text:
I'd love to get a picture of a triangulated torus in here but I've tried for hours and can't do it.
I'd love to get a picture of a triangulated torus in here but I've tried for hours and can't do it.
For someone studying math on their own far from academe, the Math Stack Exhange is a blessing. There must be people who look at it all day, answering questions as they arise, presumably accumulating some sort of points (either real points or virtue points). Usually answers to my questions are answered by these people within a day (often hours). But not this one.
"It is clear to me that a surface of constant negative Gaussian curvature satisfies the hyperbolic axiom (more than one 'straight' line not meeting a given 'straight' line). Hartshorne (Geometry: Euclid and Beyond p. 374) defines the hyperbolic plane as the Hilbert plane + the hyperbolic axiom.
I'd like a proof that this axiomatic definition of the hyperbolic plane implies a surface of constant negative Gaussian curvature. "
Clearly a highly technical question. So why bore you with this? Because no answer was quickly forthcoming, I asked one of my math professor friends about it. His response is informal, to the point, and more importantly, shows how a first rate mathematician thinks and explains things. I assure you that this guy is a big name in mathematics — full prof for years and years, author of several award winning math books etc. etc. He's also a very normal appearing and acting individual, and a very nice guy. So here goes.
" Proving that the axiomatic definition of hyperbolic geometry implies constant negative curvature can be done but requires a lot of work. The first thing you have to do is prove that given any two points p and q in the hyperbolic plane, there is an isometry that takes p to q. By a basic theorem of Gauss, this implies that the Gaussian curvature K is the same at p and q. Hence K is constant. Then the Gauss-Bonnet Theorem says that if you have a geodesic triangle T with angles A, B, C, you have
A+B+C = pi + integral of K over T = pi + K area(T)
since K is constant. This implies K area(T) = A+B+C-pi < 0 by a basic result in hyperbolic geometry. Hence K must be negative, so we have constant negative curvature.
To get real numbers into the HIlbert plane H, you need to impose "rulers" on the lines of H. The idea is that you pick one line L in H and mark two points on it. The axioms then give a bijection from L to the real numbers R that takes the two points to 0 and 1, and then every line in H gets a similar bijection which gives a "ruler" for each line. This allows you to assign lengths to all line segments, which gives a metric. With more work, you get a setup that allows you to get a Riemannian metric on H, hence a curvature, and the lines in H become geodesics in this metric since they minimize distance. All of this takes a LOT of work.
It is a lot easier to build a model that satisfies the axioms. Since the axioms are categorical (all models are isomorphic), you can prove theorems in the model. Doing axiomatic proofs in geometry can be grueling if you insist on justifying every step by an axiom or previous result. When I teach geometry, I try to treat the axioms with a light touch."
I responded to this
"Thanks a lot. It certainly explains why I couldn't figure this out on my own. An isometry of the hyperbolic plane implies the existence a metric on it. Where does the metric come from? In my reading the formula for the metric seems very ad hoc. "
He got back saying —
"Pick two points on a line and call one 0 and the other 1. By ruler and compass, you can then mark off points on the line that correspond to all rational numbers. But Hilbert also has an axiom of completeness, which gives a bijection between the line and the set of real numbers.
The crucial thing about the isometry group of the plane is that it transitive in the sense of group actions, so that if something happens at one point, then the same thing happens at any other point.
The method explained in my previous email gives a metric on the plane which seems a bit ad-hoc. But one can prove that any two metrics constructed this way are positive real number multiples of each other. "
A med school classmate who graduated from the University of Chicago was fond of saying — that's how it works in practice, but how does it work in theory?
Exactly the opposite happened when I had to do some programming. It shows the exact difference between computer science and mathematics.
Basically I had to read a large textEdit file (minimum 2 megaBytes, Maximum 8) into a Filemaker Table and do something similar 15 times. The files ranged in size from 20,000 to 70,000 lines (each delimited by a carriage return). They needed to be broken up into 1000 records.
Each record began with "Begin Card Xnnnn" and ended with "End Card Xnnn" so it was easy to see where each of the 1000 cards began and ended. So a program was written to
l. look for "Begin Card Xnnn"
2. count the number of lines until "End Card Xnnn" was found
3. Open a new record in Filemaker
4. Put the data from Card Xnnnn into a field of the record
5. Repeat 1000 times.
Before I started, I checked the program out with smaller sized Files with 1, 5, 10, 50, 100, 200, 500 Cards.
The first program used a variable called "lineCounter" which just pointed to the line being processed. As each line was read, the pointer was advanced.
It was clear that the runTime was seriously nonLinear 10 cards took more than twice the time that 5 cards did. Even worse the more cards in the file the worse things got, so that 1000 cards took over an hour.
Although the logic of using an advancing pointer to select and retrieve lines was impeccable, the implementation was not.
You've really not been given enough information to figure out what went wrong but give it a shot before reading further.
I was thinking of the LineCounter variable as a memory pointer (which it was), similar to memory pointers in C.
But it wasn't — to get to line 25,342, the high level command in Filemaker — MiddleValues (Text; Starting_Line ; Number_of_Lines_to_get) had to start at the beginning of the file, examine each character for a character return, keep a running count of character returns, and stop after 25,342 lines had been counted.
So what happened to run time?
Assume the LinePointer had to read each line (not exactly true, but close enough).
Given n lines in the file — that's the sum of 1 to n — which turns out to be n^2 + n. (Derivation at the end)
So when there were 2*n lines in the file, the runtime went up by over 4 times (exactly 2^2 * n^2 + 2n)
So run times scaled in a polynomial fashion k * n lines would scale as k^2 * n^2 + k * n
At least it wasn't exponential time which would have scaled as 2^k
How to solve the problem ?
Think about it before reading further
The trick was to start at the first lines in the file, get one card and then throw those lines away, starting over at the top each time. The speed up was impressive.
It really shows the difference between math and computer science. Both use logic, but computer science uses more
Derivation of sum of 1 to n.
Consider a square n little squares on a side. The total number of squares is n^2. Throw away the diagonal giving n^2 – n. The number of squares left is twice the sum of 1 to n – 1. So divide n^2 – n by 2, and add back n giving you n^2 + n
The terms Limit point, Cluster point, Accumulation point don't really match the concept point set topology is trying to capture.
As usual, the motivation for any topological concept (including this one) lies in the real numbers.
1 is a limit point of the open interval (0, 1) of real numbers. Any open interval containing 1 also contains elements of (0, 1). 1 is entangled with the set (0, 1) given the usual topology of the real line.
What is the usual topology of the real line? (E.g. how are its open sets defined) It's the set of open intervals) and their infinite unions and their finite intersection.
In this topology no open set can separate 1 from the set ( 0, 1) — e.g. they are entangled.
So call 1 an entangled point.This way of thinking allows you to think of open sets as separators of points from sets.
Hausdorff thought this way, when he described the separation axioms (TrennungsAxioms) describing points and sets that open sets could and could not separate.
The most useful collection of open sets satisfy Trennungsaxiom #2 — giving a Hausdorff topological space. There are enough of them so that every two distinct points are contained in two distinct disjoint open sets.
Thinking of limit points as entangled points gives you a more coherent way to think of continuous functions between topological spaces. They never separate a set and any of its entangled points in the domain when they map them to the target space. At least to me, this is far more satisfactory (and actually equivalent) to continuity than the usual definition; the inverse of an open set in the target space is an open set in the domain.
Clarity of thought and ease of implementation are two very different things. It is much easier to prove/disprove that a function is continuous using the usual definition than using the preservation of entangled points.
Organic chemistry could certainly use some better nomenclature. Why not call an SN1 reaction (Substitution Nucleophilic 1) SN-pancake — as the 4 carbons left after the bond is broken form a plane. Even better SN2 should be called SN-umbrella, as it is exactly like an umbrella turning inside out in the wind.
In the Cambridge Mass of the early 60's the name Norbert Weiner was spoken in hushed tones. Widely regarded as a genius tutti the assembled genii of Cambridge, that was all I knew about him aside from the fact that he got a bachelor's degree in math from Tufts at age 14. As a high school student I tried to read Cybernetics, a widely respected book he wrote in 1948, and found it incomprehensible.
Surprisingly, his name never came up again in any undergraduate math courses, graduate chemistry and physics courses, extensive readings on programming and computation (until now).
From PNAS vol. 114 pp. 1281 – 1286 '17 –"In their seminal theoretical work, Norbert Wiener and Arturo Rosenblueth showed in 1946 that the self-sustained activity in the cardiac muscle can be associated with an excitation wave rotating around an obstacle. This mechanism has since been very successfully applied to the understanding of the generation and control of malignant electrical activity in the heart. It is also well known that self-sustained excitation waves, called spirals, can exist in homogeneous excitable media. It has been demonstrated that spirals rotating within a homogeneous medium or anchored at an obstacle are generically expected for any excitable medium."
That sounds a lot like atrial fibrillation, a serious risk factor for strokes, and something I dealt with all the time as a neurologist. Any theoretical input about what to do for it would be welcome.
A technique has been developed to cure the arrhythmia. Here it is. "Recently, an atrial defibrillation procedure was clinically introduced that locates the spiral core region by detecting the phase-change point trajectories of the electrophysiological wave field and then, by ablating that region, restores sinus rhythm." The technique is now widely used, and one university hospital (Ohio State) says that they are doing over 600 per year.
"This is clearly at odds with the Wiener–Rosenblueth mechanism because a further destruction of the tissue near the spiral core should not improve the situation." It's worse than that because the summary says "In the case of a functionally determined core, an ablation procedure should even further stabilize the rotating wave"
So theory was happily (for the patients) ignored. Theorists never give up and the paper goes on to propose a mechanism explaining why the Ohio State procedure should work. Here's what they say.
"Here, we show theoretically that fundamentally in any excitable medium a region with a propagation velocity faster than its surrounding can act as a nucleation center for reentry and can anchor an induced spiral wave. Our findings demonstrate a mechanistic underpinning for the recently developed ablation procedure."
It certainly has the ring of post hoc propter hoc about it.
I'ver written about Urysohn's Lemma before and a copy of that post will be found at the end. I decided to plow through the proof since coming up with it is regarded by Munkres (the author of a widely used book on topology) as very creative. Here's how he introduces it
"Now we come to the first deep theorem of the book,. a theorem that is commonly called the "Urysohn lemma". . . . It is the crucial tool used in proving a number of important theorems. . . . Why do we call the Urysohn lemma a 'deep' theorem? Because its proof involves a really original idea, which the previous proofs did not. Perhaps we can explain what we mean this way: By and large, one would expect that if one went through this book and deleted all the proofs we have given up to now and then handed the book to a bright student who had not studied topology, that student ought to be able to go through the book and work out the proofs independently. (It would take a good deal of time and effort, of course, and one would not expect the student to handle the trickier examples.) But the Uyrsohn lemma is on a different level. It would take considerably more originality than most of us possess to prove this lemma."
I'm not going to present the proof just comment on one of the tools used to prove it. This is a list of all the rational numbers found in the interval from 0 to 1, with no repeats.
Munkres gives the list at its start and you can see why it would list all the rational numbers. Here it is
0, 1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5 . . .
Note that 2/4 is missing (because 2 divides into 4 leaving a whole number). It would be fairly easy to write a program to produce the list, but a computer running the program would never stop. In addition it would be slow, because to avoid repeats given a denominator n, it would include 1/n and n-1/n in the list, but to rule out repeats it would have to perform n-2 divisions. It it had a way of knowing if a number was prime it could just put in 1/prime, 2/prime , , , (prime -1)/n without the division. But although there are lists of primes for small integers, there is no general way to find them, so brute force is required. So for 10^n, that means 10^n – 2 divisions. Once the numbers get truly large, there isn't enough matter in the universe to represent them, nor is there enough time since the big bang to do the calculations.
However, the proof proceeds blithely on after showing the list — this is where the strangeness comes in. It basically uses the complete list of rational numbers as indexes for the infinite number of open sets to be found in a normal topological space. The proof below refers to the assumption of infinite divisibility of space (inherent in the theorem on normal topological spaces), something totally impossible physically.
So we're in the never to be seen land of completed infinities (of time, space, numbers of operations). It's remarkable that this stuff applies to the world we inhibit, but it does, and anyone wishing to understand physics at a deep level must come to grips with mathematics at this level.
Here's the old post
Urysohn's Lemma
The above quote is from one of the standard topology texts for undergraduates (or perhaps the standard text) by James R. Munkres of MIT. It appears on page 207 of 514 pages of text. Lee's text book on Topological Manifolds gets to it on p. 112 (of 405). For why I'm reading Lee see https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/.
Well it is a great theorem, and the proof is ingenious, and understanding it gives you a sense of triumph that you actually did it, and a sense of awe about Urysohn, a Russian mathematician who died at 26. Understanding Urysohn is an esthetic experience, like a Dvorak trio or a clever organic synthesis [ Nature vol. 489 pp. 278 – 281 '12 ].
Clearly, you have to have a fair amount of topology under your belt before you can even tackle it, but I'm not even going to state or prove the theorem. It does bring up some general philosophical points about math and its relation to reality (e.g. the physical world we live in and what we currently know about it).
I've talked about the large number of extremely precise definitions to be found in math (particularly topology). Actually what topology is about, is space, and what it means for objects to be near each other in space. Well, physics does that too, but it uses numbers — topology tries to get beyond numbers, and although precise, the 202 definitions I've written down as I've gone through Lee to this point don't mention them for the most part.
Essentially topology reasons about our concept of space qualitatively, rather than quantitatively. In this, it resembles philosophy which uses a similar sort of qualitative reasoning to get at what are basically rather nebulous concepts — knowledge, truth, reality. As a neurologist, I can tell you that half the cranial nerves, and probably half our brains are involved with vision, so we automatically have a concept of space (and a very sophisticated one at that). Topologists are mental Lilliputians trying to tack down the giant Gulliver which is our conception of space with definitions, theorems, lemmas etc. etc.
Well one form of space anyway. Urysohn talks about normal spaces. Just think of a closed set as a Russian Doll with a bright shiny surface. Remove the surface, and you have a rather beat up Russian doll — this is an open set. When you open a Russian doll, there's another one inside (smaller but still a Russian doll). What a normal space permits you to do (by its very definition), is insert a complete Russian doll of intermediate size, between any two Dolls.
This all sounds quite innocent until you realize that between any two Russian dolls an infinite number of concentric Russian dolls can be inserted. Where did they get a weird idea like this? From the number system of course. Between any two distinct rational numbers p/q and r/s where p, q, r and s are whole numbers, you can always insert a new one halfway between. This is where the infinite regress comes from.
For mathematics (and particularly for calculus) even this isn't enough. The square root of two isn't a rational number (one of the great Euclid proofs), but you can get as close to it as you wish using rational numbers. So there are an infinite number of non-rational numbers between any two rational numbers. In fact that's how non-rational numbers (aka real numbers) are defined — essentially by fiat, that any series of real numbers bounded above has a greatest number (think 1, 1.4, 1.41, 1.414, defining the square root of 2).
What does this skullduggery have to do with space? It says essentially that space is infinitely divisible, and that you can always slice and dice it as finely as you wish. This is the calculus of Newton and the relativity of Einstein. It clearly is right, or we wouldn't have GPS systems (which actually require a relativistic correction).
But it's clearly wrong as any chemist knows. Matter isn't infinitely divisible, Just go down 10 orders of magnitude from the visible and you get the hydrogen atom, which can't be split into smaller and smaller hydrogen atoms (although it can be split).
It's also clearly wrong as far as quantum mechanics goes — while space might not be quantized, there is no reasonable way to keep chopping it up once you get down to the elementary particle level. You can't know where they are and where they are going exactly at the same time.
This is exactly one of the great unsolved problems of physics — bringing relativity, with it's infinitely divisible space together with quantum mechanics, where the very meaning of space becomes somewhat blurry (if you can't know exactly where anything is).
Interesting isn't it?
In the grad school course on abstract algebra I audited a decade or so ago, the instructor began the discussion about tensors by saying they were the hardest thing in mathematics. Unfortunately I had to drop this section of the course due a family illness. I've written about tensors before and their baffling notation and nomenclature. The following is yet another way to look at them which may help with their confusing terminology
First, this post will assume you have a significant familiarity with linear algebra. I've written a series of posts on the subject if you need a brush up — pretty basic — here's a link to the first post — https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/
All of them can be found here — https://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/.
Here's another attempt to explain them — which will give you the background on dual vectors you'll need for this post — https://luysii.wordpress.com/2015/06/15/the-many-ways-the-many-tensor-notations-can-confuse-you/
To the physicist, tensors really represent a philosophical position — e.g. there are shapes and processes external to us which are real, and independent of the way we choose to describe them mathematically. E. g. describing them by locating their various parts and physical extents in some sort of coordinate system. That approach is described here — https://luysii.wordpress.com/2014/12/08/tensors/
Zee in one of his books defines tensors as something that transforms like a tensor (honest to god). Neuenschwander in his book says "What kind of a definition is that supposed to be, that doesn't tell you what it is that is changing."
The following approach may help — it's from an excellent book which I've not completely gotten through — "An Introduction to Tensors and Group Theory for Physicists" by Nadir Jeevanjee.
He says that tensors are just functions that take a bunch of vectors and return a number (either real or complex). It's a good idea to keep the volume tensor (which takes 3 vectors and returns a real number) in mind while reading further. The tensor function just has one other constraint — it must be multilinear (https://en.wikipedia.org/wiki/Multilinear_map). Amazingly, it turns out that this is all you need.
Tensors are named by the number of vectors (written V) and dual vectors (written V*) they massage to produce the number. This is fairly weird when you think of it. We don't name sin (x) by x because this wouldn't distinguish it from the zillion other real valued functions of a single variable.
So an (r, s) tensor is named by the ordered array of its operands — (V, …V,V*, …,V*) with r V's first and s V* next in the array. The array tells you what the tensor function must be.
How can Jeevanjee get away with this? Amazingly, multilinearity is all you need. Recall that the great thing about the linearity of any function or operator on a vector space is that ALL you need to know is what the function or operator does to the basis vectors of the space. The effect on ANY vector in the vector space then follows by linearity.
Going back to the volume tensor whose operand is (V, V, V) and the vector space for all 3 V's (R^3), how many basis vectors are there for V x V x V ? There are 3 for each V meaning that there are 3^3 = 27 possible basis vectors. You probably remember the formula for the volume enclosed by 3 vectors (call them u, v, w). The 3 components of u are u1 u2 and u3.
The volume tensor calculates volume by ( U crossproduct V ) dot product W.
Writing the calculation out
Volume = u1*v2*w3 – u1*v3*w2 + u2*v3*w1 – u2*v1*w3 + u3*v1*w2 – u3*v2*w1. What about the other 21 combinations of basis vectors? They are all zero, but they are all present in the tensor.
While any tensor manipulating two vectors can be expressed as a square matrix, clearly the volume tensor with 27 components can not be. So don't confuse tensors with matrices (as I did).
Note that the formula for volume implicitly used the usual standard orthogonal coordinates for R^3. What would it be in spherical coordinates? You'd have to use a change of basis matrix to (r, theta, phi). Actually you'd have to have 3 of them, as basis vectors in V x V x V are 3 places arrays. This gives the horrible subscript and superscript notation of matrices by which tensors are usually defined. So rather than memorizing how tensors transform you can derive things like
T_i'^j' = (A^k_i')*(A^k_i') * T_k^l where _ before a letter means subscript and ^ before a letter means superscript and A^k_i' and A^k_i' are change of basis matrices and the Einstein summation convention is used. Note that the chance of basis formula for tensor components for the volume tensor would have 3 such matrices, not two as I've shown.
One further point. You can regard a dual vector as a function that takes a vector and returns a number — so a dual vector is a (1,0) tensor. Similarly you can regard vectors as functions that take dual vectors and returns a number, so they are (0,1) tensors. So, actually vectors and dual vectors are tensors as well.
The distinction between describing what a tensor does (e.g. its function) and what its operands actually are caused me endless confusion. You write a tensor operating on a dual vector as a (0, 1) tensor, but a dual vector is a (1,0) considered as a function.
None of this discussion applies to the tensor product, which is an entirely different (but similar) story.
Hopefully this helps
Mathematical talent varies widely. It was a humbling thing a few years ago to sit in an upper level college math class on Abstract Algebra with a 16 year old high school student taking the course, listening with one ear while he did his German homework. He was later a double summa in both math and physics at Yale. So do mathematicians think differently? A fascinating paper implies that they use different parts of their brain doing math than when not doing it. The paper has one methodological flaw — see if you can find it.
[ Proc. Natl. Acad. Sci. vol. 113 pp. 4887 – 4889, 4909 – 4917 '16 ] 15 expert mathematicians and 15 nonMathematicians with comparable academic qualifications were studied (3 literature, 3 history 1 philosophy 2 linguistics, 1 antiquity, 3 graphic arts and theater 1 communcation, 1 heritage conservation — fortunately no feminist studies). They had to listen to mathematical and nonMathematical statements and decide true false or meaningless. The nonMathematical statements referred to general knowledge of nature and history. All this while they were embedded in a Magnetic Resonance Imager, so that functional MRI (fMRI) could be performed.
In mathematicians there was no overlap of the math responsive network (e.g. the areas of the brain activated by doing math) with the areas activated by sentence comprehension and general semantic knowledge.
The same brain networks were activated by all types of mathematical statement (geometry, analysis, algebra and topology) as opposed to nonMathematical statement. The areas activated were the dorsal parietal, ventrolateral temporal and bilateral frontal. This was only present in the expert mathematicians (and only to mathematical statements) These areas are outside those associated with language (inferior frontal gyrus of the left hemisphere). The activated areas are also involved in visual processing of arabic numbers and simple calculation. The activated areas in mathematicians were NOT those related to language or general knowledge.
So what's wrong with the conclusion? The editorialist (pp. 4887 – 4889) pointed this out but I thought of it independently.
All you can say is that experts working in their field of expertise use different parts of their brain than they use for general knowledge. The nonMathematicians should have been tested in their field of expertise. Shouldn't be hard to do.
Have you ever had the pleasure of taking a course from someone who wrote the book? I did. I audited a course at Amherst from Prof. David Cox who was one of three authors of "Ideals, Varieties and Algorithms" It was uncanny to listen to him lecture (with any notes) as if he were reading from the book. It was also rather humbling to have a full professor correcting your homework. We had Dr. Cox for several hours each weak (all 11 or 12 of us). This is why Amherst is such an elite school. Ditto for Princeton back in the day, when Physics 103 was taught by John Wheeler 3 hours a week. Physics 103 wasn't for the high powered among us who were going to be professional physicists (Heinz Pagels, Jim Hartle), it was for preMeds and engineers.
Dr. Cox had one very useful pedagogical device — everyone had to ask a question at the beginning of class, Cox being of the opinion that there is no such thing as a dumb question in math.
Well Dr. Cox and his co-authors (Little and O'Shea) got an award from the American Mathematical sociecty for their book. There's an excerpt below. You should follow the link to the review to see what the three look like along with two other awardees. http://www.ams.org/publications/journals/notices/201604/rnoti-p417.pdf. Go to any midsize American city at lunchtime, and you'd be hard pressed to pick four of the five out of the crowd of middle aged men walking around. Well almost — one guy would be hard to pick out of the noonday crowd in Williamsburg Brooklyn or Tel Aviv. Four are extremely normal looking guys, not flamboyant or bizarre in any way. This is certainly true of the way Dr. Cox comports himself. The exception proving the rule however, is Raymond Smullyan who was my instructor in a complex variables course back in the day– quite an unusual and otherworldly individual — there's now a book about him.
Here's part of the citation. The link also contains bios of all.
"Even more impressive than its clarity of exposition is the impact it has had on mathematics. CLO, as it is fondly known, has not only introduced many to algebraic geometry, it has actually broadened how the subject could be taught and who could use it. One supporter of the nomination writes, "This book, more than any text in this field, has moved computational algebra and algebraic geometry into the mathematical mainstream. I, and others, have used it successfully as a text book for courses, an introductory text for summer programs, and a reference book."
Another writer, who first met the book in an REU two years before it was published, says, "Without this grounding, I would have never survived my first graduate course in algebraic geometry." This theme is echoed in many other accounts: "I first read CLO at the start of my second semester of graduate school…. Almost twenty years later I can still remember the relief after the first hour of reading. This was a math book you could actually read! It wasn't just easy to read but the material also grabbed me."
For those with a taste for statistics, we note that CLO has sold more than 20,000 copies, it has been cited more than 850 times in MathSciNet, and it has 5,000 citations recorded by Google Scholar. However, these numbers do not really tell the story. Ideals, Varieties, and Algorithms was chosen for the Leroy P. Steele Prize for Mathematical Exposition because it is a rare book that does it all. It is accessible to undergraduates. It has been a source of inspiration for thousands of students of all levels and backgrounds. Moreover, its presentation of the theory of Groebner bases has done more than any other book to popularize this topic, to show the powerful interaction of theory and computation in algebraic geometry, and to illustrate the utility of this theory as a tool in other sciences."
I'm been through the first 200 pages of Dill's Book "Molecular Driving Forces (2003)" which is all about thermodynamics and statistical mechanics, things that must be understood to have any hope of understanding cellular biophysics. There are a lot of variables to consider (with multiple names for some) and they fall into 7 non mutually exclusive types.
Here they are with a few notes about them
l. Thermodynamic State Variables: These are the classics — Entropy (S), Internal Energy (U), Helmholtz Free Energy (F), Gibbs Free Energy (G), Enthalpy (H).
All are continuous functions of their Natural Variables (see next) so they can be differentiated. Their differentials are exact.
2. Natural variable of a thermodynamic state variable — these are defined as continuous variables which when an extremum (maximum, minimum) of the state variable using them is found, the state function won't change with time (e.g. is at equilibrium). Here they are for the 5 state functions. T is Temperature, V is Volume, N is number of molecules, S and U are what you think, and p is pressure
State Name State Function Natural Variables
Helmholtz Free Energy— F —T, V, N
Entropy —S —U, V, N
Internal Energy —U — S, V, N
Gibbs Free Energy— G —T. p, N
Enthalpy — H —S, p, N
Note that U and S are both state variables and natural variables of each other. Note also (for maximum confusion) that Helmholtz free energy is not H but F, and that H is Enthalpy not Helmholtz Free energy
3. Extensive variable –can only be characterized by how much there is of it. This includes all 5 thermodynamic state variables (F, S, U, G, H) alone with V volume, and N number of molecules. Extensive variables are also known as degrees of freedom.
4. Intensive variable — temperature, pressure, and ratios of State and Natural variables (actually the derivative of a state variable with respect to a natural variable — temperature is actually defined this way ( partial U / partial S)
5. Control variables — these are under the experimenter's control, and are usually kept constant. They are also known as constraints, and most are intensive (volume isn't). Examples constant temperature, constant volume, constant pressure
6. Conjugate variables. Here we need the total differential of a state variable (which exists for all) in terms of its natural variables to under stand what is going on.
Since U is a continuous function of each of S, V, and N
we have
dU = (partial U/ partial X) dS + (partial U / partial V) dV + (partial u / partial N ) dN
= T dS – p dV – mu dN ; mu is the chemical potential
So T is conjugate to S, p is conjugate to V, and mu is conjugate to N ; note that each pair of conjugates has one intensive variable (T, p, mu) and one extensive one ( S, V, N). Clearly the derivatives ( T, p, mu) are intensive.
7. None of the above — work and heat (q)
Thermodynamics can be difficult to master unless these are clear. Another reason is that what you really want is to maximize (S) or minimize (U, H, F, G) state variables — the problem is you have no way to directly measure the two crucial ones you really want (U, S) and have to infer what they are from various derivatives and control variables. You can measure changes in S and U between two temperatures by using heat capacities. That's just like spectroscopy, where all you measure is the difference between energy levels, not the energy levels themselves. But it is the minimum values of U, G, H, F and maximum values of S which determine what you want to know.
There's more to come about Dill's book. I've found a few mistakes and have corresponded with him about various things that seem ambiguous (to me at least). As mentioned earlier, in grad school 44 years ago, I audited a statistical mechanics course taught by E. Bright Wilson himself. I never had the temerity to utter a word to him. How things have changed for the better, to be able to Email an author and get a response. He's been extremely helpful and patient.


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|Property Type: Detached Single Family
|Year Built:
|Square Feet:
|6 Full/2 Half
|HOA Fees:
|Not available
|Lot Size:
|0.2 acres
|Lot Dimensions:
Subdivision: No Recorded Project
Located on the beautiful Panama City Beach. This new construction home is a 6 bedroom 6 bathroom home with a private pool. This home will sleep 18 guest. Sit and relax on the beautiful decks this home has to offer. Every level offers stunning views of the beach a perfect place to sit watch the tide roll in. Located near the famous Pineapple Willies and all the great local Panama City Beach hot spots.
Waterview: Gulf
Waterfront: Gulf
Parking Features: Carport, Carport Attached
Utilities: Electric, Phone, Public Sewer, Public Water
Energy: AC - Central Elect, Heat Cntrl Electric
Lot/Parcel Access: City Road
Zoning: Resid Single Family
|Elementary School:
|Breakfast Point
|Middle School:
|Breakfast Point
|High School:
Here you can request more information about this listing or schedule a showing. Simply fill out this short form with your name, email address, and phone number (optional), as well as your question, and a response will be returned to you shortly.
Items marked with a red star (*) are required.
This listing provided courtesy of Counts Real Estate Group Inc.
This information was last updated on 4/19/2018 8:49:40 AM.
All information in this listing is approximate. Buyers are expected to verify any specific information prior to signing any contract. Neither Brandon Jordan, ERA American Real Estate, ECAR, nor any other Realtor is liable for any inaccuracies herein.
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text: Disasters and Accidents Disasters and Accidents Conductor, Passenger From Derailed Amtrak Train Allege Inadequate Training In Lawsuits By Tom Banse • Jan 3, 2018 Austin Jenkins / Northwest News Network Listen Listening... / 1:07 A conductor who was at the front of the Amtrak train that derailed north of Olympia is alleging in a lawsuit filed Wednesday that the crew received inadequate training. An injured passenger made similar claims in a separate lawsuit filed hours later. The December derailment killed three people, seriously injured scores more and closed southbound Interstate 5 for two-and-a-half days. Landslide Threat Leads To Evacuations Near Rattlesnake Ridge By Anna King • Jan 3, 2018 Anna King / Northwest News Network Listen Listening... / 0:55 Officials in Yakima County, Washington, are strongly urging residents living below a shifting mountainside near Union Gap to evacuate. A huge crack that appeared on Rattlesnake Ridge last year is beginning to widen. Railway Curve Where Amtrak Train Wrecked Once Considered For Straightening By Tom Banse • Dec 22, 2017 WSDOT Listen Listening... / 1:06 The curve where an Amtrak train derailed on Monday killing three people was once on a list to be straightened. But that project was dropped from the list of improvements eventually made due to its high cost. Automatic Braking Tech To Be Operational Before Trains Resume Derailment Route By Tom Banse • Dec 22, 2017 WSDOT / Flickr Listen Listening... / 1:07 Washington state's Department of Transportation has unilaterally decided on a pre-condition before passenger trains resume service on the Point Defiance Bypass. That's where Monday's deadly derailment happened. Amtrak Train Derailment Survivors Give Notice Of Lawsuit By Tom Banse • Dec 21, 2017 Austin Jenkins / Northwest News Network Listen Listening... / 1:07 Six survivors of Monday's train derailment south of Tacoma have put Amtrak on notice that they plan to file a lawsuit. It's the first, probably inevitable, sign that various parties affected by the deadly train wreck are lawyering up. Freeway Closed By Deadly Train Derailment In Washington State Reopens By Tom Banse • Dec 20, 2017 WSDOT Listen Listening... / 1:06 Southbound Interstate 5 where Monday's Amtrak derailment sent train cars crashing down onto the freeway has re-opened. This comes as a big relief to drivers who endured hours-long detours or postponed travel between the Seattle-Tacoma area and points south. Part Of I-5 Closed By Train Derailment To Reopen Wednesday Evening By Tom Banse • Dec 20, 2017 Tom Banse / Northwest News Network Listen Listening... / 1:00 The Washington State Department of Transportation just announced that two through lanes of Interstate 5 will reopen around 5 p.m. Wednesday at the scene of Monday's deadly train derailment south of Tacoma. The southbound freeway has been closed for two-and-a-half days. NTSB Says Engineer Of Derailed Amtrak Train Had Practiced Route Before By Tom Banse • Dec 19, 2017 Tom Banse / Northwest News Network Listen Listening... / 1:01 Federal investigators from the National Transportation Safety Board expect to stay at least another week in Western Washington gathering evidence from Monday's deadly Amtrak train derailment. Investigators have not yet interviewed the hospitalized train engineer. After Two Amtrak Derailments, Pierce County Executive Has 'Serious Questions' By Austin Jenkins • Dec 19, 2017 Austin Jenkins / Northwest News Network Listen Listening... / 1:09 Monday's deadly train accident near DuPont, Washington, was the second derailment of an Amtrak Cascades in five months. That's raising questions about the safety of the popular commuter rail line that runs between Vancouver, BC, and Eugene, Oregon. Inslee Vows To Make Passenger Rail Service Safe By Austin Jenkins • Dec 19, 2017 Austin Jenkins / Northwest News Network Listen Listening... / 0:35 Washington Gov. Jay Inslee is vowing to make passenger rail service safer. The Democrat made his comments Tuesday at the scene of Monday's derailment of an Amtrak Cascades train in Pierce County. Pages« first ‹ previous 1 2 3 4 5 6 7 8 9 … next › last »


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text: Thïêñ Thïêñ is 25 years old male from Dung, Viet Nam. He is interested in meet new people. 25, male, Can Tho Likes: 5x Facebook profile Who s/he liked? Send message Favorite May she/he ask you for a date? Never Maybe Sure Rocky Rock Good food Battlefield 3 Giorgio Armani Pasta 80's Tennis Sex on the Beach Books Relationship: ShowSexuality: ShowHeight: ShowWeight: ShowBody type: ShowEye color: ShowHair color: ShowLiving: ShowKids: ShowSmoking: ShowDrinking: Show Report photo Sign up to see all photos Facebook Twitter Google+ Pinterest Similar users Adrian 22, Columbia 40 Miguel 21, Longview 23 Xbox 360 Brandon 24, Fresno 64 Dean 27, Olive Branch 88 chav 26, Mount Hope 42 Jared 22, Decatur 122 Toñito 25, Apopka 224 Ian 28, Orlando 109 Marqual 23, Quarryville 61


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There either is or is not, that's the way things are. The colour of the day. The way it felt to be a child. The saltwater on your sunburnt legs. Sometimes the water is yellow, sometimes it's red. But what colour it may be in memory, depends on the day. I'm not going to tell you the story the way it happened. I'm going to tell it the way I remember it.
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The holiday season is bringing more than cheer for actress Jessica Capshaw. The Grey's Anatomy star announced on Monday that she and her husband, Honest Company co-founder Christopher Gavigan, are expecting their fourth child. "The holidays have certainly arrived … All of our lives are being added to by the lights, the music and cheer!!" she captioned a photo of her family at a Christmas tree farm on Instagram. "This year our family has reason to further broaden our smile … baby number four is on the way!! I am a lucky, lucky girl…"
The quickest way to tell whether an actor is serious about his or her career is to look at their "headshot".Is it a professional photo? Or is it a snapshot taken on prom night? If you're using a snapshot, it's like showing up to race the Indy 500 in a VW Beetle. Nobody is going to take you seriously. I'm sorry, people… you can send me as much angry e-mail as you want about this, but it's true! Don't tell me you're destined to be a major Hollywood star but you cannot scrape together the cash to get professional headshots.
The Grinch from Dr. Seuss' How the Grinch Stole Christmas! The Musical stole the spotlight at the Empire State Building's 86th Floor Observatory December 14, 2012, to promote the musical's limited holiday engagement. Discover the magic of Dr. Seuss' classic holiday tale as it comes to life on stage. Featuring the hit songs "You're A Mean One, Mr. Grinch" and "Welcome Christmas," The Grinch discovers there's more to Christmas then he bargained for in this heart-warming holiday classic.
All actors and actress | all rights reserved | photos by googler | design by en000t


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Strategic acquisitions in a burgeoning e-commerce sector in India are expected to up the ante higher for retail companies. Amid intense speculation an
Through initial coin offerings (ICO), startups across the world have raised nearly US$4 bn by issuing digitized tokens in place of funding via crypto
At the opening keynote of CES 2018, Intel treated the audience with a remarkable range of data-driven innovations. Intel highlighted how data is bring
Apple plans to make a shift from the Intel chips it had been using until now to chips manufactured by Apple itself. A similar shift had taken place in
Qualcomm, a U.S.-based semiconductor company, unveiled a US$44 bn agreement targeted to acquire NXP Semiconductors on Tuesday, which is said to be its


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This is my video on Dark Beasts, i show my set-up and method of killing them alongside showing loot from 1 prestige! (60,000) Then I go into an xp and gp breakdown at the end of the video for the time i spent doing this! Enjoy
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text:
Definition of 'epergne'
an ornamental centrepiece for a table: a stand with holders for sweetmeats, fruit, flowers, etc
Collins English Dictionary. Copyright © HarperCollins Publishers
Word origin of 'epergne'
Webster's New World College Dictionary, 4th Edition. Copyright © 2010 by Houghton Mifflin Harcourt. All rights reserved.
Word origin of 'epergne'
Word Lists
Trends of 'epergne'
Used Rarely. epergne is in the lower 50% of commonly used words in the Collins dictionary
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Nearby words of 'epergne'
Definition of epergne from the Collins English Dictionary
Fed up with the great British weather? Here's how to afford that dream summer escape
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The Aquaculture Market in Vietnam to Grow at a CAGR of 5.73% during the period 2016-2020.
Aquaculture Market in Vietnam Market Research Report covers the present scenario and the growth prospects of the Aquaculture Market in Vietnam Market for 2016-2020. To calculate the market size, the report considers the revenue generated from the sales of Aquaculture Market in Vietnam globally. Aquaculture Market in Vietnam Market, has been prepared based on an in-depth market analysis with inputs from industry experts. The report covers the market landscape and its growth prospects over the coming years and discussion of the key vendors effective in this market.
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Buy the BI - Microsoft acquires Datazen
Software giant buys mobile analytics firm
Microsoft has announced it has acquired business intelligence (BI) player Datazen Software, in a move it claims will bolster its BI offering. The newly acquired company - founded in 2002 and headquartered...
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4 Bedrooms Detached House for sale
Whitchurch Road, Bishopsworth, Bristol, BS13 7RT
First listed £405,000 20th March 2018
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Ultimate headshot/face plant fails | funny fail compilation
A facebuster, also known as a faceplant, is generally a takedown move in professional wrestling in which an attacking wrestler forces his/her opponent down to the mat face-first without involving a headlock or facelock. A standard facebuster, also known as a jumping facebuster, involves the wrestler grabbing hold of the opponents head / hair and dropping down to their knees, forcing the opponents face into the mat.
The attacking wrestler places an opponent in an Argentine backbreaker rack, where the opponent is held face-up across both the shoulders of the wrestler, from here the wrestler falls sideways while still holding the opponents head with one arm and flipping the opponents legs over with the other, driving them down to the mat face-first. (source :
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Susie was absolutely wonderful and patient with the children. My kids are already talking about their next party with Susie Q The Clown. I will definitely be booking her again. Thank you!
We thoroughly enjoyed having Bella at our son's 2nd birthday party. Thank you! She made the party a ton more fun for the children and even some of the adults got thier faces painted! :-) I highly recommend Bella the Clown! She's fantastic!
Lisa was great! we hired her for my daughter's birthday party and she really captured all kids attention and was willing to even stay extra few minutes to finish the magic show! would recommend her to anyone looking for a clown and balloon twisting art!
Maggie the clown was wonderful! She had all of kids and adults attention. Wonderful job on ballon animals and face painting. Highly recommend to have her at your event.
It was great having sparkles clown at our party. The kids were very happy and she kept different age groups engaged in the activities.
Met him and his pals at the Rainforest Café in Burlington. I saw kids smiling after they got their balloons and faces painted. What a guy! =)
I would love to entertain for your event! I currently entertain at a local restaurant weekly for kids night and have a wonderful time. I twist balloons, do magic, paint faces, give tattoos and more. I am a member of the Clowning for Kidz Foundation and the World Clown Association.
We cater to parties, corporates, picnics, school and library shows, childcare centers, and all events imaginable! We offer the following services: - Live, silly songs with guitar or ukulele - Comedy magic shows - Balloon twisting - Juggling and balancing - Puppetry - Ventriloquism - Fancy, full-face face painting - Storytelling ...and more! We have 25 years experience clowning around all of Connecticut, Southern Massachusetts, Western Rhode Island, and Eastern New York. We are members of Clowns of America International and insured members of World Clown Association.
Consider hiring this company if you are looking for entertainers. They have years of experience in the the children's entertainment industry. They provide quality services at fair prices.
Hire the professionals of Alfred's Partytime for reliable event entertainment services. They have a team of reputable face painters who also provide quality juggling and mind-blowing magic tricks.


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Charity Brown - Ontario
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We are a Kilmainham family run business specialising in all aspects of natural stone pavers construction.
With over 20 years experience, we have earned a big reputation for our service and quality. We only use top quality materials and our team is of a very high standard.
Whatever your requirement, a stunning new driveway, a beautiful new patio or garden seating area, we are able of offer a wide variety of colours, styles and patterns to ensure a unique finish that will not only look great, but will withstand the elements for years to come.
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Your Kilmainham, professional natural stone pavers.
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Common Sense Tips for Maintaining Your Driveway
Driveways are among the most used locations of your house, often ignored up until they stain or split. Provide the attention they are worthy of by making repair work quickly. Not just can you extend your driveway's life and appealing look well into the future, you can conserve money in the long term.
Think about these seven ideas to help you care for your driveway:
1. Avoid cracks
For driveways, avoidance means sealing concrete yearly and sealing asphalt every few years to prevent water from permeating, freezing and splitting. Tree and shrub roots can trigger cracks by rising from underneath. The solution is to get rid of trees near drives or have their roots cut away.
2. Fill cracks
If cracks or holes do appear, patch them immediately. Utilize a masonry chisel to get rid of loose products and brush out the particles before using a crack filler and patching compound. After the substance dries, seal the whole driveway Guaranteeing your driveway is smooth and level may also help in reducing tripping accidents when visitors visit your home.
3. Reduce water on the driveway.
Clear a two to three-inch strip around the edges of the driveway to offer a runoff area for snow and water to minimize its opportunities of permeating the surface. Also, make sure that downspouts empty into the backyard rather than onto the driveway.
4. Do not salt cement
Cement salt and chemical de-icers can penetrate surfaces, triggering cracks. Both should be repaired. Instead of salting, utilize a snowblower or shovel to eliminate snow and utilize options like sand, coffee premises, alfalfa meal or cat litter to make surface areas less slippery. To melt ice, consider spraying the surface with sugar beet juice, which reduces the melting point of ice and is considered efficient in temperatures as low at -20 ° F.
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Raise the blade of your snow rake enough to ensure it does not scrape and damage the driveway surface. If shoveling, use a plastic shovel for irregular surfaces to decrease its danger of capturing on the driveway. Then, either push snow to the side and shovel that off or remove snow a couple of inches at a time while working your way earthward.
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7. Clean your driveway
Removing spots caused by motor oil, radiator fluid and similar substances affects more than your drive's appearance. Motor fluids can penetrate up to a quarter inch in concrete and can soften asphalt. Use non-clumping cat litter to soak up fresh oil, followed by grease-cutting dishwashing cleaning agent and warm water. For older stains, try scrubbing the stain consistently with grease-cutting, biodegradable cleaners. Be sure to utilize a regular scrub brush. A wire brush can damage the driveway's surface area.
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Useful Links: Garden & Landscape Designers Association, The National Gardening Association, Royal Horticultural Society of Ireland.
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From furniture to flatware, all orders over $49.00 ship free* 2-Day Delivery on Best Sellers
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Features: -Give a Hoot collection.-Made of 100% cotton.-Machine washable.-Product Type: Hand towel.-Color: White.-Material: 100% Cotton.-Hypoallergenic: No.-Antimicrobial: No.-Organic: No.-Pill Resist.. See Full Details »
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1869 10C J-710, BNClick the image to load the highest resolution version.
1870 Copper Pattern Half Dollar, Judd-9351870 Half Dollar, Judd-935, Pollock-1041, R.7, PR 63 Red and Brown PCGS. A seated figure of Liberty faces left on the obverse with 13 stars surrounding. The date is below, a free-standing liberty pole is behind, and LIBERTY is inscribed on a scroll crossing a small shield. The reverse is the same type that was used to coin regular-issue Seated halves. Struck in copper with a reeded edge. Both sides exhibit a slightly subdued appearance with deep brick-red colorations. A few small toning spots are also noted overall. (NGC ID# 29RR, PCGS# 60935)
Learn more at the Newman Numismatic Portal at Washington University in St. Louis.
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NEW YORK (CBSNewYork/AP) — Deron Williams returned to the Brooklyn Nets in a reserve role and Pablo Prigioni started for the Knicks as both New York teams got point guards back Monday.
Williams sat out the past five games after receiving treatment on his ankles. He has missed 16 games this season because of ankle problems.
Coach Jason Kidd kept Shaun Livingston as his starter so Williams could work his way back into shape off the bench.
Prigioni has missed 16 games since breaking his right big toe Dec. 18. Coach Mike Woodson paired him with Raymond Felton in a two point-guard lineup that was effective last season but hasn't been used much in this one.
You May Also Be Interested In These Stories
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- Andujar, Gregorius Lead Yankees To Big Win Over Twins
(TM and © Copyright 2014 CBS Radio Inc. and its relevant subsidiaries. CBS RADIO and EYE Logo TM and Copyright 2014 CBS Broadcasting Inc. Used under license. All Rights Reserved. This material may not be published, broadcast, rewritten, or redistributed. The Associated Press contributed to this report.)


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Lakeview Holiday Cottages
Looking to stay in a holiday cottage with a lake view? Country Cottages Online feature a great range of lakeside and lakeview holiday cottages where you can gaze out over the water during a self-catering holiday.
Wake up to a Cottage with a Lake View
Waking up in a holiday cottage with a lake view can be such a joy! Just imagine sitting out and enjoying a morning coffee whilst gazing out over the glistening waters of a lake and watching ducks happily paddle around or the fish jump. Lake views can be equally appealing of an evening where you can watch the sunset over the water.
Lakeview Cottages throughout the UK
Lakeview holiday cottages aren't just found in famous regions like the English Lake District and instead can be found throughout the UK. There are numerous holiday developments where lodges or cottages are centred around one big fishing lake or a small collection of fishing lakes in areas like Lincolnshire, East Anglia and the South West amongst others.
Wherever you choose to holiday in the UK or Ireland, we hope you find the best holiday cottage with a lake view for you.
Lake View, West Down, Devon, England
3 bedroom Holiday Bungalow, West Country.
Stylish 3 bedroom country cottage with countryside and lake views, with fishing lakes on site, relax and unwind in the North Devon countryside, just 15 minutes' drive from Woolacombe beach.
Sleeps from 2 to 6 people in 3 bedrooms
Number of bathrooms: 2
Pet Friendly: No
Rating: Not assessed
|from 30 April 2018 to 04 May 2018; from 07 May 2018 to 21 May 2018; from 29 June 2018 to 13 July 2018;
Kate's Cabin, Eshott, Northumberland, England
2 bedroom Pine Lodge, North England.
Beautiful 2 bedroom pet-friendly log cabin with lake views nestled on Felmoor Park in the Northumberland countryside.
Special features: Gold Award Winning Cottage.
Sleeps from 2 to 4 people in 2 bedrooms
Number of bathrooms: 1
Pet Friendly: Yes (2 dogs welcome)
Quality Rating: (4)
High View Park Lodge, White Cross Bay, Cumbria, England
3 bedroom Wooden Holiday Chalet, Lake District National Park.
Beautiful 3 bedroom Cumbrian holiday lodge with distant lake views, set next to woodlands close to Lake Windermere, one of the best areas in the Lake District National Park, shared leisure facilities on site include an indoor swimming pool, Jacuzzi, relaxing sauna and gym.
Special features: free internet access, pool ().
Sleeps from 2 to 6 people in 3 bedrooms
Number of bathrooms: 2
Pet Friendly: No
Quality Rating: (5)
|from 30 April 2018 to 18 May 2018; from 25 May 2018 to 26 May 2018; from 01 June 2018 to 04 June 2018;
Lake View Holiday Barn, Crediton, Devon, England
3 bedroom Converted Barn, West Country.
With a lake view as well as stunning countryside views out to Dartmoor National Park, this 3 bedroom barn conversion features a modern interior and is conveniently located for the Tarka Trail, visiting Exeter and exploring this gorgeous area of rural Devon.
Special features: free internet access.
Sleeps from 2 to 5 people in 3 bedrooms
Number of bathrooms: 1
Pet Friendly: No
Quality Rating: (5)
|from 27 April 2018 to 13 May 2018; from 17 May 2018 to 25 May 2018; from 01 June 2018 to 29 June 2018;
Lake View House, Farndon, Cheshire, England
5 bedroom Rural Retreat, Heart of England.
Ideal for groups of friends or family, this 5 bedroom holiday house boasts idyllic lake views over the coarse fishing lake in the garden, plus enjoys access to the excellent leisure facilities at the neighbouring caravan park where there is an outdoor swimming pool, a games room with pool and table tennis, plus a tennis court, all just 10 miles from historic Chester.
Special features: free internet access, tennis court, games room.
Sleeps from 4 to 10 people in 5 bedrooms
Number of bathrooms: 3
Pet Friendly: Yes (1 dog welcome)
Quality Rating: (5)
|from 30 April 2018 to 04 May 2018; from 07 May 2018 to 11 May 2018; from 11 June 2018 to 15 June 2018;
Bullrush Lodge, Tattershall, Lincolnshire, England
2 bedroom Log Cabin, East Midlands.
With lake views and a generous decked veranda, this 2 bedroom holiday lodge is a great place to relax by the waterside, around mid-way between Lincoln and Boston, and there are some excellent seasonal shared leisure facilities on site including an indoor swimming pool.
Special features: pool ().
Sleeps from 3 to 6 people in 2 bedrooms
Number of bathrooms: 2
Pet Friendly: No
Quality Rating: (4)
|from 27 April 2018 to 12 May 2018; from 19 May 2018 to 26 May 2018; from 02 June 2018 to 16 June 2018;
Lakeview Lodge, Melton Mowbray, Leicestershire, England
2 bedroom Wooden Holiday Lodge, The Vale of Belvoir.
Get away from it all for a fishing holiday or a relaxing rural retreat break to this stylish 2 bedroom wooden holiday lodge in the Vale of Belvoir, just 15 minutes drive from Melton Mowbray, this lodge with its lake views over the fishing lake and south facing veranda is a real treat.
Sleeps from 1 to 5 people in 2 bedrooms
Number of bathrooms: 1
Pet Friendly: No
Rating: Not assessed
Sunridge TreeHouse, Yealmpton, Devon, England
3 bedroom Holiday Treehouse, West Country.
Luxurious Devon holiday treehouse with superb woodland and lake views and a superb modern interior, complete with luxury hot tub, elevated deck and private fishing lake, new and available from April 2015.
Special features: free internet access, games room, outdoor hot tub.
Sleeps from 1 to 6 people in 3 bedrooms
Number of bathrooms: 2
Pet Friendly: Yes (6 dogs welcome)
Quality Rating: (5)


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