text: Daily Marital Interaction Quality and Carotid Artery Intima-Medial Thickness in Healthy Middle-Aged AdultsJoseph, Nataria Tennille PhD; Kamarck, Thomas W. PhD; Muldoon, Matthew F. MD; Manuck, Stephen B. PhDPsychosomatic Medicine: June 2014 - Volume 76 - Issue 5 - p 347–354 doi: 10.1097/PSY.0000000000000071 Original Articles Abstract Author InformationAuthors Article MetricsMetrics Objective To examine the association between marital interaction quality during daily life and subclinical cardiovascular disease (CVD). Studies have shown that marital status and quality of marriage are associated with cardiovascular health. However, little is known about the role of marital interaction quality during daily life in contributing to these effects. Methods The sample consisted of 281 healthy, employed middle-aged adults who were married or living with a partner in a marital-like relationship (mean age = 42.0 years, 88% white, 52% men). Marital interaction quality was assessed using hourly real-time ecological momentary assessments for 4 days, with participants rating their current or recent partner interactions on positive and negative characteristics (e.g., agreeableness and conflict). Carotid artery intima-medial thickness (IMT) was assessed using ultrasound imaging. Results Adjusting for demographics, positive marital interaction was inversely associated with IMT (b = −0.02, F(1,275) = 9.18, p = .002), and negative marital interaction was positively associated with IMT (b = 0.02 F(1,275) = 10.29, p = .001). These associations were not accounted for by behavioral and biological CVD risk factors and were consistent across age, sex, race, and education. The associations were also independent of marital interaction frequency, nonmarital social interaction quality, and personality factors. Global reports of marital quality, in contrast, were not associated with IMT. Conclusions Marital quality as measured during real-time interactions between partners was associated with subclinical CVD in healthy middle-aged adults. This study supports the use of real-time social interaction assessment for characterizing links between social relationships and cardiovascular health. From the Departments of Psychiatry (N.T.J), Psychology and Psychiatry (T.W.K), Medicine (M.F.M), and Psychology (S.B.M), University of Pittsburgh, Pittsburgh, Pennsylvania. Nataria T. Joseph is now at VA HSR&D Center for the Study of Healthcare Innovation, Implementation and Policy, VA Greater Los Angeles Healthcare System. This work does not necessarily represent the views of the Department of Veterans Affairs or of the US government.Address correspondence and reprint requests to Nataria T. Joseph, PhD, VA Greater Los Angeles Healthcare System, 11301 Wilshire Blvd, Los Angeles, CA, 90073. E-mail: email@example.com Received for publication August 27, 2013; revision received March 26, 2014. Copyright © 2014 by American Psychosomatic Society


text:
Date 9/11/2001 - Location: Baghdad, Iraq
- "How long do you think before they bomb us?"
- "But it wasn't us. It can't be us…" I rationalized.
- "It doesn't matter. It's all they need."
And it was true. It began with Afghanistan and then it was Iraq.
Four years later and the War on Terror (or is it the War of Terror?) has been won:
Al-Qaeda – 3,000
America – 100,000+
This is a rip-off the story I have just read at Baghdad Burning. Follow the link to read the full story as I'm sure it will make you think.

subset: commoncrawl


text:
Police have arrested Douglas L. Hassam, 46, of 45 High Ave. in Pembroke for allegedly being the driver who drove off after stricking a 76-year-old woman with his car in the parking lot of the Tedeschi's convenience store in Halifax on Dec. 2. The arrest was made the day after Halifax police released surveillance footage from the store. Police said the release led to a flood of calls.
Police have arrested the man they believe struck a 76-year-old woman with his car in the Tedeschi's convenience store parking lot Dec. 2 and then fled the scene.
Halifax police picked up Douglas L. Hassam, 46, of 45 High Ave. in Pembroke on Thursday. They believe he was the driver in a hit-and-run accident that severely injured Florence Pelaquin of Halifax, who was walking when she was struck. Surveillance video from the Tedeschi's store – released to the media this week – helped police identify Hassam, Halifax police Sgt. Ted Broderick said.
Hassam is charged with leaving the scene of an accident after personal injury, driving without a license, and driving an uninsured, unregistered vehicle.
Pelaquin, was taken by helicopter to a Boston hospital for treatment of a severe head injury and a broken leg. She has since returned to her home, and her condition is improving, according to her husband, Louis Pelaquin.
Hassam does not have a valid driver's license, and he was driving a car with an expired registration, police said.
Hassam has two other warrants for motor-vehicle violations, Broderick said.
On Wednesday, police released surveillance footage from the Tedeschi's store, which is on Route 106. The man believed to be Hassam was wearing a NASCAR jacket and hat. Witnesses told police that the driver got out of the Ford Taurus he was driving after hitting Pelaquin, looked at her and then got back in the car and drove away.
Broderick said his department was inundated with calls within the first hour of the surveillance video's release. A member of Hassam's family contacted police Wednesday night to confirm his identity, Broderick said.
Broderick said his department worked with Pembroke police to help track Hassam down.
Broderick said he waited several weeks before releasing the surveillance footage because he did not want to scare away the hit-and-run driver.
"Although the video was good, we didn't want to release it until our own leads were exhausted," he said.
READ MORE about this issue.


text:
Ford IDS 108.06 Native Install with Remote Support
Contacted obdexpress.co.uk about IDS 108 install and activation over TeamViewer. After agreeing on a price, he checked my system for compatibility and shortly thereafter began the install. There was a problem with my computer causing the install to have to be done twice, and he went above and bey Continue Reading →


text:
St. Louis County Schools
St. Louis County School District's declining enrollment, rising heating costs, aging buildings and less curriculum choices were a focus on the consolidation plan.
The District's School Board approved the consolidation plan reducing the seven K-12 schools to five:
- South Ridge School (new) combining the AlBrook, Alborn, Cotton, and surrounding communities.
- North Woods School (new) combining Cook, Orr, and surrounding communities.
- Cherry School (addition and interior renovation)
- Babbitt-Embarrass School (interior renovation).
- Tower-Soudan School modified as a K-6 elementary school
St. Louis County's new schools were designed to accommodate 600+/- PK-12 students. The facility planning process involved multiple meetings with staff, administrators, and input committees to set a program to best meet everyone's needs at a cost-effective budget. The design and programming process allowed for utilization of spaces for multi-use occupancy to minimize classroom downtime and maximize use of square footage.
Student safety and supervision, event management, and community usability were accomplished goals of these two new facilities.
Renovation/addition to existing K-12 facility
- Construct new three-station gym with locker rooms
- Addition/expansion of kitchen and commons space
- Public toilet access to gymnasium
- Event management building security
- Controlled office access for security
- Upgrade ventilation/electrical system
- New technology upgrades
- Relocate/construct new bus garage facility
- Renovate classrooms
- Parking lot and sidewalks
Renovation of existing schools on two sites
- Classroom technology/ventilation/heating upgrades
- Replace finishes throughout facility
- Provided security upgrades
- Upgrade accessibility needs
- Parking lot and sidewalks
ISD 2142 St. Louis County Schools


text:
The Galois Connection between Syntax and Semantics
- James Knight
- 3 years ago
- Views:
1 The Galois Connection between Syntax and Semantics Peter Smith University of Cambridge Note: The material in Chs. 2 and 3 now appears, in revised form, as (stand-alone) chapters in my Notes on Category Theory, freely available at
3 Contents Preface page ii 1 Partially ordered sets Posets introduced Partial orders and strict orders Maps between posets Compounding maps Order similarity Inclusion posets as typical 10 2 Galois connections Galois connections defined An alternative definition The relation between adjoints Fixed points and closures One way a Galois connection can arise 20 3 The Galois connection between syntax and semantics Some reminders Making the connection Drawing the consequences Triviality? More consequences Posets as sets 28 i
4 Preface In his Dialectica paper Adjointness in foundations (1969), F. William Lawvere writes of the familiar Galois connection between sets of axioms and classes of models, for a fixed [signature]. But even if long familiar folklore to category theorists, the idea doesn t in fact seem to be that widely known. The ideas here are pretty enough, elementary enough, and illuminating enough to be worth rehearsing briskly in an accessible stand-alone form. Hence these brisk notes. If you are already happy with the idea of a poset and ideas about mappings from one poset to another, proceed immediately to Chapter 2! ii
5 1 Partially ordered sets This chapter introduces a class of structures the posets which crop up in all sorts of contexts. Of course, given that this type of structure is indeed exemplified all over the place, there won t be a great deal we can say about the general case. But still, our discussion will give us a chance to introduce some key ideas like the idea of an order-isomorphism. 1.1 Posets introduced Definition A partially ordered set P = P, henceforth, a poset is a set P, carrying an ordering relation which is reflexive, anti-symmetric and transitive. That is to say, for all x, y, z P, (i) x x; (ii) if x y and y x, then x = y; (iii) if x y and y z, then x z. Why talk here of a partial order? Because there is no requirement that a given element of the poset P be order-related to any element other than itself. But do note equally that our definition doesn t actually preclude a poset s ordering relation being total: that is to say, it may be the case that for every x, y in the carrier set, either x y or y x. Perhaps we should really call our orders at least partial : but plain partial is absolutely standard. Let s immediately draw out a simple consequence of this definition, one which will help to fix ideas about the kind of ordering we are talking about here: 1
6 2 Partially ordered sets Theorem A poset contains no cycles. By a cycle, we mean a chain of elements p 0, p 1, p 2,..., p α P, all different, such that p 0 p 1 p 2... p α p 0 (to use an obvious notational shorthand). Proof Suppose there is such a cycle. By transitivity, p 0 p 1 and p 1 p 2 implies p 0 p 2. By transitivity again, p 0 p 2 together with p 2 p 3 implies p 0 p 3. Keep on going in the same way until you ve derived p 0 p α (if α is infinite, our informal argument here is infinitary!). But we are already given p α p 0. So by anti-symmetry we can conclude p α = p 0, contradicting the assumption that the elements of the cycle are all distinct. To put this graphically, suppose we draw a diagram of a poset. Put a dot for each distinct element, and a directed arrow from the dot corresponding to p to the dot corresponding to p just when p p. Then, because of transitivity, dots connected by a chain of arrows will also be directly connected. But the diagram will have no loops where we can follow a linked chain of directed arrows around and end up where we started except for the trivial self-loops that correspond to the truth of each p p. Here, then, are just a few examples of posets, starting with some based on familiar number systems before broadening out to give some non-numerical examples. (1) The natural numbers N with read as the natural less than or equals to order relation. (2) N with the converse ordering i.e. we take the order relation to be greater than or equals to. (3) N with the reordering defined as follows in terms of the natural ordering : (a) if m is even and n is odd, then m n; (b) if m and n are both odd, then m n iff m n; (c) if m and both n are both even, then m n iff m n. In other words, we put all the even numbers in their natural order before all the odds in their natural order. (4) N, = is also a poset, since identity is trivially the limiting case of a partial ordering (the smallest partial ordering on any set, where each element is order-related only to itself).
7 1.1 Posets introduced 3 (5) N \ {0} ordered by divisibility: so we put m n iff m divides into n without remainder. (Here N \ {0} is of course the set of natural numbers with 0 removed.) (6) N \ {0, 1} ordered by the same divisibility relation. (7) The real numbers R with the natural less than or equals to ordering. (8) R with the converse ordering. (9) The set R 2 of ordered pairs of reals r, s, taken with the lexicographic ordering, i.e. putting r, s r, s just if either r < r or r = r s s. (10) Any set Σ of functions f: R R, putting f g iff for all x R, f(x) g(x). (11) Choose an interpreted formal language L. Let α be the equivalence class containing all the L-wffs logically equivalent to α, and let E be the set of all such equivalence classes. Let be the relation that holds between the equivalence classes α, β E if α β. Then it is easy to check that E, is a poset. (If L comes equipped with a proof-system, then we could similarly define a poset in terms of a syntactic consequence relation rather than semantic consequence.) (12) A downward-branching binary tree has a top node, and every node has zero, one or two children (think of truth-trees ). Take N to be set of nodes. Let s say that node n is a descendant of n if n is a child of n, or is a child of a child of n, or is a child of a child of a child.... Put n n iff n = n or n is a descendant of n (in other words, n is on the same branch of the tree as n and n is at least as high on the branch as n ). Then N, is a poset. (13) Just for modal logicians: let W be a set of possible worlds in a Kripke frame for intuitionistic logic, and let be the accessibility relation between worlds. Then W, is a poset. And so it goes. Posets are ubiquitous. We ll mention just two more cases for the moment: (14) The powerset P(A) of some set A, with the members of P(A) ordered by set-inclusion, is a poset. (15) In fact, take any set M P(A), i.e. take any arbitrary collection of subsets of a set A. Then M, too is a poset. Call these inclusion posets. A bit later, Theorem will tell us that inclusion posets are in a good sense typical of all posets.
8 4 Partially ordered sets Note immediately that some of our example posets in fact have total orderings, while others have incomparable elements, i.e. elements x, y in the carrier set such that neither x y nor y x. Exercise: which examples are which? Note too the elementary point illustrated by the relationship between examples 5 and 6, and by the relationship between examples 14 and 15. Having worked out that the divisibility relation indeed gives us a partial ordering on the naturals starting from 1, we don t have to do more work to show that the restriction of that relation to the numbers from 2 onwards is also a partial ordering. Likewise, having worked out that is a partial order on the carrier set P(A), we immediately see it remains a partial order when restricted to some M P(A). This observation evidently generalizes to give us a trivial mini-theorem: Theorem If P = P, is a poset then so is P = P, for any P P, where is the relation restricted to P. In such a case, we can say that P is a sub-poset of P. A further, but equally elementary, point is illustrated by the relationship between examples 1 and 2, and between 7 and 8. This too we could dignify as a mini-theorem (whose proof is another trivial exercise): Theorem If P = P, is a poset then so is P op = P, op, where op is the opposite or converse of, i.e. for all x, y P, x op y iff y x. We ll say that P op here is the order-dual of P. Of course, if a poset s order relation is naturally represented by e.g., we ll want to represent the converse relation by : so the order dual of R, is R,. Likewise, if a poset s order relation is naturally represented by, the converse relation which orders its dual will be represented by. We ll use this sort of obvious notation in future. 1.2 Partial orders and strict orders What we ve just defined are sometimes also called weakly ordered sets. By contrast, Definition A strictly ordered set Q = Q, is a set Q, carrying a relation which is irreflexive, asymmetric and transitive. That is to say, for all x, y, z Q,
9 1.2 Partial orders and strict orders 5 (i) x x; (ii) if x y then y x; (iii) if x y and y z, then x z. For example, both N, < and R, <, with < interpreted in the natural way both times, are strictly ordered sets. So is N \{0}, where m n iff m n and m divides n without remainder. That latter example emphasizes that a strict order need not be a total order (so you need to watch the jargon here). Should we care about developing a theory of strictly ordered sets as well as a theory of partially ordered sets? Yes and no. For note we have the following three-part theorem, whose proof can again be left as an exercise. Theorem (i) Suppose P = P, is a partially ordered set, and for x, y P put x y = def x y x y. Then P = P, is a strictly ordered set. (ii) Conversely, suppose Q = Q, is a strictly ordered set, and for x, y Q put x y = def x y x = y. Then Q + = Q, is a partially ordered set. (iii) The result of going from a partially ordered set to its strictly ordered correlate then back again returns us to the original structure: i.e. (P ) + = P. Similarly (Q + ) = Q. When we have a partially ordered set we can therefore immediately define a counterpart strictly ordered set (by ignoring identities); and likewise when we have a strictly ordered set we can define a counterpart partially ordered set (by adding identities). So, yes, if we are interested in partially ordered sets at all, we ll probably care equally about their strictly ordered counterparts. But no, because of the simple definitional link, we don t really need a separate theory to treat the strictly ordered sets. We can develop the theory of partially and strictly ordered sets treating either notion as the more basic and the other as derived. It is conventional and convenient to take the idea of posets as the fundamental one. Henceforth, it will occasionally be notationally convenient to use x y as an abbreviation for x y x y (and likewise, x y as an abbreviation for x y x y, etc.).
10 6 Partially ordered sets 1.3 Maps between posets Next, we ll define three different, increasingly constrained, types of map that can relate two posets. Here s the first pair of notions we want: Definitions Suppose that P = P, and Q = Q, are two posets. And let f: P Q be a map between their carrier sets. Then (i) f is monotone iff, for all p, p P, if p p then f(p) f(p ); (ii) f is an order-embedding iff for all p, p P, p p iff f(p) f(p ). So a monotone mapping f preserves partial orderings. But note, it can so to speak squeeze them up. In other words, if p p we won t get the reversal f(p) f(p ), but we might have f(p) = f(p ). An order-embedding however preserves more: it puts a faithful copy of P s ordering inside Q. Let s make that last point more precise, via a theorem, a definition, and another theorem. First, then, we remark that: Theorem An order-embedding f: P Q between P = P, and Q = Q, is an injective (one-one) map. Proof Suppose f(p) = f(p ). Then, by the reflexivity of, we have both f(p) f(p ) and f(p ) f(p). Since f is an order-embedding, it follows by definition that p p and p p. And since is anti-symmetric, it follows that p = p. So, contraposing, if p p then f(p) f(p ), so f is one-one or injective. That theorem prompts a further key definition: Definition As before, suppose that P = P, and Q = Q, are two posets, and let f: P Q be a map between their carrier sets. Then f is an order-isomorphism iff f is a surjection (is onto) and is an order-embedding. Note that, given our last theorem, an order-isomorphism f between P and Q is one-one and onto. Hence, isomorphism is indeed an apt label. An order-isomorphism f makes one poset a copy of the other, as far as its order-structure is concerned.
11 1.3 Maps between posets 7 And now we can sharpen our claim that, if f is an order-embedding between P and Q, then f maps P to a copy embedded inside Q. Stating the result with a slight abuse of notation, we have: Theorem If f: P Q is an order-embedding between P = P, and Q = Q,, then f is an order-isomorphism between P and the poset f(p ),. f(p ) is, of course, the image of P under f, i.e. {q ( p P )f(p) = q}. And the (very) slight abuse of notation comes in not explicitly marking that the order-relation in f(p ), is the strictly speaking the relation in Q, restricted to f(p ). But here and henceforth we ll allow ourselves that degree of notational slack. The proof of our theorem is trivial: for f is, of course, onto f(p ), and is still an order-embedding. Let s use our definitions, then, in a few examples: (1) Take the pair of posets R +,, N,, where R + is the set of positive reals, and the order relations are the natural ones. Let f: R + N map a real r in R + to the natural number corresponding to r s integral part. Then f is monotone, but not an embedding (since e.g. the reals 10 and 11 get mapped to the same natural number, i.e. 3). (2) That first example reminds us that a monotone map typically doesn t preserve all order information. For an extreme example to hammer home the point, take R +, again and the one-element poset {0}, = (check that that is a poset!). Then the function f that maps every element of R + to 0 is monotone, but it s a forgetful map that throws away all non-trivial order information. (3) Take the posets R +,, N again, but this time consider a map g: N R + going in the opposite direction which takes a natural number to the corresponding integral real. Then g is an order-embedding but not a full isomorphism: g just finds a copy of the naturals embedded inside R +. (4) Consider the poset R,, and its order-dual R,. Then the map h: R R defined by h: x x is an order-isomorphism. (5) But don t be misled by that last example there isn t always an order-isomorphism between a poset and its order-dual. Consider, for example, N, and its order-dual N,. An orderisomorphism would have to map 0 which comes first in the order to some element which similarly comes first in the order.
12 8 Partially ordered sets But there is no such element in N. So there can be no orderisomorphism between N, and N,. 1.4 Compounding maps Another key result is that monotone maps can be composed together to get another monotone map: Theorem If f: P Q is a monotone map from P = P, to Q = Q, and g: Q R is a monotone map from Q to R = R,, then g following f, i.e. the composite map g f: P R, is a monotone map from P to R. Proof Need we spell this out? If p p then f(p) f(p ), since f is monotone. And if f(p) f(p ) then g(f(p)) g(f(p )), since g is monotone. So if p p then g(f(p)) g(f(p )), hence g f is monotone. It is also worth remarking on the following easy result: Theorem Composition of monotone maps is associative: in other words, if f: P Q, g: Q R, h: R S are monotone, then (f g) h and f (g h) are monotone and equal. Which leads to a brief aside. For it is, of course, trivial that the identity map Id P : P P, which maps every p P to itself, is a monotone map from P = P, to itself. Hence if you happen to know what a category is you ll see from our last two theorems that it s immediate that, if we take posets as objects and monotone mappings between them as the associated arrows/morphisms, then we get a category Pos of posets. However, although we ll sometimes be in the close neighbourhood of some category-theoretic ideas, e.g. in the next chapter, we ll not explicitly discuss categories in this book. End of aside! To continue: Theorem tells us that monotone maps compose. But so do order-embeddings and (crucially) isomorphisms: Theorem If f: P Q is an order-embedding (isomorphism) from P to Q, and g: Q R is an order-embedding (isomorphism) from Q to R, then g f: P R, is an order-embedding (isomorphism) from P to R.
13 1.5 Order similarity 9 Proof To show order-embeddings compose, just change the if s to if and only if s in the proof of Theorem And to show order-isomophisms compose, we just need to remark that if f and g are onto, so is g f. 1.5 Order similarity To repeat, an order-isomorphism between posets is indeed a one-to-one correspondence between their carrier sets, one that preserves all the basic facts about the way that the elements are ordered. So let s say that Definition The posets P and Q are order-similar (in symbols, P = Q), iff there is an order-isomorphism from P to Q. And we can readily check that order-similarity is properly so-called: Theorem Order-similarity is an equivalence relation. Proof It is trivial that order-similarity is reflexive. Theorem tells us that order-isomorphisms compose, and hence that order-similarity is transitive. So we just need to check for symmetry. In other words, we just need to confirm that if f: P Q between P and Q, then there is an order-isomorphism between Q and P. To show that, (i) define f 1 : Q P to be such that f 1 (q) = p iff f(p) = q, and then (ii) check this is indeed an order-isomorphism between Q and P. Let s have just a very small handful of examples: (1) We have already seen that N, and N, are of different order-types. Likewise there can be no order-isomorphism between N, and N,, where is the evens-before-odds ordering. For in N,, 1 has an infinite number of predecessors, and there is no element in N, with the same order property. Hence N,, N,, N, and indeed N, = are posets with the same carrier set but of different order-types. However, N,, N \ {0}, and E, are all order-similar (where N \ {0} is the set of naturals with 0 deleted, and E is the set of even numbers). (2) If Q + is the set of positive rational numbers, and their natural ordering, then Q +, is plainly not order-similar to N,. For, in their natural ordering, the rationals are dense between any two different ones there is another and not so for the natural
14 10 Partially ordered sets numbers. However we can also put a different order on the positive rationals. Map each q Q + \{0} one-to-one to the pair of naturals m, n where q = m/n and the fraction is in lowest terms. Do a standard kind of zig-zag enumeration of such pairs m, n where m, n have no common divisors other than 1. In other words, say m, n precedes m, n if m+n m +n or else if m+n = m +n and m m. And now put q q in Q + when either q = 0 or the counterpart pair for q is no later than the counterpart pair for q in the zig-zag enumeration. Then Q +, = N,. (3) The examples just mentioned involve posets whose ordering relation is total (i.e., for every x, y in the carrier set, either x y or y x). Plainly, no poset which isn t totally ordered can be order-similar to one that is. For an example of a pair of posets which are not totally ordered but which are still order-similar, consider the following trivial example. (i) The inclusion poset P({0, 1, 2}),. (ii) The poset whose carrier set is {1, 2, 3, 5, 6, 10, 15, 30}, with the numbers ordered by divisibility. It can again be left as an exercise to specify an order-isomorphism from (i) to (ii) there is more than one! 1.6 Inclusion posets as typical Posets, as we said before, are ubiquitous. Evidently they can come in all manner of sizes and shapes. And they can be built up from all manner of elements, related in a huge variety of ways. It might seem quite hopeless to try to put any organization at all on the mess here. But we can in fact tidy things up just a bit. For remember, any collection of subsets of some set A, ordered by, is a poset. We dubbed this sort of poset an inclusion poset. And we can now show: Theorem Every poset is order-isomorphic to an inclusion poset. NB, an inclusion poset can be based on any collection of subsets of a set A. The case where the poset is based on the set of all subsets of some A is a very special one. The theorem does not say that every poset is similar to one of the very special cases! Proof Take the poset P = P,. For each p P, now form the set π p = {x x P x p}. Let Π be the set of all π p for p P. Then Π is a set of subsets of P, and P π = Π, is of course an inclusion poset.
15 1.6 Inclusion posets as typical 11 Further, P is order-isomorphic to P π. For consider the function f: P Π which maps p P to π p Π. This map is evidently a oneone correspondence between P and Π. Moreover, by definition p p iff π p π p. So we are done. Hence, if what we really care about is just the order-type of various posets i.e. we don t care about what sort of elements the carrier-set has, and are just highlighting the kind of ordering that is being put on a set of elements then, without loss of generality, we can concentrate on inclusion posets as typical.
16 2 Galois connections Even inclusion posets come in all sizes and shapes. Which means that there isn t a great deal more of interest that can be said about posets in general. Hence we are soon going to move on to talk about ordered sets which have rather more additional structure. Still, before we do so, we need to explore another common relation that can obtain between posets, one which is weaker than order-isomorphism but which is still surprisingly rich in implications. The ideas here are particularly cute, and (among other things) they give us a lovely illustration of the way that modern mathematics likes to embed familiar facts into much more abstract general settings. 2.1 Galois connections defined Definition Let P = P, and Q = Q, be posets. And suppose f : P Q and f : Q P are a pair of functions such that for all p P and all q Q, (G) f (p) q iff p f (q). Then the pair f, f form a Galois connection between P and Q. If f, f is such a connection, f is said to be the left adjoint of the corresponding f, and f is the right adjoint of f. Think: f appears to the left of its order sign in (G), and f to the right of its order sign. Alternatively, the terminology lower adjoint vs upper adjoint is used. Think: f appears on the lower side of its ordering sign, and f on the upper side. (I have the star lower for the lower adjoint. Annoyingly, some authors use f and f the other way about.) Talk of adjoints is carried over from category theory. Category theorists in turn 12
17 2.1 Galois connections defined 13 There are plenty of serious mathematical examples (e.g. from number theory, abstract algebra and topology) of two posets with a Galois connection between them. But we really don t want to get bogged down in unnecessary mathematics at this stage; so for the moment let s just give some relatively simple cases, including a couple of logical ones. (1) Suppose f is an order-isomorphism between P and Q, so the inverse function f 1 is one too. Then f, f 1 is a Galois connection. For, trivially, f(p) q iff f 1 (f(p)) f 1 (q) iff p f 1 (q). (2) Let P = N, and Q be Q +,, where in each case the order relation is standard. Put f : N Q + to be the standard embedding of the natural numbers into the rationals; and let f : Q + N map a positive rational q to the natural corresponding to its integral part. Then (G) trivially holds, and f, f is a Galois connection between the integers and the rationals in their natural orderings. (3) Take an arbitrary poset P = P,, and put I = {0}, = (so I is a singleton poset with the only possible order relation). Let f be the trivial function i that maps each and every member of P to 0. And let f be some function that maps 0 to a particular member of P. Then f is right adjoint to f, i.e. the pair f, f form a Galois connection between P and I, just if, for every p P, f (p) = 0 iff p f (0), i.e. just if for every p, p f (0), i.e. just if f (0) is the maximum of P in the obvious sense. Dually, if we put f to be the same flattening function i, then f is left adjoint to it just if f (0) is the minimum of P. (4) Our next example is from elementary logic. Recall example 11 from Section 1.1. Let L be a formal language equipped with a proof-system (it could be classical or intuitionistic, or indeed any other logic with well-behaved conjunction and conditional connectives). Then α is the class of L-wffs interderivable with α; E seem to have borrowed the term from the old theory of Hermitian operators, where in e.g. a Hilbert space with inner product,, the operators A and A are said to be adjoint when we have, generally, Ax, y = x, A y. The formal analogy is evident. The first discussion of such a connection and hence the name is to be found in Evariste Galois s work in what has come to be known as Galois theory, a topic far beyond our purview here. Though you might like to find out about Galois s short life from the historical sketch in Stewart s Galois Theory (1989).
18 14 Galois connections is the set of all such equivalence classes; and α β iff α β. E = E, is a poset. Now consider the following two functions between E and itself. Fix γ to be some L-wff. Then let f be the function that maps the equivalence class α to the class (γ α), and let f be the function that maps the equivalence class α to (γ α). Now, we have (γ α) β if and only if α (γ β). So (γ α) β if and only if α (γ β). So f, f is a Galois connection between E and itself. ( Conjunction is left adjoint to conditionalization. ) (5) Our last example for the moment is again from elementary logic. Let L now be a first-order language, and Form( x) be the set of L-wffs with at most the variables x free. We ll write ϕ( x) for a formula in Form( x), ϕ( x) for the class of formulae interderivable with ϕ( x), and E x for the set of such equivalence classes of formulae from Form( x). With as in the last example, E x, is a poset for any x. Now we ll consider two maps between the posets E x, and E x,y,. In other words, we are going to be moving between (classes of) formulae with at most x free, and formulae with at most x, y free (where y isn t among the x). First, since every wff with at most the variables x free also has at most the variables x, y free, there is a trivial map f that sends ϕ( x) E x to the same element ϕ( x) E x,y. Second, we define the companion map f that sends ϕ( x, y) E x,y to yϕ( x, y) E x. Then f, f is a Galois connection between E x and E x,y. That is to say f ( ϕ( x) ) ψ( x, y) iff ϕ( x) f ( ψ( x, y) ). For that just reflects the familiar logical rule that ϕ( x) ψ( x, y) iff ϕ( x) yψ( x, y) so long as y is not free in ϕ( x). Hence universal quantification is right-adjoint to a certain trivial operation. And we can similarly show that existential quantification is left-adjoint to the same operation (exercise: spell out that last claim). Our first family of cases, then, shows that Galois connections exist
19 2.2 An alternative definition 15 and are at least as plentiful as order-isomorphisms. The second and third cases show that posets that aren t order-isomorphic can in fact still be connected. The third and fifth cases gives a couple of illustrations of how a more contentful notion (taking maxima, universal quantification) can be regarded as adjoint to some relatively trivial operation. The fourth case shows that even when the Galois-connected posets are isomorphic (in this case trivially so, because they are identical!), there can be functions f and f which aren t isomorphisms but which also go to make up a connection between the posets and since γ was arbitrary, this case also shows that there can be many different connections between the same posets. And the fourth and fifth cases already show that Galois connections are of interest to logicians. Finally in this section, let s show that connections are well-behaved, in the sense that connections compose to give new connections. In other words, we have: Theorem Let P = P,, Q = Q,, and R = R, be posets. Let suppose f, f is a Galois connections between P and Q, and g, g is a Galois connections between Q and R. Then g f, f g is a Galois connection between P and R. Proof By assumption, we have (G 1 ) f (p) q iff p f (q) and (G 2 ) g (q) r iff q g (r). So (G 2 ) gives us g (f (p)) r iff f (p) g (r). And (G 1 ) gives us f (p) g (r) iff p f (g (r)). Whence, as we want, g (f (p)) r iff p f (g (r)). 2.2 An alternative definition So far, so good. We ll meet a rather more telling example of a logically significant Galois connection in the next chapter. But first let s explore the general case in a bit more detail. Here s a simple but illuminating result: Theorem Suppose P = P, and Q = Q, are posets, and f : P Q and f : Q P are a pair of functions between their carrier sets. Then f, f is a Galois connection if and only if (i) f, f are both monotone, and (ii) for all p P, q Q, p f (f (p)) and f (f (q)) q.
20 16 Galois connections Proof (Only if ) Suppose f, f is a Galois connection. As a particular case of (G) from Defn we have f (p) f (p) iff p f (f (p)). Since is reflexive, the l.h.s. of that biconditional holds. So p f (f (p)). Similarly for the other half of (ii). Now, suppose also that p p. Since we ve shown p f (f (p )), we have p f (f (p )). But by (G) we have f (p) f (p ) iff p f (f (p )). Whence, f (p) f (p ) and f is therefore monotone. Similarly for the other half of (i). (If ) Now assume (i) and (ii) hold, and suppose f (p) q. Since by (i) f is monotone, f (f (p)) f (q). But by (ii) p f (f (p)). So, since is transitive, p f (q). Which establishes that if f (p) q then p f (q). The proof of the other half of the biconditional (G) is dual. This theorem can evidently be used to justify an alternative definition of a Galois connection as, simply, a pair of maps for which conditions (i) and (ii) hold. 2.3 The relation between adjoints Our next theorem tells us that the relation between the adjoint members of a Galois connection is rigid in the sense that if f, f is to be a connection, then f fixes what f uniquely has to be, and conversely f fixes what f has to be. Theorem If f, f 1 and f, f 2 are Galois connections between P, and Q,, then f 1 = f 2. Likewise, if f 1, f and f 2, f are Galois connections between the same posets, then f 1 = f 2. Because (i) holds, connections defined our way are sometimes called monotone Galois connections. But now recall that if Q = Q, is a poset, so is its dual Q op = Q, op. Suppose then that f, f is a (monotone) connection between P = P, and Q = Q, so that f (p) q iff p f (q). Then, trivially, there is an antitone Galois connection between P and Q op, meaning a pair of functions f, f such that q op f (p) iff p f (q). The only point of footnoting this here is to remark that, (a) you might well encounter the idea of a connection defined the antitone way (which indeed it how it appears in Galois s work); indeed (b) some authors only call the antitone case a Galois connection, reserving the term adjunction for connections defined our way. But (c) since a monotone connection between P and Q is an antitone connection between P and Q s dual, we just don t need to fuss about the difference, and without loss of generality can we can concentrate entirely on connections presented the monotone way.
CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimesMore information
Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statementMore information
Solutions to In-Class Problems Week 4, Mon.
Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics for Computer Science September 26 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised September 26, 2005, 1050 minutes SolutionsMore information
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,More information
Sets and functions. {x R : x > 0}.
Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.More information
MAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to takeMore information
Students in their first advanced mathematics classes are often surprised
CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter areMore information
13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385
mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infiniteMore information
3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)More information
Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.
M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also considerMore information
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainlyMore information
An Innocent Investigation
An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every numberMore information
4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2:
4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger setsMore information
Sets and Cardinality Notes for C. F. Miller
Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the useMore information
Introduction to mathematical arguments
Introduction to mathematical arguments (background handout for courses requiring proofs) by Michael Hutchings A mathematical proof is an argument which convinces other people that something is true. MathMore information
2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.
2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true thenMore information
COLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5More information
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
It is time to prove some theorems. There are various strategies for doing
CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, itMore information
September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the fourMore 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 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largelyMore information
Applications of Methods of Proof
CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets areMore information
1.3 Induction and Other Proof Techniques
4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.More information
Mathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,More information
A Mathematical Toolkit
A Mathematical Toolkit Philosophy 389 1 Naive Set Theory Cantor, in his 1895 work Contributions to the Founding of the Theory of Transfinite Numbers, gives the following definition of a set: By a set [Menge]More information
11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(More information
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
Introducing Functions
Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1More information
A brief history of type theory (or, more honestly, implementing mathematical objects as sets)
A brief history of type theory (or, more honestly, implementing mathematical objects as sets) Randall Holmes, Boise State University November 8, 2012 [notes revised later; this is the day the talk wasMore information
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 correspondenceMore information
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, calledMore information
Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.
MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But weMore information
Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets.
Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets. Logic is the study of reasoning. A language of propositions is fundamental to this study as well as trueMore information
Chapter 7. Functions and onto. 7.1 Functions
Chapter 7 Functions and onto This chapter covers functions, including function composition and what it means for a function to be onto. In the process, we ll see what happens when two dissimilar quantifiersMore information
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: email@example.com Course website: http://www.loria.fr/~blackbur/courses/mathMore information
Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducingMore information
1.7 Graphs of Functions
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent mostMore information
Cartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth specialMore information
Section 3. Compactness Theorem
Section 3 Compactness Theorem 1 / 24 Compactness Theorem The most important result in model theory is: Compactness Theorem Let T be a theory in language L. If every finite subset of T has a model, thenMore information
If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this questionMore information
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,More information
Logic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logicalMore information
Section 3 Sequences and Limits
Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.More information
ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, sinceMore information
diffy boxes (iterations of the ducci four number game) 1
diffy boxes (iterations of the ducci four number game) 1 Peter Trapa September 27, 2006 Begin at the beginning and go on till you come to the end: then stop. Lewis Carroll Consider the following game.More information
Likewise, we have contradictions: formulas that can only be false, e.g. (p p).
CHAPTER 4. STATEMENT LOGIC 59 The rightmost column of this truth table contains instances of T and instances of F. Notice that there are no degrees of contingency. If both values are possible, the formulaMore information
Math 3000 Running Glossary
Math 3000 Running Glossary Last Updated on: July 15, 2014 The definition of items marked with a must be known precisely. Chapter 1: 1. A set: A collection of objects called elements. 2. The empty set (More information
CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic calledMore information
CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduceMore information
Deductive Systems. Marco Piastra. Artificial Intelligence. Artificial Intelligence - A.A Deductive Systems [1]
Artificial Intelligence Deductive Systems Marco Piastra Artificial Intelligence - A.A. 2012- Deductive Systems 1] Symbolic calculus? A wff is entailed by a set of wff iff every model of is also model ofMore information
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 00More information
Lecture 17 : Equivalence and Order Relations DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notionMore information
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 rangesMore information
Notes 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 factorMore information
(Refer Slide Time: 1:41)
Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You mustMore information
Elementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,More information
The Consistency of the Continuum Hypothesis Annals of Mathematical Studies, No. 3 Princeton University Press Princeton, N.J., 1940.
TWO COMPUTATIONAL IDEAS Computations with Sets Union, intersection: computable Powerset: not computable Basic operations Absoluteness of Formulas A formula in the language of set theory is absolute ifMore information
So let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbersMore information
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verifyMore information
DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available freeMore information
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amountsMore information
Relations Graphical View
Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binaryMore information
Non-deterministic Semantics and the Undecidability of Boolean BI
1 Non-deterministic Semantics and the Undecidability of Boolean BI DOMINIQUE LARCHEY-WENDLING, LORIA CNRS, Nancy, France DIDIER GALMICHE, LORIA Université Henri Poincaré, Nancy, France We solve the openMore information
vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws
Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,More information
Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.More information
COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloringMore information
In mathematics you don t understand things. You just get used to them. (Attributed to John von Neumann)
Chapter 1 Sets and Functions We understand a set to be any collection M of certain distinct objects of our thought or intuition (called the elements of M) into a whole. (Georg Cantor, 1895) In mathematicsMore information
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule forMore information
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 categoryMore information
TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axiomsMore information
Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition
Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computingMore information
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specializeMore information
2. Propositional Equivalences
2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional Equivalences 2.1. Tautology/Contradiction/Contingency. Definition 2.1.1. A tautology is a proposition that is always true. Example 2.1.1. p p DefinitionMore information
3. Equivalence Relations. Discussion
3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,More information
Logic and Incidence Geometry
Logic and Incidence Geometry February 27, 2013 1 Informal Logic Logic Rule 0. No unstated assumption may be used in a proof. 2 Theorems and Proofs If [hypothesis] then [conclusion]. LOGIC RULE 1. The followingMore information
3.2 The Factor Theorem and The Remainder Theorem
3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomialMore information
Domain Theory: An Introduction
Domain Theory: An Introduction Robert Cartwright Rebecca Parsons Rice University This monograph is an unauthorized revision of Lectures On A Mathematical Theory of Computation by Dana Scott [3]. ScottMore information
Introduction to computer science
Introduction to computer science Michael A. Nielsen University of Queensland Goals: 1. Introduce the notion of the computational complexity of a problem, and define the major computational complexity classes.More information
Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of usMore information
Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes)
Dedekind s forgotten axiom and why we should teach it (and why we shouldn t teach mathematical induction in our calculus classes) by Jim Propp (UMass Lowell) March 14, 2010 1 / 29 Completeness Three commonMore information
INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28More information
Iterated Dynamic Belief Revision. Sonja Smets, University of Groningen. website: http://www.vub.ac.be/clwf/ss
LSE-Groningen Workshop I 1 Iterated Dynamic Belief Revision Sonja Smets, University of Groningen website: http://www.vub.ac.be/clwf/ss Joint work with Alexandru Baltag, COMLAB, Oxford University LSE-GroningenMore information
Model Theory of Fields with Operators
Piotr Instytut Matematyczny Uniwersytetu Wroc lawskiego PhDs in Logic VIII, Darmstadt, May 9-11, 2016 Beginnings May 9: Model theory of fields Initial setting A first-order language L; First order L-formulas,More information
x if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about theMore information
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 areMore information
Sets, Relations and Functions
Sets, Relations and Functions Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org email@example.com ugust 26, 2014 These notes provide a very brief background in discreteMore information
CHAPTER 2. Logic. 1. Logic Definitions. Notation: Variables are used to represent propositions. The most common variables used are p, q, and r.
CHAPTER 2 Logic 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A proposition is a declarative sentence that is either true (denoted either T or 1) or false (denoted either F or 0). Notation:More information
CHAPTER 3. Sequences. 1. Basic Properties
CHAPTER 3 Sequences We begin our study of analysis with sequences. There are several reasons for starting here. First, sequences are the simplest way to introduce limits, the central idea of calculus.More information
o-minimality and Uniformity in n 1 Graphs
o-minimality and Uniformity in n 1 Graphs Reid Dale July 10, 2013 Contents 1 Introduction 2 2 Languages and Structures 2 3 Definability and Tame Geometry 4 4 Applications to n 1 Graphs 6 5 Further DirectionsMore information
Chapter 7: Products and quotients
Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: ProductsMore information
TREES IN SET THEORY SPENCER UNGER 1. Introduction Although I was unaware of it while writing the first two talks, my talks in the graduate student seminar have formed a coherent series. This talk can beMore information
Basic Set Theory. 1. Motivation. Fido Sue. Fred Aristotle Bob. LX 502 - Semantics I September 11, 2008
Basic Set Theory LX 502 - Semantics I September 11, 2008 1. Motivation When you start reading these notes, the first thing you should be asking yourselves is What is Set Theory and why is it relevant?More information
1. R In this and the next section we are going to study the properties of sequences of real numbers.
+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of realMore information
Chapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applicationsMore information
8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that expressMore information
Math 582 Introduction to Set Theory
Math 582 Introduction to Set Theory Kenneth Harris email@example.com Department of Mathematics University of Michigan January 7, 2009 Kenneth Harris (Math 582) Math 582 Introduction to Set Theory JanuaryMore information
A Propositional Dynamic Logic for CCS Programs
A Propositional Dynamic Logic for CCS Programs Mario R. F. Benevides and L. Menasché Schechter {mario,firstname.lastname@example.com Abstract This work presents a Propositional Dynamic Logic in which the programs areMore information
Review for Final Exam
Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in ChaptersMore information
Lecture 13 - Basic Number Theory.
Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denotedMore information
! " # The Logic of Descriptions. Logics for Data and Knowledge Representation. Terminology. Overview. Three Basic Features. Some History on DLs
,!0((,.+#$),%$(-&.& *,2(-$)%&2.'3&%!&, Logics for Data and Knowledge Representation Alessandro Agostini firstname.lastname@example.com University of Trento Fausto Giunchiglia firstname.lastname@example.com The Logic of Descriptions!$%&'()*$#)More information


text:
LONDON (CBS.MW) -- Manchester United uk:mnu said it has terminated discussions with Tampa Bay Buccaneers owner Malcolm Glazer over his current proposals due to the highly leveraged nature of his takeover proposals. It said: "The Board has decided to inform all shareholders that it would regard an offer which it believes to be overly leveraged as not being in the best interests of the Company. Accordingly the Board has determined that it would not support any offer underpinned by the structure which has been outlined in relation to the Glazer family's potential offer." Manchester United said it intends to have further discussions with the Glazer family "to seek to develop a long term solution that brings stability."
Have breaking news sent to your inbox. Subscribe to MarketWatch's free Bulletin emails. Sign up here.


text:
Chapter Reports: Copper Country Quilters
2016 Reports
September 2016
The Copper Country Quilters were busy having fun this summer and sewing up a storm. In May president, Sally Hatfield presented us with a Crayon Challenge for the summer. This is the first time we had tried this so really weren't sure of the response. We were very surprised and pleased. After our August business meeting the Challenge projects were revealed! It was a blind reveal and we voted on our favorite and believe me it was a challenge to pick one favorite. Our ladies were so creative in the projects that were completed; from credit card holders to tea cozies to paper pieced masterpieces and everything in between. Jan Pederson's Splendid Sampler in Violet and Brick Red garnered the most votes from the 16 entries. What fun…. maybe we can do this again next year!
CCQ along with the Copper Thimbles and Apache Sew and Sews are continuing to plan for the 2017 Spring meeting to be held in Globe. The theme is Travel AZ…destination Copper Country. Plan for the weekend as Globe will be hosting the annual Historic Home and Building Tour and it will be the final weekend of CCQ Pieces of Friendship XXVII quilt show. A busy time in Globe.
Our meetings are held at 9:30 a.m. the 2nd, 3rd, and 4th Monday each month at the First Christian Church on Broad Street in Globe. Come and visit if you are in town.
August 2016
CCQ started the summer with two in house classes. Sally Hatfield led us in a mug rug class after the June business meeting which we may add to our quilt show boutique next year. Lynne Barte showed us how to do rope bowls last month during our regularly scheduled 'project day'. Several members made a variety of bowls. Our show and share times have indicated that the members are staying out of the heat and completing a variety of projects. We started a Crayon Challenge in May with several members participating. Completed projects will be revealed at the August business meeting.
Copper Country met with the Copper Thimbles from Kearney last month and plans are in the works for the next AQG Spring Meeting to be held in Globe. Get ready to "Travel AZ … destination Copper Country" in April.
Our July meeting is transition time. Committee positions will be turned over to new leadership in some instances and remain the same for others. Sally Hatfield will continue as our Chapter Chair with Shirlee Aqainess taking over as treasurer, Jan Pederson will continue in the secretary position. Lynne Bartee, with the help of Sharon Beydler, will coordinate all activities with Cobre Valley Center for the Arts. Kaye Ratcliff will join Darlene Martin and Laverna Andress in accessing the community needs for quilt donations and organizing our community service meetings. Remaining positions, including Projects and Programs, will remain in the same capable hands of other members that have been serving for previous years.
Activities planned for the coming year include a president's challenge, a mystery quilt project, speakers and a quilt documentation day.
Copper country Quilters meet on the 2nd, 3rd, and 4th Monday of each month at the First Christian Church on Broad St. Come and join us the next time you are visiting Globe on a Monday. Visitors are always welcome.
May 2016
Happy Birthday to us! Copper Country Quilters celebrated our 30th year as a member of the Arizona Quilters Guild on the closing weekend of our quilt show Pieces of Friendship XXVI. It was a great weekend in Globe with the annual Home and Building Tour which included a stop at the quilt show. Visitors were treated to birthday cake, coffee and punch to mark our 30 years as an AQG guild.
The viewer's choice award was presented to Jan Pederson's Amazon Star, AKA "What was I Thinking?". Runner up was Jan's hand quilted "Ruffled Roses." On Sunday we drew for the winner of our opportunity quilt, Cara Coles del mar. The winner was Tim Stickler from Miami, Az. Our busy birthday weekend started Friday with the Merry Go round class presented by teacher Reeze Hanson, speaker at the AQG Spring meeting on Saturday April 9. We had 12 ladies in the class and had a great time learning the techniques used in making a variety of patterns using hexagons. We were unable to attend Spring Meeting this year because of our quilt show and participation the home tour activities, but did enjoy Reeze's trunk show at the end of class so we didn't miss out completely. Mark your calendars for Spring meeting 2017 which will be hosted by Copper Country and Copper Thimbles. The meeting will coincide with Globe Home Tour weekend again so plan on spending the weekend with us here in Globe.
After our regularly scheduled business meeting in April, we ended our friendship block exchange coordinated by member Audrey Longhurst. 16 ladies participated and it was fun to see the variety of fabrics used in our common churn dash blocks. Copper Country Quilters have been meeting at the First Christian on Broad Street for over 25 years. We meet on the 2nd, 3rd, and 4th Monday's of the month at 10:00 a.m. Come and join us if you are visiting Globe.
April 2016
Copper Country Quilters have had a busy few weeks. Our annual quilt show, "Pieces of Friendship XXVI", opened in February and the Globe Chamber of Commerce asked us to extend the dates of the show to April 10, 2016 which is the last day of the Globe Historic Home and Building Tour weekend. By extending our show into April we become a scheduled stop on the home tour and we get to show off the beautiful Center for the Arts as well as our quilts.
Member Audrey Longhurst distributed background fabric at our January meeting to be used in a friendship block exchange within the chapter. Each member is making 16 churn dash blocks using the fabric of their choice and the common background. We will exchange the blocks in April. In the meantime, Audrey is coordinating our 2017 opportunity quilt.
In February we hosted AQG traveling teacher Barb Janson for the second time. This time around she presented her thread play class which was a new technique to the majority of the participants. We will be hosting Reeze Hanson on April 5th here in Globe prior to the Spring Meeting in Sedona.
Our members are busy working at the quilt show every day at the Center for the Arts in downtown Globe as well as working on individual quilting projects and all the other activities they are involved in within our community.
2013-2015 Chapter Reports (member login requried)


text:
Fifteen is a surprising age for a boy. Manhood is just out of reach and childhood is forever gone and there are many a lessons to be learned. My experience would be no different for as I remember this is where I was to learned to practice what I preached."
"Boy you ain't rich. You's poor trash, you had told me that fateful day. So why you go and buy a pair of twenty-dollar tennis shoes ? Didn't you realized I could've used that money to feed us?" You inquired angrily. But I had countered with the fact that I hadn't spent all of it–that fateful day. And that it was mine in the first place. I had worked for it. But you had gone on to say that day :
"I'm cutting my looses with you boy; you are insulting and self-centered and think of no body but yourself; and your arrogance has gotten us both fired with that toy gun stunt you pulled at the Dumones. You have got to go.You have got to get out of my house."
Those words had a certain modernity about them that day. It was the fortnight of the second quarter by earth's calendar; and I was to learn about failure, about success, desire, and fear, and about loss. For you it was a matter of survival–others depended on you and I was dead weight.
Well today I'm cutting my losses with you. No more nice TV's. No more casino trips for you and your girl friends and no further humiliation from you for me. We are parting ways. Finished with each other. I won't come to see you and you want call me no more. And just for the record, that story about a million dollar lottery ticket is the worst, ever, in the history of Mother stories. Something you may want to consider revising. Most of our problems stem from your idiosyncrasies. They have become unmanageable. I'm at wits-end with you.
But in more ways than not–this memory has a certain modernity about it too; the kind that only life itself can explain. It speaks of a time past and a time whose come. Of misplaced values and an expediency that any modern-day person should understand. The elderly and their care is best done by professionals.There is just so much a single person can do to accommodate their needs.
Charles Baudelaire is credited with coining the term "modernity" (modernité) in his 1864 essay "The Painter of Modern Life," to designate the fleeting, ephemeral experience of life in an urban metropolis, and the responsibility art has to capture that experience. In this sense, it refers to a particular relationship to time, one characterized by intense historical discontinuity or rupture, openness to the novelty of the future, and a heightened sensitivity to what is unique about the present (Kompridis 2006, 32–59).
As an analytical concept: modernity is closely linked to the ethos, and the philosophical, and the political and moral, and the intellectual undercurrents that intersect with the well-being of a person. For us Mrs Jones, it was about survival and respect and rights-of-passage. All of which we were ill prepared. We had parted ways before and understood that sometimes it's for the best.


text:
Explore Tallahassee
Explore Gadsden County
Explore Wakulla County
Explore Bradfordville
Explore Woodville
JAL Realty, LLC is a regional real estate brokerage company located in Tallahassee, Florida. Our sole purpose is to provide first-rate residential and commercial sales and commercial leasing. We consider good communication to be of prime importance to our customers.
Your home value should not be left to some computer algorithm. Submit a little information and we will get you a comprehensive report that not only provides you with a home value but explains why your home is worth what it is.
By submitting this information, you provide express written consent to JAL Realty, LLC and our affiliates (the "Company") sending you future information, contacting you for telemarketing purposes using the information (including any phone numbers) you have or will provide us, and monitoring and recording these communications to assure the quality of service or for other reasons. This means the Company may contact you via residential number, mobile number and/or text messaging (including use of automated dialing equipment and pre-recorded calls). You are not required to give your consent as a condition of making a purchase with the Company. Additional information can be found in our Privacy Policy.
9904 Waters Meet Drive 4
Tallahassee, FL 32312
411 S Shelfer Street xx
Quincy, FL 32351
Tallahassee, FL 32312
4307 Carnwath
Tallahassee, FL 32303
233 Clark Drive
Panacea, FL 32346
170 Linzy Store Road
Crawfordville, FL 32327
20 Sandpiper Lane
Shell Point, FL 32327
1106 Richardson Road
Tallahassee, FL 32301
000 SE Hammerly Street 0
Lee, FL 32059
3115 Galimore Street
Tallahassee, FL 32305
1914 Golf Terrace Drive
Tallahassee, FL 32301
Lot 9D March Road
Tallahassee, FL 32311
Type anything you want here and we will see what we can find in the property descriptions. Maybe a name of a golf course you want to live by or a term like fixer upper.


text: HEY ladies I need your opinions. I want to get a new hobo bag for fall/winter and I have my eye on a few .. but I need your opinions. I'm awful with decisions. All I have right now is a large soho green twill tote and a hot pink signature C large hobo I'd love to get a few, but I'm eighteen and I'm a lifeguard. So needless to say my budget isn't too high I'm looking at a few on Ebay, so I'll probably purchase them from there - I live in the middle of Wisconsin with no outlet/stores nearby. first .. TIE DYE? I think I like the brown better, but I'm contemplating. I like the style (I know it's a unique taste ) OKAY, so there's those. I like them both a lot. I don't have a brown bag, so that'd be nice, but I think white would look nice, too. AGH! I like the tie-dye satchels, too, but I do like shoulder bags. And one more option. I really like the khaki scribble hobo as well. OKAY GIRLS, give me your opinions. Especially if you have one of the bags. You guys are the experts.


text:
The thief walked into the room cautiously. There are always traps in places like this. The party always sent him in first to make sure it is safe. Like a good thief is expendable. Of course a good thief would never fall prey to a trap unless it was exceptional.
Scanning the walls, looking for clues in the flicker of the torch he carried. He craned his head to check for odd cracks or spaces that were abnormal. Then he checked the floor. He did not see anything to cause him to suspect a trap in the walls, floor, or ceiling of the room. He beckoned the party to enter but cautioned them not to touch anything else.
He turned his attention to the empty sconces on the walls. They were a good place to hide a trigger. Inexperienced treasure hunters assumed they would open secret doors. More often than not they triggered traps that would lead to an early retirement… of sorts. The sconces were safe.
Next was the decrepit table and chair in one of the corners. It is suspicious that they were still together. Every other piece of furniture they had encountered was in ruin. These looked like they might still burn.
He could see the joints were reenforced with bits of metal and nails. The back legs of the chair were bolted to the floor with hinges and the table just looked out-of-place in relation to the chair.
He brought his findings to the party who were standing just inside the door of the room. The idea is to tie a good solid rope to the chair and rig a set of pulleys. They would be safely in the hall when they pulled the rope and triggered the trap.
The thief did all the set up. Gingerly tying the rope to the chair. Then running it through some block pulleys he had tied to the sconces. Clever, he thought to himself. I am clever.
He ran the rope through a hole they made in the door, to help prevent any accidentally damage from arrows or other physically damaging traps. The door would also give them time before any gases might reach them.
Closing the door carefully, the thief pulled the rope tight…
Nothing happened.
The thief tried again with more strength. Still nothing.
He passed the rope to the warrior, the strongest of the group. The rope was pulled tight.
So tight the thief thought the rope would break.
Then the rope jerked and the warrior stumbled backwards a half step.
A groan was heard from behind the door.
It had worked!
Seconds passed and stretched into minutes. No thuds, hisses, or other common trap sounds were heard. Nothing near them had changed either.
The thief looked at the party. They all nod their heads.
Cautiously the thief opened the door. Then stepped into the room. The room was the same. The chair was titled back with the front legs slightly off the ground. The table was still slightly off. The walls and sconces were intact and in the same positions.
What had they triggered?
He quickly turned around and walked back to the door and the party just beyond.
As he reached for the door something felt off. Instead of jerking it open he pulled the rope back through the hole and looked outside.
A thin green mist had settled in the hall and his companions were slumped on the floor, Overcome by the poisonous gas.
A good thief would never fall prey to a trap unless it was exceptional. This was exceptional and his party had paid the price.
How was he going to get out and back to town?


text: Hello.. I have three windows machines that I am connecting to (one I am pulling info off and then it is going to see the salvage yard). Here is my question: Under the sharing heading I can see two of the machines I want to connect to. Machine 1 and 2. But I can't see Machine 3. If I turn off Machine 1 I can then see Machine 2 and 3. If I start up VMware I can only see the vm machine and say machine 2. Does this make sense. My Mac is only allowing me to see two at a time. Is there a way to see more? If so, what do I do in order for this to happen? Thanks


text: |Wee Willie Harris: Here's a bill from The Ritz, Wigan from 1958 or 1959 before Cliff Richard's then backing band The Drifters had become The Shadows. Apparently billed as 'Britain's wild man of rock 'n' roll' in the 1950s. He had limited commercial success (none of his singles charted), but was popular through live performances and TV appearances. His mention in "Reasons To Be Cheerful" led to a comeback in the 70s and in 2000 he even released an album entitled 'Twenty Reasons To Be Cheerful'. Right I'm off to Wigan, looks like a good bill and only 6 bob for a decent seat.


text: For my birthday extravaganza my friend brought me to do some fun things that I have never done before in Portland, Maine. First we went to this nice little restaurant called The Bayou. I had jambalaya and cornbread and just about died from how good it was, I would highly recommend it. Then we went to Portland Head Light. It was absolutely beautiful. We also visited the Goddard Mansion while we were there. Lastly we went to Get Air as I mentioned in my previous post. I truly love the people I have met and become friends with this year.


text:
We don't use a formal Christmas tree in this house so I have two very small ones that I put lights and garland on and set them on the hearth of the fireplace along with all of our Christmas stockings. So here are Addie and Roscoe patiently awaiting Santa Paws
And here is a photo of some of the Christmas decorations. I don't have too many, but what I have is enough. I don't know where people store all this stuff! And yes, those are CDs and there are plenty more that you don't see. Tom likes music very much and has a very large and ecletic collection of music.
And here is a gift my brother (who is a hydrogeology professor in California) and sister-in-law gave to us. Actually, they gave us this vase and a pen/pencil holder that I did not photograph. This is really cool. What you are seeing are fossil gastropods formed between 15 million and 40 million years ago that were found in Pakistan and cut and polished by Pakistani craftsmen. I guess he gets these at a conference he attends and I think they are great and I was glad to be the recipient.