# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

17 Pages

### regularity

Course: MATH 550, Fall 2009
School: McGill
Rating:

Word Count: 1666

#### Document Preview

Regularity Szemeredis Lemma One of the most important tools in dense combinatorics. Message: every graph G is the approximate union of constantly many random-like bipartite graph. The number of parts depends only on the error of the approximation constant but not the size of G! For disjoint subsets X, Y V , |E(X, Y )| d(X, Y ) := |X| |Y | is the density of the pair (X, Y ). A pair (A, B) of disjoint subsets...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Canada >> McGill >> MATH 550

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Regularity Szemeredis Lemma One of the most important tools in dense combinatorics. Message: every graph G is the approximate union of constantly many random-like bipartite graph. The number of parts depends only on the error of the approximation constant but not the size of G! For disjoint subsets X, Y V , |E(X, Y )| d(X, Y ) := |X| |Y | is the density of the pair (X, Y ). A pair (A, B) of disjoint subsets A, B V is called -regular pair for some > 0 if all X A, and Y B with |X| |A| and |Y | |B| satisfy |d(X, Y ) d(A, B)| . Remark Just like in a random bipartite graph... 1 Szemeredis Regularity Lemma A partition {V0 , V1, . . . , Vk } of V is called an -regular partition if (i) |V0| |V | k (iii) all but at most 2 of the pairs (Vi , Vj ), with 1 (ii) |V1| = = |Vk | i < j k2 , are -regular V0 is the exceptional set Regularity Lemma (Szemeredi) > 0 and integer m 1 integer M = M (, m) such that every graph of order at least m admits an -regular partition {V0, V1 , . . . , Vk } with m k M . Was devised to prove that dense sets of integers contain an arithmetic progression of arbitrary length. 2 History of Szemeredis Theorem Szemeredis Theorem (1975) For any integer k 1 and > 0 there is an integer N = N (k, ) such that any subset S {1, . . . , N } with |S| N contains an arithmetic progression of length k. Was conjectured by Erdos and Turan (1936). Featured problem in mathematics, inspired lots of great new ideas and research in various elds; Case of k = 3: analytic number theory (Roth, 1953; Fields medal) First proof for arbitrary k: combinatorial (Szemeredi, 1975) Second proof: ergodic theory (Furstenberg, 1977) Third proof: analytic number theory (Gowers, 2001; Fields medal) Fourth proof: fully combinatorial (Rodl-Schacht, Gowers, 2007) Fifth proof: measure theory (Elek-Szegedy, 2007+) One of the ingredients in the proof of Green and Tao: primes contain arbitrary long arithmetic progression 3 Proof of the Erdos-Stone Thm Erdos-Stone Theorem. (Reformulation) For any > 0 and integers r 2, t 1 there exists an integer N = N (r, t, ), such that any graph G on n N ver1 tices with more than 1 r1 + n edges con2 tains Trt,r . Proof strategy: Based on an -regular partition, build a regularity graph R of G. (Regularity Lemma) Show that R contains a Kr (Turans Theorem) Show that Kr R Trt,r G 4 Regularity graph Given -regular partition P = {V0, V1 , . . . , Vk } of G, m k M (, m), dene the regularity graph R = R(P, d) V (R) = {V1, . . . , Vk } ViVj E(R) if (Vi , Vj ) is -regular pair with density d(Vi , Vj ) d Goal Choose , m, d such that most edges of G go between the sets Vi and Vj with ViVj E(R) How many edges are not at the right place? n2 < n2 < k m # of edges incident to V0 : at most n n = n2 # of edges inside Vi: at most k n/k 2 # of edges between non-regular pairs: k at most 2 n 2 < n2 k # of edges between pairs of density < d: at most 2 k d n dn2 2 k 5 Regularity graph contains an r-clique Conclusion: If , m, and d is chosen such that 1 < m 2 then most edges of G go between sets Vi and Vj with ViVj E(R). d + 2 + 1 most means at least 1 r1 n + 2 n2 2 On the other hand: # of edges of G going between sets Vi and Vj with ViVj E(R): Hence 1 1 r1 1 1 r1 2 at most |E(R)| n k n 2 n 2 + n |E(R)| 2 2 k k + k2 |E(R)| 2 2 Choose m = m() such that 1 ex(m, Kr ) 1 r1 m + 2 m2 2 Then Turans Theorem R contains a Kr 6 Finding Trt,r There are r classes Vi1 , . . . , Vir such that (Vij , Vi ) is an -regular pair of density at least d, for every 1 j < r. Let n = |Vij |. Then n n 1 n. k k We nd a Trt,r in G[Vi1 Vir ]. Lemma Let (A, B) be an -regular pair with d(A, B) d Let Y B be a subset with |Y | |B|. Then |{v A : dY (v) < (d )|Y |}| < |A|. Proof. Otherwise the subsets Y B and {v A : dY (v) < (d )|Y |} A would contradict the -regularity of (A, B). 2 For a set S V let (S) = vS N (v) denote the set of common neighbors of the vertices of S. 7 Finding Trt,r (d )t1 n n (r 1)t n t n S1 V1, |S1| = t |Vi (S1 )| (d )t n for i = 2, 3, . . . , r (d )2t1 n n (r 2)t (d )t n t n S2 V2, |S2| = t |Vi (S1 S2)| (d )2tn for i = 3, . . . , r . . . (d )(r1)t1 n n t (d )(r2)tn t n Sr1 Vr1, |Sr1| = t r1 |Vr (i=1 Si)| (d )(r1)tn 8 Finding Trt,r r1 Sr NVr (i=1 Si), |Sr | = t and thus G[S1 Sr ] contains a Trt,r provided (d )(r1)t n t Strongest of the blue conditions: (d )(r1)t1 Lets not forget: 1 d + 2 + < m 2 Choose, for example: 6 m d=6 = min d t(r1) , 1 2 t(r1) Green conditions are satised by choosing a large enough threshold vertex number N = N (r, t, ). r, t, ; m, d, ; N We also needed large m earlier for using Turans Theorem. 9 The Erdos-Turan conjecture A set S of positive integers is k-AP-free if {a, a + d, a + 2d, . . . , a + (k 1)d} S implies d = 0. sk (n) = max{|S| : S [n] is k-AP-free} How large is sk (n)? Could it be linear in n? Erdos-Turan Conjecture (Szemeredis Theorem) For every constant k, we have sk (n) = o(n). Construction (Erdos-Turan, 1936) s3 (n) log 2 log 3 . n S = {s : there is no 2 in the ternary expansion of s} S is 3-AP-free. For n = 3l , |S [n]| = 2l Roths Theorem (1953) s3(n) = o(n). 10 Applications of the Regularity Lemma Removal Lemma For > 0 = () such that the following holds. Let G be an n-vertex graph such that at least n edges has to be deleted from G 2 to make it triangle-free. Then G has at least n tri3 angles. Proof. Apply Regularity Lemma (Homework). Roths Theorem For > 0 N = N ( ) such that for any n N and S [n], |S| n, there is a three-element arithmetic progression in S. Proof. Create a tri-partite graph H = H(S) from S. V (H) = {(i, 1) : i [n]} {(j, 2) : j [2n]} {(k, 3) : k [3n]} (i, 1) and (j, 2) are adjacent if j i S (j, 2) and (k, 3) are adjacent if k j S (i, 1) and (k, 3) are adjacent if k i 2S 11 Roths Theorem Proof contd (i, 1), (i + x, 2), (i + 2x, 3) form a triangle for every i [n], x S. These |S|n triangles are pairwise edge-disjoint. At least n2 18 |V (H)| edges must be removed 2 from H to make it triangle-free. Let = 18 provided by the Removal Lemma. There are at least |V (H)| triangles in H. 3 S has no three term arithmetic progression {(i, 1), (j, 2), (k, 3)} is a triangle iff ji = kj S. Hence the number of triangles in H is equal to n|S| n2 < 6n , provided n > N ( ) := 1 . 2 3 12 Behrends Construction Construction (Behrend, 1946) s3(n) n 1O 1 log N . Construct set of vectors = (a0 , a1, . . . , al1): a Vk = { Z a Z where = a l : a 2 d = k, 0 ai < for all i < q}, 2 l1 2 i=0 ai . Interpret a vector {0, 1, . . . , d 1}l as an integer a written in d-ary: l1 n = a i=0 ai di . Let Sk = {n : Vk } a a 13 Claim Sk [dl ] is 3-AP-free for every k. Proof. Assume n + n = 2n. a c b Then ai + bi = 2ci for every i < l, because ai + bi and 2ci are both < d (so there is no carry-over) So + = 2. But a b c 2 = 2 = 2 k = + + , c c a b a b and equality happens only if and are parallel. Since a b they are of the same length, we conclude = . 2 a b Take the largest Sk . Bound its size by averaging: {0, 1, . . . , d 1}l 2 < ld2 , a a so there is a k for which | ...

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

McGill - ARCH - 671
Thesis Abstract My question is one that asks if architecture can take a more active role in the passing of our culture; generally, figuratively, and more specifically, by encouraging and strengthening the dissemination of a story or group of stories.
McGill - ARCH - 671
Hard-boiled Beauty and the End of the WorldCandace Wiersema Supervisor: Michael JemtrudApril 01, 2008 (Fifth Draft) Haruki Murakamis fiction, The Hardboiled Wonderland and the End of the World is a point of departure for revisiting historical and c
McGill - ARCH - 671
(un)Folding : memory, time and placeLife is serious, art serene. Architecture does not build for the sake of the engaged or detached spectator watching a play, but rather for people who experience, in spaces, the seriousness of life. Herzog &amp; de Me
McGill - ARCH - 671
A House of Upbringing Squamish Nation Elementary SchoolEurocentric educational practices have been forced upon the Aboriginal peoples of British Columbia for more than a century. These practices were the cornerstone of a national policy that expli
McGill - ARCH - 671
McGill University School of Architecture Architectural Journalism 301-679A Prof. David Covo Prof. Howard DaviesThesis Project: Interaction Connection IntegrationPoetics of TimeagingcaredignityindependencelifememoryThesis Presentatio
McGill - ARCH - 671
a synthesis of architecture, landscape, infrastructure and public art in an eventful public spaceSCAPE, a term introduced by Rem Koolhaas, implies a reading of the urban territory as landscape[it] is an idiom for the edgeless city, in which the dis
McGill - ARCH - 671
reFashioning Chabanel: an intervention in Cit de la ModeFashion in its search for newer and more diaphanous fabrics, and architecture in its search for more and more attenuated enclosures are destined to become one. -Martin Pawley-The design of Mo
McGill - ARCH - 671
Montreal African Diaspora Cultural CentreMaster of Architecture, McGill University&quot;The management of a theatre may impose restrictions and make rules as to the place which each person should occupy during a representation. Therefore, when a colour
McGill - ARCH - 671
The Child, the Home and the CommunityWendy SchusterJanuary 22, 2008 (First Draft) There must be a balance between the role of the child at the private level of the home and at the public level of the community. The project will be a housing complex
McGill - ARCH - 671
Haunted Architecture: Ghosts Guiding DesignAnna Sampson Supervisor: David CovoApril 1, 2008 Every site is haunted by its history and memory; some are said to be haunted by the spirits of the dead. Artists will live and work in an isolated and haunt
McGill - ARCH - 671
The Branded ExperienceWelland SinApril 1st, 2008This project will adopt the framework of branding as a creative process and explore its effects on Montreal's continuous interior network. The contemporary global city is experienced as a sequence o
McGill - ARCH - 671
Le Corbusiers Poeme de langle droit: A Machine for Sacred Architecture in Our Contemporary EpochNathan KastenApril 1, 2008 (Intermediate Draft) In our contemporary epoch where culture no longer finds its orientation strictly through religion, archi
McGill - ARCH - 671
Historical ContinuityMatthew WiviottJanuary 21, 2008 (First Draft) The subject of my project will be the historical fabric of the city of Montreal and the architectural forms that have been left in its wake. The site will be the Allen Memorial Inst
McGill - ARCH - 671
Parallel NarrativesMatthew WiviottApril 1, 2008 (Second Draft)a space for the encounter of history Daniel Libeskind contributing to a richer world that can speak about the human condition and its mysteries. Alberto Perez-Gomez This thesis seeks t
McGill - ARCH - 671
ABSTRACT Spaces of flow in a new Place-des-ArtsCe projet consistera crer une plateforme interactive reliant plusieurs espaces de flux connexes au Quartier des spectacles et au Montral sous-terrain. Cette nouvelle organisation spatiale sera non seu
McGill - ARCH - 671
Revealing Roof SpacePhilippe Ashby April 1st, 2008The extensive landscape of unused flat roofs offers enormous potential to accommodate a range of activities that can be connected to form a rediscovered layer of the city. In this context, the thes
McGill - ARCH - 671
The Empty Square or an Occupant without a PlacePer KefgenApril 1, 2008What role does the public park play within the contemporary North American city? Montrals 192-hectare territory known as Saint-Michel Environmental Complex [figure 1] is a test
Concordia Canada - ENGR - 371
Tests of Hypotheses IIAssignment #12Probability and StatisticsFollowing two questions are copied from the exercises at the end of sections 9.3 &amp; 9.4.1 of the text book. To solve these problems, study the material of these sections in detail.=
McGill - ARCH - 671
Drawing, Collaging and Modelling from Ghost StoriesWhen we listen to a ghost story, we create images of the characters, site and architecture within our own imaginations. In ghost stories, we must imagine things that we do not have precedent imager
Concordia Canada - ENGR - 371
Tests of Hypotheses IAssignment #11Probability and StatisticsFollowing three questions are copied from the exercises at the end of sections 8.6, 8.7, 9.1, &amp; 9.2 of the text book. To solve these problems, study the material of these sections in
Concordia Canada - ENGR - 251
Thermodynamics IFINALENGR 251/2 Fall 07CONCORDIA UNIVERSITY FACULTY OF ENGINEERING AND COMPUTER SCIENCE DEPARTMENT OF MECHANICAL ENGINEERINGCourse Number SectionThermodynamics IExamination DateENGR 251TimeAll SectionsNumber of Pages
McGill - ARCH - 671
Genetic ArchitectureCarla SmileyJanuary 21, 2008 (First Draft)Retail Genomics is developing faster than the legislation to regulate it is.Ethicists from alldisciplines are faced with an enormous challenge, and the response will require a high
McGill - ARCH - 671
Movement as a Continuum of Space: Learning from Contemporary DanceFranois Saint HilaireApril 1, 2008 (Second Draft)Abstract Using a technique derived from the work of the choreographer Pina Bausch, an architectural proposition is developed to exa
McGill - ARCH - 671
Theatre as a Cultural and Spiritual Symbol: on the Importance of a National Theatre in QubecFranois Saint HilaireJanuary 21, 2008 (First Draft)Abstract An architectural proposition is developed for a National Theatre in Qubec as a response to the
McGill - ARCH - 671
Historic Town Waterfront RevivalWhat is it that draws us to the water? Of course we need it for survival, we use it for defense, industry, and recreation. What is it that makes the view of the waterfront more valuable than the view onto the city? I
McGill - ARCH - 671
Community ParticipationEach architectural and planning project has a variety of different players interested in the outcome. Clients, government ofcials, historic and community organizations are just some of the people who can inuence a design in ma
McGill - ARCH - 671
Market Influenced: Advertising as a Design MethodologyWelland SinJanuary 21, 2008 (First Draft)Advertising is a creative field that seeks to balance the needs of the consumer and the needs of the corporation. The values and methodologies of adver
McGill - ARCH - 671
Master of Architecture Thesis Proposition Draft March 17, 2008 By Nathan KastenLe Corbusiers Poeme de langle droit: A Machine for Sacred Architecture in Our Contemporary EpochAbstract:In our contemporary epoch where culture no longer finds its o
Toledo - PSY - 324
Term Paper Instructions The main purpose of the paper is to examine original research findings related to the topic of your term paper. The topics were very broad and it is helpful to focus on a specific topic within the topic area of your choice. Fo
Toledo - PSY - 230
Term Paper Instructions The main purpose of the paper is to apply basic concepts that you are learning in this course to the individual difference dimension that you chose from the list in the course outline. The paper focuses on individual differenc
Toledo - PSY - 230
Personality and Social Relationships Personality and Social Relationships1Social Relationships are an important part of people's lives. As we will see later, for many people maintaining good social relationships is one of their most important mot
McGill - ARCH - 528
An Introduction to Indigenous African Architecture Labelle Prussin The Journal of the Society of Architectural Historians, Vol. 33, No. 3. (Oct., 1974), pp. 182-205.Stable URL: http:/links.jstor.org/sici?sici=0037-9808%28197410%2933%3A3%3C182%3AAITI
McGill - ARCH - 303
COHOUSING RESIDENCE AT POINT ST. CHARLESSt. Charles PlaceRAND ABBAS.9734085.U2 STUDIO.301-384B.PROF. SIJ PKES.APRIL 10 . 2000ST. MADELINE WING SECTIONAL PERSPECTIVEWELLINGTON ST. ELEVATIONST. MADELAINE ELEVATIONST. MAD
McGill - BRUG - 105
body {font-family: Arial, Helvetica, Sans-Serif; font-size: 10pt}th {font-family: Arial, Helvetica, Sans-Serif; font-size: 10pt;}td {font-family: Arial, Helvetica, Sans-Serif; font-size: 9pt;}form {font-family: Arial, Helvetica, Sans-Serif; fo
McGill - C - 550
Borehole GeophysicsElectrical resistivity and gas hydrates Rt = a Rw -m Sw-n (EQ 6.20)Archies law is used to estimate concentrations of hydrocarbon (incl. gas hydrate) in the pore volume. Archies relationship works well for pore-filling hydrate i
McGill - C - 666
Abundant pyroclastic komatiitic volcanism in the 3.53.2 Ga Barberton greenstone belt, South AfricaM. Thompson Stiegler1, Donald R. Lowe1, Gary R. Byerly21Department of Geological and Environmental Sciences, Stanford University, Stanford, Californ
McGill - C - 350
CONVERGENT PLATE BOUNDARIES ISLAND ARCS A typical section across an island arc is shown below. Trench depths may be up to 11 km but are typically about 9 km. The &quot;inner slope&quot; is between about 8 and 20. The outer slope is shallow (ca. 5). Island arcs
McGill - C - 570
Toledo - PHL - 245
Natural Deduction: Recap Natural deduction allows us to construct strict proofs of the validity of arguments, using a limited set of simple rules. The rules are chosen on the basis of three main criteria: They are intuitively simple and obvious.
Toledo - PHL - 245
From Last Week To test an argument for validity, we thus construct a truth table: PQ P Q / P / ~ Q T T T T FT P T T F T TF ~QModern Symbolic LogicWeek 4: Argument Forms and Natural Deduction Reading: 6.6, 7.1F T T F F FF FFT TFHere, the
Toledo - PHL - 255
Answer KeySCIENCE &amp; PSEUDO-SCIENCEPHL255S - SPRING 2006 MIDTERM TESTPart 1: Definitions (4 points each)In 1-2 sentences, define each of the following terms, as they have been used in this course. 1. &quot;Fallible&quot; To say that an observation statemen
Toledo - PHL - 247
Burden of Proof In legal and moral reasoning, and elsewhere, &quot;burden of proof&quot; is an important notion. In a dispute, the side with the burden of proof must give a positive argument proving that their position is the correct one, or they lose the di
Toledo - PSY - 320
1 Self-Esteem Self-esteem is one of the popular topics in social and personality psychology. Over 6,000 articles have self-esteem in the title. Most researchers studying self-esteem do not relate it to attitudes. However, a closer look at measures of
Toledo - PHL - 245
Five More Replacement RulesTransposition (Trans) (pq) (~q~p) Material Implication (Impl) (pq) (~p q) Exportation (Exp) [(p q)r)] [(p(qr)] Material Equivalence (Equiv) (p q) [(pq)(qp)] (p q) [(pq) (~p~q)] Tautology (Taut) p (pp) p (p p)Modern
McGill - TOPICS - 460
Biology of Lactation 342-460B GalactopoiesisGalactopoeisisDefinition Galactopoeisis is defined as the maintenance of lactation once lactation has been established. The changes in mammary cell numbers (by growth or by cell death) and in milk yield
McGill - TOPICS - 450
Dairy Production 342-450A Milk Yield &amp; CompositionFactors Affecting Milk YieldUnder normal situations, milk production increases during the first six weeks of lactation and then gradually decreases. The actual amount of milk produced during the la
McGill - TOPICS - 450
Department of Animal ScienceTHE MAMMARY GLANDDairy Cattle Production 342-450APage 1 of 13Anatomy of the Mammary GlandWhat is a mammary gland? The mammary gland is the milk secreting structure, which includes a teat, a duct system and lobes
McGill - TOPICS - 450
Department of Animal ScienceFEEDING &amp; METABOLISM OF DIETARY PROTEINDairy Cattle Production 342-450APage 1 of 11Protein Fractions of Ruminant FeedsThese are some of the important protein fractions and definitions that are commonly used in rum
McGill - TOPICS - 450
Dairy Cattle Production 342-450A Calf managementDepartment of Animal ScienceCALF FEEDING &amp; MANAGEMENTDairy Cattle Production 342-450Page 1 of 24Dairy Cattle Production 342-450A Calf managementGoals for a successful calf management system
McGill - TOPICS - 450
Dairy Cattle Production 342-450A Milk BiosynthesisDepartment of Animal ScienceBIOSYNTHESIS OF MILK COMPONENTSDairy Cattle Production 342-450APage 1 of 9Dairy Cattle Production 342-450A Milk BiosynthesisIntroduction: Precursors of milk com
McGill - TOPICS - 450
Dairy Production342-450A Feeding during lactationDepartment of Animal ScienceFEEDING THE DAIRY COW DURING LACTATIONDairy Cattle Production 342-450APage 1 of 8Dairy Production342-450A Feeding during lactationFeeding the Dairy Cow during La
McGill - TOPICS - 450
Dairy Cattle Production 342-450A The Dry CowDepartment of Animal ScienceMANAGING THE DAIRY COW DURING THE DRY PERIODDairy Cattle Production 342-450APage 1 of 11Dairy Cattle Production 342-450A The Dry CowBackground The dry period is a crit
McGill - TOPICS - 450
Dairy Cattle Production Metabolic disordersDepartment of Animal ScienceMETABOLIC DISORDERS IN DAIRY COWSDairy Cattle Production 342-450APage 1 of 14Dairy Cattle Production Metabolic disordersIntroduction: Management of the dry cow plays an
McGill - TOPICS - 450
Department of Animal ScienceMASTITIS IN DAIRY COWSDairy Cattle Production 342-450APage 1 of 12Introduction: Mastitis is the most common and most costly disease of dairy cattle. In herds with out an effective mastitis control program, about 40
McGill - TOPICS - 450
Dairy Production 342-450A Lactation CurveLactation Curve Background Milk production starts at a relatively high rate and the amount secreted continues to increase for about 3-6 weeks. After a peak is attained, milk production gradually declines Hig
McGill - TOPICS - 450
Dairy Cattle production 342-450A Body Condition ScoreDepartment of Animal ScienceBODY CONDITION SCOREDairy Cattle Production 342-450APage 1 of 9Dairy Cattle production 342-450A Body Condition ScoreBackgrounds: Body-condition scoring is a
McGill - TOPICS - 450
Department of Animal ScienceTHE MILKING MACHINEDairy Cattle Production 342-450APage 1 of 12BACKGROUND: The milking machine is essentially a system through which two fluids, air and milk have to be transported. The milking machine installation
McGill - TOPICS - 460
Biology of Lactation 342-460B MammogenesisDevelopment of the Mammary Gland (Mammogenesis)Definition Mammogenesis is the term use to describe the development of the mammary gland. Mammary development begins when the animal is an early fetus and pro
UC Davis - ECS - 189
RNA Structure PredictionHierarchical organization of RNA moleculesPrimary structure:5'ACCACCUGCUGA 3'Secondary StructureTertiary structure:Hierarchical organization of RNA moleculesPrimary structure:5' to 3' list of covalently linked nu
McGill - TOPICS - 460
Biology of Lactation 342-460B Biological Function of LactationBiological Functions of LactationDefinition of Mmammals The production of milk by the female parent following parturition (or in some cases hatching of the young) is the only feature by
McGill - TOPICS - 450
Dairy Cattle Production 342-450A Raising Replacement HeifersDepartment of Animal ScienceRAISNG THE DAIRY HEIFERDairy Cattle Production 342-450Page 1 of 14Dairy Cattle Production 342-450A Raising Replacement HeifersBackground In this sectio