The Rearrangement Inequality

A favourite result of many students doing olympiad inequality problems is the so-called Rearrangement Inequality. This is a mathematical formulation of the idea well-known to even the smallest of child that if you prefer cakes to carrots then if you are offered two of one and one of the other, you should take two of the one you prefer!

At a more formal level, it says that given two strings of non-negative numbers

a_1\le a_2,\le \ldots\le a_n, \quad b_1\le b_2\le \ldots\le b_n,

if you want to form a sum of products of pairs, like

you get the largest result if you take

Formally, for any permutation \sigma \in S_n,

a_1b_1+\ldots+a_nb_n\ge a_1b_{\sigma(1)}+\ldots+a_nb_{\sigma(n)}\ge a_1b_n+\ldots+a_nb_1.

That is, you multiply the largest terms in each sequence together.

The notation to describe to equality case is a bit annoying. Essentially, the sums are equal if and only if the summands exactly correspond. If the sequences are strictly increasing, then equality holds only if the permutation \sigma=\text{id}.

This result is nice because, although it is rarely explicitly useful, it goes in a different direction from the standard scheme of results strengthening AM-GM, Cauchy-Schwarz and so on, and is in some sense more intuitive than these more well-known inequalities, at least in the form presented in an olympiad context.

I was thinking about this partly because it’s a nice result in its own right, but also because it came up in a research problem to do with comparing the expected likelihood of different tree isomorphism classes arising in an inhomogeneous, but relatively well-behaved, random graph model. The probability of forming a given tree is a homogeneous multivariate polynomial in the ages of the vertices that would form the tree. It is then necessary to integrate over the joint distribution (which fortunately is a product in the limit) of the ages of the vertices. I was playing around with this by considering what seemed to be the extreme cases: the star and the path. I was working with the relatively simple case n=4, and it struck me that perhaps the polynomial for the star was always at least as large as that for the path. This would be convenient as it would avoid the need for a horrific-looking integral calculation. This turned out to be true. My first method was a heavy but uncontroversial convexity and stationary point argument, but I found a pair of vectors embedded in the desired inequality on which I could deploy rearrangement.

Anyway, I thought I should be able to come up with a nice proof, and I think this is one. I think this is particularly nice because it is a demonstration that one can do a proof by induction without explicitly inducting on the natural numbers.

We begin with a base case, which is the theorem for n=2, even though we will not be doing induction in the canonical way. We are required to prove that given

a_1\le a_2,\quad b_1\le b_2,

a_1b_1+a_2b_2\ge a_1b_2+a_2b_1,

since these are the only available permutations. Moving some terms around gives

(a_2-a_1)(b_2-b_1)\ge 0,

which is true by construction, and so the n=2 result follows.

We now move straight to the general n case. We focus on the left of the two inequalities in the statement of the result, since the other will follow by an identical method, applied in reverse. We consider the case where \sigma is a transposition. For example, we might consider 12435. When we write out the result we want:

a_1b_1+a_2b_2+a_3b_3+a_4b_4+a_5b_5\ge a_1b_1+a_2b_2+a_3b_4+a_4b_3+a_5b_5,

we realise that many of the terms cancel, and the content of the theorem reduces to the n=2 case we have already dealt with. Obviously, this holds equally well whenever \sigma is a transposition. Similarly, if \sigma is a product of two disjoint transpositions, which means that two disjoint pairs of elements are interchanged, we can apply the n=2 case twice, then add on the extra terms to get the result.

In fact, we can do much better than this, by using the fact that any permutation can be expressed as a product of transpositions. We need to be careful about the risk of asserting that every time we multiply the permutation \sigma by a transposition, the value of the associated sum-product expression gets smaller. While the idea is correct, this cannot be generally true. After all, applying the same transposition twice returns us to the identity permutation!

We can nonetheless say something useful. If we start with a permutation

and we interchange the ith and jth elements, to get,

then the product sum corresponding to \tau is less than or equal to the product sum corresponding to \sigma if $\sigma(i)\leq sigma(j)$, under the implicit assumption that i<j. In other words, we can prove the rearrangement inequality for any permutation \sigma that can be obtained from the identity by repeatedly interchanging elements that are initially in increasing order. Essentially, we have defined a partial ordering on the set of permutations.

It suffices to check that all permutations have this property. In fact, this is relatively easy. We can move element n to its required position in \sigma by successively swapping with (n-1), (n-2), etc. If we set this up as an inductive argument, we can finish by applying the hypothesis to the remaining (n-1) elements, which are in the same order as the identity permutation on [n-1].

So we have proved the left-hand side of the Rearrangement Inequality. In fact, this partial ordering framework makes it clear how to prove the right-hand side. By an identical argument, we can get from any permutation to the reverse identity by a similar set of operations.

The Top-to-Random Shuffle II

In the last post, I introduced the top-to-random shuffle. In particular, we considered why this sort of procedure was important as an alternative to choosing an ordering afresh, and how to we would go about measuring how close we had got to randomness.

In this post, I want to develop the second of these points. The intuition might be that we can get very close to uniform randomness if we repeat the shuffle often enough. Recall this means that even if we choose our bet in a really complicated and careful way, we still couldn’t make much profit by knowing the actual distribution of the ordering. But we might also suspect that the pack will never be exactly random, in the same way that the distribution of the proportion of heads seen on a repeatedly-flipped coin will eventually get very close to 1/2, but will not be exactly 1/2.

This intuition is extremely sensible, and in general is true. It is a nice fact, however, that it fails for the top-to-random shuffle, where we do in fact get to a uniformly random deck. Recall that we approximated how long it would take to get to a state that was roughly random by calculating the time taken for the original bottom card to rise to the top of the deck. This time was:

where n is the total number of cards. A further shuffle is required to send the original card back into the pack somewhere. The claim is that the pack is now uniformly random. Note that if we ever actually use this, we have to be careful because although we can calculate the average time at which this event happens, the time itself is random. Rather than worry about that, let’s see why it is true.

As a motivating example, let’s assume that the pack was originally in the order 1,2,3,…,n, and consider the final relative order of the cards numbered 1 and 2. There is some small probability that on the first go (and then again on the second go possibly also) that the card number 1 stays put. Let’s ignore that possibility and progress to the first time that the 1 has been moved into the interior of the pack, so the 2 is now on top. When we do this, we are choosing a number between 2 and n uniformly at random, and moving the card numbered 1 to this position. Let’s call this number X.

Now, when we try to shuffle the 2, we are choosing a number between 1 and n uniformly at random, and moving the card numbered 2 to this position. Let’s call this number Y. Under exactly what circumstances does 2 end up above 1? The clearest example is if Y=X. Then card 2 has been moved to the position previously occupied by card 1. So card 1 moves up a position (since the card 2 is no longer on the top). The final configuration therefore includes card 1 directly above card 2. So we can say

\mathbb{P}(\text{2 above 1})=\mathbb{P}(Y<X),

where the fact that the inequality in the second probability is strict is important. To calculate this second probability, we want to exploit the symmetry of the situation. The only problems are that the case X=Y is not symmetric as then 1 ends up above 2 as described above, and also that X cannot be 1. So we account for these separately. Note

\mathbb{P}(Y=1)=\frac{1}{n},\quad \mathbb{P}(Y=X)=\frac{1}{n}.

The second result holds overall since it holds whenever we condition on a particular value of X. These events are also disjoint. Then

\mathbb{P}(Y<X)=\mathbb{P}(Y=1)+\mathbb{P}(Y<X | Y>1, X\neq Y)\mathbb{P}(Y>1,X\neq Y)

= \frac{1}{n}+\frac{1}{2}(1-\mathbb{P}(Y>1,Y\neq X))

=\frac{1}{n}+\frac{1}{2}(1-\frac{1}{n}-\frac{1}{n}) = \frac{1}{2}.

In summary, the cards 1 and 2 will be in uniformly random order. We might like to extend this idea, but it will get complicated when we add card 3 to the mix, as it is possible (if unlikely) that 1 and 2 will be further mixed before this. This shouldn’t affect the result much, but it will get complicated to define the notation required to carry this sort of argument all the way up to the nth card.

Even using induction is not going to make life substantially easier. Knowing that once we have inserted cards 1,2,…,k into the pack they are in uniformly random order is not enough to make inference about what happens once we put k+1 into the pack. We have to know something about the current positions of 1,2,…,k. For example, if one of these cards is definitely on the bottom of the pack, then the probability that k+1 ends up last among 1,2,…,k+1 is 1/n rather than 1/k+1 as it should be. So in fact we would have to control an annoying amount of information jointly.

In the argument we attempted above, we were looking at the first times some card k got folded back into the pack. Note that this division of time is different to the one we were using for the coupon collector approach to the mixing time in the previous post. Let’s try to use that instead here.

Now we consider the times at which a card is moved below card n. We deliberately decline to say what these cards are. But rather, we want to prove that, conditional on the cards below n being A_k=\{a_1,\ldots,a_k\}, the ordering of these is uniform on S_{A_k}, that is, every possibility is equally likely. Now this is easy to prove by induction. For, by conditioning on A_k and a_{k+1} being the new card to be moved below n, we are conditioning on the set of cards below n now being A_{k+1}=A_k\cup\{a_{k+1}\}. The position of the new card is uniformly random within this, by construction of the top-to-random shuffle, and so the new arrangement is uniformly random on the (k+1)! possibilities.

To see why we have proved the original result we wanted, note that this argument works at the time when the original bottom card is now at the top. So the remaining cards are uniformly randomly ordered. Inserting card n at random gives an arrangement that is uniformly random overall. So as we suggested before, working out how long it takes to get close to randomness in this case reduces to working out how long it is before the original bottom card hits the top and is re-inserted, as at that point, the pack genuinely is uniformly random.

The Contour Process

As I explained in my previous post, I haven’t been reading around as much as I would generally like to recently. A few days in London staying with my parents and catching up with some friends has therefore been a good chance to get back into the habit of leafing through papers and Pitman’s book among other things.

This morning’s post should be a relatively short one. I’m going to define the contour process, a function of a (random or deterministic) tree, related to the exploration process which I have mentioned a few times previously. I will then use this to prove a simple but cute result equating in distribution the sizes of two different branching processes via a direct bijection.

The Contour Process

To start with, we have to have a root, and from that root we label the tree with a depth-first labelling. An example of this is given below. It is helpful at this stage to conceive this process as an explorer walking on the tree, and turning back on themselves only when there is no option to visit a vertex they haven’t already seen. So in the example tree shown, the depth-first exploration visits vertex V_2 exactly four times. Note that with this description, it is clear that the exploration traverses every edge exactly twice, and so the length of the sequence is 2n-1, where n is the number of vertices in the tree since obviously, we start and end at the root.

Another common interpretation of this depth-first exploration is to take some planar realisation of the tree. (Note trees are always planar – proof via induction after removing a leaf.) Then if you treat the tree as a hedge and starting at the root walk along, following the outer boundary with your right hand, this exactly recreates the process.

The height of a tree at a particular vertex is simply the graph distance between that vertex and the root. So when we move from one vertex to an adjacent vertex, the height must increase or decrease by 1.

The contour process is the sequence of heights seen along the depth-first exploration. It is therefore a sequence:

0=h_0,h_1,\ldots,h_{2n-1}=0,\quad h_i\geq 0,

and such that |h_{i+1}-h_i|=1.

Note that though the contour process uniquely determines the tree structure, the choice of depth-first labelling is a priori non-canonical. For example, in the display above, V_3 might have been explored before V_2. Normally this is resolved by taking the suitable vertex with the smallest label in the original tree to be next. It makes little difference to any analysis to choose the ordering of descendents of some vertex in a depth-first labelling randomly. Note that this explains why it is rather hard to recover Cayley’s theorem about the number of rooted trees on n vertices from this characterisation. Although the number of suitable contour functions is possible to calculate, we would require a complicated multiplicative correction for labelling if we wanted to recover the number of trees.

The only real observation about the uses of the contour process at this stage is that it is not in general a random walk with IID increments for a Galton-Watson branching process. This equivalence is what made the exploration process so useful. In particular, it made it straightforward, at least heuristically, to see why large trees might have a limit interpretation through Brownian excursions. If for example, the offspring distribution is bounded above, say by M, then the contour process certainly cannot be a random walk, as if we have visited a particular vertex exactly M+1 times, then it cannot have another descendent, and so we must return closer to the root at the next step.

I want to mention that in fact Aldous showed his results on scaling limits towards the Continuum Random Tree through the contour process rather than the exploration process. However, I don’t want to say any more about that right now.

A Neat Equivalence

What I do want to talk about is the following distribution on the positive integers. This comes up in Balazs Rath and Balint Toth’s work on forest-fires on the complete graph that I have been reading about recently. The role of this distribution is a conjectured equilibrium distribution for component size in a version of the Erdos-Renyi process where components are deleted (or ‘struck by lightning’) at a rate tuned so that giant components ‘just’ never emerge.

This distribution has the possibly useful property that it is the distribution of the total population size in a Galton-Watson process with Geom(1/2) offspring distribution. It is also the distribution of the total number of leaves in a critical binary branching process, where every vertex has either two descendents or zero descendents, each with probability 1/2. Note that both of these tree processes are critical, as the expected number of offspring is 1 in each case. This is a good start, as it suggests that the relevant equilibrium distribution should also have the power-law tail that is found in these critical branching processes. This would confirm that the forest-fire model exhibits self-organised criticality.

Anyway, as a sanity check, I tried to find a reason why, ignoring the forest-fires for now, these two distributions should be the same. One can argue using generating functions, but there is also the following nice bijective argument.

We focus first on the critical Geometric branching process. We examine its contour function. As explained above, the contour process is not in general a random walk with IID increments. However, for this particular case, it is. The geometric distribution should be viewed as the family of discrete memoryless distributions.

This is useful for the contour process. Note that if we are at vertex V for the (m+1)th time, that is we have already explored m of the edges out of V, then the probability that there is at least one further edge is 1/2, independently of the history of the exploration, as the offspring distribution is Geometric(1/2), which we can easily think of as adding edges one at a time based on independent fair coin tosses until we see a tail for example. The contour process for this random tree is therefore a simple symmetric random walk on Z. Note that this will hit -1 at some point, and the associated contour process is the RW up to the final time it hits 0 before hitting -1. We can check that this obeys the clear rule that with probability 1/2 the tree is a single vertex.

Now we consider the other model, the Galton-Watson process with critical binary branching mechanism. We should consider the exploration process. Recall that the increments in this process are given by the offspring distribution minus one. So this random sequence also behaves as a simple symmetric random walk on Z, again stopped when we hit -1.

To complete the bijective argument, we have to relate leaves in the binary process to vertices in the geometric one. A vertex is a leaf if it has no offspring, so the number of leaves is the number of times before the hitting time of -1 that the exploration process decreases by 1. (*)

Similarly for the contour process. Note that there is bijection between the set of vertices that aren’t the root and the set of edges. The contour process explores every edge exactly twice, once giving an increase of 1 and once giving a decrease of 1. So there is a bijection between the times that the contour process decreases by 1 and the non-root vertices. But the contour process was defined only up to the time we return to the root. This is fine if we know in advance how large the tree is, but we don’t know which return to the root is the final return to the root. So if we extend the random walk to the first time it hits -1, the portion up until the last increment is the contour process, and the final increment must be a decrease by 1, hence there is a bijection between the number of vertices in the Geom(1/2) G-W tree and the number of times that the contour process decreases by 1 before the hitting time of -1. Comparing with (*) gives the result.

Mixing Times 6 – Aldous-Broder Algorithm and Cover Times

So, removing the conditioning, we have:

and so the telescoping sum gives us Matthews’ result.

How to Prove Fermat’s Little Theorem

The following article was prompted by a question from one of my mentees on the Senior Mentoring Scheme. A pdf version is also available.

Background Ramble

When students first meet problems in number theory, it often seems rather different from other topics encountered at a similar time. For example, in Euclidean geometry, we immediately meet the criteria for triangle similarity or congruence, and various circle theorems. Similarly, in any introduction to inequalities, you will see AM-GM, Cauchy-Schwarz, and after a quick online search it becomes apparent that these are merely the tip of the iceberg for the bewildering array of useful results that a student could add to their toolkit.

Initially, number theory lacks such milestones. In this respect, it is rather like combinatorics. However, bar one or two hugely general ideas, a student gets better at olympiad combinatorics questions by trying lots of olympiad combinatorics questions.

I don’t think this is quite the case for the fledgling number theorist. For them, a key transition is to become comfortable with some ideas and notation, particularly modular arithmetic, which make it possible to express natural properties rather neatly. The fact that multiplication is well-defined modulo n is important, but not terribly surprising. The Chinese Remainder Theorem is a `theorem’ only in that it is useful and requires proof. When you ask a capable 15-year-old why an arithmetic progression with common difference 7 must contain multiples of 3, they will often say exactly the right thing. Many will even give an explanation for the regularity in occurrence of these which is precisely the content of the theorem. The key to improving number theory problem solving skills is to take these ideas, which are probably obvious, but sitting passively at the back of your mind, and actively think to deploy them in questions.

Fermat’s Little Theorem

It can therefore come as a bit of a shock to meet your first non-obvious (by which I mean, the first result which seems surprising, even after you’ve thought about it for a while) theorem, which will typically be Fermat’s Little Theorem. This states that:

\text{For a prime }p,\text{ and }a\text{ any integer:}\quad a^p\equiv a\mod p. (1)

  • Students are typically prompted to check this result for the small cases p=3, 5 and 7. Trying p=9 confirms that we do really need the condition that p be prime. This appears on the 2012 November Senior Mentoring problem sheet and is a very worthwhile exercise in recently acquired ideas, so I will say no more about it here.
  • Note that the statement of FLT is completely obvious when a is a multiple of p. The rest of the time, a is coprime to p, so we can divide by a to get the equivalent statement:

\text{For a prime }p,\text{ and }a\text{ any integer coprime to }p:\quad a^{p-1}\equiv 1\mod p. (2)

  • Sometimes it will be easier to prove (2) than (1). More importantly, (2) is sometimes easier to use in problems. For example, to show a^{p^2}\equiv a \mod p, it suffices to write as:

a^{p^2}\equiv a^{(p-1)(p+1)+1}\equiv (a^{p-1})^{p+1}\times a\equiv 1^{p+1}\times a \equiv a.

  • A word of warning. FLT is one of those theorems which it is tempting to use on every problem you meet, once you know the statement. Try to resist this temptation! Also check the statement with small numbers (eg p=3 ,a=2) the first few times you use it, as with any new theorem. You might be surprised how often solutions contain assertions along the lines of

a^p\equiv p \mod (a-1).

I must have used FLT dozens of times (or at least tried to use it – see the previous remark), before I really got to grips with a proof. I think I was daunted by the fact that the best method for, say, p=7, a careful systematic check, would clearly not work in the general case. FLT has several nice proofs, and is well worth thinking about for a while before reading what follows. However, I hope these hints provide a useful prompt towards discovering some of the more interesting arguments.

Induction on a to prove (1)

  • Suppose a^p\equiv a\mod p. Now consider (a+1)^p modulo p.
  • What happens to each of the (p+1) terms in the expansions?
  • If necessary, look at the expansion in the special case p=5 or 7, formulate a conjecture, then prove it for general p.

Congruence classes modulo p to prove (2)

  • Consider the set \{a,2a,3a,\ldots,(p-1)a\} modulo p.
  • What is this set? If the answer seems obvious, think about what you would have to check for a formal proof.
  • What could you do now to learn something about a^{p-1}?

Combinatorics to prove (1)

  • Suppose I want a necklace with p beads, and I have a colours for these beads. We count how many arrangements are possible.
  • Initially, I have the string in a line, so there are p labelled places for beads. How many arrangements?
  • Join the two ends. It is now a circle, so we don’t mind where the labelling starts: Red-Green-Blue is the same as Green-Blue-Red.
  • So, we’ve counted some arrangements more than once. How many have we counted exactly once?
  • How many have we counted exactly p times? Have we counted any arrangements some other number of times?

Group Theory to prove (2)

This is mainly for the interest of students who have seen some of the material for FP3, or some other introduction to groups.

  • Can we view multiplication modulo p as a group? Which elements might we have to ignore to ensure that we have inverses?
  • What is \{1,a,a^2,a^3,\ldots\} in this context? Which axiom is hardest to check?
  • How is the size of the set of powers of a modulo p related to the size of the whole group of congruences?
  • Which of the previous three proofs is this argument is most similar to this one?
  • Can you extend this to show the Fermat-Euler Theorem:

\text{For any integer }n,\text{ and }a\text{ coprime to }n:\quad a^{\phi(n)}\equiv 1 \mod n,

where \phi(n) counts how many integers between 1 and n are coprime to n.

Branching Processes and Dwass’s Theorem

This is something I had to think about when writing my Part III essay, and it turns out to be relevant to some of the literature I’ve been reading this week. The main result is hugely helpful for reducing a potentially complicated combinatorial object to a finite sum of i.i.d. random variables, which in general we do know quite a lot about. I was very pleased with the proof I came up with while writing the essay, even if in the end it turned out to have appeared elsewhere before. (Citation at end)

Galton-Watson processes

A Galton-Watson process is a stochastic process describing a simple model for evolution of a population. At each stage of the evolution, a new generation is created as every member of the current generation produces some number of `offspring’ with identical and independent (both across all generations and within generations) distributions. Such processes were introduced by Galton and Watson to examine the evolution of surnames through history.

More precisely, we specify an offspring distribution, a probability distribution supported on \mathbb{N}_0. Then define a sequence of random variables (Z_n,n\in\mathbb{N}) by:

where (Y_k^n,k\geq 1,n\geq 0) is a family of i.i.d. random variables with the offspring distribution Y. We say Z_n is the size of the nth generation. From now on, assume Z_0=1 and then we call (Z_n,n\geq 0) a Galton-Watson process. We also define the total population size to be

noting that this might be infinite. We refer to the situation where X<\infty finite as extinction, and can show that extinction occurs almost surely when \mathbb{E}Y\leq 1, excepting the trivial case Y=\delta_1. The strict inequality parts are as you would expect. We say the process is critical if \mathbb{E}Y=1, and this is less obvious to visualise, but works equally well in the proof, which is usually driven using generating functions.

Total Population Size and Dwass’s Theorem

Of particular interest is X, the total population size, and its distribution. The following result gives us a precise and useful result linking the probability of the population having size n and the distribution of the sum of n RVs with the relevant offspring distribution. Among the consequences are that we can conclude immediately, by CLT and Cramer’s Large Deviations Theorem, that the total population size distribution has power-law decay in the critical case, and exponential decay otherwise.

Theorem (Dwass (1)): For a general branching process with a single time-0 ancestor and offspring distribution $Y$ and total population size $X$:

\mathbb{P}(X=k)=\frac{1}{k}\mathbb{P}(Y^1+\ldots+ Y^k=k-1),\quad k\geq 1

where Y^1,\ldots,Y^k are independent copies of Y.

We now give a proof via a combinatorial argument. The approach is similar to that given in (2). Much of the literature gives a proof using generating functions.

Proof: For motivation, consider the following. It is natural to consider a branching process as a tree, with the time-0 ancestor as the root. Suppose the event \{X=k\} in holds, which means that the tree has k vertices. Now consider the numbers of offspring of each vertex in the tree. Since every vertex except the root has exactly one parent, and there are no vertices outside the tree, we must have Y^1+\ldots+Y^k=k-1 where Y^1,\ldots,Y^k are the offspring numbers in some order. However, observe that this is not sufficient. For example, if Y^1 is the number of offspring of the root, and k\geq 2, then we must have Y^1\geq 1. Continue reading

The Sample Space for a Die

Also featuring: a non-Lebesgue measurable set and Dynkin’s Lemma.

At the National Maths Summer School last week, the senior students and I spent a while talking about probability space, and in particular, when it was reasonable to assign a probability to a potential event. We considered rolling a standard die, and the probabilities \mathbb{P}(D\in\{\}), the empty event, and \mathbb{P}(D=7). Though it is tempting to conclude that the latter must be zero, in the end we decided that it should not actually be defined at all.

Why? Well, if we accept \mathbb{P}(D=7)=0, then by extension we must accept \mathbb{P}(D=137) and \mathbb{P}(D=\$1.50) also both exist and are zero. What have we gained? In reality very little. But the cost is this: we might define an event to be any subset of the sample space. Before, our sample space was \Omega=\{1,2,3,4,5,6\}, and so there are exactly 64 events, including the possibly counter-intuitive empty event \{\}. This is finite, which is always nice. With the extra events, however, we must extend the sample space to \Omega=\{1,2,3,4,5,6,7,\ldots, where “…” means “the rest of the universe”. This is a fairly exotic mathematical object, and really has no place in any sensible discussion.

This reminded me of one of my favourite results from Part II Probability and Measure. Of course, for uncountable sample spaces, we cannot necessarily assume all subsets of \Omega are measurable. Instead we build up a sigma-algebra of measurable sets, most importantly for Lebesgue measure on \mathbb{R}. An immediate question to ask is: are all subsets of \mathbb{R} Lebesgue-measurable?

And the answer is ‘no’. Why? The standard counterexample is as follows. Consider Lebesgue measure on the unit interval U=\mathbb{R}/\mathbb{Z}, with endpoints identified. Now consider the rationals in U, which are actually a subgroup \mathbb{Q}\leq U, with uncountably many cosets. Pick an element from each coset (*). Call this sets A. Then, working modulo 1, U=\cup_{q\in\mathbb{Q}\cap U}A+q. If A is Lebesgue-measurable, then so is A+q, and \mu(A+q)=\mu(A) (**). Combining these two results, using countable additivity:

\mu(U)= (0 if \mu(A)=0, \infty otherwise).

This is a contradiction, and hence we conclude that A is not Lebesgue-measurable.

Remark on (*): This relies on the Axiom of Choice. In fact, the existence of non-Lebesgue measurable sets MAY be equivalent to AC.

Remark on (**): I was suddenly unsure that this was obvious. I mean, this is such a weird set that it is in fact not measurable: why should its hypothetical measure be translation invariant? It is tempting to argue vaguely, by saying that the construction of Lebesgue measure is invariant under translation at all steps. As so often with elementary measure theory, recourse to Dynkin’s Lemma is more reliable.

Let D be the collection of measurable sets whose measure is invariant under translation. By definition, D is invariant under translation (of its elements). D certainly contains all intervals in U, which is a pi-system generating \mathcal{B}([0,1]). But, in a classic proof by suggestive notation, we can check that D is a d-system. The presence of the empty sets is clear. If we have B\subset A, both in D, then also A\backslash B\in D, as the translates of x\in A\backslash B must be in A, but cannot be in B, as B is translation invariant. Finally, given A_1\subset A_2\subset\ldots\subset D, then x\in\cup A_i\Rightarrow x\in A_n for some n, so all of x’s translates are in A_n, and hence in \cup A_i.

Now we can deploy Dynkin’s Lemma. D must be the sigma-algebra of all measurable sets, as we wanted.

NMSS 2012 – Strong Law of Large Numbers for a Coin Flip

The 2012 National Mathematics Summer School, held at Queens’ College affiliated to the University of Birmingham, and run by the United Kingdom Mathematics Trust, is drawing to a close today. I gave a problem-based talk on Probability to two groups of 20 junior students (15/16 year olds selected based on strong performance in national competitions for their agegroups), and a lecture to the six senior students (some of 2011’s strongest and most enthusiastic junior students) on the SLLN for the simplest non-trivial random variable imaginable: a coin flip.

In case any of the students, or indeed anyone else, is interested, a text of the problems, and the worked solutions that took up the majority of the lecture will be available here for a short while. Do email me if there are any questions!

Senior Probability Solutions. [Link removed. Email me if interested]

Remarkable fact about Brownian Motion #4: The Dirichlet Problem

So this property of Brownian Motion is so elegant, in my opinion, that when I was recently asked what my ‘favourite theorem’ was, I suggested this. With this result, we can use this probabilistic structure to specify solutions to an important PDE, with boundary conditions, over a large class of domains.

Given a domain D, Laplace’s equation is: \Delta u=0 on D, and u=f on the boundary dD, where f is any continuous function defined there. This PDE arises wherever the notion of potentials is defined, for example electromagnetism, fluids and thermodynamics.

Theorem: Given suitable regularity conditions on D to be discussed later, Laplace’s equation has a unique solution, given by:

Notation: First, what does this mean? Define T_D:=\inf\{t:B_t\not\in D\}, to be the time at which a Brownian Motion leaves the domain D. This is a stopping time, and so will be suitable for application of the Strong Markov Property. \mathbb{E}_x means that we are taking expectation with respect to a BM started at x. So informally, we are defining u(x) as: start a BM at x; see where it hits the boundary of D; record the value of f at that point. Then set u(x) to be the expected value of this process.

Existence: First, we are going to check that the solution conjectured is a solution. We will need a lemma:

Lemma: A locally-bounded function u satisfies \Delta u=0 on a domain D if and only if it has the property that for every closed ball \bar{B(x,r)}\subset D we have:

where \sigma_{x,r} is the surface area measure on the boundary S(x,r) of the ball radius r centred on x. Essentially, this says that u(x) is equal to the average value of u on a ball around x.

Proof of Theorem: First, existence. Set u as specified in the statement of the theorem. Given a Brownian Motion started at x, we have stopping times T_r<T_D corresponding to the hitting times of the ball radius r around x and the boundary dD. The domination condition holds by continuity provided B(x,r) is contained within D. So we may apply the Strong Markov Property:

By definition, the left hand expression is u(x). But also, because the distribution of B_{T_r} is uniform on S(x,r), the right hand side is equal to:

and so by the lemma, this guarantees that the function u is harmonic on the interior of D.

The lemma can also be used to show uniqueness. Continue reading

Uncountable Infinity and Cantor’s Diagonal Argument

In my last post, I talked about why infinity shouldn’t seem terrifying, and some of the interesting aspects you can consider without recourse to philosophy or excessive technicalities. Today, I’m going to explore the fact that there are different kinds of infinity. For this, we’ll use what is in my opinion one of the coolest proofs of all time, originally due to Cantor in the 19th century.

Last time we talked about sets which have infinite size. In particular, a set S is called countable, if its elements can be paired up with the set of positive integers. This is equivalent to saying that you can find a way to list the elements of S, in such a way that every element comes up eventually. It might initially seem as if every set should be countable, but consider the reals between 0 and 1. If we were to put them in a list, what should the first element in the list be; and what should the second be; and so on? Is it obvious that we can make this work? But if we can’t make it work, how can possibly prove that it is impossible, because there are so many potential ways to list the elements?

The Rationals

As a warm up, consider the rationals: \mathbb{Q}=\{\frac{p}{q}:p,q\text{ integers with no common factor}\}. Is this set countable? For simplicity, let’s just think about the rationals between 0 and 1. What is the most sensible way to list them? What about starting with 0 and 1, then 1/2, then the ones with 3 as denominator, ie 1/3 and 2/3. Can we carry on with strategy? First we need to check exactly what we are doing: we are ordering the rationals by increasing size of the denominator, then within that, by increasing size of the numerator. What about 2/4? That isn’t a rational as we’ve described them, so we can just ignore it.

With this algorithm is that every such rational will turn up eventually. We’ve therefore shown that this set of rationals is countable. It’s hard to give a precise ‘formula’ for how the list works. To work out where 2/11 appears, you need to know about the prime factors of the integers less than 11, and keep track of lots of things. It turns out that 2/11 appears 35th in the list, but the easiest way to deduce this is to check up to that point in the list. The important fact is that it must appear at some point, and only once.

The Reals

Theorem (Cantor): The set of real numbers between 0 and 1 is not countable.

Proof: This will be a proof by contradiction. That means, we will assume that the set is countable, then derive a false statement. From this, we can deduce that the set cannot be countable.

Suppose it is countable. This means we can write all the reals in a list. Write them in binary, so it probably looks something like this:

and so on. Continue reading


The Hidden Lives of America’s Poor and Middle Class The Hidden Lives of America’s Poor and Middle Class This series explores how current programs and policies for helping families escape poverty, build stability, move up the ladder, and invest in the future need to change.

Research from the US Financial Diaries project illuminates the prevalence of volatile incomes and the problems they can cause. But volatile incomes aren’t all bad. Many households need lump sums for large purchases, and spikes in income can make pulling together that lump sum easier than slow and steady saving.

Many households get a large income spike during tax time—either from a tax refund or, for lower-income households, the Earned Income Tax Credit. Other research indicates that a large number of low-income households use this income to purchase things like appliances or to make a down payment on a vehicle.

Vehicles, in fact, are one of the most common purchases. It’s easy to overlook the importance of reliable transportation for a family trying to move out of poverty. A reliable car is especially important for lower-income families in rural America, where work, health care, childcare, and basic necessities such as healthy food are always miles away. Affordable, flexible, reliable transportation is often also the catalyst for getting a better-paying job, or simply earning more from a current one by working overtime, picking up extra hours at short notice, or taking a second shift. In the United States, areas with adequate public transportation to meet these needs are few and far between, making cars a vital tool for financial stability and upward mobility.

Car dealers and auto lenders know that people commonly use tax refunds to purchase vehicles, and they roll out the red carpet to attract buyers who may not have the necessary lump sum to purchase a vehicle any other time of the year. Of course, that makes lower-income households and households with volatile incomes vulnerable to poor decision-making. It’s human nature to struggle with making good decisions when you have a big income spike—that money doesn’t have to literally be in your pocket to start burning a hole. A purchase that would seem unaffordable any other time of the year can feel within reach. And making a decision about the right vehicle—in terms of quality, reliability, safety, and affordability—is difficult at any time.

Too often, low- and moderate-income households have limited options for acquiring a reliable, affordable vehicle (that lump sum from tax time isn’t enough to buy a quality vehicle outright) through conventional financing, and therefore they resort to predatory dealers and lenders. That means higher interest rates and fees, and a higher purchase cost—it’s expensive to be a poor car buyer. And if the vehicle breaks down, it adds to the expense and income volatility they are already struggling with.

We at Freedom First Credit Union set out to develop an automobile financing program specifically to address the challenges faced by these households, particularly because our service area includes a largely rural, five-county area around Roanoke, Va., with little to no access to public transportation.

In creating our program Responsible Rides, we balanced several competing design parameters:

1. Target lower-income households but make qualification realistic for them. To qualify for Responsible Rides, an applicant’s income cannot be more than 300 percent above the US poverty level for their family size (for example, $72,000 for a family of four). In general, they have to have a credit score no lower than 550 (the national average score is 687).  But if their credit score is below the minimum, we allow for compensating factors and give applicants a chance to explain why their credit score is below the minimum We also consider alternative ways of showing credit worthiness, such as a history of regular, prompt payment of bills.

When Paula Jones came to us, for example, her credit was in tatters. Jones made ends meet with two different nursing jobs, and had been struggling to get her finances in order after several years of caring for her aging parents before their deaths last year. She got behind on her own bills trying to keep up with those of her parents, and unpaid medical bills from a hospital stay for debilitating back spasms were the last straw. She wouldn’t normally have qualified for anything other than a sub-prime auto loan, but we got to know her and made a decision based on more than her credit score. It was clear from her attitude and hard work that she was credit-worthy. “[The program] just opened a door I didn’t know was there,” she said.

2. Make loans a stepping stone to greater financial stability for the borrower but also sound lending decisions for us. Of course, we don’t approve everyone. All participants must provide a hardship letter that explains their circumstances and describes why transportation is essential to their household. The household can’t have more than $1,500 in unpaid collections (excluding medical collections) or have filed for bankruptcy in the last 12 months. And they have to have worked for the same employer for the last 90 days.

A loan doesn’t end when the application is approved—that’s just the start. Most of these households will face a financial crisis during the average term of the loan—60 months. We want them to have the tools necessary to navigate through the next hardship, and we want them to know we will be there with them. Borrowers must complete a financial education counseling program, including one-on-one time with our certified financial advisor and group meetings. The topics covered include both traditional financial education—setting financial goals, creating spending plans, understanding credit scores, etc.—and less common topics like a one-hour class on vehicle maintenance and coaching on choosing auto insurance.

3. Trust borrowers but help them make good buying decisions. One of the main reasons Responsible Rides is successful is that we actively help borrowers choose a vehicle. Our program director helps applicants figure out what they need in a vehicle and what price they should expect to pay based on the local market. We make loans only for vehicles purchased from an approved dealership list that we’ve vetted to make sure our customers are treated well.

Since its inception, Responsible Rides has made $1.9 million in affordable loans and enabled 377 individuals to purchase a reliable vehicle. The average loan size is $10,500, the average rate is 13.85 percent, and our delinquency ratio at year-end last year was 3.12 percent.

But of course, households experiencing a high degree of income and expense volatility need more than just auto loans. We’ve also been experimenting with an alternative to payday loans in our community. Borrowers can get short-term credit to help them through a crisis, but their repayments include a savings deposit they can access only after they’ve repaid their loan. The idea is to create a new emergency savings fund to provide a cushion for weathering the next short-term cash crunch without resorting to credit at all. Based on the results, we’re expanding into a “payday consolidation” loan to help families who are caught in a recurring cycle of payday loans. We are hopeful these new programs will make as big of an impact on the lives of low- and middle-income households as Responsible Rides has.

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Recent Homes For Sale in Alpine CA (California Terrace)

About Alpine

Alpine is a hidden gem located about 30 miles East of El Cajon. Alpine is a unique area of San Diego that has all the amenities of a large city but with a small town feel. Alpine is at the foot of the Cuyamaca Mountains and is famous for the surrounding apple orchards. The population of Alpine is sitting around 17,000 but rising as more and more people are falling...

Recently sold in complex


The San Remo - Cemento Stingray

San Remo, on the Italian coast is also known as the City of Flowers for the way it blooms in summer. It is a Mediterranean resort town with gorgeous villas known for their Riviera grandeur. Look beyond the resorts and hotels to find the little visited old town with winding roads that come down the hillside. At the base is 25 kilometers of coastal walking and biking lanes. Head back up the hill into San Remo for a well earned and romantic meal at one of the exquisite hotels.

Dimensions: 11.5"L x 8"H x 1.5"D (23" shoulder drop)

Details: Stingray embossed Italian calf skin. 100% cotton lining. Italian leather interior. Signature wood inlaid zipper puller and hardware. Removable wallet is snake-embossed Italian goat skin with a hand-painted grey/cemento ombre detail.

made in la


Oncology Programs 2017-06-02T11:40:41+00:00

Cancer is a group of more than 100 diseases whereby cell division and growth is abnormal and uncontrolled. In very simplified terms, normally cells divide, grow and die in a very regulated and controlled fashion and as needed by the body. Cancer develops when cells start to grow in an unconstrained and unsuppressed manner due to damage or alteration in the genetic makeup of the cell. That is, rather than dying and being replaced by new healthy cells, cancer cells continue to grow and multiply forming more and more abnormal cells that create lumps or masses of tissue called tumours.  Not only can tumours release hormones that have a negative effect on the body but they can also replace normal tissue and interfere with the digestive, nervous and circulatory system. Metastatic cancer is when cancer cells spread to other parts of the body and start to develop new tumours with cells from the primary tumour. Thus, if cancer originates in the breast and metastasizes to the lung, the cancer cells in the lungs are breast cancer cells.    However, not all tumours are considered harmful. Benign tumours are noncancerous and often not life-threatening though some might grow large enough to cause local pressure and may be uncomfortable. Also, some cancers, such as leukemia do not usually form tumours rather cancer does not allow normal blood forming and blood function.

Treatment In Cuba

Cuba has long been recognized as a leader in the area of health care and has become world renown in medicine and medical training, with a large number of  scientists  who are well respected by foreign peers for the developments of such vaccines as meningitis B and hepatitis B. The Cuban government and Cuban researchers have in the last 30 years pushed biotechnology and immunotherapy (also known as biotherapy or biological therapy) to the forefront. According to the World Health Organization, Cuban researchers in biotechnology have made very important advancement in the development of new cancer prevention tools and treatments with emphasis on transferring cancer from a deadly disease to a chronic one. It is believed that cancer cells are naturally and frequently found in the body but are usually wiped out by the immune system and only become life threatening when the immune system cannot differentiate between healthy and unhealthy cells  Immunotherapy treatment for cancer stimulates the immune system to fight cancerous cells and to control cancer growth. It has a further benefit in that it aids chemotherapy and radiotherapy by making them more effective and less toxic hence increasing patient’s life expectancy and improving quality of life.

By 2013, Cuba biotechnology centers has 1200 international patents (including Nimotuzumab or CIMAher for head and neck cancer as well as the gastrointestinal (GI) tract cancer), CIMAVAX-EGF and Racotumomab or Vaxira (for non-small cell lung cancer). It also markets pharmaceutical products and vaccines in more than 50 countries and is currently testing 90 new products in more than 60 clinical trials.

  • Nimotuzumab (CIMAher)

For cancers that have an overexpression of EGFR (epidermal growth factor receptor) targeting therapy with anti-EGFR antibodies remain the only non-chemotherapy treatment against it. One such drug that has been gaining international recognition is Nimotuzumab developed by the Center of Molecular immunology in Cuba. Though it’s not the only one of it’s kind, it is safe to say that researchers at the center were globally the first to research and develop anti-EGFR antibodies.

Nimotuzumab is a humanized monoclonal antibody which is an antibody made by combining human antibody with a mouse or rat monoclonal (clone from a single cell) antibody. Unlike its competitors, Nimotuzumab has a greater number of human like epitopes (the part of the antigen -in this case, cancer cells- that the antibodies bind to) attached to the mouse or rat monoclonal antibody. It has specific ability to recognize, bind and disable the EGFR receptors. Since the overexpression of EGFR is confined for the most part to the tumour cells, and since Nimotuzumab is mainly active at the site of the tumour side effects are very minimal, so much so that it is the first cancer vaccine to be used on children.

Nimotuzumab is administered as monotherapy or in combination with radiotherapy and chemotherapy, increasing survival time and quality of life. It is a very well tolerated medication even with prolonged treatment and is approved for the treatment of patients with advanced cancer of head and neck as well as high-grade malignant glioma in adults and children and in advanced esophageal cancer.

  • CimaVax-EGF

CimaVax-EGF is the world’s first stage 3 and stage 4 non-small cell lung cancer (NSCLC) vaccine developed in Cuba by the Center for Molecular Immunology and the Center for Genetic Engineering and Biotechnology. It works by stimulating the immune system to produce antibodies that are specifically targeted for and bind to epidermal growth factor (EGF) a protein found naturally in the body and is responsible for signaling cells to divide and grow. In certain types of cancers the body produces too much EGF forcing the cells to grow and divide uncontrollably. CimaVax-EGF is not a preventive vaccine, rather it acts by restraining the growth and division of cancer cells slowing down its development converting a late stage lung cancer into a chronic disease.

CimaVax-EGF is given after chemotherapy and radiation have been administered. It is well tolerated and has few side effects

This vaccine is in the trial process in the Uk, Canada, China and Peru and is in the process of approval in Argentina, Brazil, Colombia and Paraguay. Furthermore, in the end of April, 2015 Roswell Park Cancer Institute, a US based cancer research center, has signed an agreement with the Molecular Immunology Center of Cuba to begin clinical trials for CimaVax-EGF.

  • Racotumomab

Racotumomab is the second non-small cell cancer (NSCLC) vaccine developed by the Center of Molecular Immunology in Cuba and has been used as the primary treatment for NSCLC in Cuba since January, 2013.

Racotumomab works by activating a cellular and humoral immune response against NeuGc GM3 ganglioside present in some tumors tissues by blocking the tumor growth and slowing down it’s development and hence increasing patient’s life expectancy and improving quality of life. Since NeuGc GM3 is not only found in lung cancer cells but also in breast cancer cells, in melanoma cells and neuroectodermal pediatric tumor cells it is hoped that this medication can be used as part of the treatment for such cancers. Racotumomab is presently being tested in a large multinational clinical trial.

Racotumomab is a well tolerated cancer treatment with the most common side effect is burning in the arm and slight pain in the injection area as well as a possibility of asthenia (physical weakness or lack of energy).

*Please note that Cuba continues to follow the standard treatment for cancer by using surgery, radiation and chemotherapy in combination with immunotherapy. Treatment however is determined by the type, stage of cancer and patient’s overall health.

Have Questions? Contact Us!

Head and Neck Cancers
Sarcomas Medical Program
Lung Cancer Medical Program


Physical Education

The high school experience represents the culmination of physical education instruction for California’s kindergarten through grade twelve students. Throughout their school years, students have experienced a planned sequence of formal physical education instructional experiences. When students reach ninth grade, they are ready to integrate all that they know with all that they can do. They become capable of higher-order thinking and of more skilled performance. The high school courses provide a blueprint for delivering content in a manner that equips students to make a successful transition from the physical education instructional program to participation in physical activity during adulthood.

The foundation for a physically active lifestyle is firmly laid so that students become independent learners who initiate and monitor their own participation in physical activity. Students are provided with opportunities to improve their health-related physical fitness as well as to understand advanced concepts related to physical fitness and physical activity. Students learn how to achieve physical fitness and the importance of maintaining an active lifestyle throughout their lifetime.

Participation in physical activity also can be an important venue for the social, psychological, and emotional development of young adults. Social skills and personal responsibility skills need to be taught and practiced with feedback. Physical education courses provide an ideal setting for adolescents to learn appropriate social interaction skills, suitable ways to express and control emotions, and opportunities to solve complex problems.

California Department of Education: Physical Education Framework for California Schools 2009


What are Geriatric Social Workers?

What Do Senior Social Workers Do?

One of the most common things a geriatric social worker does is to help clients deal with complicated government programs that they may not realize are out there and able to aid them. These programs include everything from social services and local community programs to healthcare providers and other legal entities.

Not only can senior social workers help their clients apply for services from a variety of public and private programs, they’re also experts who are able to help resolve any issues that may occur in the delivery of those services, including social services and obtaining funding for elder care. Social workers for seniors are able to direct older adults to a number of beneficial programs, such as helping seniors gain access to local senior transportation, food delivery services like Meals on Wheels, in-home assistance services, and many other programs.

Additionally, geriatric social workers often provide counseling services that can help the elderly cope with many end-of-life issues. This counseling can make the transition from their home to a long-term care environment as smooth as possible. It can also reduce the burden of dealing with a variety of common issues facing the elderly, including the creation of living wills and DNR orders.

Making Families Stronger

There’s nothing wrong with getting a little help when you need it. With the aid of geriatric social workers, older adults have the opportunity to remain in their own home longer, enjoying significantly safer lives. For any family providing care for an elderly family member, consulting with a social worker can make social services and elderly parents a much easier match to make.


Google Street View is an excellent way to watch your neighborhood change. In fact, we've conducted our own informal surveys of urban transformation in Detroit, San Francisco, and Brooklyn. While our investigations were based on casual observation, now a pair of sociologists from Harvard are using Google Street View data to measure gentrification—and predict if those trends will continue.

As part of a new study published in American Sociology Review, Jackelyn Hwang and Robert J. Sampson tracked the movement of middle- and upper-middle-class residents into Chicago neighborhoods that have previously shown signs of economic decline. But Hwang and Sampson are not studying migration patterns specifically—rather, they're examining the real-life results. Using Google Street View, they were able to virtually tour thousands of streets, looking for visual signs of reinvestment and renewal from 2007 to 2009.

The fact that Hwang and Sampson have codified these features into an instant assessment system using only Google's images is what's so interesting. Instead of the cold hard data of a Census report, Hwang and Sampson have been able to create a more qualitative, nuanced definition of gentrification based on observation. They managed to isolate three different groups of indicators found on Google Street View images—structural mix, beautification, and lack of disorder and decay—to rank each block. Hwang herself hand-coded about 2,500 blocks for all these indicators (you can see her entire coding guide here), and by the end she was able to "read" a block in about a minute.

Examples of various stages of gentrification from the study. Block face 1 for example, scored lower due to sandblasted brick, boarded up windows, and graffiti.

It's fascinating to see the familiar hints of gentrification divided up into these metrics. The structural mix includes the details that Hwang and Sampson are able to see based on architectural cues. For example, "sleek, geometric, glass or steel exterior materials" and "reconstructed or restored porches and balconies" mean that a building has been renovated or is entirely new, while "ornate detailing" and "aluminum siding" are signs that the property is older and has not been taken care of. They also look at the presence of particular land uses like converted industrial buildings (warehouses turned into lofts, for example) and luxury high-rises.

The study also takes into consideration the details around the properties themselves, where things like speed bumps and bike lanes are proof of public investment and therefore boosted the block's rating. The presence of trash and graffiti, both easily seen from Google's cameras, resulted in a lower score.

One indicator that had to be thrown out due to the time period—remember, this was the start of the real estate bust—were "for sale" signs. "We just assumed that newly built houses for sale were developers promoting gentrification, but as I researched Chicago's real estate market more, I saw that they'd overspeculated and overbuilt these houses," says Hwang. "They're for sale because they couldn't sell them to anyone."

Comparing gentrification indicators using traditional methodology in 1996 and the new Google Street View method

After assigning each block a "degree of gentrification," Hwang and Sampson compared the 2007-2009 study with a similar one in 1996. In fact, one of the reasons they chose Chicago is because it had been so well-documented by similar pre-Street View research. Sampson had even led a study in 1995 focusing on crime and disorder where he drove around in SUVs and filmed city blocks with a camcorder. Compared to 1996, they could clearly see the patterns of gentrification intensifying in certain neighborhoods and sometimes spreading to adjacent ones.

But in several cases, even after accounting for variables like transportation, crime, and even perception, gentrification did not grow in neighborhoods as predicted, says Hwang. "They all had showed signs of gentrification or were next to ones that did, and you'd think it would have spread—but it didn't."

Hwang and Sampson looked into more detailed information about those specific neighborhoods to see what they had in common. When cross-referencing their predictors for gentrification with actual population and city data, they found some unsettling trends. While they could clearly see which blocks had showed early signs of gentrification, the process was only likely to continue if at least 35 percent of the residents were white. Even if some gentrification signs were present in neighborhoods that were 40 percent or more black, the process slowed down or eventually stopped.

While this does seem to reinforce the stereotypical view of gentrification (white residents moving in to a neighborhood will change its fate), Sampson says it proves something slightly different. "Predominantly black, seriously discriminated-against neighborhoods in Chicago and many other American cities aren't reaping the same benefits from the transformation of cities," he says. "So rather than saying you need whites, I think what is needed—and this has always been the case—is some concerted effort to rethink urban policy."

Besides the obvious discussions about race and economic policy that this study is sure to provoke, it also makes a case for this new way to measure "gentrification," which is so often seen as a bad word. When defined by Hwang and Sampson's methods, gentrification is really about urban investment, which almost always yields a positive transformation for local residents. Using these images, it's a transformation you can see—or not see—for yourself. [Harvard Gazette]


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Dark aether

Dark Aether is a parallel dimension created on the planet Aether by a Leviathan impact. The dark dimension is the home of the Ing, a hostile species Samus Aran encounters in Metroid Prime 2: Echoes.

Details of Dark Aether's formation are not mentioned throughout Echoes. The Leviathan and the natural energy harvested by the Luminoth are somehow responsible for its creation. Landmarks on Aether and Dark Aether tend to mirror each other. A select few objects will actually move due to events in the other dimension.

Since Aether is split, its planetary energy is split as well. If one dimension gains control of all of the energy, the other perishes. A conflict over the energy provoked a lengthy war between the Luminoth and Ing, with the Luminoth on the losing side by the time Samus arrived. Energy is exchanged via the Energy Transfer Module, which binds to Samus after she defeats a Dark Alpha Splinter. Due to this bind and Samus's lack of a functional ship, she agrees to help the Luminoth leader, U-Mos, reclaim the lost energy in the different regions of Dark Aether. To access Dark Aether, Samus must use portals found scattered throughout Aether. These can either occur naturally or can be mechanically generated by Luminoth technology. The Ing can travel between Light and Dark Aether in a gaseous form, but must quickly possess a living creature to survive on Light Aether's surface.

Thanks to the efforts of Samus Aran, the Ing were defeated and the planetary energy they had harvested was transferred back to Light Aether. The removal of the energy from Dark Aether caused the bleak dimension to collapse into utter oblivion.

Ad blocker interference detected!


Meryl Streeps Globes speech was beautiful, heartfelt and played right into Donald Trumps hands

Image: GETTY

Meryl Streep’s speech at Sunday’s Golden Globes hit all the right notes, setting the internet on fire for most of Monday.

As for the subject of Streep’s words, Donald Trump threw gasoline on the fire by doing what he always does: tweeting.

For people who have been watching Trump battle with his critics for more than a year, the whole situation was rather predictable. It quickly devolved into a petty, partisan fight over old news (Trump’s mocking of a disabled journalist) and whether it was a celebrity’s place to comment on politics.

It was and still is, based on a cursory look at what’s floating around on social media the perfect story to distract from the ongoing calamity that is the Trump administration confirmation process.

It has never been more important to pick battles. With Trump’s inauguration less than two weeks away, there are any number of stories of massive importance that need the public’s attention. There’s Trump’s business interests that could theoretically put him in violation of the U.S. Constitution the very moment he is sworn in.

Trump has thrived for months on attacks from journalists, celebrities and other people that he has grouped together as the “elites” who have ruined the country for hardworking, blue-collar Americans. Monday was no different, with plenty on the conservative side (even some who aren’t Trump supporters) declaring that Streep’s speech was why Trump won. It is a familiar and tired pattern.

Unfortunately, Monday was a particularly important day. This week, Trump’s nominations for his Cabinet are due to be put through the ringer in confirmation hearings, which could even be delayed due to a lack of important disclosures and vetting. News also broke on Monday that Jared Kushner would be a senior adviser to the president despite being Trump’s son-in-law, federal nepotism laws and his own potential conflicts due to business interests. Late on Monday, USA TODAY reported Rex Tillerson, Trump’s secretary of state nominee, had been CEO when Exxon had conducted business with Iran, Syria and Sudan through subsidiaries despite U.S. sanctions against those countries.

Streep was spot on with her points. The problem is that they were centered on issues that are more or less part of the past. Voters elected Trump despite the fact that he mocked a reporter on stage at a rally. Yes, it’s worth bringing up, but at this point it’s more of a distraction from the issues at hand than anything else. Streep’s call to action to donate to the Committee to Protect Journalists is also an admirable one. It would have been better, however, to tell people to call their elected officials and demand that they properly vet his cabinet nominees.

Its unreasonable to expect every individual to fully appreciate just how twisted the Trump media cycle has become, but for those who have come to comprehend the magnitude of manipulation we are now facing, it’s tough to watch. Its overwhelming; we need a breather. So it is easy and downright pleasant to get sucked into the feel-good Streep-fest.

Sadly, it’s also the kind of thing that Trump thrives on.

Read more:


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'we have to allow the police to gather all the facts just before unnecessary speculation and await the facts right before determining its impact on our place.

Probe: law enforcement forensic officers Focus on the scene late in to the night during the hrs after the horrific stabbing in Woolwich

A plot to attack United kingdom and US targets having a 'filthy bomb' and fuel-crammed limousines that noticed al Qaida operative Dhiren Barot sentenced to thirty yrs in jail in 2006, and seven other Gentlemen associated with the plot jailed the subsequent 12 months.

This spectacular Victorian type terrace residence is beautifully taken care of by The existing Chiswick Park operator. It provides a wealth of Area to get a rising file...

The Group provides a  supportive surroundings, exactly where specific wants are recognised and companions are inspired to use the talents they have already got, as well as building new types.

‘that they had no intention of running off or leaving or anything at all. In my opinion, they had been watching for the law enforcement to reach to generally be shot via the police. which is The one thing I am able to Feel. It really is horrendous whatever they had been performing to that man.’

A law enforcement spokesman said: “law enforcement are investigating a suspected arson at the previous Little Plumstead healthcare facility which occurred while in the early Chiswick Park hours of yesterday morning.

Weapons: Forensic specialists bag up proof on the scene from the aftermath of the horrific killing yesterday

A set of nineteenth-century docks remain, on the website in their sixteenth-century predecessors, plus the afterwards growth on the Dockyard while in the early Victorian period of time is represented in a bunch of buildings along with the get more info dockyard chimney, a distinguished landmark on Woolwich Church Avenue.

The fire were taken care of as “potentially suspicious” but law enforcement have now verified it's getting dealt with as arson.

Shooters Hill as a place has an extended historical past. The name 'Shooters' is derived within the time archers practised on the positioning many hundreds of yrs before golfing was played...

At ISO three,two hundred we see Olympus utilizing additional sharpening and raising distinction, plus more luminance and chroma sound show up while in the shadows. element retention, however, is about even between The 2.

The grip is really nice, with an excellent leather-based-like texture. It really is hard rubber, but nevertheless delicate to the touch, and heat. Just the proper Mix for a camera prone to get many use.

for hundreds of years the area amongst the Thames and The existing city centre is dominated by docks, warehouses and factories. The Royal Dockyard set up here by King Henry VIII within the sixteenth century was the catalyst for Woolwich's eventual advancement into a armed forces garrison city. Its gun wharf was expanded by the Board of Ordnance within the seventeenth century into what grew to become the Royal Arsenal, a substantial weapons manufactory.

a method utilized to unlock the plot and vital components of a story. These elements can be represented visually through several graphic organizers showing the start, Center, and close of a story.

Phoneme identity On this exercise, small children learn to recognize the identical Appears in different phrases. (Teacher: What audio is the same in take care of

Phonemic recognition is the ability to discover, think of, and perform with the individual Seems in spoken text. An illustration of how commencing audience exhibit us they have phonemic consciousness is combining or blending the separate Appears of a phrase to convey the term ("/c/ /a/ /t/ - cat

Tubes tend to be created possibly of metal or aluminium, although There exists composite scaffolding, which utilizes filament-wound tubes of glass fibre in scaffolding a nylon or polyester matrix, due to high expense of composite tube, it is often only utilized when You will find a threat from overhead electric powered cables that can't be isolated. If metal, They can be both 'black' or galvanised. The tubes appear in a variety of lengths and a normal diameter of forty eight.

Onsets and rimes are elements of monosyllabic text in spoken language. These models are more compact than syllables but could possibly be greater than phonemes. An onset may be the initial consonant audio of the syllable (the onset of bag

two or three nights back, an extremely potent windstorm blew through San Francisco like I do not ever don't useful content forget occurring like it ahead of. Our condominium building is just throughout the street from A different creating that's 5 tales higher and currently loaded with scaffolding top to base for outdoor painting function.

Get most effective estimate we have been properly rated amongst the most effective service suppliers of Scaffolding Inspection Services. The available service is rendered by our dexterous industry experts making use of hottest equipment and revolutionary technology. The service features a comprehensive inspection Resolution before the voluminous generation, staying away from any flaws and wasting of Substantially time and cash.

(Most corporations will not likely do this nowadays and just do the job about it producing an pointless nuisance) The crew had The complete place scaffolded in a well timed manner and as I viewed I could inform they understood just what they were being executing. I'd personally hugely advise security service scaffolding to your scaffolding wants!

Specialist branches supply a complete variety of modular aluminium obtain towers, minimal degree obtain and Secure entry methods. scaffolding

A pump-jack is usually a kind of portable scaffolding method. The scaffold rests on supports connected to two or even more vertical posts. The person raises the scaffolding by pumping the foot pedals on the supports, like an auto jack.[10]

(the teacher points out to students why the system can help comprehension and when to use the method), modeling

very good foundations are crucial. Often scaffold frameworks would require a lot more than easy foundation plates to safely and securely have and unfold the load. Scaffolding can be utilized without base plates on concrete or identical difficult surfaces, Despite the fact that foundation plates are usually suggested. For surfaces like pavements or tarmac foundation plates are needed. For softer or even more doubtful surfaces sole boards look at this must be employed, beneath a single typical a sole board should be at the least 1,000 square centimetres (160 in2) without dimension below 220 millimetres (eight.

Decoding is the opportunity to translate a phrase from print to speech, generally by using knowledge of audio-symbol correspondences. it's scaffolding also the act of deciphering a whole new word by sounding it out.

g., ham- the /m/ blends Along with the /a/ to distort the vowel). this method is termed coarticulation. as a consequence of coarticulation, some children have trouble hearing the individual sounds in text as well as the thought of phonemes needs to be explicitly brought for their consideration by means of instruction.

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The World We Knew

is not the world our children will know.
How can we help them adjust
to ... what we cannot understand?

Have you looked at our library
of free eBooks?

Can you to eat
TWENTY vegetables

jobs, eBooks, Jack Canfield, podcast, audio, mp3, Chris Howard, employment, security, Tony Robbins, newsletter, partner, profit, health, environment, Bob Proctor, nutrition, corporations, insurance, wellness, concentrated, plants, ideal-business, water, Wozniak, microbusiness, alkaline, free, trial, norisk, free-shipping, Ralph Waldo Emerson, stockmarket, stock, securities, video, wave, wav, eBooks wmv, asf, Walace Wattles, microclustered, James Allen, charged, disinfectant, safe, effective, questions, pleasant, healthy, wise, income, antacid, blood, ERW, EOW, REDOX, reduced, oxygen, hydrogen, molecule, electrolysis, pressure, diabetes, neuropathy, natural, electrode, batch, tank, Robert Kiyosaki, membrane, convenient, Robert Ringer, easy, clean, Jim Collins, pure, holisitic, energy, dc,


When color space and design space come together to produce simple and classic.

"SInce a three-dimensional object casts a two-dimensional shadow, we should be able to imagine the unknown four-dimensional object whose shadow we are.” ― Marcel Duchamp

Our History

Markoz has always been drawn to glass both from an artistic sense and from an engineering sense.   He was introduced to glassblowing by his daughter during a father-daughter field trip.    He fell in love with the medium.   Markoz has been blowing glass for more than 7 years.   He has won two international design awards.

A piece of blown glass is different from other art as glass is such a common material in human lives.   If you place a piece of glass art in front of any person, they will have an opinion and often a suggestion.  Whereas if you place an image of a Picaso in front of the same person, they may be without opinion or suggestion.

For me,  glass blowing is always about making art and not the individual piece.  425Glass:

Markoz at the bench.

425Glass - Organically Blown Glass(tm)


Introduction: Paper Fortune Teller

This thing is sick

Step 1: Materials

All you need for this project is scissors and a piece of paper

Step 2: Fold

Fold the paper to make a square. Do this by taking a corner and folding it across the paper as shown. Make sure the edges line up.

Step 3: Cut

Cut away the paper that will not be part of your square. You can discard that extra piece.

Step 4: Fold

When you unfold your triangle you should have a square. Make another crease to form an "x" across your paper. This is done by folding from corner to corner.

Step 5: Fold

Take each corner and fold it into the center.

Step 6: Fold

Flip over the piece of paper. Once again fold each corner into the center of the paper.

Step 7: Fold

Fold your square in half to form a rectangle shape.

Step 8: Complete

Place your fingers underneath each square. Then push your fingers up and towards the center to give it shape. You can now write inside this device to tell fortunes

tomatoskins (author)2017-02-08

What a wonderful fortune teller!

About This Instructable

More by AndrewHaselhan:Paper Fortune Teller
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Sublimation Printing Solution For Roland Printer

Transfer printing is a decentralized transfer of digital printing, for polyester, chemical fiber, nylon, spandex, chiffon and other textile fabrics, inkjet printers are used to print a variety of pieces of paper on the transfer paper, and then through Heat transfer machine high temperature heating, the transfer paper on the pattern transfer to a variety of textile fabrics on a printing method.

Roland rt640

Transfer printing is characterized by a simple process, less process, high efficiency, no pollution, less investment. After the transfer of high color adhesion, graphic clear, colorful, safe and environmentally friendly, suitable for all kinds of cloth cloth textile fabric production. The way in which the cloth is to be transferred is a thermal transfer printer, a thermal transfer ink, a thermal transfer paper, and a thermal transfer machine.

Roland sublimation transfer program:Recommended models: Roland XF640, Roland RF640, Roland RT640

roland rt640

Roland Digital Printing Machine Features: XF640 is the latest gold double print seven generations of head staggered arrangement, the highest speed can be printed to 102 square meters per hour, at high speed can still maintain high-quality output, to meet the needs of large customers. Roland RF640 is a gold seven generations of head, variable droplet control, 1.6M wide, can achieve 33 square meters per hour high-speed mode. Roland RT740, gold 7 lead, 1.8M wide, can achieve 6-8 color printing top quality print.

ink cartridge

Best Ink: Imported Italian J-TECK sublimation transfer ink


Las Vegas Entrepreneur Takes on Overpriced Burial Caskets

Rest in Peace for Less With Caskets Made in China

Make Your Own Coffin

Affordable Funeral Services

I am frustrated with how some people take advantage of others who are facing a difficult time in their lives when coping with the deal of a loved one. People have enough to deal with when a loved one dies they should have honest and well meaning advice to help guide them. This site aims to do that as part on my new site on money matters in general.

I actually remember hearing about the pressure tactics used by some to get people to pay way more for a coffin than they would chose to given all their other priorities. The prices are often incredibly high and the tactics used to pressure people into paying more by making it seem like avoiding wasting money on a funeral casket is somehow a sign of not enough love for the deceased I find disheartening. For my father’s funeral our friend made my father’s casket himself and some of those he worked with dug his grave. How much more loving a gesture can there be? That is far more special I think than spending huge amounts of money.

I can’t remember exactly how it happened. I can’t remember if he said he wanted a simple pine box coffin first, I think he did. It is the type of thing he definitely would have wanted – a simple coffin. He wouldn’t want to waste money on a fancy casket. He spent money much more sensibly in my opinion, on things like travel, saving for retirement and paying for his kids education. What is right for each person is up to them and those that love them, but equating the fanciness of the casket with how much the person is loved and respected is wrong.

A funeral is a difficult time. And a meaningful time. And a time when people are venerable. I hope this site can help people find honorable solutions that are also affordable and fitting for their needs.


Trident of the Deep

Sea Spear

weapon (melee)

Body Slot: Held
Activation: Free (see text)
Weight: 7 lbs.
+2 trident throwing, distance, returning trident.
The spear’s wielder is treated as unarmored when making swim checks and is immune to pressure damage from deep water. It also grants its bearer the abilities to breathe underwater and to communicate with any water-breathing race. These effects are constant and require no activation.

The Trident of the Deep is a Undine weapon. Forged in their capital, it was made as a gift for friends of the Undine race.

Trident of the Deep

Alfheim (underconstruction) Tian Urvogel7


5 Plants For Your Bedroom That Will Cure Insomnia

Numerous people nowadays suffer from insomnia, sleep apnea, or night terrors. Fortunately there is a simple and natural solution to improve your sleep and finally get the rest you need. In order to get some rest all you have to do is to place one of the following plants into your bedroom to get a calm and relaxing atmosphere to enjoy.

White Jasmine

It’s not just about the smell, this plant has potent healing properties. Jasmine oil has been used for hundreds of years in parts of Asia as a natural remedy for depression, anxiety, emotional stress, low libido and insomnia.

Snake Plant

The interesting thing about this plant is that it releases more oxygen during the night, even more than during the day. That’s why it improves the quality of the air and makes the breathing better.

What we know is that the scent of lavender eases anxiety and insomnia.

It also slows the heart rate and provides a calm atmosphere in the bedroom, perfect for rest and sleep with its relaxing properties.

English Ivy

English Ivy appears to be one of the easiest plants to grow and it will clean and purify our indoor air by removing toxins in a natural way. The evergreen climbing vine that is most commonly found outside, can substantially reduce the amount of mold in the air of your home by up to 94%.

Aloe Vera

Aloe Vera has incredible benefits that most people aren’t aware of.  One of the reasons it’s such a good air purifying plant is that, unlike most plants, it actually releases oxygen and absorbs carbon dioxide at night.

Source: Healthy Food House

(1370 Posts)


Letting agent couple who failed to register over £40,000 of deposits escape jail

A married couple who ran a lettings agency where they failed to protect tenants’ deposits have escaped jail.

George and Kim Overill ran Oakpark Properties in Harwich, Essex, and pleaded guilty to two counts each of contravention of professional diligence as directors and a further two each of fraudulent trading.

The couple failed to register 73 deposits totalling £42,794.

Prosecuting, Alison Lambert said that the Overills “stuck their heads in the sand and were not dealing with the problems”.

The court heard that Mrs Overill took exotic holidays and built a home extension, while staff raised concerns about the firm’s accounts and the opening of a new branch in Clacton.

One of the company’s book keepers was only asked at interview if she knew how to work Excel, and noticed that money coming into the company was not enough to cover its outgoings.

Mrs Lambert said: “She said Mrs Overill dealt with the deposits and that there was no system letting her know which ones where coming in and out other than to tell her verbally.”

In 2014, the company decided to merge with another letting agent, Valentines Properties, but that fell through once they too realised that accounts did not add up.

In mitigation, Stephen Dyble said the couple had never intended to commit fraud when setting up the company, and that the monies owed have been paid back.

His Honour Judge Christopher Morgan told the defendants: “Those who rent properties understand the reasons behind tenant deposit schemes.

“They are there to protect them. Tenants might have had to save for them or borrow money to raise them. For those receiving them, they may seem like insignificant amounts, but for many they represent many months of hard saving and are a lot of money to those individuals.

“You as letting agents sit in a position of trust and responsibility to tenants and landlords, and you have breached that trust by your actions.”

The couple were given an 18-month sentence suspended for two years. They were also ordered to complete 180 hours of unpaid work and have been banned from being company directors for two years.

Cllr Susan Barker, Essex County Council councillor for Culture, Communities and Customer, said: “The outcome of this prosecution has been a success for Trading Standards in Essex. We have worked closely with partners to ensure that the defendants were brought to justice.

“Going forward we will continue to work hard with our partners to clamp down on fraudulent agencies that are unfairly taking advantage of tenants. My congratulations to everyone involved in the case.”

There will be a hearing on November 16 to decide compensation and Proceeds of Crime financial orders.

Mortgages – Find cheapest rates from 0.98%

Written by: Houseladder


Fellbach Germany Germany

Fellbach Tourist Map

Fellbach is a mid-sized town on the north-east Border of Stuttgart in Baden-Württemberg, Germany. With a population of approximately 43,700 as of December 2003, it is the second largest town in the District Rems-Murr-Kreis. The area of the town is 27.7 km2 (10.7 sq mi). Fellbach was first mentioned as Velbach in 1121. It was called Vellebach in 1357 and the name Fehlbach was used in around 1800. On 14 October 1933, it was declared a city. After World War II it reached a population of more than 20,000 in 1950 and therefore received the status "Große Kreisstadt". Fellbach has 3 main districts: Fellbach, Schmiden (since 1 January 1973) and Oeffingen (since 1 April 1974).


Saturday, December 31, 2011

Hugo Porta vs New Zealand - 1978

He has gone down in rugby history as the first player to be considered world class that Argentina ever produced. His achievements for Los Pumas were truely remarkable as he was able to transform Argentina from being considered a minnow rugby nation to setting the side on its path to where it is today. He was Argentina´s first  member of the IRB Hall of Fame and is universally considered as one of the greatest flyhalves of all time. His goalkicking abilities made his such a crucial player as any time Argentina were inside an opponents half Porta could be trusted to inflict pain on the opposition with a kick. Teams had to be on the constant alert as he was also very talented with dropkicks.

Porta made his international debut as a 20 year old vs Chile in 1971 and also faced Brazil, Paraguay and Uruguay that year. The following year he was a part of the Argentina team which defeated South Africa (the team was known as the South African Gazelles) 18-16 in one of two matches. In 1982 he then kicked 21 points to defeat South Africa in Bloemfontein for a team known as the South American Jaguares due to the controversy surrounding apartheid. But it was his European tour in 1976 that first really saw his name underlined as a player of note. Argentina had been leading against Wales, then the best European side, before going down 20-19 following two late tries to the home side.

Argentina´s famous 24-13 victory over Australia in 1979 was the match that have been said to have put Argentina on the map. The best player on the field was Hugo Porta as the flyhalf converted two tries, kicked a dropgoal and a penalty. Bigger things were yrt to come, however, as in 1983 Porta helped Argentina make history as Los Pumas crushed Australia 18-3 in Brisbane.  Two years latter Porta kicked Argentina into a 21-21 draw vs New Zealand. Porta played in the first two Rugby World Cup´s but was not able to make a notable contribution as Argentina´s only success was a win vs Italy in 1987.

While it took his success´ in the early 1980´s for Portta to become a household name, he had already made it known to onlookers that there was something brewing. Such an example was his performances in Argentina´s 1978 tour of New Zealand. The mighty All Blacks were too strong but Los Pumas did better than their reputation suggested and Porta proved to be a world class talent. He landed kicks from halfway and did what Jonny Wilkinson made himself famous for - dropkicking goals off both feet and doing so with little time or space. Truely a rugby legend, Porta paved the way for the future of Los Pumas as his career inspired many future stars.

1 comment:


A day at the Races

What better way to spend the weekend, but at the Goodwood 73 Members meeting…
What better way of getting there… but in our Trident.

A great event, packed with some stunning cars, including an ex lightweight works Grantura.
..maybe next year we could roll the 69 Tuscan out?

11070241_1087726557910144_830995237750878956_n EX WORKS GRANT - 1112984948062_n 988903_1087726704576796_5850410083319616009_n


Girax - The Ringmaster

The charming goblin manager of The Circus of Wonders

Girax was the fast-talking, cigar-chomping, top hat-wearing, cane-swinging ringmaster of the Circus of Wonders. He had somehow conned our heroes into joining his circus full time, preforming menial tricks.

He tried to stop our heroes from escaping the circus, but Konstrukt ended up stopping Girax instead, with a well-placed arrow in the chest.

Girax - The Ringmaster

The Dawn Isles hermanlilleng hermanlilleng