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## Thursday, May 31, 2012 ... /////

### Matrix theory: a novel alternative to second quantization etc.

I still consider Matrix theory (BFSS, 1996) one of the most conceptually original developments in theoretical physics of the last 20 years.

It is a relatively unusual way to describe physics in the 11-dimensional asymptotically flat vacuum of M-theory – and in other sectors of string theory. The physical phenomena are completely equivalent as in other descriptions; but the way how they're encoded in the mathematics looks very different.

There are several natural pedagogical ways to get to Matrix theory and I will try to sketch the following three of them:

1. try to study some obviously beautiful quantum mechanical models in extreme limits (the infinite number of colors) and try to find a simplified description of this limit;
2. start with M-theory whose explicit equations weren't known before BFSS 1996 and transform it via dualities and tricks into something that you may describe;
3. try to invent a completely new framework (different from quantum field theory and second quantization) to describe multiparticle states and interactions between the particles, among other things, that may lead to the same kind of physics.
It's the last approach that was used in the title of this blog entry.

Some hindsight is needed to describe the situation in this way and some of this hindsight is often missing in the pioneering papers; BFSS 1996 is no exception. But we have it now and we may look at the situation from various perspectives and avoid some speculative comments that were discussed at the beginning but that were found to be invalid.

The third approach seems to be the most conceptually valuable one but let me start from the first approach.

Maximally supersymmetric gauge quantum mechanics

This year, the $\NNN=4$ gauge theory in $d=4$ celebrated its 35th birthday. It's obviously a beautiful theory, the most supersymmetric non-gravitational theory in $d=4$ you may find. It may be derived as a dimensional reduction of the 10-dimensional supersymmetric theory; as a low-energy limit of the dynamics of D3-branes; and in many other ways.

It has many cool symmetries including the $SL(2,\ZZ)$ S-duality group (which is able to exchange electricity and magnetism, the weak coupling and the strong coupling, electrically charged particles with magnetic monopoles), the superconformal symmetry, the dual superconformal symmetry, their infinite-dimensional extended and completed union, the Yangian, and others. For those reasons, various scattering amplitudes, while nonzero, dramatically simplify relatively to the form you would expect in a field theory with a similar Lagrangian but fewer supercharges. Many of those features may be exposed in the twistor-based approaches.

As I have mentioned above, this 4-dimensional theory may be viewed as the dimensional reduction of the 10-dimensional gauge theory whose action looks like this:$\eq{ S &= \int \dd^{10} x\, \LL,\\ \LL &={\rm Tr}\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i\bar\Psi D^\mu \gamma_\mu \Psi \right] }$ It's an ordinary Yang-Mills Lagrangian for a gauge field associated with a gauge group we will take to be $U(N)$, i.e. $SU(N)\times U(1)$ including the $U(1)$. The unitary groups are the least complicated infinite family of simple compact Lie groups, and for a Majorana-Weyl (real-and-chiral) fermion field (covariant derivatives have to be used). This innocent combination is enough to produce an exactly supersymmetric classical (or effective) field theory. It's non-renormalizable in $d=10$ but its dimensional reductions to $d\leq 4$ are renormalizable.

By dimensional reductions, we mean that the "excessive" dimensions are compactified on a torus – they are made periodic – and the radii of the torus are sent to zero. For this reason, finite-energy excitations are forced to be constant in these dimensions. Moreover, the rotational symmetry between the compactified and uncompactified dimensions becomes badly broken. It is no longer sensible to consider the components of the gauge field $A_\mu$ along the compactified dimensions $\mu$ to be components of a gauge field. It's better to consider them scalars.
Off-topic but fun: A reader has pointed out that in the new version of the Hartle-Hawking-Hertog paper, your humble correspondent and HB are thanked for the innocent sign error whose impact so far looks isolated (but I still believe that there must be some other errors unless the paper shows some really important loophole in the Ehrenfest "theorem" way of thinking about the evolution in quantum gravity).
For example, if we dimensionally reduce from $d=10$ to $d=4$, six of the components of the gauge field (whole matrices in the adjoint of the gauge group) will become six scalars: they're the source of the $SO(6)\sim SU(4)$ "R-symmetry" of the $d=4$ gauge theory. Once we get some scalars, some gauge couplings for fermions also produce the Yukawa couplings as a result. It's convenient to decompose the fields under the unbroken $SO(d-1,1)$ i.e. $SO(3,1)$ Lorentz symmetry acting on the large dimensions which is a subgroup of the original $SO(9,1)$ symmetry.

This 2007 model was called Toyota Matrix M-theory. No kidding. Those East German engineers who refused the German reunification plan to produce a competing car, The Trouble With Trabant: Not Even Wrong.

The 35th anniversary article discussed the reduction to $d=4$ which gives the prettiest "descendant" of the 10-dimensional gauge theory, one that has the most amazing properties at the quantum level. But we may dimensionally reduce the theory to lower dimensions, too. For example, we may reduce it to $d=1$ spacetime dimension. That's the minimum you may naively think of even though the reduction to $d=0$ may actually be important, too.

What does it mean to have a $d=1$ quantum field theory? It has one spacetime dimension. We want at least one dimension of time. So a simple counting, $1-1=0$, implies that time is the only dimension on which our fields depend. What does it mean? Well, it means that it's not a full-fledged "quantum field theory" anymore. It's a quantum mechanical model. Instead of fields such as $\Phi^6(x,y,z,t)$, we have "fields" that may be renamed as $X^6(t)$. They're quantum observables. You see that the field content is completely analogous to the textbook models of non-relativistic quantum mechanics.

In similar contexts, "quantum mechanics" is often used as a synonym for a 1-dimensional quantum field theory. (Of course, "quantum mechanics" also means – and mainly means – the general framework of modern physics given by the Copenhagen school's postulates that all quantum theories, including higher-dimensional quantum field theories and string theory, obey.)

You might argue that quantum mechanics has advantages over higher-dimensional quantum field theories. It doesn't need any renormalization and similar stuff. The wave function is just $\psi(x_1,x_2,\dots)$ and obeys some partial differential equations in a finite number of variables. There's no room for infinities. Everything looks simple. Everything that annoys you about regularization, renormalization, renormalization group etc. is fundamentally absent here. (Well, it's not quite true because some of these things have nontrivial analogues in $d=1$ but it's still true that you may choose a fundamentally unequivocal, finite description of quantum mechanical models.)

Fine, so what is the quantum mechanical model we are considering here? Its Lagrangian looks like this:$\eq{ S &= \int \dd t\,\LL\\ L &= \frac{1}{2g}{\rm Tr} \zav{ \dot X^i \dot X^i + 2\theta^T\dot\theta+\frac{1}{2}[X^i,X^j]^2-2\theta^T \gamma_i[\theta,X^i] } }$ You see that the Lagrangian contains some Klein-Gordon kinetic terms for the nine scalars $X^i$, Dirac-like kinetic terms for the 16 components of the fermions $\theta$, a quartic commutator-squared-based potential for $X^i$ that arises from the quartic terms in $F_{\mu\nu}F^{\mu\nu}$ in the 10-dimensional gauge theory, and a Yukawa term. All the terms are traces over the $N\times N$ matrices, just like in any other version of the $U(N)$ gauge theory with fields in the adjoint.

I should be more explicit in explaining how many degrees of freedom the theory actually has. Well, all the fields transform as Hermitian matrices – i.e. adjoint representation – under the $U(N)$ gauge group. Moreover, the bosonic fields $X^i$ arising from $A_\mu$ carry an extra index $i=1,2,\dots,9$ which corresponds to the 9 spatial dimensions we dimensionally reduced. The remaining component $A_0$ may be (but doesn't have to be) set to $A_0=0$ by the gauge redundancy.

So the fields $X^i$ are nine Hermitian $N\times N$ matrices. Because we ultimately want to study the quantum theory, all the components of these matrices are operators. Much like in normal non-relativistic quantum mechanics, we find out that there are canonical momenta $\Pi^i$, also nine Hermitian matrices, and they have the commutators$[X^i_{kl},\Pi^j_{mn} ] = i\hbar \delta^{ij} \delta_{nk}\delta_{lm}$ where the pairing of the indices is determined by the $U(N)$ symmetry: just appreciate that in the index pairs $kl,mn$, one index is "lower" and one is "upper" (we suppressed the difference to streamline the notation only) and each Kronecker delta-symbol has to have one upper index and one lower index, too. I've restored $\hbar$ to emphasize that this is a "modest" extension of the undergraduate quantum mechanics models.

Now, we must also discuss the fermionic degrees of freedom $\theta$. They arose from the Majorana-Weyl (real chiral) spinor of the 10-dimensional gauge theory so they include 16 real components that now transform as a 16-component real spinor of $SO(9)$, the manifest rotational symmetry acting on the scalars $X^i$ as well. When promoted to the adjoint of $U(N)$, each component becomes a Hermitian matric again. Much like in the usual quantization of Dirac fields, these degrees of freedom are Grassmann i.e. fermionic variables that are canonical momenta to themselves (because the kinetic term in the Lagrangian only contains one time derivative):$\{ \theta^a_{kl},\theta^b_{mn} \} = \hbar \delta^{ab}\delta_{nk}\delta_{lm}$ where $a,b=1,2,\dots, 16$ and the remaining commutators (all the anticommutators have already been written because they're only appropriate for pairs of fermionic objects) vanish.

So except for the high number of dimensions indicated by the indices such as $i,j$ that take 9 values and except for the extra degeneracy of the coordinates given by the gauge indices $k,l,m,n=1,2,\dots,N$, and except for the extra fermionic variables, this is a pretty normal quantum mechanical model similar to the non-relativistic toy models you know from your undergraduate course of quantum mechanics (or its equivalent if you are self-taught).

We really want to say that the state of the physical system described by this $U(N)$ quantum mechanical model may be encoded in a wave function $\psi(x,\theta)$. How many variables does it depend on?

Well, there are $9N^2$ real components in the matrices $X^i_{mn}$: note that the Hermiticity reduces the number of real (i.e. Hermitian, when treated as operators on the Hilbert space) components $2N^2$ exactly to one-half of that. The canonical momenta $\Pi^i_{mn}$ are just $-i\hbar$ times the derivatives with respect to the variables $X^i_{mn}$. And then we have the fermionic coordinates; out of the $16N^2$ components of $\theta^a_{mn}$, one-half of them are taken to be the coordinates and the remaining one-half may be taken to be the canonical momenta. So we have $8N^2$ Grassmann variables.

Because the Taylor expansion in the Grassmann variables terminates, we may rewrite the wave function in terms of components and their number is simply $2^{8N^2}$ because each of the $8N^2$ variables is either present or absent in the given monomial in the power law expansion. For example, for $N=5$, we have $200$ fermionic coordinates and the number of component functions is therefore $2^{200}\sim 10^{60}$. Each of these component functions depends on $9N^2=225$ real bosonic variables.

To solve the quantum mechanical model for $N=5$ by the brute force, you simply calculate some coupled linear partial differential equations for $10^{60}$ functions that depend on 225 variables. It's a trivial task you've been doing in your kindergarten, at least in principle, and you may totally avoid any complications with regularization and renormalization.

I am partly kidding because the number of components and coordinates looks horrifyingly high and for realistic simulations of the large $N$ limit, you really don't want $N=5$ but something like $N=100$. But it's just a problem for the naive brute-force approaches. If you understand the model well and you know some maths, you may find methods to calculate properties of the model which are doable on one line or two. Or several pages. It shouldn't shock you that the naive brute-force simulation is neither the only way nor the recommended way to attack a seemingly mathematically difficult problem although this claim might be controversial among many laymen including physics fans.

If it makes you happier, the number of independent variables on which the wave function for a physical state depends is effectively $8N^2$ and not $9N^2$ because all physical states have to be invariant under the $N^2$ generators of $U(N)$; well, only $N^2-1$ of these conditions are nontrivial. After all, it's a gauge theory and therefore this gauge-invariance condition for the physical states comes from varying the action with respect to $A_0$: the corresponding charge $Q=J_0$ – which is an $N\times N$ matrix – has to vanish.

Fine, we know a certain supersymmetric quantum mechanical model – a quantum field theory in 0+1 dimensions – that may be in principle studied. Remarkably enough, the union of the Hilbert spaces from all these $U(N)$ models for all integer values of $N$ gives you the Hilbert space of M-theory in 11 dimensions! It contains everything you expect in a consistent theory of quantum gravity, including gravitons and their superpartners, their interactions, gravitational force obeying the equivalence principle, the 10+1-dimensional Lorentz symmetry, evaporating black holes that preserve the information, M2-branes and M5-branes (and strings and D-branes in compactified, stringy versions of the matrix model, including the right stringy interactions), and other things.

The quantum mechanical model was known before it was conjectured by BFSS that the model was relevant for M-theory in 11 dimensions. It was known as the description of D0-branes, point-like particles moving in the 10-dimensional type IIA string theory, at low energies. Just like D3-branes in type IIB string theory give rise to the $d=4$ 35-years-old gauge theory at long distances, a similar statement holds for the D0-branes.

However, you should only say that you qualitatively understand what the model describes for a finite and small enough value of $N$. When some parameters such as $N$, the size of the matrices, are sent to infinity, the most important (and lowest-energy) objects and processes in the theory may describe something that admits (if not requires) a qualitatively different language. In this case, the particles originally viewed as D0-branes in a 10-dimensional string theory (where they're just a part of physics and interact with other objects) actually become the complete description of an 11-dimensional theory of quantum gravity, namely M-theory.

This conclusion may look shocking – how an ordinary undergraduate quantum mechanical model (which isn't even a quantum field theory, so we don't see how it could contain multiparticle states with indistinguishable particles and/or Lorentz invariance) could contain all the marvels of quantum gravity including Lorentz invariance, graviton scattering, equivalence principle, black holes? But remarkably enough, it does.

I will postpone the explanation why the matrix model includes the right building blocks of M-theory and why there's no detectable contradiction to the final section of this blog entry – about the conceptual leap. But before we get there, let us look at a proof that the matrix model is correct.

Seiberg's derivation of Matrix theory in DLCQ

The October 1997 proof was originally presented by Nathan Seiberg and, in a slightly less complete version (but more extended in some other directions we're not interested in here), by Ashoke Sen. So why is the matrix model correct?

Consider M-theory in the asymptotically flat 11-dimensional Minkowski spacetime parameterized by $X^0\dots X^{10}$. We want to find an explicit Lagrangian that describes all the objects in this 11-dimensional theory of quantum gravity – which may be obtained as the strong coupling limit of type IIA string theory or heterotic-E string theory.

It's useful to single out two light-like directions or coordinates,$X^\pm = \frac{X^{0}\pm X^{10}}{\sqrt{2}},$ and use $X^\pm$ instead of $X^0$ and $X^{10}$. The light-like coordinates are very useful in relativity. For example, the expression $t^2-z^2$ appearing in the invariants may be rewritten as $(t+z)(t-z)$: the difference of two terms may be written as one term that factorizes. Correspondingly, it's useful to consider the light-like components of the momentum etc. such as $P^+$.

The states in M-theory have continuous, non-negative values of $P^+$. However, it's very convenient to put the theory in a certain "box" for the momentum to become discrete. However, we don't want the box to matter so its size has to be sent to infinity. More explicitly, we want to make the following identification of the light-like coordinate $X^-$:$X^- \approx X^- + 2\pi R.$ As long as $R$ is huge – imagine that it is hundreds of billions of light years (although the citizens of the 11-dimensional empire don't necessarily use these units for the distances in their completely different world) – the identification doesn't affect the local physics. If a copy of your local experiment is happening hundreds of billions of light years away from you, it shouldn't matter in your lab. We implicitly use some "modest version" of locality in M-theory here.

However, the change of the $X^-$ coordinate isn't a pure translation in space; it contains a translation in time by the same amount. In fact, you should be worried about the consistency of this light-like compactification. If we made this periodic identification for $X^0$, a time-like coordinate, we would create closed time-like curves which would make the theory inconsistent. On the other hand, a spatial compactification of $X^{10}$ would be harmless. What about the marginal case, the light-like compactification?

Well, we may take it as the limit of spatial compactifications that are OK which should be enough for you to believe that it's OK, too. So we're taking an M-theoretical spacetime and identify points that differ by something like the vector$(M,0,0,0,0,0,0,0,0,0,M+\epsilon)$ in the usual coordinates $X^{0\dots 10}$. It's a nearly null interval but it's a little bit spacelike because of the extra $\epsilon$ in the last coordinate. We may scale the huge $M\sim R$ and $\epsilon$ in such a way that the the proper length of the vector above is tiny, much shorter than the 11-dimensional Planck length.

However, if that's so, we may use the Lorentz symmetry of the 11-dimensional M-theory – an assumption supported by lots of evidence, including the Lorentz symmetry of the 11-dimensional supergravity, the low-energy limit of M-theory – and boost the vector above to a simple vector of the form$(0,0,0,0,0,0,0,0,0,0,\epsilon')$ whose last spatial coordinate is much shorter than the 11-dimensional Planck length. Note that the momenta of all the particles in the original M-theory get boosted by a correspondingly dramatic boost. But we know the spacetime in which these boosted objects propagate: it's M-theory compactified on a very short spatial circle given by $\epsilon'$. But it's nothing else than type IIA string theory at a weak coupling!

Moreover, the null component of momentum $P^+$ which is the complementary variable to $X^-$ is quantized because $X^-$ is compact, $P^+ = \frac{N}{R},\quad N\in\ZZ$ We wanted to describe objects with a fixed and finite momentum in the 11-dimensional Planck units, i.e. $P^+$ is finite, and because $R$ needs to be sent to infinity, the integer $N$ has to be sent to infinity for these interesting states, too.

However, as we boosted the light-like interval defining the light-like compactification to a spatial one, we have a new interpretation for $N$: it's the number of units of momentum in the direction of the short compactified coordinate $X^{10}\sim X^{10}+\epsilon'$ we mentioned at some point above. But the momentum in the extra 11th dimension is nothing else than the number of D0-branes. So the original state of M-theory is equivalent, via this boost and in the limit, to a state involving $N$ D0-branes in type IIA string theory.

When you analyze the right role of the $R\to\infty$, $\epsilon'\to 0$ limit, the relevant estimated magnitude of the energy of the D0-branes (as a function of the finite momenta and energies in the original M-theory spacetimes), the precision we need for this energy in the limit, and the states and objects that may be neglected in this limit because they're infinitely times heavier, you will find out that what you need to describe the original states with $P^+=N/R$ in the M-theory spacetime is nothing else than the low-energy limit of the dynamics of $N$ D0-branes in type IIA string theory at a vanishingly low coupling (because of the small $\epsilon'$).

It's nothing else than the supersymmetric quantum mechanics model we have already written above; for some time, the model was also referred to as the DKPS model because of a pre-matrix-theory paper that studied it.

So the Hilbert space of the modestly light-like-compactified M-theory is the direct sum of the Hilbert spaces of the $U(N)$ matrix quantum mechanical models for $N$ D0-branes. The Hamiltonian of the matrix model gets directly translated as the light-like Hamiltonian (generator of evolution in a light-like direction) in M-theory. The relevant value of $N$ for the generic states in M-theory with finite energies is the $N\to\infty$ limit.

As we have seen, there exists a proof that the BFSS matrix model is an equivalent description of M-theory. The proof only relies on the Lorentz invariance of M-theory; its well-known relationship with type IIA string theory; some elementary derivable properties of D0-branes in type IIA string theory; and the irrelevance of the near-light-like compactification if the radius is sufficiently long.

But we may still be shocked by the claim that this seemingly naive non-relativistic quantum mechanical model with 9 bosonic matrices $X^i$ describes an 11-dimensional, and not just 10-dimensional, theory that is moreover Lorentz-invariant in the large $N$ limit and includes gravity, gravitons, gravitinos, multiparticle states with indistinguishable particles, membranes, fivebranes, black holes, and their interactions including all the right loop corrections that could be derived from effective quantum field theories, among many other things.

Can we see that those things are present in the theory? This brings me to the final "megapoint" I want to make, namely that Matrix theory is a really cool and original way to mathematically describe many processes that used to be described – and are still being described, in most cases – by totally different mathematics.

All roads lead to string theory: describing quantum gravity in totally new, revolutionary yet consistent languages

In order to understand what kind of physical phenomena Matrix theory allows and implies, it's useful to start with small values of $N$. For such small values, the M-theoretical physics does depend on the details of the light-like compactification; we're away from the large $N$ limit. However, many things acquire their expected properties already for small $N$.

Start with $N=0$. The degrees of freedom are $0\times 0$ matrices. In other words, there are no degrees of freedom. There is a unique wave function (up to normalization) that depends on them, namely $\psi() = 1$. This state vector may be identified with the vacuum state of M-theory as it carries no energy or momentum.

That was easy: too easy. Let's continue with $N=1$. That's the last matrix model that will be easy and fully solvable, of course. This model should describe all states in M-theory that have $P^+=1/R$, the minimal positive value of the light-like momentum. We are dealing with a $U(1)$ gauge theory in 0+1 dimensions. The bosons $X^i$ and $\theta^a$ are $1\times 1$ matrices so they're not really matrices at all.

Because $U(1)$ is an Abelian group (you know it from electromagnetism), all the commutator terms vanish. So the Hamiltonian is just quadratic, namely$P^0_{\rm BFSS} = \frac{(\Pi^i)^2}{2}.$ That's it. It's just like the non-relativistic kinetic energy – for a particle in 9 spatial dimensions. No potential because the potential terms vanish. No terms from fermions, either. You might be puzzled what this non-relativistic formula has to do with the relativistic physics in 10+1 dimensions. The answer is that the right interpretation of the energy in the quantum mechanical model is $P^-$, a light-like component of the energy-momentum vector that is treated as energy in the light cone quantization.

Massless particles' energy and momentum obeys $(P^0)^2-(P^i)^2 - (P^{10})^2 = 0, \quad i=1,2,\dots ,9$ which may be rewritten, using the light-like components as$2P^+ P^- - (P^i)^2 = 0, \quad P^- = \frac{(P^i)^2}{2P^+}$ which has the simple quadratic, non-relativistic form! So the light-like description which is the ultimate limit of the "infinite momentum frame" which described ultrarelativistic particles is mathematically analogous to non-relativistic physics. Note that $2P^+$ is just a constant in the sector with $P^+=1/R$ i.e. $N=1$.

You could even say that the light-like momentum component $P^+$ plays the role of the non-relativistic mass $m$. Also, if you use the light-like coordinates, you may find a copy of the Galilean group that is embedded, without any deformation, right into the Lorentz group. At low speeds, relativity reduces to non-relativistic physics approximately; but at the speed of light, a copy of non-relativistic physics is embedded in the relativistic one exactly!

We see that the $N=1$ model describes particles with arbitrary values of the nine "totally transverse" components of the momentum $P^i={\rm Tr}(\Pi^i)$. The tenth component treated as a spacelike one, the light-like component $P^+$, is equal to $1/R$ which is a fixed constant. That's OK because we're considering a sector of the Hilbert space only; the states with higher values of $P^+$ will be found in the models with higher values of $N$.

The remaining component, i.e. the other light-like component $P^-$, is treated as energy and the BFSS Hamiltonian gives us the right dispersion relations already for $N=1$. We shouldn't forget about the fermionic degrees of freedom. There are 16 Hermitian objects $\theta^a$ transforming as the real spinor of $SO(9)$. They're serving as momenta to themselves so you may combine them into 8 Grassmann variables and their 8 Grassmann derivatives.

By Taylor expanding the wave function in the 8 Grassmann variables, you get $2^8=256$ components. That's exactly the number of polarizations in the graviton supermultiplet of 11-dimensional supergravity. Recall that these get decomposed to 128 fermionic components (of the gravitino) and 128=44+84 bosonic components including the physical polarizations of the graviton as well as those of the C-field three-form potential.

One may actually define the 32 real supercharges we expect in M-theory. 16 of them are nothing else than the 16 supercharges of the maximally supersymmetric gauge theory; the remaining 16 of them are trivial or kinematical and these generators are given simply by ${\rm Tr}(\theta^a)$. This decomposition of supercharges to the "complicated dynamical" ones (one-half) and the "trivial kinematic" ones (the other half) is a consequence of the light cone treatment. To get from the maximally supersymmetric gauge theory to the maximally supersymmetric supergravity, we needed to double the number of supercharges (from 16 to 32) and the existence of the kinematic or trivial supercharges did the job for us.

We have seen that already the non-interacting $N=1$ model knows about the number (and representations under rotations) of the components of the supergraviton multiplet and the right dispersion relation (relationship between energy and momentum). That's quite a success for such a simple model. Note that we only have one particle; our derivations tell us that there can't exist any multiparticle or otherwise complicated states that would have the minimal value of the longitudinal momentum.

Screws and Nuts by Mandrage, a Pilsner band, has been among the Czech radios' top 3 songs for more than half a year. The observation that men and women are like screws and nuts obviously has a sexual connotation but most physicists don't appreciate that the Czech word for nuts in this sense, "matice", is the same word as one for "matrices", and moreover, matrix strings are screwing around the matrices as the monodromy introduces a permutation. This breathtaking ignorance of rudimentary Czech among most physicists and their lacking sense of humor has led to the widespread but flawed terminology "matrix strings" for what should actually be called "screwing strings". ;-)

But the real fun only starts with $N=2$ when we obtain the first real matrices. Let us talk about all the values $N\gt 1$ simultaneously. The first observation we want to make is that the $U(N)$ matrix model does contain states composed of several objects.

In quantum field theory, one may create multiparticle states as$\ket\psi = a^\dagger_\text{here} a^\dagger_\text{there}\ket 0.$ One simply acts with several creation operators at different places to create several objects. As long as the places are sufficiently distant, the Hilbert spaces associated with the places are independent and the whole Hilbert space isn't far from a tensor product of the Hilbert spaces associated with the individual regions.

However, the matrix model isn't a quantum field theory. It doesn't have any creation and annihilation operators for particles. How can it possibly contain multiparticle states? The right indistinguishability conditions and statistics? And other things? It looks like some fully well-defined but seemingly non-relativistic undergraduate quantum mechanical model.

The matrices are the answer. A funny thing about large (or any) matrices is that they may take special matrix values that are block-diagonal. Imagine that the matrices for all $X^i$ and similarly $\theta^a$ have the following form:$X^i = \pmatrix{ \text{WOLF}^i & \heartsuit^i \\ \heartsuit^{i,\dagger} & \text{BUNNY}^i }$ Here, WOLF is a square matrix that contains the information about positions of particles that make up an object called WOLF. BUNNY is a similar square matrix whose size may be the same or different. And the hearts are some generally rectangular off-block-diagonal blocks.

We see that large enough matrices are capable of including several objects, such as WOLF and BUNNY. The longitudinal momentum $P^+$ is linked to the linear size of the matrices so $P^+$ of the WOLF-BUNNY composite states is simply the sum of the values of $P^+$ that these two animals would carry separately. And indeed, if we set $\heartsuit=0$ for a while, the Hamiltonian for the large matrices simply reduces to the sum of the Hamiltonians for both animals: just recall how a trace of a block-diagonal matrix behaves. The two animals evolve independently.

What about the heart? Because of the commutator terms in the Lagrangian, the off-diagonal elements of $X^i$ in the off-block-diagonal rectangles $\heartsuit$ (and their Hermitian conjugate ones) actually acquire a mass. By a mass, we mean the term in the Hamiltonian that you would associate with a massive field such as the W-boson field, $m^2 \cdot \heartsuit^\dagger \heartsuit/2$. The value of the mass is actually proportional to the distance between WOLF and BUNNY, assuming that their internal sizes are much smaller than their relative distance.

In other words, these off-block-diagonal matrix elements become harmonic oscillators with huge frequencies. The further WOLF and BUNNY are, the higher frequencies we face, the further the equally spaced energy levels of the harmonic oscillators are from each other, and the more we may neglect the possibility that the harmonic oscillator could actually be found in an excited state. We may approximate the $\heartsuit$ harmonic oscillators by assuming that they're in the ground state most of the time (or, classically, at the $\heartsuit=0$ point). Only when WOLF and BUNNY get close enough to each other, a significant chance that the harmonic oscillator gets excited emerges. The virtual effects of this harmonic oscillator – its propagators – transmit influences between WOLF and BUNNY. Of course, the interaction between them is love, a reason I picked $\heartsuit$. ;-)

Once again, we see that the matrices may have a block-diagonal form. The blocks on the diagonal remember the coordinates of particles in the subsystems while the off-block-diagonal blocks have $X$ mostly equal to zero but their ability to get excited or nonzero is actually the one and only source of the interactions between the subsystems!

I won't be proving that the interactions induced in this way are exactly those you expect from M-theory, e.g. that they reduce to supergravity at very low energies. But it's true. Instead, let us discuss some simpler but still very interesting issues.

We said that there were many harmonic oscillators with frequencies scaling with the distance between WOLF and BUNNY. Don't they give us huge zero-point energies $\hbar\omega/2$ that would also depend on the WOLF-BUNNY distance which would drive WOLF and BUNNY towards each other with a constant force? The answer is that such terms really do arise but they get exactly canceled!

The other compensating sources we haven't considered are the fermions. The off-block-diagonal elements of those 16 $\theta^a$ also behave as harmonic oscillators, but the fermionic ones. The zero-point energies will scale as $-\hbar\omega/2$ where $\omega$ is again proportional to the BUNNY-WOLF distance. And if you're careful about all the factors, you will find out that those 16 Hermitian fermions exactly cancel the bosonic oscillators from 8 matrices $X^i$ and that's exactly the right number because the ninth one is the longitudinal one, parallel to the BUNNY-WOLF separation, and it doesn't get massive at all. Supersymmetry actually plays a key role in making the individual separated objects independent and keeping their force low at very high separation. It's rather unlikely to find a non-supersymmetric matrix model that would have the same desirable properties.

That's great. We have multi-particle states. But do we also have gravitons and gravitinos with $P^+\gt 1/R$? The answer is Yes. They have $P^+=N/R$ and are represented by the $N\times N$ blocks. In other words, we may find such a graviton as the only object in a state of the $U(N)$ matrix model. Now, $U(N)$ and its adjoint representation roughly decomposes to $SU(N)\times U(1)$. The $U(1)$ part behaves much like the $N=1$ model, except that the value of $P^-$, our Hamiltonian, gets correctly rescaled: recall that $P^-$ is inversely proportional to $P^+$ and that's what a calculation yields (after the $P^i$ momentum is fairly divided among the $N$ entries to keep the trace constant).

The relative, interacting, $SU(N)$ degrees of freedom describe the relative coordinates of those $N$ D0-branes. And one may prove via index theorems that this model has exactly one state of a vanishing energy: there must exist a mathematically fascinating ground state wave function with a vanishing energy that solves a rather complicated differential equation in $9(N^2-1)$ bosonic variables even though no one can write how this wave function looks like too explicitly (the most understandable argument in favor of the state's existence that I know is based on the interpolation between the BFSS model and screwing string theory which is "more solvable" in the weakly coupled limit). When this state is tensor-multiplied with the degrees of freedom of the $U(1)$ model, we get the right single-graviton or single-gravitino states with all the required polarizations and all the required values of the momenta.

As you have noticed, the matrix model differs from non-relativistic multiparticle quantum mechanics by having the off-diagonal entries of all the matrices $X^i_{mn}$. Otherwise the diagonal entries $X^i_{nn}$ (no summing) behave in a very similar way to $X^i_n$ in non-relativistic quantum mechanical models where $i$ labels a direction in space and $n$ labels a particle. But the BFSS matrix model also yields the off-diagonal entries, whole matrices! There is a sense in which this extension of position operators to matrices is analogous to the conceptual transformation that was imposed by the quantum revolution. But we're doing it at another level: each matrix entry $X^i_{mn}$ in the BFSS model is an operator acting on the Hilbert space; we're just saying that there are many such operators so that one may organize them into new matrices whose matrix indices remotely resemble the indices labeling one of many particles in multiparticle quantum mechanics. But there are two such indices for each matrix of position operators!

Another thing you could worry about is the following issue: are the gravitons indistinguishable particles? And do they have the right statistics that depends on the spin? Once again, the answer is Yes. But the reasons are technically very different from those in quantum field theory. In quantum field theory, the multiparticle states are created by the creation operators constructed from the quantum fields. There's only one field for each particle species (and it commutes or anticommutes with itself at spatial separations) so the resulting states inevitably end up being symmetric or antisymmetric wave functions for commuting and anticommuting field operators, respectively.

How can the same symmetry and antisymmetry of the wave functions be realized in the matrix model? The answer is the $U(N)$ gauge symmetry. Some time ago, I have mentioned that the physical states also have to be gauge-invariant: the BFSS matrix model is a gauge theory, after all. We said that this effectively reduced the number of bosonic coordinates from $9N^2$ to $8N^2$ – well, it is really $8N^2+1$.

But we're dealing with a pretty large system and many interesting things follow from the $U(N)$ symmetry, too. For example, consider a system of $N$ gravitons such that each of them has the minimum longitudinal momentum $P^+=1/R$. A funny thing of the $U(N)$ group is that it has an $S_N$ subgroup, the permutation group. The physical states must be invariant under it, too. If you consider states that are only supported by simultaneously diagonalizable matrices, the invariance under $S_N$ condition simply means that the wave functions have to be symmetric! And if the graviton-like particles are actually gravitinos, you will get the antisymmetry because of some extra permutations of fermions that the permutation operator induces. You will get the right indistinguishability of the particles with otherwise identical quantum numbers. And the right spin-statistics relationship for the symmetry or antisymmetry emerges, too!

Needless to say, this (anti)symmetrization also applies to particles with larger values of $P^+=N/R$, represented by larger blocks: the permutation group exchanging equally large blocks is a subgroup of $U(N)$, too.

It's kind of incredible: the $S_N$ permutation of particles was a purely discrete operation, a bookkeeping device, in quantum field theory. But in the BFSS matrix model, it is actually enhanced to a much more nontrivial structure, a whole continuous $U(N)$ group. When particles of the same kind are far from each other, the $U(N)$ symmetry is broken to $S_N$ and the (anti)symmetry of the wave functions is the only residual condition. But Matrix theory shows that if the particles are coincident (or at least very close), the permutation group gets enhanced to a whole unitary group!

You may have believed that Nature is hiding some cool symmetries but they are broken to smaller symmetries. But would you be able to invent – by pure thought – that an important example of this phenomenon is a secret $U(N)$ symmetry that is broken to the permutation group exchanging particles? Nature and mathematics are clearly more creative than we are. We were literally forced to discover the matrix description of indistinguishable particles (much like if we observed it experimentally); attempts by arrogant but limited self-described "seers" to social engineer it in a man-made way would have almost certainly failed. Humans are pretty smart but there are just many clever mysteries and mechanisms that are much more likely to be discovered only once our skulls hit them.

This is the kind of a revolutionary description that was promoted from the very title of this blog entry. You could look for theories extending general relativity that are as similar to non-relativistic quantum mechanics as possible. If you were creative and lucky, you would ultimately introduce matrices of degrees of freedom, with the right commutators, and a simple enough supersymmetric Hamiltonians would be surprisingly found to describe not only a Lorentz-invariant theory in the large $N$ limit but actually a theory that has all the wonders of a consistent theory of quantum gravity.

For a while, you would think that you have found a completely new framework to understand quantum gravity, a competitor of string/M-theory. However, at some time, you would realize that it's actually exactly equivalent to string/M-theory or at least one of its Hilbert space's superselection sectors! That's not a coincidence; despite the great diversity and amazing phenomena how various things are related and represented, all these structures are just projections of a single theory as long as they are consistent. As Joe Polchinski once stated, all roads lead to string theory.

A moving human constructed out of screwing string theory (sometimes incorrectly called "matrix string theory").

Because this blog entry has gotten pretty long, I will wrap it up at this point. Sometimes in the future, I plan to explain why other objects with the right features – including (spherical, toroidal, and other) membranes, $E_8$ gauge bosons (and whole gauge supermultiplets) on Hořava-Witten domain walls, and strings with the right perturbative string interactions in the weakly coupled limit (in compactified versions of the BFSS matrix model, something I was fortunate to discover) emerge out of the matrix model description of string/M-theory.

So far, I thank you for your patience if you managed to penetrate up to this point...


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# Is BMI (Body Mass Index) a Weight Gain Indicator in Pregnancy?

Created by Łucja Zaborowska, MD, PhD candidate
Reviewed by Rijk de Wet
Last updated: Sep 23, 2022

Is BMI relevant in pregnancy? Does weight gain have anything to do with our health? How can we improve our fertility? We'll answer these and many other questions in the article below. ⤵

## What is BMI?

Body mass index (BMI) is calculated with a simple equation that uses both height and weight of a person:

$\text{BMI} = \frac{ \text{weight [kg]} }{ (\text{height [m]})^2 }$

We use it to assess the weight and proportions of a given person. BMI allows us to determine whether a person is underweight, within normal range, overweight, or obese. The index is easy to use and widely applied in almost all medical sciences.

However, it does come with a few limitations:

• BMI cannot be used for pregnant women or people with developed muscle tissue. The additional weight in both groups comes from different processes than simply putting on weight.
• BMI should be modified for people with specific disabilities, e.g., missing a limb.
• BMI in children shouldn't be used directly, but only via percentiles (try the child BMI percentile calculator).

## Is BMI relevant during pregnancy?

Yes, BMI is relevant, but only if we think about the pre-pregnancy BMI. Both your pre-pregnancy weight and height play a vital role in assessing your recommended pregnancy weight gain.

Why is it important, you may ask?

High maternal weight may lead to serious pregnancy complications, such as high blood pressure or gestational diabetes. Overweight and obese women risk having a more traumatic birth; the excessive weight gain may result in a greater occurrence of congenital anomalies and stillbirths.

Women who gain less weight than recommended also bear some risk of adverse pregnancy outcomes, including low birth weight, premature births, and increased mortality of neonates.

The gestational weight gain differs depending on the BMI and ranges between 28–40 lb for underweight women to only 11–20 lb for patients with obesity. For a complete list of recommendations for weight gain, visit the How to maintain pregnancy weight article.

## What BMI is safe for pregnancy?

The safest pre-pregnancy body mass index (BMI) is the one within normal range: 18.5–24.9. The maternal weight gain for this BMI group should oscillate between 25–35 pounds (11.3–15.9 kg) for single and 37–54 pounds (16.8–24.5 kg) for twin pregnancy.

Maternal and neonatal outcomes are influenced by both being underweight or obese in pregnancy. Being underweight does slightly less harm than being obese. One thing is certain — women of normal weight have lower chances for many disorders of pregnancy, such as fetal complications or blood-clotting disorders. And, of course, weight gained during pregnancy might also be very challenging to lose.

## Can BMI affect fertility?

Yes, BMI does affect fertility.

• Obesity and being overweight affect the function of ovaries and may result in a lower amount of hormones circulating in the bloodstream. Being in these BMI categories may cause problems with getting pregnant and complicate the initial IVF therapy.

• Extremely low BMI may also harm your potential to conceive — a certain amount of fat tissue is required to produce pregnancy hormones. Women who weigh very little tend to lose their periods and stop ovulating.

Bear in mind that there are tight associations between pre-pregnancy BMIs and a higher risk of complications typical for gestation. Both underweight and overweight women may see worse pregnancy outcomes compared to normal-weight women.

To summarize, keeping a healthy weight gain both before and during pregnancy is one of the best investments you could offer to yourself and your future child!

## Does the obese BMI during pregnancy increase health risks?

Yes. High BMI increases health risks in both pregnant and non-pregnant women. The list of possible complications of gaining too much weight is long, and it includes disorders for both mother and child:

• Hypertensive disorders of pregnancy, including dangerous conditions such as preeclampsia and eclampsia;
• Gestational diabetes;
• Congenital anomalies;
• Delivery of large infants;
• Miscarriage;
• Stillbirth (see the stillbirth risk calculator);
• Sleep apnea; and
• C-section complications, e.g., wound infections.
Łucja Zaborowska, MD, PhD candidate
Before pregnancy
Height
ft
in
Weight
lb
BMI
You are...
Weight during pregnancy
Week
Twins
no
Min. weight
kg
Max. weight
kg
Min. weight gain
kg
Max. weight gain
kg
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format: plain


text:
# Difference fun

Logic Level 2

$\Large \square - \square$

The above 2 boxes are to be filled in with 2 distinct integers between 1 and 100 (inclusive).

If the resultant number is positive but no more than 10, how many pairs of such integers can there be?

×


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# Ideal Gas Law Calculator

Created by Gabriela Diaz
Last updated: Jun 24, 2022

If you're studying ideal gases, the ideal gas law calculator may be helpful. Are you wondering how to find the pressure of a gas? With this tool, you'll be able to quickly determine this and other properties such as the volume, mass, or temperature of an ideal gas.

• What is an ideal gas;
• The ideal law gas equation;
• The universal gas constant; and
• A numerical example of how to calculate the pressure of a gas.

## What is an ideal gas?

An ideal gas is a theoretical model of a gas that allows simplifying the study of real gases through the ideal gas law equation (PV = nRT). At very low densities all gases behave very similarly to an ideal gas. This low-density condition corresponds to gases at higher temperatures and lower pressures.

Given the simplicity of the state equation, it results convenient to approximate real gases to an ideal gas. Many gases, including nitrogen, oxygen, hydrogen, argon, helium, and even heavier gases like carbon dioxide, and mixtures like air, can be treated as ideal gases by employing the ideal gas law with a small margin of error.

The above explains why we use the ideal gas approximation. Let's now take a look at what the main characteristics of an ideal gas are:

• The gas consists of a large number of molecules that are randomly moving at high speeds;
• When compared to the volume occupied by the gas, the volume of these molecules is considerably small (particles);
• The distances between the molecules are relatively large (low density);
• Because the gas particles are widely spaced and moving at high speeds, they only interact via elastic collisions; and
• Potential energy is considered insignificant compared to the kinetic energy of molecules.

Why not take a look at to find out how to calculate the density of different substances.

## Ideal gas law equation

A state equation is any equation that relates pressure, temperature, and specific volume. The ideal gas law is one of the simplest state equations used in thermodynamics. It's used to calculate and predict gases' properties such as pressure (p), temperature (T), or volume (V).

The ideal gas law is a result of empirical research that showed that the volume of a gas, under specific pressure and temperature conditions, is inversely proportional to its pressure and directly proportional to its temperature:

$p \, V = n \, R \,T$

where:

• $p$ - Absolute pressure of the gas usually in kPa or psia;
• $V$ - Volume occupied by of the ideal gas in m3 or ft3;
• $n$ - Number of moles in mol or kmol;
• $R$ - Ideal gas constant; and
• $T$ - Absolute temperature of the gas in Kelvin K or Rankine R.

To find any of these values, simply enter the other ones into the ideal gas law calculator. For example, if you'd like to determine the volume $V$ of gas, you'd need to input the values of $p$, $n$, and $T$.

The universal gas constant or ideal gas constant $R$ has been measured for various gases under ideal conditions of high temperatures and low pressures. It has been found to have the same value for all substances: 8.3144626 J/(mol·K).

The ideal gas equation can also be expressed in the following ways:

\begin{aligned} p \, v &= R_s \, T \\ p \, V &= m \, R_s \, T \\ P \, \overline{v} &= R \, T \end{aligned}

where:

• $R_s$ - Specific gas constant in J/(kg·K) or BTU/(lb·R);
• $m$ - Mass in kg or lb;
• $v$ - Specific volume in m3/kg or ft3/lb; and
• $\overline{v}$ - Specific molar volume in m3/mol or ft3/kmol.

From the ideal gas law, other gas laws are derived. That is the case of the Boyle's law, Charles's law, and Gay-Lussac's law.

💡 When a gas is close to its condensation point, it does not satisfy the ideal gas equation. This would be the case of a vapor.

## How to find the pressure of a gas with the ideal gas law calculator

Imagine that we are interested in determining the pressure of 140.5 kg of air that occupies a volume of 120 m3, at 25 °C. Let's see how to calculate the pressure of the gas using the ideal gas law calculator:

1. On the Volume (V) row, enter the volume of the gas. By clicking on the unit's dropdown menu, you can choose between different volume units.
2. Next, we are going to input the mass of air. Click on the Advanced mode button, and the Total mass of the gas (m) field will appear. Here input the value of the mass, 140 kg.
3. Proceed to enter the value of the substance's Molar mass (M). For air, this value corresponds to 28.9647 g/mol or 0.0289647 kg/mol.*
4. Finally, in the Temperature (T) field, enter the temperature, 25 ° C.
5. Once these are entered, the calculator will display the value of Pressure (P), 100.2063 kPa,

🙋 Use the Advanced mode of the calculator to input or get the values of the total mass of gas, molar mass, and ideal gas constant.

Gabriela Diaz
Pressure (p)
Pa
Volume (V)
cu ft
Amount of substance (n)
mol
Temperature (T)
°F
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text:
# Verify using Mathematical Induction

#### Guilmon

##### New member
Verify that

1(1!)+2(2!)+...+n(n!) = (n+1)! - 1

is true using induction

This problem has me stumped...

#### MarkFL

Staff member
Re: Mathematical Induction

Hello and welcome to MHB, Guilmon!

Have you verified this is true for the base case $P_1$?

#### Prove It

##### Well-known member
MHB Math Helper
Re: Mathematical Induction

Verify that

1(1!)+2(2!)+...+n(n!) = (n+1)! - 1

is true using induction

This problem has me stumped...
If you theoretically could line up a row that is infinitely long, then you know that any domino will push the next one over, as long as the first domino is pushed. Mathematical induction works the same way. You need to prove a statement for a base case (which is equivalent to pushing the first domino), and then you need an inductive step, which is to prove that IF the statement is true for an arbitrary case, THEN the statement will be true for the next (which is equivalent to any domino pushing the next one over).

So how do you think you would go about proving the base case?

#### chisigma

##### Well-known member
Re: Mathematical Induction

Verify that

1(1!)+2(2!)+...+n(n!) = (n+1)! - 1

is true using induction

This problem has me stumped...
With simple steps You can verify that is...

$(n+1)!-1 = (n+1)\ n! -1 = n\ n! + n! -1$ (1)

What does You suggest (1)?...

Kind regards

$\chi$ $\sigma$

#### Guilmon

##### New member
The nth case I understand. It is the n+1th case that has me stumped.

#### MarkFL

Staff member
After having verified $P_1$ is true, I would next state the induction hypothesis $P_k$:

$$\displaystyle \sum_{i=1}^k(i\cdot i!)=(k+1)!-1$$

Now, our goal is to algebraically transform $P_k$ into $P_{k+1}$.

What do you think we should do to both sides of $P_k$ to meet that goal?

Last edited:

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
The nth case I understand. It is the n+1th case that has me stumped.
I am still not sure that you understand how induction works and what has to be proved (not how it is proved). There is no separate nth and (n+1)th cases in a proof by induction.

To make sure we are on the same page, I recommend starting from the very beginning: identifying the property P(n) that you need to prove for all n. Without this step, everything else is useless. Note that P(n) has to be true or false for each concrete n; in particular, P(n) cannot be a number such as 1(1!)+2(2!)+...+n(n!). Identifying P(n) in simple cases is easy: just remove the words "for all n" (if they are present) from the claim you are asked to prove. In this case, P(n) is the equality 1(1!)+2(2!)+...+n(n!) = (n+1)! - 1.

Next, you need to know how to write P(0) (or P(1)) and P(n+1). To do this, replace n with 0 (or 1, or n+1, respectively) everywhere in P(n).

Now you should be able to write the base case P(0) (or P(1)) and the induction step: "For all n, P(n) implies P(n+1)". Write both claims explicitly, replacing P by its definition. Only when you do this, you'll know what you need to prove. How to prove it is the following step.


text:
# The Myth of Ability

PUBLIC LECTURE - ICE-EM

#### by John Mighton

Institution: creator of the JUMP (Junior Undiscovered Math Prodigies) project in Canada
Date: Mon 23rd May 2005
Time: 6:00 PM
Location: Theatre A, Elisabeth Murdoch, The University of Melbourne

Abstract: The International Centre of Excellence for Education in Mathematics (ICE-EM) and the University of Melbourne Department of Mathematics and Statistics are sponsoring a free public lecture with John Mighton, creator of the JUMP (Junior Undiscovered Math Prodigies) project in Canada and author of the best-selling book, The Myth of Ability: Nurturing Mathematical Talent in Every Child.

Abstract: It is a common perception in Australian society that you either have an aptitude for mathematics or you donâ€™t. As a result, children who start to struggle and fall behind in mathematics classes often feel that theyâ€™re no good at mathematics and never will be. John Mighton reveals that even students who canâ€™t do much beyond simple counting can become confident at mathematics. His idea is simple, brilliant and deeply unorthodox. He realised students were failing in a conventional classroom situation because they often perceived themselves as being no good at mathematics and stopped trying. So he started a volunteer teaching program, which he named JUMP (Junior Undiscovered Math Prodigies). He persuaded some friends to volunteer as tutors for primary school students. Many of the tutors themselves had struggled with mathematics. When they began applying Mightonâ€™s teaching principles, what they found astonished them all.

About the speaker: Awarded an NSERC fellowship for research in mathematics at the Fields Institute, John Mighton has taught at McMaster and the University of Toronto. He is also a Governor General's Award-winning playwright and appeared in the Academy Award-winning film Good Will Hunting. He lives in Toronto. His groundbreaking book The Myth of Ability was first published in Canada, where it has been a runaway success. It will be on sale at the lectureâ€”paperback $23.00. For More Information: Parking is available on University grounds for$4 per car. Please alert your local school and invite friends who may be interested. See www.ice-em.org.au/pdfs/John_Mighton_flyer.pdf for more information. Print out the flyer from the web, or contact Edwena Dixon at 8344 1777 for a copy.


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[–] 176 points177 points

sorry, this has been archived and can no longer be voted on

DOES NOBODY ELSE CARE THAT THIS LOOKS LIKE IT'S HORRIBLE FOR THE CHARGER?!

[–] 16 points17 points

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I have a feeling this is one of those 4chan "let's pretend this is a good idea so a bunch of people break their electronics" pranks.

Can't open your beer? How about potentially breaking your expensive charger to do it, even though there are probably dozens of other things you can use, like the edge of the table that's RIGHT THERE?

[–] 2 points3 points

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ive opened several beers with one, works pretty well.

[–] 0 points1 point

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It isn't even a beer, just a coke. Complete waste of a perfectly good charger.

[–] 69 points70 points

sorry, this has been archived and can no longer be voted on

[–] 20 points21 points

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he is way too damned happy for that context

[–] 58 points59 points

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he is way too damned alive for that context

[–] 9 points10 points

sorry, this has been archived and can no longer be voted on

He's happy cause he's about to make a killing selling new chargers to all those clever little coke drinkers out there.

Or well, he would be, if he wasn't like, dead, n junk.

[–] 1 point2 points

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Ohh I get it now, ha! thanks random explaining guy!

Ninja edit: ^ not sarcasm

[–] 0 points1 point

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Just doing my job, citizen.

[–] 0 points1 point

sorry, this has been archived and can no longer be voted on

A "ninja edit" is when you change something before anyone can see what you changed.

Your edit was just a normal edit.

[–] 1 point2 points

sorry, this has been archived and can no longer be voted on

Oh. I always thought a ninja edit was something you edit in before the 3 minute threshold where reddit marks a * to indicate you've edited your post. That made sense at the time because people would say one thing, have someone reply to them, then change their original comment from underneath them without any edit marks appearing, making the replier look silly.

That seemed like a ninja-maneuver to me.

[–] 0 points1 point

sorry, this has been archived and can no longer be voted on

Huh. Maybe that's what it is. I hope I didn't come across as rude! What you said makes sense, too, though. I've only ever seen it used the way I mentioned but considering what you said, I might never know if I have seen one that way.

[–] 0 points1 point

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Nah man you didn't come across as rude.

Also, I would probably bet on you being right, not me. I just thought that's what it might have been, but you actually had a definition ready to go.

[–] 0 points1 point

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He died???

[–] 1 point2 points

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Well, it's funnier than cancer

[–] 0 points1 point

sorry, this has been archived and can no longer be voted on

[–] 0 points1 point

sorry, this has been archived and can no longer be voted on

Are you using RES on an iPhone? I have a feeling that I'm missing something here

[–] -1 points0 points

sorry, this has been archived and can no longer be voted on

It's a photoshopped screenshot from his computer

[–] 5 points6 points

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Don't care. Have a couple dead ones (frayed cords/damaged magsafe tips) that I'm gonna use like this from now on (instead of them sitting in a drawer)

[–] 0 points1 point

sorry, this has been archived and can no longer be voted on

[–] 2 points3 points

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Spare parts :D

The cords are just damaged, the power supply themselves are fine. I could get another cord for them, I just haven't.

See here for a little test of how it holds up as the most expensive bottle opener: http://www.reddit.com/r/lifehacks/comments/1t05ni/a_coworker_put_an_apple_charger_to_good_use/ce39lqp

[–] 25 points26 points

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Calm down

[–] 31 points32 points

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Well his name is Crazymike...

[–] 4 points5 points

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What? the $120 bottle opener? [–] 6 points7 points sorry, this has been archived and can no longer be voted on$79 from Apple.

[–] 4 points5 points

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I think the joke is that it's for and Apple product, so who cares.

[–] 7 points8 points

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Well the guy that is using the said apple product to work on some html in the background.

[–] 1 point2 points

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Its only an apple device.. no charge no harm

[–] 0 points1 point

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No. I have played with that part of the charger and it's actually pretty sturdy. Way less bad for it than it seems.

[–] 0 points1 point

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I have several chargers. Some of them have a metal knob thingy that holds the plug on. I bet that wouldn't break easily.

[–] 2 points3 points

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How much? I'll put $50 down that says it won't last ten attempts. [–] 16 points17 points sorry, this has been archived and can no longer be voted on Done. I'll open 10 beers between me and my GF today and if it breaks, oh well (I have old non-working ones) 1 and 2 were just cracked. I don't expect money, just want to see if it lasts. Here it is after 2 beers: http://imgur.com/AmKsMSH (it pushed right back into place, so it would still slide together). No change after the third (girlfriend drinks a little slower than me). Still no change on the fourth (she's really slow, I might be on my own here haha). 5 and 6, again no change, I haven't pushed the metal bit back in since the first 2, it just hasn't come out any further. 7 and 8, again no change, but was a little harder to push back in than the first time I tried. Got a good buzz going. now we're at 9. Still hasn't ripped that metal nub out. I'm genuinely surprisesd. ....andddd 10 (7 hours later). again, no change. I'd post another picture but it would look exactly the same. I was still able to push it back into place. Well this was anti-climatic. I was hoping to post some charger carnage. I'm leaving it next to my beer fridge to use until it breaks. [–] 3 points4 points sorry, this has been archived and can no longer be voted on I like you for doing this. I'm worried about how it will end if that is after two. [–] 1 point2 points sorry, this has been archived and can no longer be voted on Dude owes you$50!

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Has it broken yet?

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It has not, I've still been using it but I've been away for the holidays so its seen less use.

[–] 92 points93 points

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Ahhh sublime text. nothing better for web dev.

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I use Komodo, always have. Is Sublime really that good that I should try switching?

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I've used it before, and I wasn't super impressed with it over Notepad++. Am I missing something?

[–] 32 points33 points

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Multiple cursors.

Believe the hype.

[–] 7 points8 points

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Is it possible to fix the Ctrl+Tab behaviour?

I love Sublime otherwise, but trying to tab between documents it's fucking impossible to get to where you expect to go. I feel like I'm going to end up in Mexico if I try too much.

[–] 0 points1 point

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Also there is Sublime Text 3

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Wait, how the hell does that work?

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For example, if you want to add a class to a series of <li> tags you can put a cursor on each line and add the class to all of the tags by typing it once and not copy-pasting.

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My god.... That sounds incredible. I have to check that out now....

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Ways to do it are:

• CTRL+click to put a cursor where you want.

• Highlight a selection (or cursor in place) and CTRL+D to select successive matches

• Highlight a selection (or cursor in place) and ALT+F3 to match all occurrences in the document.

You're welcome.

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Sublime text is extensible without needing a compiler, unlike Notepad++, Sublime text has a pretty large set of options in programmer friendly config files, but most of all, I felt that sublime text just felt cleaner and less clunky than using Notepad++. You lose a bit of real estate up at the top in notepad++ as well.

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The configuration is overwhelming though.

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The configuration is overwhelming

Maybe you should switch to EMACS...

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I use PHP Storm right now, but I did try out Sublime Text 2 for a while, I have that feeling like I'd like to switch back to it, but I'm not convinced yet.

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Fair enough, though I got used to it after a little bit. I really rely on the folder view, scriptable actions, fake intellisense, and the scrollbar thumbnail, at this point. I probably wouldn't be able to go back to notepad++.

There are two versions of sublimetext though, have you tried version 2?

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I did try Sublime Text 2, but right now I use PHP Storm, if I was comparing Notepad++ and SublimeText I'd definitely use Sublime Text. But for now PHP Storm seems to be much easier to use with GIT being in it and all kinds of Watchers for converting my LESS files to CSS and so on

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Yea that sounds better for web development. I do application stuff with lots of plugins, so I like Sublimetext :)

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I didn't configure anything until I found something I didn't like, it works pretty well out of the box and can be made to do most stuff you need and shit you never needed before

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What languages is it best for?

I do HTML, PHP, JS, LESS (CSS) and so on.

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That sounds like pretty much everything I use it for. By default it has some code completion and syntax highlighting, but so many plugins etc once you feel comfortable. I still haven't fully set mine up, but it can integrate builds and get plugins like integrated ftp etc. Its pretty sweet.

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• If you're working in node, python, ruby, or other noncompiled languages, you can run a repl from within sublime text itself.

• All of the color schemes are fully user editable.

• the whole multiple cursors thing is actually magical, you can position them on multiple lines, after every instance of a given search, etc.

• it has support for emmet.io text expansion

• it has built in code snippets for every language

• with certain plugins, you can get ide like syntax that provides autocomplete for previously written functions

• you can update git repos from inside sublime text itself

• you can push to github gist from inside sublime text itself

• it has a nice little markdown preview plugin that I use to take my notes in, it support mathjax and as a result LaTeX style mathematical syntax

• If you have javascript in an html document, it knows and will syntax highlight the js correctly

• when working on larger projects, the sidebar allows you to keep track of all files in your (say, django) project, and move between them. AS a result, you don't need to open a file manager to search for files, and if you're making a small edit, like changing a urls.py file or adding some additional js file to a head, you don't need to open the file, you can just search for the file from within ST itself, without ctrl+o or anything, and then you make the change, save and close the file. Its glorious.

• on big projects, you can search every open file, and I think you can search every file in the sidebar, but I forget how to do that

• when working on big projects, you can customize syntax rules, highlighting rules, shortcuts, open files, languages, etc. to that specific project, and have the project state saved, and that's all done in a user editable file, meaning you can also push it to github or dropbox or wherever, and have other people access the file as well.

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BOOM.

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Yes

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Same here except I use Coda 2.

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Same here except I use Brackets.

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Total: USD $70 back to notepad++ [–] 13 points14 points sorry, this has been archived and can no longer be voted on No, it's completely free to use just every like 20 saves you have to put up with a dialog box. [–] 4 points5 points sorry, this has been archived and can no longer be voted on then i will give it a try. thanks. [–] 0 points1 point sorry, this has been archived and can no longer be voted on As someone who saves extremely frequently as a habit, damnit. [–][deleted] sorry, this has been archived and can no longer be voted on [deleted] [–] 1 point2 points sorry, this has been archived and can no longer be voted on Sublime has a built-in plugin to emulate vim commands. If I'm not doing something quickly in the terminal, I prefer Sublime over MacVim. [–] 1 point2 points sorry, this has been archived and can no longer be voted on I'm personally a huge fan of Brackets. It's awesome. And free. [–] 1 point2 points sorry, this has been archived and can no longer be voted on HOLY TITS. I have been doing Learn Ruby the Hard Way and have been using Notepad++ and then using the terminal to run my .rb files. I downloaded Sublime Text 2 and it ran right on the bottom. Awesome possum. [–] 0 points1 point sorry, this has been archived and can no longer be voted on Geany baby [–] 0 points1 point sorry, this has been archived and can no longer be voted on First thing I thought when I saw that picture. Did a little mental jig, followed by a grin. [–] 0 points1 point sorry, this has been archived and can no longer be voted on Vi or take a hike... [–] -1 points0 points sorry, this has been archived and can no longer be voted on On non-Windows platforms, yes. On my Mac I have yet to come across a better web dev tool. On Windows, however, Visual Studio nowadays (VS2013) is kicking some pretty mean ass for web dev. [–] 33 points34 points sorry, this has been archived and can no longer be voted on Dat colorful code. [–] 23 points24 points sorry, this has been archived and can no longer be voted on It's called sublime text. It's color coordinated for whichever type of code you write so you know exactly where a new line starts and what the arguments are at a glance. [–] 18 points19 points sorry, this has been archived and can no longer be voted on You can get the same color scheme (monokai) for other editors though. [–] 4 points5 points sorry, this has been archived and can no longer be voted on Nice. I had no idea. This looks great in notepad++. You just make my life a lot easier. Have an upvote. [–] 2 points3 points sorry, this has been archived and can no longer be voted on Wow, I didn't realize Notepad++ had this built in, either. Thanks! [–] 5 points6 points sorry, this has been archived and can no longer be voted on It's called syntax highlighting and is a dominant feature is pretty much every IDE/code editor... [–] -5 points-4 points sorry, this has been archived and can no longer be voted on sublime text isn't just a color, it's a full text editor that is full of community made features such as closing parentheses automatically, a bar on the right that shows your entire code and allows you to quickly access a line when you may have 10,000 lines, etc. it's just very useful. [–] 3 points4 points sorry, this has been archived and can no longer be voted on Pfff.. hipsters with their colours. Casuals... [–] 5 points6 points sorry, this has been archived and can no longer be voted on pfff.. I code with typewriters. No color coding there. [–] 0 points1 point sorry, this has been archived and can no longer be voted on How do you compile? [–] 0 points1 point sorry, this has been archived and can no longer be voted on snail mail it to the compiler obviously. [–] 1 point2 points sorry, this has been archived and can no longer be voted on Right! I thought I recognized it. [–] 0 points1 point sorry, this has been archived and can no longer be voted on You're saying that like it's a new thing. [–] 2 points3 points sorry, this has been archived and can no longer be voted on But those k&r braces... :( [–] 0 points1 point sorry, this has been archived and can no longer be voted on Shots fired. [–] 26 points27 points sorry, this has been archived and can no longer be voted on What is that horrible looking meat-brick on the plate [–] 19 points20 points sorry, this has been archived and can no longer be voted on rugbrød [–][S] 13 points14 points sorry, this has been archived and can no longer be voted on Translation: Rye bread (bread made from rye, sometimes from sour dough, often filled with grain). [–] 0 points1 point sorry, this has been archived and can no longer be voted on And you've chickpeas with rice and salad there? [–][S] 2 points3 points sorry, this has been archived and can no longer be voted on Wasn't my plate. I prefer bacon. [–] 5 points6 points sorry, this has been archived and can no longer be voted on Actually looks tasty. [–] 4 points5 points sorry, this has been archived and can no longer be voted on 'meat-brick' is a comical way of describing what you see, reminded me that Germans have a thing called 'meat-cheese' (Fleischkäse), which is essentially meatloaf. [–] 4 points5 points sorry, this has been archived and can no longer be voted on Who doesn't have a bottle opener? [–] 2 points3 points sorry, this has been archived and can no longer be voted on First World Ingenuity [–] 2 points3 points sorry, this has been archived and can no longer be voted on Am I the only one curious about what OP is eating? Suddenly I'm hungry [–] 2 points3 points sorry, this has been archived and can no longer be voted on What one of these chargers look like after 2 beers cracked: http://i.imgur.com/AmKsMSH.jpg (don't worry, it's already broken) Not done with all 10 yet but there hasn't been any change on 3 through 9. [–] 1 point2 points sorry, this has been archived and can no longer be voted on My problem here is that it's a glass coke bottle and he decided not to use a hard edge of something as opposed to using the charger. [–] 1 point2 points sorry, this has been archived and can no longer be voted on I'm intrigued with whats on the monitor. Tell me more about it? Too me it looks like HTML, but am i wrong or? [–][S] 0 points1 point sorry, this has been archived and can no longer be voted on You are correct. From left to right: HTML, CSS, JS [–] 1 point2 points sorry, this has been archived and can no longer be voted on Again, out of curiosity, what do you do for a living? Web Development i assume? [–][S] 0 points1 point sorry, this has been archived and can no longer be voted on I do technical project management and web development (both frontend and backend) in an advertising agency. So: A lot of consulting with clients and developing campaigns [–] 1 point2 points sorry, this has been archived and can no longer be voted on GUYS I HAVE AN IDEA OF A GREAT USE FOR A MAC CHARGER, WAIT FOR IT...... charge your mac with it. That's putting it to "good use". [–] 5 points6 points sorry, this has been archived and can no longer be voted on When did this sub become "look at this guy actually using a lifehack we've seen 1,000 times"? [–] 3 points4 points sorry, this has been archived and can no longer be voted on When did this sub become "ZOMG Apple chargers" [–] 0 points1 point sorry, this has been archived and can no longer be voted on Glass coke bottles are twist off Edit: Sorry I'm American [–] 10 points11 points sorry, this has been archived and can no longer be voted on Not Mexican ones [–] 0 points1 point sorry, this has been archived and can no longer be voted on That's an American one. [–][S] 1 point2 points sorry, this has been archived and can no longer be voted on This was shot in Denmark, where the glass bottles are not twist off. I did try to twist one once. Do not recommend. [–] 0 points1 point sorry, this has been archived and can no longer be voted on I am assuming this is what the life of a coder looks like? [–] 2 points3 points sorry, this has been archived and can no longer be voted on [–] 0 points1 point sorry, this has been archived and can no longer be voted on then im afraid to see what else you have in store. [–] 0 points1 point sorry, this has been archived and can no longer be voted on OP needs a beer. [–] 0 points1 point sorry, this has been archived and can no longer be voted on LET: no open beverages near the computer. [–] 0 points1 point sorry, this has been archived and can no longer be voted on$90 Bottle Opener, nice.

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/u/scarlac what screen size is that? Been looking for something that could accommodate three columns of code comfortably.

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Regular 27" Apple "Thunderbolt Display". If you want a 27" screen there's plenty of cheaper alternatives. ... But if you already own a Mac with thunderbolt, there are some conviniences to be had.

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Neato, a few more times and he'll have an $85 bottle opener. [–] 0 points1 point sorry, this has been archived and can no longer be voted on OP works at Google. [–][S] 0 points1 point sorry, this has been archived and can no longer be voted on OP works at an advertising agency. [–] 0 points1 point sorry, this has been archived and can no longer be voted on HTML? [–][S] 1 point2 points sorry, this has been archived and can no longer be voted on HTML. [–] 0 points1 point sorry, this has been archived and can no longer be voted on That looks like a mighty fine lunch for work. [–] 0 points1 point sorry, this has been archived and can no longer be voted on Ooooooo sublime text editor, nice [–] 0 points1 point sorry, this has been archived and can no longer be voted on [–] 0 points1 point sorry, this has been archived and can no longer be voted on You must have a really nice cafeteria at your work. [–] 0 points1 point sorry, this has been archived and can no longer be voted on Wouldn't it be simpler just to carry a pocket knife of some sort? Especially since they are useful in so many other ways than just "bottle opener." In fact, "pocket knife" should be life-hack number 1. [–] 0 points1 point sorry, this has been archived and can no longer be voted on Yeah, this barely works. There are thousands of better ways of opening a bottle. [–] 0 points1 point sorry, this has been archived and can no longer be voted on What website you working on? Just curious [–] 0 points1 point sorry, this has been archived and can no longer be voted on Snob developer alert. Chickpeas and mexican sugarcane glass bottle coke. Flying 3 vim tabs and a Mac. [–] 0 points1 point sorry, this has been archived and can no longer be voted on Or you could, y'know, charge your apple device. That's a good use too. [–] 0 points1 point sorry, this has been archived and can no longer be voted on Html with sublime? [–] 0 points1 point sorry, this has been archived and can no longer be voted on oh cmon, we know it wouldnt work, that flimsy plastic wont break anything [–][deleted] sorry, this has been archived and can no longer be voted on [deleted] [–] 13 points14 points sorry, this has been archived and can no longer be voted on The ultimate mindfuck: programming on a mac and actually using a windows phone. [–] 2 points3 points sorry, this has been archived and can no longer be voted on fuck, that's me... I just bought a windows phone to try out something new. But now i'm too poor to buy something better... [–] 1 point2 points sorry, this has been archived and can no longer be voted on Is it that bad? I never even had a chance to give it a shot. I've literally only seen one of them IRL, and that was just a lady taking it to the T-Mobile store for some reason. Closest thing after that is one of my college buddies on facebook that works for Microsoft making posts about it. Granted it's not bulletproof logic, but I just feel like if there's a product from a company that big that nobody will use, it just can't be that good. [–] 2 points3 points sorry, this has been archived and can no longer be voted on Well, I bought the Lumia 800 when it was new, that's like two years ago. And i'm stuck on WP7.8. The new Windows Phones are using WP8 which is a lot more user friendly and has a lot more features. Right now WP7.8 isn't really supported anymore(Microsoft says they still do but there is nothing that they do for the OS) and there are no apps being made for it anymore. So there's no Vine, no Instagram and no Tumblr app. There is Facebook, Twitter and Whatsapp of course. WP8 is the future of Windows Phone and it has all apps that all major mobile OS-s have. The OS runs really good though, super smooth and and it's speed is still the same as when it was brand new. I'm not sure if this is the same for WP8 though since it's on a different kernel or something techy stuff like that. My final conclusion is that I wont get a Windows Phone anymore, probably because I like iOS too much and I work a lot on my MacBook Pro. But if you buy medium to low end smartphones I can recommend Windows Phones that cost under$200. If you want a real good solid high end smartphone I recommend to go iOS or Android.

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I got an iphone for work last year and felt like it was pretty 'eh', and I upgrade to a Droid this summer - fucking awesome.

iOS is fucking awful for everyone except the users. I just boycott it out of principle for the way Apple handles it - nobody in history has treated developers worse. Their apps are always buggy and unstable - owing primarily to how they handle their updates. Mandating that everything that goes on from their phones has to come from them - it's really bullshit

As far as handsets go, Apple's phones are pretty nice, but having an actual selection is much more preferable IMO. Overall, for the functionality and the price, every phone on the market beats Apple by a landslide.

I was just intrigued to hear an actual testimony from a Windows Phone user - never heard one before.

[–] 2 points3 points

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Well, it looks as if he's doing some web development. It's pretty normal for a web development company, or just a single web developer, to have access to multiple devices with multiple (mobile) operating systems and browsers on them.

This is to test out the product being made on multiple platforms to make sure that it runs smooth and looks good on every single one of them.

[–] 0 points1 point

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I'm more interested in the keyboard choice. Why use a crappy shallow thing like that when you can get much better. This guy needs a mechanical keyboard.

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Eh, they suit a purpose. Most people who have an android phone don't use most of its features or processing power.

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If anything, I thought that was awesome. Best of both worlds for some.

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Why hate on Windows phones? They're awesome.

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Because they're not.

[–] 1 point2 points

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Oh, I didn't know people don't like them. I like to switch between mine and my droid every other day. It doesn't have the functionality of android but I like the new windows look. Everyone is entitled to their opinion though!

[–] -1 points0 points

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On the list of things I want from an electronic device function is very high while looks sits quite a bit lower on the list.

[–] 0 points1 point

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Most expensive bottle opener ever.

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According to Google, THESE are the world's most expensive bottle openers.

[–] -1 points0 points

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Nope, this is the most expensive bottle opener.

[–] 0 points1 point

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Are you insulting my bottle openers? I have one made from gold and sterling silver.

[–] 0 points1 point

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EDIT: And why are you dont use SCSS? ^

[–][S] 2 points3 points

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EDIT: And why are you dont use SCSS? ^

It was not my screen but I've been on projects for the client: We need to deliver the code in HTML/CSS/JS as per their specification. They also have restrictions on what libraries to use, how the code is delivered, etc.

[–] 0 points1 point

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You heard of SCSS? It renders it in normal CSS but it is way easier to code with it.

example (who never heard of scss):

.class1 {
.class2 {
font-weight: bold;
.class3 {float:left;}
}
}

renders in:

.class1 .class2 {font-weight: bold;}
.class1 .class2 .class3 {float:left;}

You also can use variables, placeholders and even can do math with it.

Here are some informations: http://sass-lang.com/

[–] -1 points0 points

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This MUST be Japan... only Japan has glass Coke bottles and shredded cabbage to go with all lunches. Plus, I can see the Japanese label on the bento box.

Hey, maybe I'm just down the hall from you. Did I pee next to you today? I know you thought I was checking, but that was just an accidental eye drift! Honest!

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Going by the code and the chunk of ryebread on his plate, I'm gonna say Denmark.

Plus, I can see the Japanese label on the bento box.

That would be a box for a Windows phone.

[–][S] 2 points3 points

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I'm gonna say Denmark.

Bingo.

[–] 0 points1 point

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Oh. I couldn't really see it, I just thought it might be so, so I went for it. Really, Denmark also loves shredded raw cabbage?? I eat it, but that shit really sucks!

[–] 1 point2 points

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We have glass coke bottles in California, too. I think they taste better than canned coke, because they're made with real sugar. I only get them once in a while though, because diabeetus.

[–] 0 points1 point

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I didn't know they still had that in the USA. But yeah, glass is cool. I have a bar and I want to serve it that way, but I can't due to lack of refrigerator space.

[–] -4 points-3 points

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Better than it's original intended use.

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This belongs in r/cooking

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God, those keyboards suck so much balls.

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As a PC personfanboy, I do agree, That apple charger does work best as a can opener.

[–] -3 points-2 points

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text:
# Remarks on the derived center of small quantum groups

@article{Lachowska2019RemarksOT,
title={Remarks on the derived center of small quantum groups},
author={Anna Lachowska and You Qi},
journal={Selecta Mathematica},
year={2019},
volume={27}
}
• Published 18 December 2019
• Mathematics
• Selecta Mathematica
Let uq(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {u}}_q(\mathfrak {g})$$\end{document} be the small quantum group associated with a complex semisimple Lie algebra g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy…
4 Citations
• Mathematics
• 2022
Let u ∨ ζ denote the small quantum group associated with a simple Lie algebra g ∨ and a root of unity ζ . Based on the geometric realization of u ∨ ζ in [8], we use a combinatorial method to derive a
• Mathematics
• 2022
It is easy to ﬁnd algebras T ∈ C in a ﬁnite tensor category C that naturally come with a lift to a braided commutative algebra T ∈ Z ( C ) in the Drinfeld center of C . In fact, any ﬁnite tensor
• Mathematics
• 2022
We consider the small quantum group u(Gq), for an almost-simple algebraic group G over the complex numbers and a root of unity q of sufficiently large order. We show that the Balmer spectrum for the
• Mathematics
• 2021
A modular category C gives rise to a diﬀerential graded modular functor, i.e. a system of projective mapping class group representations on chain complexes. This diﬀerential graded modular functor

## References

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### Almost cocommutative Hopf algebras

• Algebra i Analiz 1(2), 30–46
• 1989

### Duke Math

• J., 69(1):179–198,
• 1993
Abstract:In this Paper we find all indecomposable components of adjoint representation of small quantum sl2 with multiplicities.
• Mathematics
• 2006
The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2pth root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of
• Mathematics
• 2006
In math.RT/0304173 the derived category of the principal block in modules over the Lusztig quantum algebra at a root of unity is related to the derived category of equivariant coherent sheaves on the
• Mathematics
• 2016
We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The exemplary case


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# Huffman tree generation if the frequency is same for all words

Can a valid Huffman tree be generated if the frequency of words is same for all of them?

Example :

Value     | Frequency
--------------------
Google    | 2
Yahoo     | 2
Microsoft | 2
Amazon    | 2

## 2 Answers

Yes. When creating a Huffman tree, if you ever find you need to select from a set of objects with the same frequencies, then just select objects from the set at random - it will have no effect on the effectiveness of the algorithm.

Of course, one might question why you're bothering to build a Huffman tree if you know all the frequencies are the same - I can tell you what the optimal encoding is.

Yes. Whenever identical frequencies occur, the Huffman procedure will not result in a unique code book, but all the possible code books lead to an optimal encoding.