analyse_freenrg is an analysis app that has been designed to analyse the output of all free energy calculations in Sire. analyse_freenrg reads in a Sire Saved Stream (.s3) file that contains a list of Sire.Analysis free energy objects (e.g. FEP, TI, Bennetts). analyse_freenrg will average and analyse these free energies according the to options you supply, e.g. assuming that the free energies are stored in freenrgs.s3, and you want to average iterations 100-200 from the simulation, and write the results to ‘results.txt’, type;
sire.app/bin/analyse_freenrg -i freenrgs.s3 -r 100 200 -o results.txt
Alternatively, if you just want to average over the last 60% of iterations, type;
sire.app/bin/analyse_freenrg -i freenrgs.s3 -o results.txt
(you can specify the percentage to average using the ‘–percent’ option)
analyse_freenrg automatically knows how many free energies are contained in the s3 file, what their types are and what should be done to analyse the results. For example, the waterswap, ligandswap and quantomm apps all output s3 files that contain FEP, Bennetts and TI free energy data, so analyse_freenrg knows automatically to perform FEP, Bennetts and TI analysis on that data and to report all of the results. analyse_freenrg also knows whether or not finite difference approximations have been used, whether forwards and backwards windows were evaluated, the temperature and conditions of the simulation etc. The aim is that it should handle everything for you, so you can concentrate on looking at the potential of mean force (PMF) or final result.
For FEP data, analyse_freenrg will return the FEP PMF across lambda, together with errors based on statistical convergence (95% standard error) and the difference between forwards and backwards free energies (if available).
For Bennetts data, analyse_freenrg will return the Bennetts Acceptance Ratio PMF across lambda, with errors based on statistical convergence (95% standard error).
For TI data, analyse_free energy will return the PMF across lambda based on polynomial fitting of the gradients and analytic integration of the resulting function. It will also return the integral across lambda using simple quadrature. Errors are based on statistical convergence (95% standard error) and on the difference between the forwards and backwards finite difference gradients (if available, and if finite-difference TI was used).
If you need more help understanding or interpreting the results of an analyse_freenrg analysis then please feel free to get in touch via the Sire users mailing list, or by creating a github issue.


At St Gregory’s, across Years 1 to 6, we use the Maths – No Problem! programme. It is a highly structured approach to teaching maths and deepens the understanding of all children. Each lesson is based around an ‘anchor task’ which the children explore using different methods. The children are encouraged to talk about their maths and explore their ideas fully, then work independently using the workbooks.
Maths — No Problem! is a series of textbooks and workbooks written to meet the requirements of the 2014 English National Curriculum. The Maths — No Problem! Primary Series was assessed by the DfE’s expert panel, which judged that it alone met the core criteria for a high-quality textbook to support teaching for mastery maths. As a result, the Maths — No Problem! Primary Series is the only recommended textbook for schools on the mastery programme.
Teaching maths for mastery is a transformational approach to maths teaching which stems from high performing Asian nations such as Singapore. When taught to master maths, children develop their mathematical fluency without resorting to rote learning and are able to solve non-routine maths problems without having to memorise procedures.
This approach may be very different to the way in which you were taught yourself. To help you support your child at home, please click on the ‘Parent Videos’ below, which will give you an insight into some of the mathematical approaches.
You can find out more about your child’s year group and what they are learning in maths by clicking on the links to the relevant schemes of work below.
We are very excited to be part of this mathematical journey!


let's think about thewaves for the electron. so here i've written outa wave, which is just the same form as the wavesi had for the photon. now let's think a little bitabout the negative energy solutions. if i let the energygo to minus energy, so i turn a positive solutioninto a negative solution. then you see that if ialso take the time to minus
the time, then, thisway it doesn't change. this changes sign, this changessign, nothing changes all together. so what this tellsus, straight away, is that these negative energysolutions are the same thing as if you have an electron. normally, electronstravel forward in time, for the positiveenergy solutions. but here, for thenegative energy solution,
t goes to minus t. so a negative energy solutionis an electron travelling backwards in time. that sounds a little bitlike science fiction, but i'll hope to convinceyou by the end of the lecture that actually, it's not. this is actuallywhat really happens. this is real fact. let's put anexclamation mark there,
because it really is kind of asurprising statement to make. now let's try to write down awave equation for the electron, like we did for the photon. so an equation thatwill give those waves, and give the correctrelation between the energy and the momentum. now, for the electron,we want to be clear about which is thepositive energy piece, and which is the negativeenergy piece, because they're
separate. and so to do that,our wave equation can only have singlederivatives in it. because as soon as wehave two derivatives, then we square everythingand we lose the sign. so if we write down a waveequation for the electron, you might think that you candifferentiate with respect to x. you can differentiatewith respect to time.
if we're only going tohave single derivatives, then it should looksomething like that. then, that will then act onthe electron wave function, since this is the electron,this is presumably the positive energy solution,so i'll put a little plus on it. and let's see what happensif that is equal to zero. well, you remember a d by dxgives you a factor of momentum. d by dt gives youa factor of energy. so this equation tells youthat p is equal to e over c.
so this is indeed thepositive energy solution. and we can write down a negativeenergy one, just the same way, by changing thesign in the middle. that changes thesign of the energy. so this is now thenegative energy solution, and this corresponds top equals minus e over c, because i changed the sign here. now that's still not goodenough for our electron, because the electron has a mass.
and so the tricky thing,is how you get the mass into this equation, sothat everything works out. and the answer, is that -which you find by staring at it for a while - is that youhave to couple these two equations togetherusing the mass. so the equation forthe positive solution, when coupled to thenegative solution, is a factor of the mass (timesa factor of c over hbar, as it turns out) - sothat multiplies psi minus.
and then you do the same thing,couple the negative energy solution to the positiveenergy solution. now, why does this work? well, we have to do alittle bit of algebra. so, to solve this, let'stake this derivative that's acting on the negativesolution and act it on the equation withthe positive solution. so to write that out,i'm going to have d by dx plus 1 over c, d bydt, acting on the whole
of the left-hand side. now, if i do that, if thisacts on the left-hand side, then to keep theequation true, i have to also have this acton the right-hand side. but if this acts onthe right-hand side, this, then that's justwhat we have here, in the second equation. so this, acting onthis, is the same thing (and we have this extrafactor of m c over hbar)
- so if i write thism c over hbar here, then this is actuallyjust equal to this. and since i put an extram c over hbar here, i've got to put one here,to keep everything true. now you see, if wecompare this with this, i have a equationjust for psi plus. now if we look atwhat that equation is, then let's just multiply outthese derivatives on the top. you get a d2 by dx squaredhere, a d2 by dt squared here,
and because thesign is different, the cross terms will cancel. so we wind up with d2by dx squared, minus 1 over c squared,d2 by dt squared, acting on psi plusis equal to the thing that we have here, whichis m c over h-bar squared, acting on psi plus. and actually, if you didthis the other way around, so you started withthe minus equation,
and worked throughthe plus equation, then you would get exactly thesame equation for the minus 1. so this actually workswhether you have psi plus, or whether you havepsi minus there. now, this equation we can solve. the solutions arejust waves, just like they were for the photons. and you rememberfor the photons, when you differentiatewith respect to x,
then you get a factor p squared. when you differentiate thewaves with respect to t, you get a factor of e, sotwice you get e squared. so we get p squared minusone on c squared e squared. and then on the right-hand side,we have m squared, c squared. and that's preciselythe relationship between energy andmomentum that we want for a massiverelativistic particle, so that we wantfor our electrons.
so you can see thatthe whole thing works. the equations that istarted from, here, these linearequations, those then, there are equations for ourelectron wave functions, the positive energy one, andthe negative energy ones. and these equations werefirst discovered by dirac, in 1928 in a very famous paper. so these things are calledthe dirac equations. so, what does all thisactually mean, physically?
what it means, isthat if you have massive relativisticparticles, like electrons - so, if the particles have mass,then besides the positive energy solutions, you must also havethe negative energy solutions. you can't get away fromthem, because they're coupled together like this. so, massive particles,e.g., electrons, imply, immediately,that there must be another sort of particlewhich has the same mass,
but is different. it's the negativeenergy particle. so if we haveelectrons, then we have to have what wecall anti-electrons. this is the negativeenergy solution. so what dirac actuallypredicted here is anti-matter. just by looking atthese wave equations, and looking at making themconsistent with relativity, they have to be wavesbecause of quantum mechanics,
then the only way you canfit those two things together if the particles have mass,is if there's antimatter. so the astonishing thing is thatdirac predicted anti-matter. now, i'd like to emphasise thatthis was a genuine prediction. sometimes when people saythat things are predicted, they really mean explained. so they're predicting somethingthat they already know about. but at the time when dirac didthis calculation, essentially the calculation thatwe've just done,
nobody had a clue that therewas anything like anti-matter. it seemed a fantastic thing,science fiction if you like - well it wasn'teven science fiction, no one had even thought of it. and so this was areal prediction. it was a prediction thatsomething strange should happen, that nobody hadever guessed would happen, and of course, at the time,nobody really believed it. now so far, we've onlydiscussed the dirac equation
in one spatial dimension. and that again, wasto make it easier. of course, thereal dirac equation is in three dimensions,spatial dimensions. and then those extrafeatures come in, just like with the photon. so if we do it inthree dimensions, then what we find is that,just like with the photon, there were two polarizations,a left-handed one,
and a right-handedone, and they carry angular momentumplus and minus hbar, you find a similarthing for the electron. so if you do all thisin three dimensions, then you find - and i'm afraidi'm not going to show you in any detail - that there aretwo different polarizations, we call them twospins, because they're intrinsic to the electron. and for the electron,there's something
rather curious, inthat the spins are not plus and minus hbar, likethey are for the photon, but they're plus orminus a half hbar. this is something that comesout of the dirac equation, when you do it in three dimensions. so for an electron, thereare actually four states. there's the positiveenergy state, left-handed. there's positive energystate, right-handed. there's the negative energystate left-handed, and also
the negative energystate right-handed. now let's think abouthow the field looks - or the waves if you like -look for the electron now, because of what theimplications are of the spin. so if you consider electronfields that go around in a circle, like we didfor the photon field, so we look at theangular momentum, then the fieldlooks like, again, a sine, the angletheta times the angular
momentum minus the energy timesthe time, divided by hbar. but now, the angular momentumis in units of h-bar over 2, not in units of hbar. now, you rememberfor the photon, it had to be inunits of h-bar, so that when we go round the circleand get back to the beginning, we get back to where we started. now, electrons dosomething really weird. for electrons, you have togo around the circle twice
to get back towhere you started. so for electrons,if you do theta, i goes to thetaplus 4 pi, then you can see this expressionwill stay unchanged - the half here cancels outthe - takes the 4 down to 2, and then you get backto where you started. if you only do thetagoes to theta plus 2 pi, so you only go aroundthe circle once, then the wave functiondoesn't go back to itself,
it goes to minus itself. so you get an extra minus sign. so if you think it's a littlebit strange that sometimes you have to go around twice in orderto get back where you started, then i have a little partytrick for you, which shows you that sometimes thisis actually necessary. so here we have a perfectlyordinary wine glass, i'll hold on the flatof my hand like that. now i'm going to turnthe wine glass around,
so you should watch carefullywhat happens to the wine glass. so first, i turn itaround once, like this. and you can see i'mnot not back to where i started, because itfeels very awkward. and then i carry on up,and turn it round again, and now i get backto where i started. now, this extra minus signthat we get for electrons is actually very important. you remember inthe last lecture,
when we were talking aboutthe double slit experiment, and we had two differentpossible things that could happen, andin order to work out the probabilityof that happening, we had to add up theamplitudes for those two different processes,and square it, and that gave usthe probability. so for photons,what we found then, was the probability - in thatcase of going through the two
slits, although it couldapply to just about any process where there aretwo ways of doing things - has to be the sum of thetwo amplitudes squared. now for electrons,i told you last time that it worked inexactly the same way. we take the two amplitudes,we put them together and square them. but, that wasn't quite true. you see, the two processeshere, because they're the same,
you get by swappingthe two photons around. now, when we turn - whenwe swap two photons around, it's a bit like takingthem round in the circle. so for the photons,we get a plus sign. but if you take twoelectrons round in a circle to swap them round, thenyou get this minus sign. you have to take themaround twice in order so for the electrons, there'sactually a minus sign here. so to sum up, we have - thisis another difference now,
between photons and electrons. for photons, we havea plus sign here. we have two differentpolarizations, which have angular momentum,plus or minus h-bar. and this is actually a genericsort of class of the way that things can happen. so objects that look likethis, you can call bosons. whereas for the electrons,we have a minus sign here. the minus sign is because thepolarisation states aren't
in units of h-bar, butin units of a half hbar. this puts them in a differentkind of class of object. these objects we call fermions. bosons are named after anindian physicist called bose, and fermions are namedafter an italian phycicist called fermi - that's wheretheir names come from. now, for fermions,like electrons, this has a veryimportant consequence. if you think about - ifyou have two electrons
in the same state, then, ifthey're in the same state, then these two amplitudesare equal to each other. and if the two amplitudes areequal, then they cancel out, so the probability is zero. so what this meansis that for fermions, then they can't bein the same state. there's no probabilityat all for them being in the same state. so they're alwaysin different states.
and this is something calledthe pauli exclusion principle. and it's because of thepauli exclusion principle that you can build up allsorts of different atoms. so that's why all thedifferent atoms exist, and why the worldaround us is so varied. because, if we have a proton,and an electron going around it, that would be hydrogen, say. if we want to addanother electron, it can't just siton top of this one,
it has to be somewhere else. so the atom looks different. because it can't sitin the same state. actually, it's notquite like that, because there's two spins. so actually, we can put twoelectrons in this orbit, one with spin up, onewith the spin down. that would give us helium. and then the nextone, lithium, has
to come in a different orbit. so that makes all the differentatoms look very different. and is one of themain things that explains the reason theworld's the way it is.


Whispered Discrete Math Problems Secrets
If essay writing you're a creative problem solver, writing code provides you with the capability to create innovative solutions to unique issues. https://www.duq.edu/about/centers-and-institutes/center-for-teaching-excellence/teaching-and-learning/strengths-and-dangers-of-essay-questions-for-exams It is crucial to understand the myriad types of issues that will be shown on the test. A couple of these problems are rather tricky to solve utilizing a computer.
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The questions are deliberate to find out when you have a fundamental understanding of elementary mathematics, and should you have the skill to explanation obviously. Not understanding the proof was not a choice. Now, a last observation.
There are a couple normal strategies for proving an implication, and a few points that are relevant to all proofs. Though many of the fundamental combinatorial results were obtained mainly by ingenuity and in depth reasoning, the modern theory has developed into a significantly deeper theory with a systematic and effective toolkit. You will find that a number of these concepts are actually identical or nearly so.
Discrete Math Problems - Is it a Scam?
The reals aren't countable. Once you receive the concepts clear, no sum will appear to be tricky to you. There are always special cases where it's possible to discover the specific eigenvalues and eigenvectors (for triangular matrices it's trivial, regardless of what the size).
In case the order doesn't matter then we've got a combination, in the event the order do matter then we've got a permutation. It can help to understand the probability of locating a given object in a variety of objects. As per the above mentioned discussion, a null set is going to be a subset of itself.
Top Choices of Discrete Math Problems
There you may also find info about obtaining the LaTeX source code in the event you wish to remix the book. The quantitative section contains 28 questions. Additional exercises appear at the conclusion of each chapter and in depth solutions are offered in the appendices.
The printing number are available at the bottom of the webpage on the opposite side of the title page. Like that it's possible to use the PDF viewer that's natively provided by your device. Another edition is presently offered.
The Awful Secret of Discrete Math Problems
The branch of mathematics that addresses discrete variables is called discrete math. When you are finished, you will know a critical part of mathematics and you'll feel justifiably good about your achievement. Learning discrete math in our internet classroom is easy and efficient.
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Discrete Math Problems Features
Instruction should be supplied at times during which the student is the most alert. Ever since then it's been used as the main text for this course at UNC, in addition to at other institutions. The expression finite mathematics may be applied to portions of the area of discrete mathematics that addresses finite sets, particularly those areas pertinent to business.
College essay help can help you save you once you're in a tight spot. Students ought to be comfortable solving problems in the regions described within this document. It may also be unhealthy when students make an effort to cheat their way to college.
Discrete Math Problems - the Conspiracy
Put simply, the complete number of individuals multiplied by the range of handshakes that each can make will be the overall handshakes. In addition, the rewards are external. Taking in an assortment of composing ability it's not overly much difficult as you believe.
What's Actually Happening with Discrete Math Problems
The very first problem I ran into was hoping to get an adequate book. Were you were ready to fall asleep before the conclusion of it. In the long term, you'll be less competent and you will imagine the best way to succeed in life is to take short cuts and cheat.
What You Don't Know About Discrete Math Problems
One just utilizes the very same solution pattern. It is a group of all elements in a special context or application. Just take a look at the straightforward logic circuit below.
In the event the incidence matrix is comparable to it, it has to be symmetric too. The majority of them seek to deal with the 4 standard regions of mathematical computation. If you're unable to get a solution, try harder.


PENINGKATAN KEMAMPUAN INVESTIGASI MATEMATIKA MELALUI PEMBERIAN PROYEK MATEMATIKA PADA SISWA KELAS XI IPA 1 SMA NEGERI 2 WATAMPONE
This study is a classroom action research wich aims at improving investigation ability of mathematics on function derivation and describing the process of mathematics learning by providing Mathematics Project. The subject of the study was 40 grade XI IPA 1 SMAN 2 Watampone of academic year 2011/2012. The improvement of mathematics investigation ability was measured by investigation ability indicators, namely (1) problem identification, (2) organization and collection of information/facts/Mathematics symbols, (3) building and implementation of planning and problem solving, (4) drawing conclution, (5) interpretation of the result of investigation, and (6) communicating the result of investigation. The study was conducted in two cycles by following the stages of classroom action research on each of the cycle: (1) the planning, explaining of the preparation based on the result of pre-action wich resulted on low investigation ability, (2) the action, the implementation from all of the planning of action constructed on the first stage, (3) the observation, observation during the action wich obtained quantitative data and qualitative data, and (4) the reflection, the stage to study and process the data obtained during the observation. The instrument of study used observation sheet and test of investigation ability of mathematics. The observation sheet was used to evaluate the implementation of learning process by giving Mathematics project assignment; whereas, test of investigation ability of Mathematics was used to evaluate the improvement of students’ investigation ability.The results obtained after giving the project assignment are (1) on cycle I, the mean score of students’ investigation ability after the learning is 57,05; (2) on cycle II, the result is 80,58. The observation sheet reveals that students have improvement on the activeness and ability in conducting investigation toward the project assignment proven by the achievement of indicators of investigation ability wich improved on each of cycle. The conclution based on the result of observation and test of investigation ability is there is adequately significant improvement of mathematics investigation ability on each of the cycle.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.


If students of the present generation are apprehensive about any subject then it is certainly Maths as most of them either abominate it or score abysmally low marks in the subject. However, it is evident that without mastering Maths students cannot perform well in academics as in higher studies Mathematics involvement is prevalent in all field e.g. Science, Commerce and other streams. and its application can be witnessed in almost all field e.g Physics economics, actuarial etc.
A good command in Maths opens a plethora of doors other than Science, Statistics, Actuary, Economics. Moreover, without it, you cannot excel other subjects as it is an integral part of other subjects. Therefore, it is very imperative to master in Math and it is only feasible if you have immense interest in Maths. But what if you do not have interest in it as most of the students find this subject tedious and difficult. but if maths taught with the right approach and lucid concept it can become most interesting subject and students find it one of the most scoring subjects as every year thousands of students score full marks in Math across all boards.
But it is unfortunate that Maths is feared in our country where some of greatest Mathematicians of all time were born e.g Aryabhata, Bhaskaracharya and Ramanujan. It is owing to unconventional teaching method that makes it boring subject.
Baidya Sapient Home Tutors is one of the fastest expanding tutoring service providers in Delhi-NCR owing to its inspiring and dedicated tutors.With over 500 experienced tutors in our network. To date we have provided tutoring services to over 1000 students in a very short span of time and this number is increasing exponentially. We have over 25 Maths tutor with experience ranging from 3 years to 15 years. If you see next Ramanujan in your child, enrol him for demo class and experience your child learning curve going up fastly. Our tutors not only cover NCERTand syllabus but also encourage students to bring daily life maths problem to them to augment their interest in the subject.


Here's the analysis I just got back from My Renowned Statisticsologist:
(A) Well, first calculate the probably of one acquaintance having an accidental death in the family. This in the long run is simply the number of families that have accidental deaths in the average week divided by the number of families (this is in the world, in the US, in your state whichever you think is most relevant).
Now, two calculate the chances of two in the same week, you need to define your terms a bit more explicitly. (1) Do you mean in one particular week (e.g., what's the probability of two such occurences in the week of March 3-9)? (2) Or do you mean at some point in your life two will happen in a week? (3) Or do you mean that given one happened, what's the probability of another happening within 7 days?
(3) is easiest because we can make a reasonable assumption about independence because we will clearly be talking about two separate events. There might be some correlation, but it's probably small. Then the answer would simply be whatever you got for (A).
(1) is next easiest. If we assume independence (a big assumption) then you just square (A). If we think there is some positive correlation, i.e., acquaintances' family members might often travel together in the same car, then this will be closer to just plain old (A). Or, if accidental death includes things like death by earthquake, then there's certainly some positive correlation. If you really mean, a safe dropping on someone's head or falling down the stairs, then these are probably close to independent and your answer is (A)^2
(2) This is the hardest. Since the chance is so small you can approximate it by 1 - ((1 - "1")^N) where "1" is the answer you get in the above paragraph and N is the # of weeks you live in a year.
Did you get all that, Brother/Sister? Well, just in case you didn't, here's the upshot: the odds are ASTRONOMICAL that my two Stenos' family members would suffer near-simultaneous mortal "accidents." Like a gajillion-and-a-half to one. So I'm going to proceed on a double-homicide theory.
And Brothers and Sisters, please do let this insider's access I've given you into my management of this crisis inform your understanding of how I intend to govern the planet, once I take over. A Benevolent and Wise World Leader does not act willy-nilly he consults with trusted and knowledgeable advisors and makes rational policy conclusions that incorporate and reflect their analytical expertise.
That, people, is how you govern.
Whether or not I decide to formalize the position of Renowned Statisticsologist er, Statistician into my regime structure, you can bet I will be relying heavily on this aforequoted gentleman's number-crunching insights well into the future.


Abstract: The original Specker-Blatter Theorem (1983) was formulated for classes of structures of one or several binary relations definable in Monadic Second Order Logic MSOL. It states that the number of such structures on the set is modularly C-finite (MC-finite). In previous work we extended this to structures definable in CMSOL, MSOL extended with modular counting quantifiers. The first author also showed that the Theorem does not hold for one quaternary relation (2003).
In this paper we show:
1) The Specker-Blatter Theorem also holds for CMSOL when hard-wired constants are allowed. The proof method of Specker and Blatter does not work in this case.
2) The Specker-Blatter Theorem does not holds already for with one ternary relation definable in First Order Logic FOL. This was left open since 1983.
Using hard-wired constants allows us to show MC-finiteness of counting functions of
various restricted partition functions which were not known to be MC-finite till now. Among them we have the restricted Bell numbers , restricted Stirling numbers of the second kind or restricted Lah-numbers . Here is an non-negative integer and is an ultimately periodic set of non-negative integers.
Lorenz Halbeisen and Salome Schumacher. Some implications of Ramsey Choice for n-element sets
Abstract: For a positive natural number n, “Ramsey Choice for n-element sets”, denoted RC(n), is the choice-principle that every infinite set X contains an infinite subset Y such that the family of n-element subsets of Y has a choice function. The relation between RC(n) and several other choice principles for families of n-elements sets will be discussed. This is joint work with Salome Schumacher.
Vince Velkey. Game proofs of Lebesgue measurability
Abstract: Infinite games can be used to prove regularity properties from determinacy. Usually, due to the Solovay unfolding phenomenon, determinacy at some level of the projective hierarchy gives the regularity property at the next; in particular, Borel determinacy gives analytic regularity.
However, in the usual proof of the key lemma for Lebesgue measurability, we use the fact that all analytic sets are Lebesgue measurable. We discuss how to avoid this problem by moving to different games characterising Lebesgue measurability.
Vasco Brattka. Continuity and computability of functions
Abstract: We discuss the relation between computability and continuity of functions. For one, there is a well-known classical result of Addison that shows that continuity is just computability with respect to some oracle, which establishes a uniform relation between the two concepts. On the other hand, it is folklore in Weihrauch complexity that the limited principle of omniscience (LPO) is the weakest single-valued discontinuous function with respect to the topological version of Weihrauch complexity, which provides another uniform characterization of discontinuity. If combined with results by Matthias Schröder on admissible representations, then this characterization of discontinuity carries over to a wide class of topological spaces. It is also well-known that LPO parallelizes to the Turing jump function. If one combines all these results in the appropriate way, then one obtains a non-uniform characterization of continuity of single-valued functions over a wide class of topological spaces in first-order terms over Turing degrees. Essentially, this characterization states that a function is discontinuous if and only if its parallelization computes Turing jumps on a Turing cone. If written out with quantifiers, then this characterization is reminiscent of the classical epsilon-delta characterization of uniform continuity for metric spaces, where epsilon and delta are now ranging over Turing degrees, the metric is replaced by the Turing degree map and pairs of points are replaced by sequences.
Lucas Wansner. Amoeba-regularity and inaccessibles
Abstract: In 1995 Judah and Repický defined a regularity property we call amoeba-regularity. Similar to the most other regularity properties, the amoeba-regular sets form a σ-algebra containing all analytic sets. However, the statement “every set in second level of the protective hierarchy is amoeba-regular” is independent from ZFC.
In this talk, we investigate the consistency strength of this statement and show that it implies that the first uncountable cardinal in V is inaccessible in L. In fact, we show if all sets in a given level of the protective hierarchy are amoeba-regular, then all sets in the same level have the Baire property in the dominating topology.
This is joint work with Raiean Banerjee (Hamburg).
Lukas Schembecker and Vera Fischer. Partitions of Baire space into compact sets
Abstract: We define a c.c.c. forcing which adds a maximal almost disjoint family of finitely splitting trees on ω (a.d.f.s. family) or equivalently a partition of Baire space into compact sets of desired size.
Furthermore, under CH we construct a Sacks-indestructible (by countably supported iteration and product of any length) maximal a.d.f.s. family.
David Binder and Thomas Piecha. Administrative Normal Form and Focusing for Lambda Calculi
Abstract: The Curry-Howard correspondence between deductive systems and computational calculi is one of the great unifying ideas. It links purely logical investigations to practical problems in computer science, in particular the design and implementation of programming languages. Many aspects of this correspondence are widely known, such as the correspondence between natural deduction for intuitionistic logic and the simply typed lambda calculus. On the other hand, the importance of the sequent calculus in proof-theoretic investigations is not yet reflected in the study of programming languages, where languages based on the lambda calculus dominate. Our contribution is a step towards bridging this gap by introducing and discussing the correspondence between two normal forms and their respective normalization procedures: administrative normal form and ANF-transformation on the one hand, and focused normal form and static focusing on the other. Though invented for different purposes, compiler optimizations in the case of the ANF-transformation and proof search in the case of focusing, they are structurally very similar. Both transformations bring proofs into a normal form where functions and constructors are only applied to values and where computations are sequentialized. In this talk we make this similarity explicit by showing how the ANF-transformation on lambda terms corresponds to static focusing of lambda-mu-mu’tilde terms.
Marcel Ertel. Paul Bernays and the philosophical significance of independence results
Abstract: We provide a reconstruction of Paul Bernays’ assessment of the philosophical significance of independence and non-conservativity phenomena in mathematics. In the first part, we sketch a systematic presentation of Bernays’ post-1931 philosophy of mathematics, a fully worked-out account of which remains a desideratum today. Particular emphasis is laid on the relationship of formal and ‘contentual’ reasoning, the differentiation of degrees of (non-)constructivity, the open-endedness of mathematical concept-formation, and the questions of evidence and objectivity in mathematics. In the second part, we investigate on this basis Bernays’ response to the Gödel-Cohen independence results in set theory, with special emphases on Bernays’ subtle assessment of the roles of first-order logic and of schemata, and on his account of the concept of a non-standard model. In this context we also discuss Bernays’ 1961 axiomatization of class theory BL (“Bernays-Levy”) based on a reflection principle, which yields the usual axioms as well as much stronger large cardinal existence theorems (below measurable cardinals), but which incurs philosophical difficulties due to it requiring the existence of non-predicative classes, conflicting with Bernays’ own account of the set-vs.-class distinction. In the final part, the results of the earlier parts are related to contemporary debates on the justification of new axioms and on the issue of truth-value (in)determinacy in set theory.
Levin Hornischer. Dynamical Systems via Domains: Semantics for Non-symbolic Computation
Abstract: Famously, domain theory provided a mathematical semantics for symbolic computation: systematically explaining code of a programming language by assigning it to the mathematical object computed by it. We lack this kind of explainability for the non-symbolic computation performed, e.g., by neural networks in artificial intelligence. This talk, which is based on my PhD thesis, suggests an extension of domain-theoretic semantics to non-symbolic computation.
Generally, non-symbolic computation can be viewed as a dynamical system: a set of (computational) states together with a dynamics describing how to move from one state to the next (program). Thus, the task is to construct, for a given dynamical system, a `domain’ that describes the behavior of the system. Domains, i.e., the object of study of domain theory, are certain partial orders whose elements intuitively are regarded as outputs of computational processes while the order describes information containment.
To build the semantics, the high-level idea is this: Given a dynamical system X, we construct a structure D that we will call the dynamical domain of X. Intuitively, D consists of `basic’ elements that represent increasingly finer observations of the system X together with the `limits’ of these basic elements. The additional domain-theoretic structure on D is such that it induces a dynamical system on these limit elements, which we call the system modeled by D. We show that this modeled system is isomorphic to X. Categorically speaking, we then prove that the construction of the dynamical domain for a system and the construction of the system modeled by a dynamical domain are adjoint.
Matteo de Ceglie. A generalisation of the set theoretic multiverse
Abstract: Ten years ago, Hamkins, 2012 changed the landscape of the foundations of mathematics, by introducing a novel conception that tried to clarify some ambiguous notions in current set theoretic practice. In particular, he provided a revolutionary interpretation for the practice of forcing: a multiverse of different set theoretic universes. Such an idea immediately sparked an intense debate in the philosophy of set theory and the foundations of mathematics. In the following years, several crucial contributions were made.
Indeed, while the general idea behind pluralism in the philosophy of mathematics is more or less the same every time, the actual mathematical details can vary enormously from one characterisation to the other. Even though all these different set theoretic multiverses share the same, general, philosophical idea, they differ wildly from the mathematical perspective. There are some proposal of assessing all these differences, but this research field is still in its infancy. In this paper, I propose a novel, more general, framework for the set theoretic multiverse: the Universal Multiverse.
Drawing from the ideas of Universal Logic, this new framework is not a new set theoretic multiverse. Instead, it is a general theory ofthe multiverse, that investigates the various set theoretic multiverses from a common and abstract point of view. This means highlighting the common features of all multiverses, studying them as mathematical structures without any other metaphysical and ontological connotation.
The first step to carry out this program is to define a method of comparing and categorising all the different multiverses. A multiverse can be characterised in several ways: the most common one is to describe it as a set of universes. For example, Steel’s set generic multiverse is the set of all set-generic extensions of a core universe, Friedman’s Hyperuniverse is the set of all countable transitive models of ZFC, etc..
Generalising on this idea, a general theory of the set theoretic multiverse is the theory of the Multiverse Operator, Mlti. This theory is analogous to Tarski’s theory of the consequence operator as introduced in Tarski, 1928. A multiverse operator is a function defined on a stage Vκ of a set theoretic universe. A multiverse operator maps each stageVκ with the set of all universes, M, that are part of the multiverse generated using Vκ as the ground universe:
Mlti : Vκ → M.
For example, the operator Mltgeneric applied to any Vκ, written Mltgeneric(Vκ), will map to Steel’s set generic multiverse. In this way we can define in very general terms what a set theoretic multiverse is: it is an ordered couple (Vκ,Mlti), where Vκ is a stage ofV and Mlti is a multiverse operator.
The general theory of the multiverse that I propose is the study of the multiverse operator in general, without considering the special feature of each of them (e.g. the difference between Mltgeneric and Mltvlogic). From this very abstract and general perspective, I claim that the general structure ⟨Vκ,Mlt⟩ forms a Tarski structure. That is, it obeys the following axioms:
1)Vκ ⊆ Mlt(Vκ);
2)Vκ ⊆ Vλ ⇒ Mlt(Vκ) ⊆ Mlt(Vλ);
3)Mlt(Mlt(Vκ)) ⊆ Mlt(Vκ).
In this paper, I will expand on the meaning of these axioms for the multiverse operator, and sketch the road forward in this research field.
Christine Gaßner. Second-Order Henkin Semantics and the Axiom of Choice
Abstract: Inspired by questions raised by Paolo Mancosu and Stewart Shapiro (2018), we will examine in more detail the meaning of the Axiom of Choice in second-order predicate logic. First, we give an overview of second-order predicate logic with comprehension axioms for predicates and briefly compare different well-known variants of the Henkin interpretation which have been described in detail by Henkin (1950), Shapiro (1991), Väänänen (2019), and others. The model-theoretic semantics mainly discussed here is based on so-called predicate structures and second-order Henkin structures, as defined by Asser (1981) on the basis of a suggestion by his student and later colleague, Schreiber. The predicates are attributes in the sense of Scholz and Hasenjaeger (1961), and the language used is a many-sorted first-order language with identity. In the second part, we summarize consequences for the relationships between different formulations of the axiom of choice. In particular, we consider second-order principles of choice introduced by Ackermann (1935), Hilbert and Ackermann (1938) and Asser (1981), which we consequently call Ackermann axioms, Hilbert-Ackermann axioms, Zermelo-Asser axioms, Russell-Asser axioms, and Asser axioms, respectively. The Fraenkel-Mostowski-Specker-Asser method used to construct second-order Henkin structures allows us to provide new insights into the relationships between these principles.
Ackermann, W.: Zum Eliminationsproblem der mathematischen Logik. Mathematische Annalen 111, 1935, pp. 61−63.
Asser, G.: Einführung in die mathematische Logik, Teil III: Prädikatenlogik höherer Stufe. Teubner, 1981.
Gaßner, C.: The Axiom of Choice in Second-Order Predicate Logic. Mathematical Logic Quarterly, 40 (4), 1994, pp. 533−546.
Henkin, L.: Completeness in the Theory of Types. Journal of Symbolic Logic 14 (3), 1950, pp. 81−91.
Hilbert, D., and W. Ackermann: Grundzüge der theoretischen Logik. Springer, 1938.
Scholz, H., and G. Hasenjaeger: Grundzüge der mathematischen Logik. Springer, 1961.
Shapiro, S.: Foundations without foundationalism: a case for second-order logic. Oxford University Press, 1991.
Väänänen, J. A.: Second-order and higher-order logic. Stanford Encyclopedia of Philosophy. August, 2019.
Juan Miguel Lopez Munive. A defense of proof-theoretic logical pluralism
Abstract: Dicher (2016) argues that a crucial element of Restall’s (2014) proof-theoretic defense of meaning-invariant logical pluralism is mistaken, namely: the sameness claim. According to this claim, inference rules that differ only structurally are identical. In this talk, my aim is to defend Restall’s proof-theoretic pluralism by showing that Dicher’s argument is unsound. To do this I apply Mares’ and Paoli’s (2014) renewal of Avron’s (1988) distinction between internal and external consequence relation and argue that the sameness claim is to be understood as holding at the level of the former. Additionally, I defend that Dicher’s purported counterexample to the sameness claim does not concern the structure of the internal consequence relations of the rules he considers. Consequently, Restall’s proof-theoretic pluralism manages to avoid Dicher’s objection.
Pedro Pinto. A generalization via proof mining methods
Abstract: Recently, an iterative method for approximating common fixed points of two nonexpansive maps was introduced. This method alternates between the well-known Halpern-type and Mann-type definitions, and was thus dubbed the alternating Halpern-Mann iteration (HM). In , the (HM) method was studied in the setting of CAT(0) spaces, and asymptotic regularity and strong convergence were established. If one restricts oneself to Hilbert spaces, the proof follows standard arguments. Such arguments are not available in the nonlinear setting of CAT(0) spaces and the use of logical tools seem to be necessary.
In this talk, we will discuss the role of logical tools in obtaining this result, namely via the application of a general macro developed in  for the elimination of certain weak compactness arguments in the context of the proof mining program.
 B. Dinis, and P. Pinto, Strong convergence for the alternating Halpern-Mann iteration in CAT(0) spaces, arXiv:2112.14525 (Submitted), 2021.
 F. Ferreira, L. Leuștean, and P. Pinto, On the removal of weak compactness arguments in proof mining, Advances in Mathematics, 354:106728 (55pp), 2019.
 L. Leustean, and P. Pinto, Rates of asymptotic regularity for the alternating Halpern-Mann iteration, arXiv:2206.02226 (Submitted), 2022.
 H. Cheval, and L. Leuștean, Quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration, Optimization Methods and Software, 2022.
 H. Cheval, U. Kohlenbach, and L. Leuștean, On modied Halpern and Tikhonov-Mann iterations, arXiv:2203.11003 (Submitted), 2022.
 U. Kohlenbach, and L. Leuștean, Eective metastability of Halpern iterates in CAT(0) spaces, Advances in Mathematics, 231(5):2526-2556, 2012.
 K. Schade, and U. Kohlenbach, Effective metastability for modified Halpern iterations in CAT(0) spaces, Fixed Point Theory and Applications, 2012:191 (19pp), 2012.
 R. I. Boț, E. R. Csetnek, and D. Meier, Inducing strong convergence into the asymptotic behaviour of proximal splitting algorithms in Hilbert spaces, Optimization Methods and Software, 34(3):489-514, 2019.
Abstract: The frame concept, developed in Artificial Intelligence, cognitive science and linguistics, has recently regained interest in philosophy and philosophy of science. There have also been attempts at modelling mathematical proofs using frames, starting from a distinction between structural frames, which capture expectations about proof methods as well as constituents of proofs and their relations, and ontological frames, which capture the mathematical structures and typical referring linguistic and formulaic expressions.
We will illustrate two points about using frames to model mathematical proofs: First, how frames can be used to explain aspects of innovation in mathematics, e.g. the discovery of zero or the transfer between different subject areas. Secondly, how frame theory relates to concepts of understanding proofs, notably Avigad’s (2008) theory.
Joost Joosten. A decidable fragment of the Quantified Provability Logic of Heyting Arithmetic
Abstract: We present a nested-implication free fragment of quantified modal logic, called QRC1, that we show to be sound and complete for Heyting Arithmetic. The only allowed modality is the diamond which is to be interpreted as consistency. Our results are obtained by importing earlier results on QRC1 for Peano Arithmetic to the setting of Heyting Arithmetic using well-known conservativity results and proving that we always stay in the required complexity classes during our Solovay construction. In addition we make some observations on properties of QRC1 including interpolation and fixpoints. We close off by showing that QRC1 also defines a fragment of True Quantified Provability Logic of Heyting Arithmetic (and Peano Arithmetic). The latter shows — using results of Montagna, Artemov and Vardanyan— that going from the full logic to the fragment supposes a jump from non-arithmetical to decidable.
This is joint work with Ana Borges de Almeida, Dick de Jongh and Albert Visser.
José Antonio Pérez Escobar and Deniz Sarikaya. Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the dichotomy
Abstract: In this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different communities, which endorse different evaluative standards for theories. We evaluate the distinction between pure and applied mathematics from a late Wittgensteinian perspective. We note that the classical exegesis of the later Wittgenstein’s philosophy of mathematics, due to Maddy, leads to a clear-cut but misguided demarcation. We then turn our attention to a more niche interpretation of Wittgenstein by Dawson, which captures aspects of the aforementioned distinction more accurately. Building on this newer, maverick interpretation of the later Wittgenstein’s philosophy of mathematics, and endorsing an extended notion of meaning as use which includes social, mundane uses, we elaborate a fuzzy, but more realistic, demarcation. This demarcation, relying on family resemblance, is based on how direct and intended technical applications are, the kind of evaluative standards featured, and the range of rhetorical purposes at stake.
Bernd Buldt. On Visser’s move on G2
Abstract: There is a long-running debate on Gödel’s second incompleteness theorem (G2: the unprovability of consistency) that began in the late 1950s when it became clear that any uncritical formulation of the theorem will suffer counterexamples (the so-called intensionality of G2). The discussion itself and suggestions for how to amend the situation have been dominated by Kreisel’s request to find a `canonical representation’ of mathematical theories and their notions of provability in order to dodge the specter of intensionality. Recently, Albert Visser injected a fresh, new idea into the discussion by suggesting that we should conceive of “the second incompleteness theorem, not as a statement concerning the relation of a theory and a sentence [i.e, the unprovable consistency statement], but concerning the relation between two theories” (Visser16, p.~0; emphasis in the original). In the talk I will discuss Visser’s move against the backdrop of the previous discussion and try to asses its promise and some of the challenges it faces.


A Fast Mass Spring Model Solver for
High-resolution Elastic Objects
Abstract: Real-time simulation of elastic objects is of great importance for computer graphics and virtual reality applications. The fast mass spring model solver can achieve visually realistic simulation in an efficient way. Unfortunately, this method suffers from resolution limitations and lack of mechanical realism for a surface geometry model, which greatly restricts its application. To tackle these problems, in this paper we propose a fast mass spring model solver for high-resolution elastic objects. First, we project the complex surface geometry model into a set of uniform grid cells as cages through *cages mean value coordinate method to reflect its internal structure and mechanics properties. Then, we replace the original Cholesky decomposition method in the fast mass spring model solver with a conjugate gradient method, which can make the fast mass spring model solver more efficient for detailed surface geometry models. Finally, we propose a graphics processing unit accelerated parallel algorithm for the conjugate gradient method. Experimental results show that our method can realize efficient deformation simulation of 3D elastic objects with visual reality and physical fidelity, which has a great potential for applications in computer animation.


The modulo function is the integer remainder of dividing var1 by var2.
The modulus operator returns the first operand modulo the second operand, that is, var1 modulo var2, in the preceding statement, where var1 and var2 are variables. .
A strict comparison (e.
let x = 5; let y = 3; let z = (x + y) * 2; // z is 16. Example 4: UNION with DISTINCT. It returns an array of information or null on a mismatch.
. Function names can contain letters, digits, underscores, and dollar signs (same rules as variables). . Practical examples and use cases. Operators take one or more operands and are capable of manipulating target values or operands.
. let x = 5; let y = 3; let z = (x + y) * 2; // z is 16.
In this example, the spread operator is used to create a new object merged that contains all the properties from obj1 and obj2.
Rules and restrictions of MySQL UNION.


Fun with numbers: fire up your brain with these addictive games
Numbers are not just for math geeks and people with useful college degrees. Number and the multitude of ways you can combine them make for a great mobile games as well. From the popular and time tested Sudoku to a more recent 2048, at least one of these games will surely grab your attention and make your brain do some work.
2048: Not just for IT maniacs
2048 is a single-player puzzle game, in which the objective is to slide numbered tiles on a grid to combine them and create a tile with the number 2048. You start with a randomly populated 4×4 grid. You move your number in any of four directions and when any two same numbers touch the result is their total sum. Once you get to 2048, the game ends. That is, if you don’t run out of moves.
The game is translated into almost every language and supports automatic saving and every version of Android above 2.3. You also get two modes – the regular one and time trial where number are added automatically after certain amount of time. You can download 2048 from Google Play Store or simply try it out in your web browser by going here.
Numbers Addict: really addicting
In this game you get to play with a single digit numbers. The player has to clear to gaming board from all bubbles which represent a certain number. The game ends only when bubbles cross the top border. The user is in control of where numbers fall and he gets to see three next numbers.
Numbers disappear in two cases. First, when there is a certain number of them on a playing field. This number is determined by a numerical value of each bubble. For number 5 there has to be at least 5 bubbles, and so on. The second way is counting of the numbers.
You can choose from 3 difficulty levels. Easy mode lets you play without any timers, so that players can enjoy the game at their own speed. Medium difficulty settings adds the timer and new bubbles are automatically displayed after five seconds.
The expert level makes you face not only the timer but also a new type of bubble. This bubble disappears only when player has a certain score which is written on the bubble.
Users say that Numbers Addict improved their memory, cognitive and concentration flexibility and stimulated brain cells that were asleep. If that sounds good to you, head over to Play Store and have some fun.
Numbers Quiz: Riddles full of numbers
Numbers Quiz is based around well known and sometimes dreaded sequences of numbers that you are probably familiar with from school. Your goal is to figure out which number is logically the next correct one in the sequence.
You start from the easiest riddles and quickly move up in difficulty. For every solved riddle you get 5 coins which you can in turn use when you get stuck and can’t solve the problem.
You can also play against your friends by simply inviting them for a math match. Download Numbers Quiz for free from Play Store.
Sudoku: The classic brain twister
The objective of Sudoku is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid contains all of the digits from 1 to 9.
The modern Sudoku only started to become mainstream in 1986, but only became officially recognized as being common in western media publications in 2005. Today, we get to enjoy this addictive and challenging game on almost all of our devices. Play Store offers a wide variety of both free and paid games. On of the most popular and well rated is simply called Sudoku Free.
With its four difficulty settings, intuitive interface and automatic saving feature you can enjoy a game of Sudoku everywhere. The only downside may be the slightly dated graphics. If that is something that bothers you too much, you can always search for another version.
Threes: Unmatched design mixed with great game-play
Threes! is an indie puzzle video game by Sirvo that remained for several weeks in top places among paid games. It is no surprise really, with the simple concept, pleasant music, intuitive control system and above all great design, Threes is one of the most likable games in Play Store.
It all starts with a simple idea that is not entirely easy to master. You move tiles on 4×4 grid and you goal is to combine addends and multiples of three. Each round you select in which direct should your tiles move and when two tiles with the same numerical value overlap they merge into a new one with a value equal to their total. Two threes make six, two sixes make twelve and the game continue in this manner all the way into a four digit numbers.
That sounds simple, right? Well, it is, but it also isn’t. Each overlap makes one tile disappear but each move also makes a new tile appear on a empty space. The games ends when all 16 tiles are full and you cannot make any moves. You can see the value of the number that is going to appear next and you have to make sure to take it into a consideration, otherwise you won’t get far in this game.
One aspect that hugely contributed to the success of this game is its graphic design. The minimalistic design is though out from every aspect. For example, numbers from 3 up have their own individual “personality” with an original graphic and sound design. The lower numbers can act a little shy, while the higher numbers are not afraid to tell you what is on their mind.
The game costs $1.99 and for that money you cannot go wrong. Grab it from Play Store and have some good fun.


Designed by experts, made for everyone. The Casio Currency calculator features an extra-large 8-digit display to keep your calculations clear when they need to be. It also features percentage calculations including profit margins, as well as a euro conversion function to aid with complex financial calculations. The calaculator comes in a brushed metal finish case.
Casio 8-digit Currency Calculator
Casio’s currency calculator features currency conversion to aid transaction efficiency. It also features a clear 8-digit display to ensure clarity, as well as percentage calculation functions.


Solid State Physics
Yuan Ming Huang
4.2.1 Dispersion relation
We start with the simplest case of one-atomic linear chain with nearest neighbor interaction (see Fig. 4).
Figure 4. Vibrations of a linear one-atomic chain (displacements).
The nth lattice point is connected with (n-1)th lattice point on the left side and the (n+1)th lattice point on the left side. If one expands the energy near the equilibrium point for the nth atom and use quasi elastic approximation, he comes to the following equations. The force exerted by the (n-1)th lattice point can be described by the force f1
The force exerted by the (n+1)th lattice point can be described by the force f2
Applying the Newtonian equation, one gets the following equation
To solve this infinite set of equations let us take into account that the equation does not change if we shift the system as a whole by the quantity a times an integer n. We can fulfill this condition automatically by searching the solution as
, where .
It is just a plane wave but for the discrete co-ordinate na. Immediately we get (see Problem 2.1)
, where .
The expression (9) is called the dispersion law. It differs from the dispersion relation for a homogeneous string, ¦Ø = sq. Another important feature is that if we replace the wave number q as
where g is an integer, the solution (8) does not change (because exp (2¦Ð i¡Á integer) = 1). Consequently, it is impossible to discriminate between q and q' and it is natural to choose the region
to represent the dispersion law in the whole q-space. So, for a linear chain, the wave number q takes L discrete values in the interval (¨C¦Ð/a, ¦Ð/a). Note that this interval is just the same as the Wigner-Zeitz cell of the one-dimensional reciprocal lattice. This law is shown in Fig. 4.
Figure 4. Vibrations of a linear one-atomic chain (spectrum).
Note that there is the maximal frequency ¦Øm that corresponds to the minimal wave length ¦Ëmin = 2¦Ð/qmax = 2a. The maximal frequency is a typical feature of discrete systems vibrations.
Because of the discrete character of the vibration states one can calculate the number of states, z, with different q in the frequency interval ¦Ø, ¦Ø+d¦Ø. One easily obtains (see Problem 2.2)
This function is called the density of states (DOS). It is plotted in Fig. 5.
Figure 5. Density of states for a linear one-atomic chain.
We shall see that density of states is strongly dependent on the dimensionality of the structure.
Now we discuss the properties of long wave vibrations. At small q we get from Eq. (9)
is the sound velocity in a homogeneous elastic medium. In a general case, the sound velocity becomes q-dependent, i. e. there is the dispersion of the waves. One can discriminate between the phase (up) and group (ug) velocities. The first is responsible for the propagation of the equal phase planes while the last one describes the energy transfer. We have
At the boundaries of the interval we get up = (2/¦Ð)u while ug = 0 (boundary modes cannot transfer energy).


>> Wednesday, 19 August 2009
What Is It?
A simple calculation that tells you the consumption of poly(ethylene terephthalate), PET by an injection molding machine.
The Calculation Is...
kg / hr = W x C x S
W = Weight of the container in kg. (e.g. 45 grams = 0.045 kg)
C = Number of injection cavities
S = Number of injection cycles per hour (shots)
(3,600 / Cycle Time (sec) for one set of preforms)
A machine with a cycle time of 14 seconds making a 35 gram bottle
W = 0.035
C = 8
S = (3,600 / 14) = 257.1
kg / hr = 0.035 x 8 x 257.1 = 72 kg / hr


Minimizing Regret ##TOP##
I decided to take the Regret Minimization Framework one step further and think about how my 80-year-old self would look back on the decisions I made with my time, and how I can alter my habits and activities to minimize future regret.
So what will your last words be? And who will be in the room to hear them? Remember how short and fragile life is. As difficult as this is to think about and say, some of you reading this email right now will not be here next year. Make every moment count with your family. Take a moment now to ponder 10 ways to minimize your regrets at the end of your life.
A wise man has the ability to visualize outcomes before they occur. Never make decisions from a self-centered perspective. Self-centered decisions often will grant you short-term gain but create long-term damage for you and everyone around you. Always hedge your bets toward the wise thing to do. You will have fewer regrets.
An important problem in sequential decision-making under uncertainty is to use limited data to compute a safe policy, i.e., a policy that is guaranteed to perform at least as well as a given baseline strategy. In this paper, we develop and analyze a new model-based approach to compute a safe policy when we have access to an inaccurate dynamics model of the system with known accuracy guarantees. Our proposed robust method uses this (inaccurate) model to directly minimize the (negative) regret w.r.t. the baseline policy. Contrary to the existing approaches, minimizing the regret allows one to improve the baseline policy in states with accurate dynamics and seamlessly fall back to the baseline policy, otherwise. We show that our formulation is NP-hard and propose an approximate algorithm. Our empirical results on several domains show that even this relatively simple approximate algorithm can significantly outperform standard approaches.
We consider combinatorial optimization problems with uncertain parameters of the objective function, where for each uncertain parameter an interval estimate is known. It is required to find a solution that minimizes the worst-case relative regret. For minmax relative regret versions of some subset-type problems, where feasible solutions are subsets of a finite ground set and the objective function represents the total weight of elements of a feasible solution, and for the minmax relative regret version of the problem of scheduling n jobs on a single machine to minimize the total completion time, we present a number of structural, algorithmic, and complexity results. Many of the results are based on generalizing and extending ideas and approaches from absolute regret minimization to the relative regret case.
We study a general adversarial online learning problem, in which we are given a decision set X' in a reflexive Banach space X and a sequence of reward vectors in the dual space of X. At each iteration, we choose an action from X', based on the observed sequence of previous rewards. Our goal is to minimize regret, defined as the gap between the realized reward and the reward of the best fixed action in hindsight. Using results from infinite dimensional convex analysis, we generalize the method of Dual Averaging (or Follow the Regularized Leader) to our setting and obtain upper bounds on the worst-case regret that generalize many previous results. Under the assumption of uniformly continuous rewards, we obtain explicit regret bounds in a setting where the decision set is the set of probability distributions on a compact metric space S. Importantly, we make no convexity assumptions on either the set S or the reward functions. We also prove a general lower bound on the worst-case regret for any online algorithm. We then apply these results to the problem of learning in repeated two-player zero-sum games on compact metric spaces. In doing so, we first prove that if both players play a Hannan-consistent strategy, then with probability 1 the empirical distributions of play weakly converge to the set of Nash equilibria of the game. We then show that, under mild assumptions, Dual Averaging on the (infinite-dimensional) space of probability distributions indeed achieves Hannan-consistency.
The regret minimization framework is really more of a tripod that balances the business, relationship management, and where you are in your professional and personal life. Entrepreneurial life is always a bit of a high wire act, but these focal points do distill the various components into a simple formula.
While our goal is to move as fast as we can and minimize regret, if you can program machine-learning to focus on a percentage of your operations, it will freeyou up for the randomness that is the most rewarding and potentially most effective areas of marketing.


What’s the Golden Ratio and How is it Used in Dental Treatment?
The Greeks said that all beauty is mathematics. It is from this basic principle that the idea of the golden ratio was born. While the golden ratio as it relates to beauty is somewhat controversial, the golden ratio has shown to be very useful in evaluation and treatment for smile design, so these concepts are still used today. The golden ratio has been described as the ratio that is most attractive to the human eye and mind.
Origins of the Golden Ratio
The golden ratio was first recorded by the Pythagoreans. The symbol of the golden ratio is the pentagon, which the Pythagorean School took as their symbol. Simplified, the premise of the golden ration is that the ratio of the smaller length to the larger length is identical to the ratio of the larger length to the total length. The golden ratio is derived from this equation as being 1:1.618.
Golden Ratio and Facial Beauty
The golden ratio has been found to be present in most things that are considered beautiful. Human DNA, flowers, and many other naturally occurring patterns and designs include the golden ratio. It is from this idea that the Greeks began measuring facial features in an effort to mathematize facial beauty.
It was found that the golden ratio could be found between facial features of women that were considered beautiful. For example, a golden relationship was found between the nose and teeth and the teeth and the chin, with the distance between the nose and teeth being one and the distance between the teeth and the bottom of the chin being 1.618 in women that were considered beautiful.
Applying the Golden Ratio to Dentistry
A golden mean gauge was developed to apply the golden ratio quickly and easily to dental esthetics. The four front teeth, from central incisor to premolar are in golden ratio to each other in ideal smiles. When these teeth are not in agreement with the golden ratio, the golden mean gauge or subsequently developed charts can be used to help dentists and dental labs understand where teeth should ideally be placed. The golden ratio can also be used to help oral surgeons and orthodontists bring jaws into alignment for esthetic purposes and function.
Relationship Between Tooth Height and Width
The golden ratio has recently been found to be present between the width and height of the central incisors. When the incisors are ideal in height and width, the width of the two teeth combined will be equal to 1.618 the size of the height of each tooth. This discovery has been helpful in the fabrication of implants and other dental treatments.


So how can we use this information-- in what we just figured out here, which is derived from the half-life-- to figure out how old this sample right over here? So we need to figure out what our initial amount is. So if you want to think about the total number of potassium-40s that have decayed since this was kind of stuck in the lava.
It might be 1 gram, kilogram, 5 grams-- whatever it might be-- whatever we start with, we take e to the negative k times 1.25 billion years. So you get the natural log of 1/2-- we don't have that N0 there anymore-- is equal to the natural log of this thing.
The natural log is just saying-- to what power do I have to raise e to get e to the negative k times 1.25 billion? k is equal to the natural log of 1/2 times negative 1.25 times 10 to the ninth power.
So you get 1 over this quantity, which is 1 plus 0.01 over the 11%. And then, if you want to solve for t, you want to take the natural log of both sides. And then, to solve for t, you divide both sides by negative k. And you can see, this a little bit cumbersome mathematically, but we're getting to the answer. The mathematics really is something that you would see in high school.
So we got the natural log of 1 over 1 plus 0.01 over 0.11 over negative k. We're just dividing both sides of this equation by negative k. So let's take the natural log of our previous answer. If you saw a sample that had this ratio of argon-40 to potassium-40, you would actually be able to do that high school mathematics.
So we could actually generalize this if we were talking about some other radioactive substance.
And now let's think about a situation-- now that we've figured out a k-- let's think about a situation where we find in some sample-- so let's say the potassium that we find is 1 milligram. And usually, these aren't measured directly, and you really care about the relative amounts.
But since floods jumble materials of different origins and ages together, that meant the scientists had to date dozens of different minerals.
The youngest crystal in the footprint layer would represent the oldest possible age for the prints; the oldest crystal in the layer above it would represent the youngest they could be.
And so we could make this as over 1.25 times 10 to the ninth. If I have a natural log of b-- we know from our logarithm properties, this is the same thing as the natural log of b to the a power.
So the negative natural log of 1/2 is the same thing as the natural log of 1/2 to the negative 1 power. Anything to the negative power is just its multiplicative inverse. So negative natural log of 1 half is just the natural log of 2 over here. It's essentially the natural log of 2 over the half-life of the substance.
In this video, I want to go through a concrete example.


There are many variations on the game of Poker. In the most basic game, “Straight Poker“, each player is dealt five cards. The player with the “best” hand wins. To determine a winner, the hands are ranked from highest to lowest, with the hands that are less common being highest. The hands that are recognized in straight poker (in no particular order) are:
Note that C = Clubs, D = Diamonds, H = Hearts and S = Spades. Example: QS = Queen of Spades.
A Straight Flush
Any five sequential cards of the same suit; note that A 2 3 4 5 is considered sequential as is 10 J Q K A
Any five sequential cards, ignoring suit
Four of a Kind
Four of the same rank, for example 7C 7D 7H 7S 3D
Three of a Kind
Three cards of the same rank and two unrelated cards, for example 9C 9H 9D QS AC
Two cards of the same rank and three other cards which do not combine with any of the others to make a different hand, for example 3C 3H AC QH 10D
Two cards of one rank, two cards of another rank and a third of a different rank
Three cards of one rank and two of another
Five non-sequential cards of the same suit, for example AD 5D 6D JD KD
Five cards which do not meet any of the combinations above, for example AH 6S QD JC 5C
How many different five card hands are possible in one deal?
How many different ways can we select a hand which is a Royal Flush? (Hint: How many choices are there for the suit? Once that has been chosen, how man choices are there for the lowest card? After that, how many choices do we have for the remaining four cards? Does the multiplication or addition priciple appy here?)
What is the probability of drawing a straight flush in one draw of a five-card poker hand? (Hint: Remember that the probability of an event occuring is the ratio of the number of ways the event can occur to the number possible outcomes. In this case, that’s the number of ways of drawing a straight flush divided by the number of ways of drawing five cards.)
By calculating the number of ways a player can draw each of the hands, rank the following hands according to their relative frequency: Straight Flush, Straight, Four of a Kind, Three of a Kind, One Pair, Two Pairs, Full House and Flush.


One of the possibilities we have available for calculations involves comparing one statistic to itself or another statistic over time. Here I'll show an example of comparing Gross Income to itself, one month earlier. This type of stat calculation is often used to track growth.
Starting at "More" and then "Statistics" from the main menu bar:
Click "Add Stat" to begin.
Set the stat name and make sure the time period matches the time period of the statistics you want to use in the calculation, then change the Values option to "Calculation" and click "Set up Data".
Now I select the first statistic I need to use in the calculation - Gross Income, and click "Start".
The first statistic I'm using now displays below under "Calculation Steps". I now select Gross Income again, but mark "How many Days ago" as 6, and click the division button.
Now if I use the regular save button, the calculation will kick in on the next time period where the Gross Income stat is entered - and Gross Income Comparison will be automatically entered for you (providing there is information from six months earlier to compare against).
The Save & Recalculate All Values button will go back through any prior weeks where Gross Income can be compared against six months earlier, and then continue forward.


For children that struggle with math, having the support of an experienced math tutor is essential. Effective math tutoring involves both teaching the solutions to math problems and the structure of how math works. This way, the math students can get a better grasp of the core concepts.
There are specific traits that anyone that wishes to become a math teacher or tutor on the side should possess for their success. Those with these traits can teach more efficiently and convey the world of math to students in the most efficient manner. Below are the common traits of a good math tutor that you need to consider.
For one to be a good teacher, they need to be fully prepared and versed on the subject they will be teaching. Any successful math tutor knows the material and is prepared to answer questions when they come up. Moreover, effective preparation allows the tutor to switch gears when needed and provide proper alternatives for problem-solving.
One an important skill that every math tutor needs to possess is the ability to explain more challenging concepts in a simple and clear manner. The use of a simple and easy to understand language can assist a student feel at ease and take the next step to solve the problem. Furthermore, a good teacher should be able to summarize the idea quickly and succinctly.
Innovative: For math tutors, innovation means coming with the different and refreshing ways to assist students to understand the concepts so they can solve the problems. A tutor can use such methods like flash cards, extra notes and other such things to shake up the sessions and help elicit better learning.
Communication: Efficient math tutoring is not merely about explaining the subject to the student but also involves to listening to them. The a math tutor can understand the students’ issues better and come up with a better plan to help them. By speaking clearly and concisely, the tutor can explain the concepts needed to assist the student. By intently listening to the student, the teacher can help them in a manner that best suits their needs.
Persistence is the crucial trait that every tutor should possess. By continuing to present the information, listening to the student and providing innovative learning ways, the tutor will eventually get to help almost everyone they teach to understand the crucial concepts of math.
Having passion for teaching, the student’s success, and the subject is a vital part of being a great math tutor. To retain the student’s interest; there is the need for the passion if teaching to be there.
A a good tutor can get information from their student to help better demonstrate their knowledge. Moreover, building an open atmosphere that allows the student to ask questions is essential in developing a good math teacher/student relationship.


Count and Color 1 through 10 - Each of the ten worksheets in this set focus on a different number from and provide students with practice counting as well as enhancement of fine motor skills when they trace and print each number and number word. Writing in Math Class Teachers incorporate writing in math class to help students reflect on their learning, deepen their understanding of important concepts by explaining and providing examples of those concepts, and make important connections to real-life applications of the math they are learning.
Number Match - Draw a line to match objects to numbers. Draw a picture and label. While important to work on, such efforts should not deny a full math education to otherwise capable students. By attending to the simple steps of self-verbalizing, they can monitor more of their attentional slips and careless errors.
Adaptive behavior of children who exhibit specific arithmetic disabilities and associated neuropsychological abilities and deficits in BP Rourke Et Neuropsychology of learning disabilities. In many school systems, special education services are provided almost exclusively on the basis of children's reading disabilities.
Three more fairies came to sit with them on the flower. Respond to the content. Students may respond to repeated failure with withdrawal of effort, lowered self-esteem, and avoidance behaviors. How would you like to react.
Count and Color Dragon Spots - Students will count to 20 and make two pictures look alike. Equally important is frequently asking students to verbalize what they are doing.
To add zest, some of the problems can be answered incorrectly and a goal can be to find the "bad eggs. Read Math Out Loud for more reflection on the connection between speaking and writing. Discuss beforehand in small groups; write in pairs or small groups; respond to writing in small groups.
Developing numeration concepts and skills. Language disabilities, even subtle ones, can interfere with math learning. Counting Practice from - Students will count items in each group and circle the number that tells how many. Counting, Reading, and Writing Numbers Within 10 Chimpanzee Numbers 5 and 6 - Students will draw bananas for the chimps and practice printing the numbers 5 and 6.
Writing About Math. Writing about math can be a very positive and fruitful learning experience.
Here's a look at some of the benefits; a variety of writing categories and topics; and suggestions for creating a positive environment for writing about math. Writing About Math. Writing about math can be a very positive and fruitful learning experience.
Here's a look at some of the benefits; a variety of writing categories and topics; and suggestions for creating a positive environment for writing about math. math writing prompts You can use these writing prompts for writing practice in preparation for the spring TELPAS.
You have the option of choosing one of the two writing prompts listed for a two-week period and allow students to complete their compositions over the two weeks listed OR you can have the students write on a prompt a week, so that they complete two prompts in two weeks.
During Math Writing, students express and articulate their thinking and understanding by working on a math problem or math concept using pictures, numbers, and words, and occasionally by creating problems of their own.
Featured Topic: Writing in Math Class. Teachers incorporate writing in math class to help students reflect on their learning, deepen their understanding of important concepts by explaining and providing examples of those concepts, and make important connections to real-life applications of the math.
Teach the topics that shape our students' lives. Whether you're looking for a text, a webinar or a grab-and-go lesson, these resources will help your students explore identity and diversity, recognize injustice and learn to take action.Math writing topics


By Jacqmin-Cadda H.
Read Online or Download Analysis of left-censored longitudinal data with application to viral load in HIV infection PDF
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Additional resources for Analysis of left-censored longitudinal data with application to viral load in HIV infection
It is an estimate of the assumed common variance of the two populations. 2 Let Multivariate test be the mean vector in sample 1 = and similarly for sample 2. We need to test the significance of the difference between and . We will consider first the case where the covariance matrix, which is assumed to be the same in the two samples, is known. Then we will consider the case where an estimate of the covariance matrix needs to be used. 2. using the chi-squared test. is unknown If the covariance matrix is not known, it is estimated using the covariance matrices within each group but pooled.
3 Reliability-Coefficient In cases where composite scales are developed, this measure is useful to assess the reliability of the scales. 9 for market research are typically not sufficient to warrant further analyses using these composite scales. 6 illustrates how to compute means and the correlation matrix for a list of variables in SAS. 7. 9. 9% of the variance. The rotated factor patterm shows that Factor 1 groups variables IT1, IT3, IT4, IT6 and IT7, while Factor 2 reflects variables IT5, IT8 and IT9.
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Analysis of left-censored longitudinal data with application to viral load in HIV infection by Jacqmin-Cadda H.


Mathematics Curriculum Statement
At Marine Park Primary we recognise that Mathematics is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. We aim to provide a high- quality mathematics education with a mastery approach so that all children:
- become fluent in the fundamentals of mathematics.
- reason mathematically.
- can solve problems by applying their mathematics. (National Curriculum 2014)
We intend on delivering a curriculum which:
- Allows children to be a part of creative and engaging lessons that will give them a range of opportunities to explore mathematics following a mastery curriculum approach.
- Gives each pupil a chance to believe in themselves as mathematicians and develop the power of resilience and perseverance when faced with mathematical challenges.
- Recognises that mathematics underpins much of our daily lives and therefore is of paramount importance in order that children aspire and become successful in the next stages of their learning.
- Engages all children and entitles them to the same quality of teaching and learning opportunities, striving to achieve their potential.
- Makes rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems.
- Provides equal opportunities for children to apply their mathematical knowledge to other subjects (cross-curricular links).
- Enable children to develop a passion for mathematics and celebrate maths in all areas of life. From home life to outdoor learning involving mathematics and allow these opportunities to be discussed and celebrated within the curriculum.
- Provide opportunities for children to apply disciplinary literacy within the subject discipline of Maths. Pupils will be taught how to read, write and talk like a mathematician. We recognise that literacy skills are both generic and subject specific and value the contribution maths can make to these skills. Pupils will have opportunities to read widely to gain knowledge and by the end of their primary education, will leave with the skills to read fluently and with confidence in maths, supporting future success at the next stage of their education, and beyond. Our Maths curriculum plays an integral part in our overarching P.A.C.E curriculum, which follows our school vision of “Education in its broadest sense has the ability to change and improve people’s lives.”
Follow the link below to read the statement in full


Dear students, Read the instructions carefully
Here you can download questions for practice of 1st Bi-Monthly.
The file includes the following:
- Theory questions from Past Papers
- Numerical Problems from Past Papers
- Extra Numerical Problems for Complete preparation
I recommend you all that after completing theory and numerical problems you should work hard in extra numerical problems, this will help you to memorize the concept and you will be able to solve these problems anytime throughout the year.
This all material is related to Chapter 2.
For Chapter 1, just prepare the following:
- The MCQs Worksheet which was taken as test. (Solution is already uploaded)
- Just read the 1st Chap notes and collect the important points only. No need of rattafication now. (*Don’t memorize the definitions of physical, base or derived quantities and etc., just go through it)
- Do read the Contribution of Muslim Scientists from the Textbook.
Start your practice now and consult your teacher in case of any confusion.
Best of Luck.


I am a first year PhD student at the Applied Algebra & Geometry Group at Czech Technical University in Prague, supervised by Prof. Tomas Pajdla jointly with Prof. Konrad Schindler via The ELLIS PhD Program. I received my Specialist degree (eq. to MSc.) in Applied Mathematics and Computer Science from Lomonosov Moscow State University, Moscow, Russia in 2014.
My research interests lie in Computer Vision, Multiple View Geometry, and Deep Learning.
E-mail: diana.sungatullina at cvut.cz


Sudoku...but with zombies
Instead of numbers, we use the dead. In Sudoku, numbers are arbitrary—there's no adding, subtracting, etc. So why not flex some logographical muscles. And why not zombies.
There are nine zombies: one for each number in a typical 9x9 Sudoku game. The challenge is to step away from numerical logic and exercise visual thinking. There are four levels of difficultly. Hints are available, and numbers can be turned on. But the only score is time, and the only competition is your will.


Colour Count Cards
This listing is for 180 Colour Count Cards, focusing on the skill of number recognition and counting numbers 1-10. The skill of colouring is encouraged using this resource as your student will colour in the correct number of images as set out by the number noted on each card.
This is a two level resource. Level one has the correct number and image amount match. Level two differentiates between number of images and the number requested to colour.
Print on black/white paper and encourage your student to colour in the correct amount.
When working with young children or children with additional needs who may find number concepts difficult, I encourage them to point at each image to count them out before we begin to colour.
54 page resource with 180 Colour Count Cards.


If one day of the year screams “Party!” in math class, that day is March 14. Each year on 3/14, teachers across the globe take a break from the normal routine to plan a special celebration in honour of Pi, or the number 3.14. March 14 also happens to be the birth date of Albert Einstein which makes the day an extra special one for planning math challenges and math fun.
We here at DGPS celebrated Pi day and Albert Einstein’s birthday on 15/5/2017. All students of the primary from Year 1-6 had to come dressed as Albert Einstein wearing wigs, moustaches, spectacles and pictures of pies (or the Pi symbol). Students were also asked to carry a hula-hoop for the celebration. Students of all ages put in a lot of effort in dressing up as young Einsteins. During the school assembly all the students dressed as Einstein were called upfront for a group photograph. There were also many competitions for different grades like Year 1-3 had the hula hoop competition and Year 6 had to try writing the full number form for Pi. Students of Year 1-3 had to carry snacks in their lunch boxes which were in circular shape denoting a pie. It was a day to be remembered and celebrated


Chessex 12mm Speckled Strawberry 36ct D6 Set (25904)
Dice are great tools for random variable determinations in Board Games, Table Top Games, Role Playing Games, Party Games, etc. Dice can also be great additions to math tutoring from simple addition/subtraction to primary multiplication/division.
Die Set Contains 36D6
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