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ℓ
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The bill here is tidal, not electromagnetic or mechanical, and it is the one place this construction is bounded directly by what has actually been observed rather than by an idealization. A rigid processor of size ℓ falling radially experiences a stretching tidal acceleration between its two ends of approximately
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Published contexts (8)
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Equation 78 · Evolutionary Physics
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The bill here is tidal, not electromagnetic or mechanical, and it is the one place this construction is bounded directly by what has actually been observed rather than by an idealization. A rigid processor of size ℓ falling radially experiences a stretching tidal acceleration between its two ends of approximately
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Article →Equation 9 · Datacenters
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where Ptx is launch power, Li are discrete losses at each connector and splice, α is the fibre’s attenuation coefficient, and ℓ is the run length. A link can pass a binary link-up test while its measured Prx sits close to a receiver’s sensitivity floor rather than comfortably above it — a connector with contamination, a bend under the cable’s minimum radius, or a transceiver aging out of spec all show up first as reduced margin, not as an outage. NVIDIA’s DGX SuperPOD cabling guide devotes explicit attention to this: cable latency, connector types, and deployment methodology are treated as first-order design inputs precisely because a marginal physical layer produces…
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Article →Equation 3 · AI Research
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where hℓ(x) is the frozen activation the network produces at layer ℓ for input x , and w, b are the probe’s own parameters, trained on labelled examples the network never saw during its own training. Nothing in that objective touches the network’s weights or its downstream computation. A probe reports only whether some linear function of this one activation predicts the label well.
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Article →Equation 29 · AI Research
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where P and N index matched positive and negative examples of the target behaviour, hℓ is the residual-stream activation at layer ℓ , and α is a coefficient the operator sets by hand. Nothing in this construction requires understanding why the network represents the concept along this direction, only that adding it produces the intended behavioural shift.
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Article →Equation 2 · AI Research
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The first concrete operation in almost any interpretability project is the least glamorous: run the model forward on a batch of inputs, and at some chosen point in the computation — a residual-stream position, an attention head’s output, a particular MLP layer — copy the activation tensor out before it is overwritten by the next step of the forward pass. This is extraction, and it produces nothing on its own beyond a large table of vectors. What turns it into evidence is probing: fitting a small, separately trained classifier to predict some property of interest directly from those vectors, while the model’s own weights stay frozen. Formally, for an activation aℓ read out at layer ℓ…
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Article →Equation 8 · AI Hardware & Semiconductors
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where Cfix collects driver, receiver, bump and pad capacitance, cw is capacitance per unit length of the conductor, ℓ is the reach, and Vsw is the signalling swing. Two assumptions are doing the work and both deserve stating. First, this treats the link as a lumped capacitance rather than a transmission line — reasonable for a few millimetres of on-package wire at moderate rates, wrong once the link is long or fast enough to need equalisation, retiming and serialisation, at which point the fixed circuit overhead dominates and the linear term stops being the whole story. Second, it ignores clocking and idle power, which for real links are not negligible.
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Article →Equation 10 · AI Hardware & Semiconductors
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Within its range of validity, the model explains the entire strategy of advanced packaging. Shorten ℓ and you reduce the wire term. Reduce the connector pitch and you shrink Cfix per wire while getting more wires per millimetre of die edge, so you can lower the per-wire data rate for the same aggregate bandwidth — and a lower rate lets you drop the swing and delete the equalisation circuitry. The gains compound, which is why the reported numbers for short-reach parallel interfaces are so far below what a long-reach serial link costs.
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Article →Equation 5 · Particle Physics
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Here s is the sagitta, B the field, ℓ the path length in the field and p the momentum [ 2 ] . Two consequences follow directly and are worth stating plainly, because they explain most of the architecture of a collider detector.
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