![]() ![]() Her thesis advisor tumbled sideways under a gray cloud tinged with orange. Gladstone threw up his forearms, a futile gesture, as Cindy pulled the bear spray trigger and applied a three-second blast. ![]() “With Frontiersman Bear Spray,” she said rising and popping the safety tab. Do you know how?”Īs Gladstone remained rooted to his chair, mouth agape, Cindy dropped her right hand into her briefcase, and extracted a black cylinder topped by nozzle and trigger. “I know, for example, that you’re not wearing your pants, that you’re about to stand and expose your genitals, because you’ve already done it.” “An implant that lets me drop back twenty seconds and adjust what’s already, allegedly, occurred.” “Would we, Professor Gladstone? Because I’ve fabricated a prototype.” Cindy tapped the left side of her jaw. “I do disagree that it can be reversed at the macro level to produce a time machine.” Gladstone leaned forward. I have a few suggestions.”Ĭindy smiled and said, “You disagree that entropy is connected to time?” “But don’t worry, we’ll hammer out a viable proposal. You don’t mind if I call you Cindy?” He sat back. He lifted the thin sheaf of papers on his desk and dropped them. Gladstone removed reading glasses, reducing his brown eyes to normal size. Theresa had laughed and said, just make sure he keeps his pants on. ![]() Earlier, Cindy had asked Theresa, the only other female doctoral candidate, about Gladstone. Cindy, briefcase clasped to her chest, positioned her rear on the edge of the office’s twill sofa. He sat behind a beige metal desk with faux oak top and beckoned with his left hand. Professor Gladstone’s age? Cindy had guessed fifties with his steel hair and bushy eyebrows, but at close range, surveying a face of furrows, upped the estimate to mid-sixties. That rise then, by symmetry, puts the baseline infinitely far below you, or at $-\infty$, and moreover due to the Archimedean nature of the real numbers, prevents you from distinguishing these finite-bin cases any more (all of them, if you check for yourself, have differential entropy $-\infty$).You can print this page by copying its URL to M. ![]() And that is what I mean by saying that to then go up into continuous distributions, you then have to "soar up high" - effectively the integral jams in an uncountable infinity and rises your reference point far above the true baseline so you can distinguish the different infinite amounts that are "above the infinite boundary of the discrete entropy". But if you continue in this way, long before you reach a truly continuous distribution you will soon "top out" (after a countably infinite number of bins have been summed) using discrete entropy, saturating at positive infinity. $$P(x) = \begin \delta(x - a_2)$ meaning "we know the particle is at either $a_1$ exactly or $a_2$ exactly but not which", then by the usual discrete entropy formula you have entropy $0.693$ nat (or $1$ shannon using $\lg$ instead of $\ln$). A simple example is precisely that which is one unit wide: To understand it more precisely, consider a uniform distribution where the differential entropy is zero. The inability to specify an absolute information in this context should be taken as based on the intuitive idea that the amount of information required to specify a specific value in an infinite continuum is itself infinite as you must distinguish one from amongst an infinitude of possibilities to achieve such a specification. you have a logarithm of a unitful quantity, which means that it will depend on the units you measure the axis $x$ in), which is not a concept that makes sense for a discrete information source. In particular, note that the differential entropy responds to a change in scale (i.e. Differential entropy should be considered as a measure of relative (privation of) information - not absolute. ![]()
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