in a natral way, i.e., in the same way or every object. Deinition 2.2. Given nctors F, G : C D, a natral transormation η : F G is a collection o arrows in D, speciically, one arrow or each object X o C, called 2 η X, sch that the ollowing diagram commtes or every X, Y C. X F X F F Y η X Y GX G GY Deinition 2.3. A natral isomorphism is a natral transormation in which every arrow is an isomorphism. η Y 2 I m switching notation slightly here, to avoid nested sbscripts.

4 4 SCOTT MESSICK 3. Limits The notion o a limit in category theory generalizes varios types o niversal constrctions that occr in diverse areas o mathematics. It can show very precisely how thematically similar constrctions o dierent types o objects, sch as the prodct o sets or grops o topological spaces, are instances o the same categorical constrct. Consider the Cartesian prodct in sets. X Y is sally deined by internally constrcting the set o ordered pairs {(x, y) x X and y Y }. Bt it can also by identiied as the set which projects down to X and Y in a niversal way, that is to say, doing something to X Y is the same as separately doing something to X and Y. Deinition 3.1. A prodct o objects A, B in a category C is an object, C, together with morphisms p : C A and q : C B, called the projections, with the ollowing niversal property 3. For any other object D C with morphisms : D A and g : D B, there is a niqe morphism : D C sch that p = and q = g. In other words, every (D,, g) actors niqely throgh (C, ). D g C q B p A Example 3.2. In the Set, the prodct is the Cartesian prodct. The projections p : A B A and q : A B B are given by p(a, b) = a and q(a, b) = q. Given another set D with arrows : D A and g : D B, the niqe arrow : D A B is given by (d) = ((d), g(d)). That this commtes and is the only arrow doing so are transparent, as or example, (p )(d) = p((d)) = p((d), g(d)) = (d). Example 3.3. In Grp, the prodct is the direct prodct o grops. The constrction is similar to set; the prodct is given by the nderlying Cartesian prodct, with a grop operation constrcted elementwise rom those o the actors. To demonstrate that the niversal arrow exists and is niqe, it sices to show that the nction given by the same constrction is in act a homomorphism, given that, becase we are in the category o grops, and g are also homomorphisms. (Trivially, the projections are homomorphisms.) This proo is straightorward: (3.4) (d 1 d 2 ) = ((d 1 d 2 ), g(d 1 d 2 )) = ((d 1 )(d 2 ), g(d 1 )g(d 2 )) = ((d 1 ), g(d 1 ))((d 2 ), g(d 2 )) = (d 1 )(d 2 ) Example 3.5. In Top, the prodct is the sal prodct o topological spaces. In act, this prodct is oten deined as the coarsest topology which makes the projections continos, which is exactly what is needed to make the analogos constrction work. paper. 3 A ormal deinition o niversal properties exists, bt is nnecessary or the prposes o this

5 LIMITS IN CATEGORY THEORY 5 Example 3.6. In a poset, the prodct is the greatest lower bond, i it exists. This provides not only an example which is very dierent rom sets, bt also one showing that the prodct doesn t always exist. Let c = glb(a, b). There is only one choice o projections. I d has arrows to a and b, it means a d and b d, so d is an pper bond. Bt c is a least pper bond, so c d. This gives the niversal arrow which easily commtes and is niqe becase arrows are scant in this category. Proposition 3.7. The prodct o any two objects in a category, i it exists, is niqe p to niqe isomorphism. Proo. Let A and B be objects in a category, and C, p C, q C and D, p D, q D be prodcts o A and B. By the niversal property, there exist niqe morphisms : C D and g : D C which commte with the projections. This gives p C g = p D and p D = p C and hence p C g = p C. Similarly, q C g = q C. Ths, g is an arrow rom C to itsel which commtes with the projections. Bt by the niversal property there can only be one morphism rom C to itsel which commtes with the projections, and the identity sices. Hence g = 1 C. The other way arond is similar. C A g B D C A 1 C g B C Remark 3.8. A prodct can be generalized in an obvios way to any nmber o actors other than two. Later I will speak o a category with all small prodcts ; this jst means the prodcts o any set o objects exists in the category. In Set it s clear that all small prodcts exist. The second most important example o a limit is an eqalizer. In sets, and in many similar categories, this is jst the sbset o the domain o two parallel arrows where those two nctions are eqal. Deinition 3.9. An eqalizer o two arrows, g : X Y in a category C is an object, E, together with a morphism e : E X sch that e = g e, with the ollowing niversal property: or any O C with a morphism m : O X sch that m = g m, there is a niqe morphism : O E sch that m = e. This eqalizer may be denoted eq(x, Y ).

6 6 SCOTT MESSICK E e X Y g m O Remark In Grp and Top, eqalizers are constrcted exactly the same way. In Grp, it can be viewed as a dierence kernel. (In act, kernels can also be viewed as limits). Now we re ready or the general notion o a limit, bt irst, it s sel to deine a cone. Notice that in the prodct and the eqalizer, the projections played the same role as the morphism e. In what ollows, J shold be thoght o as a small category, sch as two discrete objects (no morphisms except identities) in the case o prodcts, or a jst a pair o objects with a pair o arrows in the case o eqalizers. The nctor F : J C shold be thoght o as a diagram o that shape in the category C. Ths the limit o the diagram is taken. Deinition Given a nctor F : J C, a cone o F is an object N C together with morphisms ψ X : N F (X) or every X J sch that or every morphism : X Y in J, the triangle commtes, i.e. F ψ X = ψ Y. ψ X ψ Y N ψ Z F (X) F (Y ) F (Z) F () F (g) Deinition A limit o a nctor F : J C is a niversal cone 4 L, φ X. That is, or every cone N, ψ X o F, there is a niqe morphism : N L sch that φ X = ψ X or every X J. The limit object may be written lim i J F (i). N ψ X ψ L Y φ X φ Y F (X) F (Y ) F () Proposition The limit o any diagram in a category, i it exists, is niqe p to niqe isomorphism. 4. Pllbacks The pllback (iber prodct) is the last limit I will deine explicitly. There are many other important limits, bt pllbacks will be my example or how all limits come rom prodcts and eqalizers. 4 The term limit is overloaded to mean either the cone, i.e., the object with the arrows, or jst the object.

7 LIMITS IN CATEGORY THEORY 7 Deinition 4.1. A pllback is a limit o a diagram o the ollowing orm: A C B. That is, a pllback is an object D with morphisms p 1 : D A and p 2 : D B which make the sqare commte and are niversal, i.e. or every other object Q with morphisms q 1 : Q A and q 2 : Q B, there is a niqe morphism : Q D which makes the diagram commte. The pllback, interpreted as the object D, may be written A C B. Q D B A Proposition 4.2. In Set, the pllback is given by the set X Z Y = {(x, y) (x) = g(y)} where : X Z and g : Y Z, together with the restricted projection maps p 1, p 2 into X and Y. Proo. Let D, q 1, q 2 be another cone o the same diagram. We have (q 1 (d)) = g(q 2 (d)) or every d D. Ths the pairs (q 1 (d), q 2 (d)) are in X Z Y. Let (d) = (q 1 (d), q 2 (d)). Proposition 4.3. In any category, a pllback can be constrcted sing a prodct and an eqalizer. C A B p2 B p 1 A p 1 The prodct A B indces two parallel diagonal arrows to C, o which the eqalizer can be taken. gp 2 C g E p 2e p 1e e A B p2 B p 1 A Now, consider any other cone commting with the pllback diagram. The niversal property o the prodct gives a niqe arrow to the prodct. By a diagram chasing argment, this arrow can be seen to eqalize the diagonal arrows, which gives a niqe arrow to the eqalizer that makes the diagram commte. Ths, the eqalizer is the pllback. p 1 gp 2 C g

8 8 SCOTT MESSICK 5. Complete Categories Deinition 5.1. A category is complete i every small 5 diagram has a limit. Theorem 5.2. Let F : J Set be any small diagram in Set. Then the limit o F is the set L = lim i J F (i) = {(x i ) i J F (i) (F )(x i ) = x cod Ar(J )} Remark 5.3. An eqivalent condition to the one given is that (F )(x i ) = (F g)(x j ) whenever, g Ar(J ) and cod = cod g. For g can be the identity, which gives the statement in the theorem, and to go the other way, note that i cod = cod g = k then (F )(x i ) = x k = (F g)(x j ). This makes it clear that an eqalizer is being taken in order to prodce the limit. Proo. The limit cone is L with restricted projection maps (p i ). Let (N, (ψ i ) i J ) be any other cone. We have (F )(ψ i (n)) = (F g)(ψ j (n)) whenever cod = cod g, or every n N. Ths the tples (ψ i (n)) i J are in L. Let (n) = (ψ i (n)) i J. Theorem 5.4. A category with all eqalizers and all small prodcts is complete. Proo. This proo is a carel and slightly clever generalization o the idea in the previos proo. Actal eqality no longer exists, so an eqalizer has to be sed to do the trick. We will take two prodcts and ind two arrows between them o which to take the eqalizer, which will be the limit. The prodct is taken irst over all objects in the diagram, then over the codomains o all arrows in the diagram, indexed by arrows. F (cod ) i F (i) F (dom ) v F p F (cod ) p F (cod ) The irst prodct with its projections (repeated i necessary) can make a cone or the second prodct in two dierent ways, as shown in the triangle on top and the sqare on the bottom o the irst igre. This gives the two niversal arrows and v as shown, which respectively make all the triangles commte, or all the sqares. 5 A diagram is small i the collection o objects is a set.

9 LIMITS IN CATEGORY THEORY 9 F (i) F (cod ) φ i E e p i i F (i) F (dom ) v F p F (cod ) p F (cod ) The eqalizer is taken as shown, and the arrows φ i = p i e are ormed to make the cone E, φ i. That it s a cone can be seen by inspecting the diagram. Given : j k, we have e = ve, p e = p ve, p k e = F p j e, phi k = F phi j. Then any other cone Q, (psi i ), gives a niqe map t to the prodct, by its niversal property, bt since it s a cone, t = vt, so there is a niqe arrow s : Q E by the niversal property o the eqalizer. 6. Another Limit Theorem The ollowing is a beatil and ascinating theorem, and in a paper o larger scope, mch more cold be done with it. Even withot the prpose o proving other theorems, it illstrates a lot o intition behind the notion o a categorical limit. Theorem 6.1. Let F : J C be a diagram in C. Then an object X C is a limit o F i and only i there is a natral isomorphism Hom C (C, X) = lim i J Hom C (C, F (i)) where the limit on the right is in Set and hence exists. Proo or prodcts. First sppose that C is the prodct A B. We need to ind a bijection o sets, Hom(D, C) = Hom(D, A) Hom(D, B). Given arrows rom D to A and B gives s a cone o F, so since C is the prodct we have a niqe arrow given by the niversal property. For the other direction, given any arrow h : D C we simply take p h, q h, so we have a bijection. To show natrality, let α : M N be an arrow in C. Then the ollowing diagram mst commte: Hom(M, C) Hom(M, A) Hom(M, B) Hom(N, C) Hom(N, A) Hom(N, B) Here the vertical arrows are jst let-composition by α and the horizontal arrows are jst elementwise right-composition by p, q. The diagram commtes by associativity. On the other hand sppose we have sch a natral isomorphism or some object C. We need to ind the projections in order to orm a prodct. To do this, we consider a particlar, enlightening case o the natral isomorphism: Hom(C, C) = Hom(C, A) Hom(C, B)

10 10 SCOTT MESSICK Even in an arbitrary category, we know Hom(C, C) has an identity element. By plgging the identity throgh the isomorphism, we get p : C A and q : C B. Now let D,, g be another cone. We can prove the niversal property by sing the ollowing commtative diagram: Hom(D, C) Hom(D, A) Hom(D, B) Hom(C, C) Hom(C, A) Hom(C, B) (, g) 1 C (p, q) We know we can constrct the vertical arrows becase the bijection gives s : D C rom (, g). The vertical arrows are let-composition by. The commtative diagram then states that p = and q = g. Notice that this theorem illstrates what a prodct morally is: an object sch that speciying a map to the object is the same as speciying a map to the actors. General proo. The general case proceeds similarly. Let L be the limit o F : J C. We need to ind a bijection o sets, Hom(X, L) = lim Hom(X, F (i)). What is an element o the limit on the right-hand side? Well, we have an explicit description o a limit in Set. Bt we have to be carel, becase the nctor here is not F itsel bt HF, where H is the Hom-nctor Hom(X, ). Then the ollowing is tre or an element (ψ i ) o the limit: (HF )(ψ i ) = ψ j whenever : i j in J. Bt we know what H does to arrows, so we have F ψ i = ψ j wheenver : i J in J. Ths, the statement that a tple o arrows (ψ i ) is in the limit o Hom-sets is exactly the statement that X, (ψ i ) is a cone o F. This gives a niqe arrow : X L and everything proceeds jst as beore. To go the other way, we again take the bijection Hom(L, L) = lim Hom(L, F (i)) and eed the identity throgh it, yielding a cone L, (φ i ) as beore, which we mst prove is niversal. Let X, (ψ i ) be any other cone. Hom(X, L) lim Hom(X, F (i)) Hom(L, L) lim Hom(L, F (i)) (ψ i ) 1 L Jst as beore, we can ind with the bijection and constrct the commtative diagram to complete the proo. (φ i )

11 LIMITS IN CATEGORY THEORY 11 Reerences [1] Mac Lane, Sanders. Categories or the Working Mathematician. Second Edition, [2] Gillo, Bert and Haris Skiadas. WOMP 2004: Category Theory. [3] Anders, Alan. DRP Notes, Winter 2007.

HOMOTOPY FIBER PRODUCTS OF HOMOTOPY THEORIES

HOMOTOPY FIBER PRODUCTS OF HOMOTOPY THEORIES HOMOTOPY FIBER PRODUCTS OF HOMOTOPY THEORIES JULIA E. BERGNER Abstract. Gien an appropriate diagram of left Qillen fnctors between model categories, one can define a notion of homotopy fiber prodct, bt

More information

Every manufacturer is confronted with the problem

Every manufacturer is confronted with the problem HOW MANY PARTS TO MAKE AT ONCE FORD W. HARRIS Prodction Engineer Reprinted from Factory, The Magazine of Management, Volme 10, Nmber 2, Febrary 1913, pp. 135-136, 152 Interest on capital tied p in wages,

More information

its heating value. As a rule the proportions are CH 4, H 2 and CO. Table 3-1 lists their heating values.

its heating value. As a rule the proportions are CH 4, H 2 and CO. Table 3-1 lists their heating values. 3. Energy Conversion 3.1 Heating vales The chemical enthalpy is converted into heat by the oxidation o the carbon and hydrogen contained in the el. I, in accordance with Figre 31, the gas is cooled to

More information

Solutions to Assignment 10

Solutions to Assignment 10 Soltions to Assignment Math 27, Fall 22.4.8 Define T : R R by T (x) = Ax where A is a matrix with eigenvales and -2. Does there exist a basis B for R sch that the B-matrix for T is a diagonal matrix? We

More information

SECTION 6: FIBER BUNDLES

SECTION 6: FIBER BUNDLES SECTION 6: FIBER BUNDLES In this section we will introduce the interesting class o ibrations given by iber bundles. Fiber bundles lay an imortant role in many geometric contexts. For examle, the Grassmaniann

More information

CHAPTER ONE VECTOR GEOMETRY

CHAPTER ONE VECTOR GEOMETRY CHAPTER ONE VECTOR GEOMETRY. INTRODUCTION In this chapter ectors are first introdced as geometric objects, namely as directed line segments, or arrows. The operations of addition, sbtraction, and mltiplication

More information

Optimal Trust Network Analysis with Subjective Logic

Optimal Trust Network Analysis with Subjective Logic The Second International Conference on Emerging Secrity Information, Systems and Technologies Optimal Trst Network Analysis with Sbjective Logic Adn Jøsang UNIK Gradate Center, University of Oslo Norway

More information

CIVE2400 Fluid Mechanics. Section 1: Fluid Flow in Pipes

CIVE2400 Fluid Mechanics. Section 1: Fluid Flow in Pipes CIVE00 Flid Mechanics Section : Flid Flow in Pipes CIVE00 FLUID MECHNICS... SECTION : FLUID FLOW IN PIPES.... FLUID FLOW IN PIPES.... Pressre loss de to riction in a pipeline..... Pressre loss dring laminar

More information

9. Quotient Groups Given a group G and a subgroup H, under what circumstances can we find a homomorphism φ: G G ', such that H is the kernel of φ?

9. Quotient Groups Given a group G and a subgroup H, under what circumstances can we find a homomorphism φ: G G ', such that H is the kernel of φ? 9. Quotient Groups Given a group G and a subgroup H, under what circumstances can we find a homomorphism φ: G G ', such that H is the kernel of φ? Clearly a necessary condition is that H is normal in G.

More information

PHY2061 Enriched Physics 2 Lecture Notes Relativity 4. Relativity 4

PHY2061 Enriched Physics 2 Lecture Notes Relativity 4. Relativity 4 PHY6 Enriched Physics Lectre Notes Relativity 4 Relativity 4 Disclaimer: These lectre notes are not meant to replace the corse textbook. The content may be incomplete. Some topics may be nclear. These

More information

Sickness Absence in the UK: 1984-2002

Sickness Absence in the UK: 1984-2002 Sickness Absence in the UK: 1984-2002 Tim Barmby (Universy of Drham) Marco Ecolani (Universy of Birmingham) John Treble (Universy of Wales Swansea) Paper prepared for presentation at The Economic Concil

More information

ASAND: Asynchronous Slot Assignment and Neighbor Discovery Protocol for Wireless Networks

ASAND: Asynchronous Slot Assignment and Neighbor Discovery Protocol for Wireless Networks ASAND: Asynchronos Slot Assignment and Neighbor Discovery Protocol for Wireless Networks Fikret Sivrikaya, Costas Bsch, Malik Magdon-Ismail, Bülent Yener Compter Science Department, Rensselaer Polytechnic

More information

Introduction to HBase Schema Design

Introduction to HBase Schema Design Introdction to HBase Schema Design Amandeep Khrana Amandeep Khrana is a Soltions Architect at Clodera and works on bilding soltions sing the Hadoop stack. He is also a co-athor of HBase in Action. Prior

More information

NOTES ON CATEGORIES AND FUNCTORS

NOTES ON CATEGORIES AND FUNCTORS NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category

More information

8 Service Level Agreements

8 Service Level Agreements 8 Service Level Agreements Every organization of men, be it social or political, ltimately relies on man s capacity for making promises and keeping them. Hannah Arendt Key Findings Only abot 20 percent

More information

Configuration Management for Software Product Lines

Configuration Management for Software Product Lines onfigration Management for Software Prodct Lines Roland Laqa and Peter Knaber Franhofer Institte for Experimental Software Engineering (IESE) Saerwiesen 6 D-67661 Kaiserslatern, Germany +49 6301 707 161

More information

2.3 Domain and Range of a Function

2.3 Domain and Range of a Function Section Domain and Range o a Function 1 2.3 Domain and Range o a Function Functions Recall the deinition o a unction. Deinition 1 A relation is a unction i and onl i each object in its domain is paired

More information

Foundations of mathematics. 2. Set theory (continued)

Foundations of mathematics. 2. Set theory (continued) 2.1. Tuples, amilies Foundations o mathematics 2. Set theory (continued) Sylvain Poirier settheory.net A tuple (or n-tuple, or any integer n) is an interpretation o a list o n variables. It is thus a meta-unction

More information

On a Generalized Graph Coloring/Batch Scheduling Problem

On a Generalized Graph Coloring/Batch Scheduling Problem Reglar Papers On a Generalized Graph Coloring/Batch Schedling Problem Giorgio Lcarelli 1, Ioannis Milis Dept. of Informatics, Athens University of Economics and Bsiness, 104 34, Athens, Greece, {glc, milis}@aeb.gr

More information

Linear Programming. Non-Lecture J: Linear Programming

Linear Programming. Non-Lecture J: Linear Programming The greatest flood has the soonest ebb; the sorest tempest the most sdden calm; the hottest love the coldest end; and from the deepest desire oftentimes enses the deadliest hate. Socrates Th extremes of

More information

Herzfeld s Outlook: Seasonal Factors Provide Opportunities in Closed-End Funds

Herzfeld s Outlook: Seasonal Factors Provide Opportunities in Closed-End Funds VIRTUS HERZFELD FUND Herzfeld s Otlook: Seasonal Factors Provide Opportnities in Closed-End Fnds When it comes to investing in closed-end fnds, a comprehensive nderstanding of the inefficiencies of the

More information

Lecture 16 : Relations and Functions DRAFT

Lecture 16 : Relations and Functions DRAFT CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

More information

SEGREGATED ACCOUNTS COMPANIES ACE CAPABILITIES: AN OVERVIEW

SEGREGATED ACCOUNTS COMPANIES ACE CAPABILITIES: AN OVERVIEW SEGREGATED ACCOUNTS COMPANIES CAPABILITIES: AN OVERVIEW SIMPLICITY OUT OF COMPLEXITY SEGREGATED ACCOUNTS CAPABILITIES Managing yor own risks jst got simpler. In recent years, increasing reglation has led

More information

A Note on Di erential Calculus in R n by James Hebda August 2010.

A Note on Di erential Calculus in R n by James Hebda August 2010. A Note on Di erential Calculus in n by James Hebda August 2010 I. Partial Derivatives o Functions Let : U! be a real valued unction deined in an open neighborhood U o the point a =(a 1,...,a n ) in the

More information

Chapter 3. 2. Consider an economy described by the following equations: Y = 5,000 G = 1,000

Chapter 3. 2. Consider an economy described by the following equations: Y = 5,000 G = 1,000 Chapter C evel Qestions. Imagine that the prodction of fishing lres is governed by the prodction fnction: y.7 where y represents the nmber of lres created per hor and represents the nmber of workers employed

More information

Designing and Deploying File Servers

Designing and Deploying File Servers C H A P T E R 2 Designing and Deploying File Servers File servers rnning the Microsoft Windows Server 2003 operating system are ideal for providing access to files for sers in medim and large organizations.

More information

2 2 Matrices. Scalar multiplication for matrices. A2 2matrix(pronounced 2-by-2matrix )isasquareblockof4numbers. For example,

2 2 Matrices. Scalar multiplication for matrices. A2 2matrix(pronounced 2-by-2matrix )isasquareblockof4numbers. For example, 2 2 Matrices A2 2matrix(prononced 2-by-2matrix )isasqareblockofnmbers. For example, is a 2 2matrix. It scalleda2 2 matrix becase it has 2 rows 2 colmns. The for nmbers in a 2 2matrixarecalledtheentries

More information

A Contemporary Approach

A Contemporary Approach BORICP01.doc - 1 Second Edition Edcational Psychology A Contemporary Approach Gary D. Borich The University of Texas at Astin Martin L. Tombari University of Denver (This pblication may be reprodced for

More information

Spectrum Balancing for DSL with Restrictions on Maximum Transmit PSD

Spectrum Balancing for DSL with Restrictions on Maximum Transmit PSD Spectrm Balancing for DSL with Restrictions on Maximm Transmit PSD Driton Statovci, Tomas Nordström, and Rickard Nilsson Telecommnications Research Center Vienna (ftw.), Dona-City-Straße 1, A-1220 Vienna,

More information

Compensation Approaches for Far-field Speaker Identification

Compensation Approaches for Far-field Speaker Identification Compensation Approaches for Far-field Speaer Identification Qin Jin, Kshitiz Kmar, Tanja Schltz, and Richard Stern Carnegie Mellon University, USA {qjin,shitiz,tanja,rms}@cs.cm.ed Abstract While speaer

More information

In this chapter we introduce the idea that force times distance. Work and Kinetic Energy. Big Ideas 1 2 3. is force times distance.

In this chapter we introduce the idea that force times distance. Work and Kinetic Energy. Big Ideas 1 2 3. is force times distance. Big Ideas 1 Work 2 Kinetic 3 Power is force times distance. energy is one-half mass times velocity sqared. is the rate at which work is done. 7 Work and Kinetic Energy The work done by this cyclist can

More information

Modeling Roughness Effects in Open Channel Flows D.T. Souders and C.W. Hirt Flow Science, Inc.

Modeling Roughness Effects in Open Channel Flows D.T. Souders and C.W. Hirt Flow Science, Inc. FSI-2-TN6 Modeling Roghness Effects in Open Channel Flows D.T. Soders and C.W. Hirt Flow Science, Inc. Overview Flows along rivers, throgh pipes and irrigation channels enconter resistance that is proportional

More information

Chapter 2. ( Vasiliy Koval/Fotolia)

Chapter 2. ( Vasiliy Koval/Fotolia) hapter ( Vasili Koval/otolia) This electric transmission tower is stabilied b cables that eert forces on the tower at their points of connection. In this chapter we will show how to epress these forces

More information

10 Evaluating the Help Desk

10 Evaluating the Help Desk 10 Evalating the Help Desk The tre measre of any society is not what it knows bt what it does with what it knows. Warren Bennis Key Findings Help desk metrics having to do with demand and with problem

More information

3 Distance in Graphs. Brief outline of this lecture

3 Distance in Graphs. Brief outline of this lecture Distance in Graphs While the preios lectre stdied jst the connectiity properties of a graph, now we are going to inestigate how long (short, actally) a connection in a graph is. This natrally leads to

More information

LOGARITHMIC FUNCTIONAL AND THE WEIL RECIPROCITY LAW

LOGARITHMIC FUNCTIONAL AND THE WEIL RECIPROCITY LAW 85 LOGARITHMIC FUNCTIONAL AND THE WEIL RECIPROCITY LAW KHOVANSKII A. Department o Mathematics, University o Toronto, Toronto, Ontario, Canada E-mail: askold@math.toronto.edu http://www.math.toronto.edu

More information

(c light velocity) (2) C

(c light velocity) (2) C ropean Jornal of Mechanical ngineering Research ol., No., pp.3, Jne 5 Pblished by ropean entre for Research Training and Development UK (www.eajornals.org) NW XPLNTION O DLTION RSULTS O HRGD PRTILS IN

More information

WHITE PAPER. Filter Bandwidth Definition of the WaveShaper S-series Programmable Optical Processor

WHITE PAPER. Filter Bandwidth Definition of the WaveShaper S-series Programmable Optical Processor WHITE PAPER Filter andwidth Definition of the WaveShaper S-series 1 Introdction The WaveShaper family of s allow creation of ser-cstomized filter profiles over the C- or L- band, providing a flexible tool

More information

Curriculum development

Curriculum development DES MOINES AREA COMMUNITY COLLEGE Crriclm development Competency-Based Edcation www.dmacc.ed Why does DMACC se competency-based edcation? DMACC tilizes competency-based edcation for a nmber of reasons.

More information

Stock Market Liquidity and Macro-Liquidity Shocks: Evidence from the 2007-2009 Financial Crisis

Stock Market Liquidity and Macro-Liquidity Shocks: Evidence from the 2007-2009 Financial Crisis Stock Market Liqidity and Macro-Liqidity Shocks: Evidence from the 2007-2009 Financial Crisis Chris Florackis *, Alexandros Kontonikas and Alexandros Kostakis Abstract We develop an empirical framework

More information

On the urbanization of poverty

On the urbanization of poverty On the rbanization of poverty Martin Ravallion 1 Development Research Grop, World Bank 1818 H Street NW, Washington DC, USA Febrary 001; revised Jly 001 Abstract: Conditions are identified nder which the

More information

Corporate performance: What do investors want to know? Innovate your way to clearer financial reporting

Corporate performance: What do investors want to know? Innovate your way to clearer financial reporting www.pwc.com Corporate performance: What do investors want to know? Innovate yor way to clearer financial reporting October 2014 PwC I Innovate yor way to clearer financial reporting t 1 Contents Introdction

More information

11 Success of the Help Desk: Assessing Outcomes

11 Success of the Help Desk: Assessing Outcomes 11 Sccess of the Help Desk: Assessing Otcomes I dread sccess... I like a state of continal becoming, with a goal in front and not behind. George Bernard Shaw Key Findings Respondents help desks tend to

More information

Position paper smart city. economics. a multi-sided approach to financing the smart city. Your business technologists.

Position paper smart city. economics. a multi-sided approach to financing the smart city. Your business technologists. Position paper smart city economics a mlti-sided approach to financing the smart city Yor bsiness technologists. Powering progress From idea to reality The hman race is becoming increasingly rbanised so

More information

TRIGONOMETRY REVIEW TRIGONOMETRIC FUNCTIONS AND IDENTITIES. November 4, 2004 15:00 k34-appa Sheet number 1 Page number 1 cyan magenta yellow black

TRIGONOMETRY REVIEW TRIGONOMETRIC FUNCTIONS AND IDENTITIES. November 4, 2004 15:00 k34-appa Sheet number 1 Page number 1 cyan magenta yellow black November 4, 4 5: k4-appa Sheet nmber Page nmber can magenta ellow black a p p e n d i a TRIGONOMETRY REVIEW TRIGONOMETRIC FUNCTIONS AND IDENTITIES ANGLES Angles in the plane can be generated b rotating

More information

Closer Look at ACOs. Putting the Accountability in Accountable Care Organizations: Payment and Quality Measurements. Introduction

Closer Look at ACOs. Putting the Accountability in Accountable Care Organizations: Payment and Quality Measurements. Introduction Closer Look at ACOs A series of briefs designed to help advocates nderstand the basics of Accontable Care Organizations (ACOs) and their potential for improving patient care. From Families USA Janary 2012

More information

Market Integration and Strike Activity

Market Integration and Strike Activity Market Integration and Strike Activity Ana Maleon FNRS and CEREC, Facltés Universitaires Saint-Lois; and CORE. Vincent Vannetelbosch FNRS and CORE, Université catholiqe de Lovain. April 2005 Abstract We

More information

Equilibrium of Forces Acting at a Point

Equilibrium of Forces Acting at a Point Eqilibrim of orces Acting at a Point Eqilibrim of orces Acting at a Point Pre-lab Qestions 1. What is the definition of eqilibrim? Can an object be moving and still be in eqilibrim? Explain.. or this lab,

More information

On the Equivalence Between a Commonly Used Correlation Coefficient and a Least Squares Function

On the Equivalence Between a Commonly Used Correlation Coefficient and a Least Squares Function On the Eqivalence Between a Commonly Used Correlation Coefficient and a Least Sqares Fnction Diane C. Jamrog George N. Phillips, Jr. Yin Zhang Janary 2003 (revised October 2003) Abstract Many objective

More information

GUIDELINE. Guideline for the Selection of Engineering Services

GUIDELINE. Guideline for the Selection of Engineering Services GUIDELINE Gideline for the Selection of Engineering Services 1998 Mission Statement: To govern the engineering profession while enhancing engineering practice and enhancing engineering cltre Pblished by

More information

The Intelligent Choice for Disability Income Protection

The Intelligent Choice for Disability Income Protection The Intelligent Choice for Disability Income Protection provider Pls Keeping Income strong We prposeflly engineer or disability income prodct with featres that deliver benefits sooner and contine paying

More information

Stock Market Liquidity and Macro-Liquidity Shocks: Evidence from the 2007-2009 Financial Crisis

Stock Market Liquidity and Macro-Liquidity Shocks: Evidence from the 2007-2009 Financial Crisis Stock Market Liqidity and Macro-Liqidity Shocks: Evidence from the 2007-2009 Financial Crisis Chris Florackis *, Alexandros Kontonikas and Alexandros Kostakis Abstract We develop an empirical framework

More information

Planning a Managed Environment

Planning a Managed Environment C H A P T E R 1 Planning a Managed Environment Many organizations are moving towards a highly managed compting environment based on a configration management infrastrctre that is designed to redce the

More information

Closer Look at ACOs. Designing Consumer-Friendly Beneficiary Assignment and Notification Processes for Accountable Care Organizations

Closer Look at ACOs. Designing Consumer-Friendly Beneficiary Assignment and Notification Processes for Accountable Care Organizations Closer Look at ACOs A series of briefs designed to help advocates nderstand the basics of Accontable Care Organizations (ACOs) and their potential for improving patient care. From Families USA Janary 2012

More information

Performing Arithmetic Operations on Round-to-Nearest Representations

Performing Arithmetic Operations on Round-to-Nearest Representations 1 Performing Arithmetic Operations on Rond-to-Nearest Representations Peter Kornerp, Member, IEEE Jean-Michel Mller, Senior Member, IEEE Adrien Panhalex Abstract Dring any composite comptation there is

More information

Deploying Network Load Balancing

Deploying Network Load Balancing C H A P T E R 9 Deploying Network Load Balancing After completing the design for the applications and services in yor Network Load Balancing clster, yo are ready to deploy the clster rnning the Microsoft

More information

Central Angles, Arc Length, and Sector Area

Central Angles, Arc Length, and Sector Area CHAPTER 5 A Central Angles, Arc Length, and Sector Area c GOAL Identify central angles and determine arc length and sector area formed by a central angle. Yo will need a calclator a compass a protractor

More information

The Intelligent Choice for Basic Disability Income Protection

The Intelligent Choice for Basic Disability Income Protection The Intelligent Choice for Basic Disability Income Protection provider Pls Limited Keeping Income strong We prposeflly engineer or basic disability income prodct to provide benefit-rich featres delivering

More information

You know from calculus that functions play a fundamental role in mathematics.

You know from calculus that functions play a fundamental role in mathematics. CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.

More information

Fibrations and universal view updatability

Fibrations and universal view updatability Fibrations and universal view updatability Michael Johnson Computing Department, Macquarie University NSW 2109, Australia mike@ics.mq.edu.au Robert Rosebrugh Department of Mathematics and Computer Science,

More information

Purposefully Engineered High-Performing Income Protection

Purposefully Engineered High-Performing Income Protection The Intelligent Choice for Disability Income Insrance Prposeflly Engineered High-Performing Income Protection Keeping Income strong We engineer or disability income prodcts with featres that deliver benefits

More information

TrustSVD: Collaborative Filtering with Both the Explicit and Implicit Influence of User Trust and of Item Ratings

TrustSVD: Collaborative Filtering with Both the Explicit and Implicit Influence of User Trust and of Item Ratings TrstSVD: Collaborative Filtering with Both the Explicit and Implicit Inflence of User Trst and of Item Ratings Gibing Go Jie Zhang Neil Yorke-Smith School of Compter Engineering Nanyang Technological University

More information

FIBER PRODUCTS AND ZARISKI SHEAVES

FIBER PRODUCTS AND ZARISKI SHEAVES FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also

More information

A Spare Part Inventory Management Model for Better Maintenance of Intelligent Transportation Systems

A Spare Part Inventory Management Model for Better Maintenance of Intelligent Transportation Systems 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 A Spare Part Inventory Management Model for Better Maintenance of Intelligent

More information

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

More information

Chapter 1. LAN Design

Chapter 1. LAN Design Chapter 1 LAN Design CCNA3-1 Chapter 1 Note for Instrctors These presentations are the reslt of a collaboration among the instrctors at St. Clair College in Windsor, Ontario. Thanks mst go ot to Rick Graziani

More information

Executive Coaching to Activate the Renegade Leader Within. Renegades Do What Others Won t To Get the Results that Others Don t

Executive Coaching to Activate the Renegade Leader Within. Renegades Do What Others Won t To Get the Results that Others Don t Exective Coaching to Activate the Renegade Leader Within Renegades Do What Others Won t To Get the Reslts that Others Don t Introdction Renegade Leaders are a niqe breed of leaders. The Renegade Leader

More information

Power functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd

Power functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd 5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts

More information

Research on Staff Explicitation in Organizational Knowledge Management Based on Fuzzy Set Similarity to Ideal Solution

Research on Staff Explicitation in Organizational Knowledge Management Based on Fuzzy Set Similarity to Ideal Solution Send Orders for Reprints to reprints@benthamscience.ae The Open Cybernetics & Systemics Jornal, 015, 9, 139-144 139 Open Access Research on Staff Explicitation in Organizational Knowledge Management Based

More information

Cosmological Origin of Gravitational Constant

Cosmological Origin of Gravitational Constant Apeiron, Vol. 5, No. 4, October 8 465 Cosmological Origin of Gravitational Constant Maciej Rybicki Sas-Zbrzyckiego 8/7 3-6 Krakow, oland rybicki@skr.pl The base nits contribting to gravitational constant

More information

Mathematics for Computer Science

Mathematics for Computer Science Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: patrick.blackburn@loria.fr Course website: http://www.loria.fr/~blackbur/courses/math

More information

Candidate: Shawn Mullane. Date: 04/02/2012

Candidate: Shawn Mullane. Date: 04/02/2012 Shipping and Receiving Specialist / Inventory Control Assessment Report Shawn Mllane 04/02/2012 www.resorceassociates.com To Improve Prodctivity Throgh People. Shawn Mllane 04/02/2012 Prepared For: NAME

More information

Inferring Continuous Dynamic Social Influence and Personal Preference for Temporal Behavior Prediction

Inferring Continuous Dynamic Social Influence and Personal Preference for Temporal Behavior Prediction Inferring Continos Dynamic Social Inflence and Personal Preference for Temporal Behavior Prediction Jn Zhang 1,2,3,4 Chaokn Wang 2,3,4 Jianmin Wang 2,3,4 Jeffrey X Y 5 1 Department of Compter Science and

More information

High Availability for Microsoft SQL Server Using Double-Take 4.x

High Availability for Microsoft SQL Server Using Double-Take 4.x High Availability for Microsoft SQL Server Using Doble-Take 4.x High Availability for Microsoft SQL Server Using Doble-Take 4.x pblished April 2000 NSI and Doble-Take are registered trademarks of Network

More information

Optimal control and piecewise parametric programming

Optimal control and piecewise parametric programming Proceedings of the Eropean Control Conference 2007 Kos, Greece, Jly 2-5, 2007 WeA07.1 Optimal control and piecewise parametric programming D. Q. Mayne, S. V. Raković and E. C. Kerrigan Abstract This paper

More information

FIBRATION SEQUENCES AND PULLBACK SQUARES. Contents. 2. Connectivity and fiber sequences. 3

FIBRATION SEQUENCES AND PULLBACK SQUARES. Contents. 2. Connectivity and fiber sequences. 3 FIRTION SEQUENES ND PULLK SQURES RY MLKIEWIH bstract. We lay out some foundational facts about fibration sequences and pullback squares of topological spaces. We pay careful attention to connectivity ranges

More information

Introducing Revenue Cycle Optimization! STI Provides More Options Than Any Other Software Vendor. ChartMaker Clinical 3.7

Introducing Revenue Cycle Optimization! STI Provides More Options Than Any Other Software Vendor. ChartMaker Clinical 3.7 Introdcing Revene Cycle Optimization! STI Provides More Options Than Any Other Software Vendor ChartMaker Clinical 3.7 2011 Amblatory EHR + Cardiovasclar Medicine + Child Health STI Provides More Choices

More information

ACTA UNIVERSITATIS APULENSIS No 15/2008 PRODUCTS OF MULTIALGEBRAS AND THEIR FUNDAMENTAL ALGEBRAS. Cosmin Pelea

ACTA UNIVERSITATIS APULENSIS No 15/2008 PRODUCTS OF MULTIALGEBRAS AND THEIR FUNDAMENTAL ALGEBRAS. Cosmin Pelea ACTA UNIVERSITATIS APULENSIS No 15/2008 PRODUCTS OF MULTIALGEBRAS AND THEIR FUNDAMENTAL ALGEBRAS Cosmin Pelea Abstract. An important tool in the hyperstructure theory is the fundamental relation. The factorization

More information

Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections

Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections Faster Inversion and Other Black Box Matrix Comptations Using Efficient Block Projections Wayne Eberly 1, Mark Giesbrecht, Pascal Giorgi,, Arne Storjohann, Gilles Villard (1) Department of Compter Science,

More information

This chapter is all about cardinality of sets. At first this looks like a

This chapter is all about cardinality of sets. At first this looks like a CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

More information

High Availability for Internet Information Server Using Double-Take 4.x

High Availability for Internet Information Server Using Double-Take 4.x High Availability for Internet Information Server Using Doble-Take 4.x High Availability for Internet Information Server Using Doble-Take 4.x pblished April 2000 NSI and Doble-Take are registered trademarks

More information

The Dot Product. Properties of the Dot Product If u and v are vectors and a is a real number, then the following are true:

The Dot Product. Properties of the Dot Product If u and v are vectors and a is a real number, then the following are true: 00 000 00 0 000 000 0 The Dot Prodct Tesday, 2// Section 8.5, Page 67 Definition of the Dot Prodct The dot prodct is often sed in calcls and physics. Gien two ectors = and = , then their

More information

Phone Banking Terms Corporate Accounts

Phone Banking Terms Corporate Accounts Phone Banking Terms Corporate Acconts If there is any inconsistency between the terms and conditions applying to an Accont and these Phone Banking Terms, these Phone Banking Terms prevail in respect of

More information

The Good Governance Standard for Public Services

The Good Governance Standard for Public Services The Good Governance Standard for Pblic Services The Independent Commission for Good Governance in Pblic Services The Independent Commission for Good Governance in Pblic Services, chaired by Sir Alan Langlands,

More information

Optimal Personalized Filtering Against Spear-Phishing Attacks

Optimal Personalized Filtering Against Spear-Phishing Attacks Optimal Personalized Filtering Against Spear-Phishing Attacks Aron Laszka and Yevgeniy Vorobeychik and Xenofon Kotsokos Institte for Software Integrated Systems Department of Electrical Engineering and

More information

Theoretical Computer Science

Theoretical Computer Science Theoretical Computer Science 410 (2009) 4489 4503 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs A push relabel approximation algorithm

More information

3. Prime and maximal ideals. 3.1. Definitions and Examples.

3. Prime and maximal ideals. 3.1. Definitions and Examples. COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,

More information

UNIT 62: STRENGTHS OF MATERIALS Unit code: K/601/1409 QCF level: 5 Credit value: 15 OUTCOME 2 - TUTORIAL 3

UNIT 62: STRENGTHS OF MATERIALS Unit code: K/601/1409 QCF level: 5 Credit value: 15 OUTCOME 2 - TUTORIAL 3 UNIT 6: STRNGTHS O MTRIS Unit code: K/601/1409 QC level: 5 Credit vale: 15 OUTCOM - TUTORI 3 INTRMDIT ND SHORT COMPRSSION MMBRS Be able to determine the behavioral characteristics of loaded beams, colmns

More information

MING-CHIH YEH LEE, B.A., M.S. A THESIS COMPUTER SCIENCE

MING-CHIH YEH LEE, B.A., M.S. A THESIS COMPUTER SCIENCE NEURAL NETWORK STRUCTURE MODELING: AN APPLICATION TO FONT RECOGNITION by MING-CHIH YEH LEE, B.A., M.S. A THESIS IN COMPUTER SCIENCE Sbmitted to the Gradate Faclty of Texas Tech University in Partial Flfillment

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic

Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic Parametric Domain-theoretic models of Linear Abadi & Plotkin Logic Lars Birkedal Rasmus Ejlers Møgelberg Rasmus Lerchedahl Petersen IT University Technical Report Series TR-00-7 ISSN 600 600 February 00

More information

A guide to safety recalls in the used vehicle industry GUIDE

A guide to safety recalls in the used vehicle industry GUIDE A gide to safety recalls in the sed vehicle indstry GUIDE Definitions Aftermarket parts means any prodct manfactred to be fitted to a vehicle after it has left the vehicle manfactrer s prodction line.

More information

1 The Concept of a Mapping

1 The Concept of a Mapping Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 1 The Concept of a Mapping The concept of a mapping (aka function) is important throughout mathematics. We have been dealing

More information

CSC 505, Fall 2000: Week 8

CSC 505, Fall 2000: Week 8 Objecties: CSC 505, Fall 2000: Week 8 learn abot the basic depth-first search algorithm learn how properties of a graph can be inferred from the strctre of a DFS tree learn abot one nontriial application

More information

Solving Newton s Second Law Problems

Solving Newton s Second Law Problems Solving ewton s Second Law Problems Michael Fowler, Phys 142E Lec 8 Feb 5, 2009 Zero Acceleration Problems: Forces Add to Zero he Law is F ma : the acceleration o a given body is given by the net orce

More information

Horses and Rabbits? Optimal Dynamic Capital Structure from Shareholder and Manager Perspectives

Horses and Rabbits? Optimal Dynamic Capital Structure from Shareholder and Manager Perspectives Horses and Rabbits? Optimal Dynamic Capital trctre from hareholder and Manager Perspectives Nengji J University of Maryland Robert Parrino University of exas at Astin Allen M. Poteshman University of Illinois

More information

Exploiting Spectral Reuse in Resource Allocation, Scheduling, and Routing for IEEE Mesh Networks

Exploiting Spectral Reuse in Resource Allocation, Scheduling, and Routing for IEEE Mesh Networks Exploiting Spectral Rese in Resorce Allocation, Schedling, and Roting for IEEE 802.16 Mesh Networks Lien-W Chen 1, Y-Chee Tseng 1, Da-Wei Wang 2, and Jan-Jan W 2 1 Department of Compter Science, National

More information

The Single Vehicle Approval Scheme

The Single Vehicle Approval Scheme GUIDE The Single Vehicle Approval Scheme A gide to the approval of special prpose light goods vehicles Saving lives, safer roads, ctting crime, protecting the environment A gide to the approval of special

More information