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# Which, if any, parts of the Locate City Bomb are dubious by RAW?

The “locate city bomb” is a somewhat-notorious example of theoretical optimization, turning a minor 1st-level divination into something quite similar to a nuclear blast that kills everything for miles around.

The typical process:

Typical process:

1. Cast locate city from Races of Destiny.

2. Use the Snowcasting feat from Frostburn to add the Cold descriptor to locate city.

3. Use the Flash Frost feat from Player’s Handbook II, which can only be applied to cold spells, to add 2 cold damage to all creatures within the area of the Snowcasted locate city.

4. Use the Energy Substitution feat from Complete Arcane, which can only be applied to a spell that have the acid, cold, electricity, or fire descriptor, to turn the cold damage to electricity damage, and change the descriptor to the corresponding one.

5. Use the Born of the Three Thunders feat from Complete Arcane, which can only be applied to an area spell with the electricity or sonic descriptor that deals hit point damage. Among other things, this forces creatures within locate city’s area to make a Reflex save.

6. Use the Explosive Spell feat from Complete Arcane, which can only be applied to a cone-, cylinder-, line-, or burst-area spell that allows a Reflex save, to force those within who fail the Reflex save to the edge of the effect, taking 1d6 damage for every 10 feet moved.

Which, if any, of these steps are dubious per the rules as written?

• What prompted this? – fectin Nov 11 '19 at 21:36
• @fectin-freeMonica It’s a common enough point of contention, but the proximate cause was this question. – KRyan Nov 11 '19 at 21:57
• Are we certain this isn't a duplicate? I'm sure that I've seen this elsewhere. Admittedly, searching "Locate City" on the 3.5e tag doesn't bring up any obvious dupes. – J. Mini Nov 12 '19 at 16:56
• @J.Mini On this site? I’m about 90% sure not. I did double-check, though it’s of course possible that I misremembered and also missed it in my search query. But I suspect you are thinking of a similar discussion on another site, since this is an argument that comes up from time to time. Or perhaps you’re thinking about one of the other questions about locate city where it was argued in comments or perhaps within an answer, but as a tangent, not the actual question. – KRyan Nov 12 '19 at 16:57
• Darn it. Here. – Hey I Can Chan Nov 13 '19 at 14:53

# The trick's limited to casters that don't prepare spells

Not dubious and more just not mentioned is that access to the locate city bomb is incredibly limited. Only casters who cast spells without preparation can drop a locate city bomb, and a locate city bomb can't be brewed into a potion, scribed onto a scroll, or put into a wand. That said, if the reader is unsurprised to learn that the locate city bomb is the exclusive province of characters both that are mid-level bards, sorcerers, and other casters that don't prepare spells and that have access to either the bard, sorcerer, or ranger spell list, then the reader should skip to this answer's Issues section.

The locate city bomb the question describes requires that the 1st-level Sor/Wiz spell locate city [div] (Races of Destiny 166) be modified by the metamagic feat Flash Frost Spell (Player's Handbook II 91). That metamagic feat can be applied "only to spells that have the cold descriptor." When it's prepared, the spell locate city doesn't have the descriptor cold, and it won't gain that descriptor until after to it has been applied the benefit of the general feat Snowcasting (Frostburn 50), and that feat's benefit applies when the spell's cast not prepared.

Thus the typical prepared caster can't prepare a flash frost locate city spell. (If he could, the spell would typically occupy a 2nd-level spell slot as the "flash frost spell uses up a spell slot one level higher than the spell’s actual level.") By extension this means that the typical prepared caster also can't prepare the locate city spell modified by the metamagic feats Born of the Three Thunders (Complete Arcane 76), Energy Substitution (electricity) (Shadowdale: Scouring of the Land 149), and Explosive Spell (CAr 79) so as to make a (potential) bomb. (If a typical prepared caster like a ranger or wizard could prepare in one of his spell slots a locate city bomb, the born of the three thunders explosive energy substitution (electricity) flash frost locate city spell would occupy a 4th-level spell slot, so you know.)

In fact, it seems that of the core classes only bards and sorcerers can cast the locate city bomb as the question describes. The bard or sorcerer takes a move action to collect some nearby snow and takes 1 round (the locate city spell's normal casting time) then 1 full-round action (the extra time a caster that doesn't prepare spells needs to apply one or more metamagic feats' benefits to a spell) to employ the feat Snowcasting while casting the locate city spell modified by the feats Born of the Three Thunders, Energy Substitution (electricity), Explosive Spell, and Flash Frost Spell, the (heavily) modified spell expending one of that sorcerer or bard's precious 4th-level spell slots.

To be clear, at its most basic, this means that a locate city bomb comes from at least a bard 10 (about 1/day) or sorcerer 8 (about 3/day). Also, since neither class receives bonus feats, the trick's game elements marshal every level-up feat the character gained during levels 1–7 plus maybe another feat in the feat Eschew Materials (Player's Handbook 94) as this selection of feats is generally suboptimal if the Snowcasting feat goes unused (see also here). If the bomb—for whatever reason—does not work, that's a lot of wasted effort and resources.

## Issues—dubious and otherwise

Below are this writer's immediate concerns with the process the question proposes.

• Does the perfect flash frost spell even exist? The Flash Frost Spell feat in isolation from a rules-as-written perspective is at least mildly dysfunctional. The benefit of the feat Flash Frost Spell "can be applied only to spells… that affect an area" and a spell so modified "deals an extra 2 points of cold damage per level of the spell to all targets in the area" (PH2 91). Most spells that have an area entry don't also have a target or targets entry. None of the spells in the Player's Handbook have both, for example. (While the 2nd-level Sor/Wiz spell shatter [trans] (PH 278) and the 3rd-level druid spell quench [trans] (PH 267–8) both offer the caster a choice between either an area or a target, neither spell has a mode that offers both an area and one or more targets.) Given the game's vastness, there's probably a spell that the feat Flash Frost Spell as written can modify, but finding such a spell won't be effortless, and the spell locate city won't be among them.

It's entirely possible, however, to read the benefit of the Flash Frost Spell feat's use of the word target as informal, but then what is dealt damage when the feat's applied to the locate city spell? Were a spell that normally deals damage modified by the feat, a reader could discern these informal targets, but as the locate city spell doesn't normally deal damage, the DM must rule what these informal targets are. Any creature in the area? All creature and unattended objects in the area? All creatures, their attended objects, and unattended objects in the area? And so on. The question asks what parts of the locate city bomb need DM adjudication, and this is a point that the DM must address as the Flash Frost Spell feat provides no guidance in this regard.

• What does extra mean? The spell modified by the feat Flash Frost Spell "deals an extra 2 points of cold damage per level of the spell" (PH2 91). There is no formal definition of extra damage when the term is used in this fashion. (The Rules Compendium, for example, consistently uses the term almost exclusively in conjunction with critical hits, a nonstarter here.)

This makes it unclear if a flash frost locate city spell deals 2 points of damage because the extra damage applies despite the underlying spell dealing no damage or deals no damage precisely because the underlying spell deals no damage. There is, so far as I can tell, no textual support for ∅+X=X besides—and I don't use the term pejoratively—common sense (i.e. adding something to nothing nets something). However, as this is a role-playing, a case can be made for ∅+X=∅. This'll take a minute.

The Dungeon Master's Guide on Effect of Weapon Size that says, "Weapons that deal less than 1 point of damage have no effect. Once a weapon only deals 1 point of damage, it’s not a weapon if it shrinks further" (28). Thus, for example, a grig ninja can only look forlornly at its sized-for-a-Tiny-creature poisoned +1 flaming shuriken (PH 117, 121) (166 gp 1 sp; 0.025 lbs.). The object, although crafted as a weapon (so it also can't be used instead as an improvised weapon), is—according to the DMG—just not a weapon, and it has no effect. Even were the grig to somehow make an attack with it, no effect—in this DM's opinion—includes things like the ninja's extraordinary ability sudden strike, any poison coating the shuriken, and the magic weapon special ability flaming. Another reader's opinion on the definition of no effect may vary, of course, but I suspect that a DM that tells a player that a monster's attack had no effect then goes on to detail how the PC has been poisoned, level drained, and improved grabbed will be seen as a bit disingenuous.

The wider view is this: If X can't do Y, then doing Y with X anyway also can't result in Z. For example, when a flash frost locate city spell is cast, because the spell normally doesn't deal any damage, the spell's effect won't generate other effects that're contingent upon the spell dealing damage. Those are just ignored.

In sum, to this reader, saying that the extra damage from the Flash Frost Spell feat applies to the spell locate city is like asking Pizza Hut to deliver a topping but no pizza: Chances are an intrepid negotiator could buy from a Pizza Hut only a topping just like a determined player could find a DM who'd agree that the +2 points of extra cold damage from the feat Flash Frost Spell can be added to the locate city spell's ∅ damage, but she might have to shop around for an accommodating Pizza Hut or an agreeable DM.

• How do metamagic benefits apply exactly? As previously discussed, there's an absolutely precise order of operations that must be adhered to if a locate city bomb is to be dropped. After realizing the benefit of the feat Snowcasting the caster of the spell locate city then applies on the fly to that spell the following metamagic feats in this exact order: first, the feat Flash Frost Spell that can only be applied to spells that possess the descriptor cold; second, the feat Energy Substitution (electricity) that changes the locate city spell's descriptor from cold to electricity; third, the feat Born of the Three Thunders that can only be applied to a spell that deals electricity or sonic damage and that adds an effect requiring a Reflex saving throw; and, fourth, the feat Explosive Spell that can only be applied to a spell that mandates a Reflex saving throw. If metamagic feats are applied in any other order the bomb is a dud.

This reader agrees with this fine answer that constant targeting isn't a thing in D&D 3.5. (Also see this answer; I mean, I, like, really agree!) However, when metamagic is applied to a spell, it's unclear if the metamagic benefits apply sequentially in the order the caster prefers or if the benefits apply simultaneously.

In a totally different context, the sometimes-suspect Main FAQ says that as "a general guideline, whenever the rules don’t stipulate an order of operations… apply them in the order that’s most beneficial to the 'controller' of the effect" (51), but this general guideline is not followed by the only example of the interaction between metamagic feats when their order application matters: the interaction between the feats Empower Spell (PH 93) and Maximize Spell (97–8). The printed benefit of the feat Maximize Spell, in part, says

An empowered, maximized spell gains the separate benefits of each feat: the maximum result plus one-half the normally rolled result. An empowered, maximized fireball cast by a 15th-level wizard deals points of damage equal to 60 plus one half of 10d6. (ibid.)

To my knowledge, no other examples exist of how to apply metamagic benefits when the order is actually important, as it is to make a locate city bomb. In other words, for a locate city bomb to go off, the DM must rule that each metamagic feat is applied in the most favorable order.

If the DM rules that each metamagic feat is applied simultaneously and separately, then the feats Flash Frost Spell and feat Energy Substitution (electricity) find the spell locate city with its descriptor cold from the Snowcasting feat acceptable, but the feats Born of the Three Thunders and Explosive Spell do not find the cold locate city spell acceptable: the cold locate city spell won't be eligible for those feats until after the other feats have been applied to the spell.

In short, this is a gray area as there is no rule-as-written order of metamagic benefits application—and the DM must rule that order of application favors the bomber for the locate city bomb to work. This general topic, by the way, is rarely broached. I found only a 2013 thread from the Rule of Cool messageboards that reaches some eyebrow-raising conclusions in support of simultaneity and a 2005 thread from Andy Collins's Tapatalk messageboards. (Unfortunately, 3.5 revision architect Collins himself doesn't participate in the latter, but the OP of the former does—in 2013! Small world.)

(So the reader's aware, besides the locate city bomb the other topic in metamagic application ambiguity is the duration of a spell modified by both the feat Extend Spell (PH 94) and the feat Persistent Spell (CAr 81). Folks apparently die on hills named 24 Hours, 24 Hours Plus Its Normal Duration, 48 Hours, and Would You Guys Keep It Down? We're Trying to Play a Game Here.)

• A circle is a burst? Although he knows he shouldn't be, this reader actually is mildly uncomfortable with the locate city spell's entry Area: 10 miles/level radius circle, centered on you meeting the requirement of the feat Explosive Spell that says that the feat's benefit "can be applied only to spells that… affect an area (a cone, cylinder, line, or burst)" (79). The entry 10 miles/level radius circle, centered on you is an Other area about which that the Player's Handbook says, "A spell can have a unique area, as defined in its description" (PH 176). Like the 2nd-level Sor/Wiz spell locate object [div] (249) and its entry Area: Circle, centered on you, with a radius of 400 ft. + 40 ft./level and the 6th-level Sor/Wiz control weather [trans] (214) and its entry Area: 2-mile-radius circle, centered on you; see text, the spell locate city just doesn't have an area that's a cone, cylinder, line, or burst.

Technically, it seems that the locate city spell is ineligible on these grounds alone to be the beneficiary of the feat Explosive Spell. (Ironically, as this fine answer mentions, equally ineligible is the 3rd-level Sor/Wiz spell fireball [evoc] (231) that has the entry Area: 20-ft.-radius spread and that has the fact that it's a darn spread mentioned in the example in the benefit of the feat Explosive spell! [Cue groans.])

• "Mission Control, am I ready for launch?" This isn't dubious per se, but it is something to be aware of. The spell locate city possesses the entries Range: 10 miles/level and Area: 10 miles/level radius circle, centered on you. Unlike a normal area spell that has its casters "select the point [within the spell's range entry] where the spell originates" (PH 175), the caster is always the center of the locate city effect. This may mean that, if a caster does manage to drop a locate city bomb, she herself may need a way to avoid the ensuing effect. (The game tends to say so when an effect is centered on the caster yet leaves the caster unaffected; cf. the safe-for-the-caster 3rd-level Sor/Wiz spell fireburst [evoc] (SpC 93) et al. and the your-psion-has-acid-resistance-right? 6th-level psion/wilder power breath of the black dragon [psychometabolism] (Expanded Psionics Handbook 81).)

In other words, a DM that sees a Sor8 drop a locate city bomb may rule that the caster rockets straight up, is dealt 42,240d6 points of damage, and is technically in space. (Also see this question.)

I know that's a lot, but there's a lot here to unpack. Your patience is appreciated. Anyway, to sum up, there are, in my opinion, a few dubious points according to the rules as written, therefore making it so the DM must rule on the following points:

1. Can the benefit of the Flash Frost Spell feat be applied to spells that possess an area entry yet that don't also possess a target entry?
2. When a spell benefits from the feat Flash Frost Spell, is the damage that's a benefit of the feat dealt no matter the spell's normal effect?
3. Are metamagic benefits applied sequentially in the order the recipient chooses?
4. When the game uses the word circle, does it actually mean sphere, and is that sphere also a burst?

If to each of those questions you said Yes, then you can drop a locate city bomb as the question describes. However, then the DM must answer the final question: Will the caster be affected by the spell?

Note: Even if the DM says yes to only #1 and #2, a Sor4 with a Snowcast flash frost locate city spell can take 1 round then a full round action to commit murder on a kingdom scale. Just sayin'. Also I have come to hate the feat Explosive Spell. Seriously, no damage cap? Really?

• I can't downvote such a good answer over it, but so far as I can tell, there is absolutely no reason whatsoever to suspect that any and all uses of the word “target” solely and exclusively refer to the “Target” entry of a spell, and I find the suggestion that it does here problematic and misleading. – KRyan Nov 22 '19 at 16:44
• @KRyan I hope the answer now addresses your concerns. – Hey I Can Chan Jan 29 '20 at 19:31
• Yup, that’s a fair point there and this answer is excellent. – KRyan Jan 29 '20 at 19:36
• +1 because I strongly approve of this answer confirming my feelings about 3.5 some 10+ years ago despite what everyone on the PbP board I was on. – Pyrotechnical Jan 29 at 14:26

RAW, the legality of the locate city bomb is ambiguous. Step 3. is the primary problem, hinging on the definition of “extra” which the rules don’t actually define. Step 3. has another possible issue, and step 6. has some maybe-issues, but those I don’t consider those RAW-valid.

1. No problem; clearly locate city is a spell that can be cast.

2. No problem; even if you argue that you need to actually have snow around you and use a move action to collect it, you can do that.

3. The Flash Frost feat describes its effect as referring to “spells that use cold and ice to damage your foes.” The problem with this argument, however, is that this is the description, prior to the actual Benefit section describing, in rules terms, what the effect is. The benefit section says only that “This metamagic feat can be applied only to spells that have the cold descriptor and that affect an area.” Locate city meets those criteria when cast with Snowcasting.

The benefits section does say that the spell deals “an extra 2 points of cold damage,” which is a fairly-common ambiguity in the rules: can you add to, gain a bonus to, gain extra of, something you didn’t have before? Should we treat the spell as having previously done 0 damage, so now it deals 0 + 2 damage, or should we treat “extra” as requiring the damage to have already been present? This is ambiguous. However, “ambiguous” is as far as it goes—the rules use “extra” a lot, and it’s often unclear, but the game never defines it. So this still could work. And ruling “extra” as always requiring some previous quantity could set a problematic precedent for a lot of other things, since the word is used a lot.

Another potential issue for Flash Frost is that we are going to be using Energy Substitution on it to change the cold descriptor, as required by Flash Frost, to the electricity descriptor. However, there really is nothing in the rules that suggests that D&D 3.5e uses “constant targeting,” that would cause us to go back and check the spell’s cold descriptor for Flash Frost after we apply it. What little we do have suggests D&D does not do this. See here for more.

4. No problem; with Snowcasting and Flash Frost, locate city is a divination [cold] spell that deals 2 cold damage, which Energy Substitution can change to a divination [electricity] spell that deals 2 electricity damage.

Note that the Deities & Demigods version of Energy Substitution, which is also the version found in the SRD, is the older version of the feat, and does not mention changing the descriptors, which would cause a problematic ambiguity. Complete Arcane has the latest version of the feat, however, and explicitly addresses this.

5. No problem; locate city affects an area, and thanks to the previous steps, ours deals damage and has the electricity descriptor. Works as written.

6. Locate city specifies its area as a 10-mile/level-radius circle, and “circle” isn’t on the list of valid area types for Explosive Spell. However, this is really part of a wider underspecification in D&D 3.5e—very few spells actually define themselves to be “bursts” as Explosive Spell indicates. Burst is simply the default kind of area, so things will just say “X-foot radius” rather than “X-foot-radius burst.” For that matter, they also will often not bother to distinguish between a burst and an emanation—unless the effect moves, the two are usually indistinguishable. Barring Explosive Spell on locate city under this rationale would suggest that it also doesn’t apply to a whole host of spells it almost-certainly was intended for—like the fireball described in the feat’s own example.

The other argument people sometimes make is that locate city specifies “circle,” rather than “sphere.” The idea isn’t that this prevents the combination, so much as makes it much less useful, as it can only hit things in one plane (the geometric term, not the cosmological one), and even then they’re just shunted above or below that and almost-certainly take no damage as they don’t move 10 feet. This is, in my opinion, truly grasping at straws, and a very bad precedent to set in the game. For one thing, it’s going to cause a lot of problems with using locate city for its straightforward purpose, locating cities, but more importantly, the game uses “square” everywhere when really it ought to be using “cube.” A whole lot of the game’s rules are written assuming 2D when things are really 3D; who knows why. But locate city is just another case of that—trying to argue more than that is just asking for trouble from other precedents.

Note that, regardless of RAW, I recommend banning the combo. It’s overpowered. That’s not the question here. Personally, I would probably nix Flash Frost from adding cold damage to any overly-large-area, non-damaging spell. I’d even allow Flash Frost on non-damage spells, so long as they’re small-ish, but not to something so large. Even without the rest, that would cause problems.

• On #3, my concern is that the benefit of the feat Flash Frost Spell says, in part, that a modified "spell deals an extra 2 points of cold damage per level of the spell to all targets in the area" (PH2 91 and emphasis mine). Since the spell locate city, by default, deals not zero but no damage then there's nothing delivering that extra 2 points of damage (maybe like somehow applying the flaming magic weapon special ability to a sock or some other can't-be-used-as-an-improvised-weapon object). Or is such an avenue simply not worth exploring? – Hey I Can Chan Nov 11 '19 at 20:24
• @HeyICanChan Good point, should address. It’s ambiguous. – KRyan Nov 11 '19 at 20:25
• (So far as I can tell, extra damage is pretty consistently used tightly in conjunction with effects that already deal damage. That is, I couldn't quickly find anywhere else where the reader would reach the conclusion that an instance of extra damage can be added to a ∅ quantity. After all, a grig ninja's shuriken deals no damage so he doesn't still deal his sudden strike damage with his shuriken. But you've addressed it, that's cool, and thank you. If I feel strongly about it later, I'll answer the question myself.) – Hey I Can Chan Nov 11 '19 at 20:44
• @HeyICanChan An example like that is precisely what I had in mind. Sudden strike, IIRC, says that if damage is reduced to 0 by damage reduction that it doesn’t apply, but this is a weapon that deals $0$ damage by default—it’s not clear to me that it shouldn’t be $0+X\text{d}6$. – KRyan Nov 11 '19 at 20:49
• Sometimes that short description is the only place some rules can be found (as per answers to this question), and, for spells, the short list description is the only place that has a superscript indicating some focus or material components that don't have prices also aren't in a spell component pouch (e.g. plane shift, simulacrum). Just sayin'. – Hey I Can Chan Nov 12 '19 at 16:37


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# The Percolation Signature of the Spin Glass Transition - Condensed Matter > Statistical Mechanics

The Percolation Signature of the Spin Glass Transition - Condensed Matter > Statistical Mechanics - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: Magnetic ordering at low temperature for Ising ferromagnets manifests itselfwithin the associated Fortuin-Kasteleyn FK random cluster representation asthe occurrence of a single positive density percolating network. In this paperwe investigate the percolation signature for Ising spin glass ordering - bothin short-range EA and infinite-range SK models - within a two-replica FKrepresentation and also within the different Chayes-Machta-Redner two-replicagraphical representation. Based on numerical studies of the $\pm J$ EA model inthree dimensions and on rigorous results for the SK model, we conclude that thespin glass transition corresponds to the appearance of {\it two} percolatingclusters of {\it unequal} densities.

Autor: J. Machta, C. M. Newman, D. L. Stein

Fuente: https://arxiv.org/

format: plain


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## Past Talks

Motion by mean curvature and coupled KPZ from particle systems
Monday, 25 May 2020.
Professor @Waseda University
Funaki was a Professor @The University of Tokyo from 1995 until 2017 and is now a Professor @Waseda University from 2017. He is a Professor Emeritus of the University of Tokyo.

#### Abstract

It's one of common interests in probability group to derive nonlinear PDEs or stochastic PDEs from microscopic interacting systems via scaling limits in space and time, usually by some averaging effect caused by the local ergodic property of the microscopic systems.

In the talk, we discuss the derivation of two objects: Motion by mean curvature (MMC) and coupled KPZ (Kardar-Parisi-Zhang) equation. The microscopic system we take is that of particles which perform random walks with interaction. To derive MMC, we allow creation and annihilation of particles. The system exhibits a phase separation to sparse and dense regions of particles and, macroscopically, the interface separating these two regions evolves under the MMC. We pass through the Allen-Cahn equation with nonlinear Laplacian at an intermediate level. On the other hand, the coupled KPZ equation is a system of nonlinear singular stochastic PDEs. It is ill-posed in a classical sense and requires renormalizations. We derive it under the fluctuation limit of multi-species weakly-asymmetric particle system of interacting random walks without creation and annihilation. The so-called Boltzmann-Gibbs principle plays a fundamental role for both.

References

The first part is joint work with S. Sethuraman, D. Hilhorst, P. El Kettani and H. Park ( arXiv:2004.05276), while the second is with C. Bernardin and S. Sethuraman ( arXiv:1908.07863).

Back to Program

The vanishing discount problem for systems of Hamilton-Jacobi equations
Monday, 18 May 2020.
Research Fellow @Tsuda University (Japan), Professor Emeritus @Waseda University (Japan), Fellow @American Mathematical Society
Professor Ishii received his Ph.D. from Waseda University in June 1975. He worked at Chuo University from 1976 until 1996, at Metropolitan University from 1996 until 2001, and at Waseda University from 2001 until 2018.

#### Abstract

In the talk, I discuss the recent developments of the vanishing discount problem. The main focus concerns that for monotone systems of Hamilton-Jacobi equations. The principal tool in the analysis for the vanishing discount is the use of Mather measures from Aubry-Mather theory or their generalizations. I explain a way of constructing Mather measures and an approach to the vanishing discount problem.

This talk bases on joint work with Liang Jin of Nanjing University of Science and Technology.

References
• [1]  Hitoshi Ishii and Liang Jin, The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 2: nonlinear coupling, arXiv:1906.07979 [math.AP], to appear in Calculus of Variations and PDE's.
• [2]  Hitoshi Ishii, The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling, arXiv:1903.00244 [math.AP].
• [3]  Andrea Davini and Maxime Zavidovique, Convergence of the solutions of discounted Hamilton-Jacobi systems, Adv. Calc. Var. Online publication (2019), DOI: 10.1515/acv-2018-0037.

Back to Program

Recent progress on a sharp lower bound for Steklov eigenvalue
Monday, 11 May 2020.
Professor
@ Xiamen University, China
Professor Xia received his PhD in July 2012 from the Albert Ludwig University of Freiburg in Germany. He was a doctoral student of Professor Guofang Wang. Professor Xia was a Postdoc Fellow at the Max Planck Institute for Mathematics and Sciences at Leipzig in Germany from 2012-2014, and Postdoc Fellow at McGill University in Montreal, Canada, from 2015-2016.

#### Abstract

Lichnerowicz-Obata's theorem says that for a closed $n$-manifold with Ricci curvature $Ric\ge (n-1)K>0$, the first (nonzero) eigenvalue is greater than or equal to $nK$, with equality holding only on a round $n$-sphere. Similar results for the first Dirichlet eigenvalue and Neumann eigenvalue have been shown by Reilly, Escobar and C.Y.Xia respectively. For the first (nonzero) Steklov eigenvalue, a conjecture has been made by Escobar saying that for a compact manifold with boundary which has nonnegative Ricci curvature and boundary principal curvatures bounded below by some $c>0$, the first (nonzero) Steklov eigenvalue is greater than or equal to $c$, with equality holding only on a Euclidean ball. This conjecture is true in two dimensions due to Payne and Escobar. In this talk, we present a resolution to this conjecture in the case of nonnegative sectional curvature in any dimensions. We will also discuss a sharp comparison result between the first (nonzero) Steklov eigenvalue and the boundary first eigenvalue.

This is a joint work with Changwei Xiong.


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# Naive Bayes Hyperparameter Tuning

Wallace: Author Detection via Recurrent Neural Networks Leon Yao Department of Computer Science Stanford University [email protected] These baselines provide an understanding of the possible predictive power of a dataset. GaussianNB(priors=None) [source] Gaussian Naive Bayes (GaussianNB) Can perform online updates to model parameters via partial_fit method. Quick Start. We now use the Sonar dataset from the mlbench package to explore a new regularization method, regularized discriminant analysis (RDA), which combines the LDA and QDA. This Training Model class only uses the Multinomial Naive Bayes implementation found in the Spark MLLib library. Distance Metrics, K Nearest Neighbors, Clustering, Decision Trees, Ensemble Methods, Dimensionality Reduction, Pipeline Building, Hyperparameter Tuning, Grid Search, Scikit-Learn In the final module, you'll learn how to use regular expressions in Python and how to manage string values, analyze text, and perform sentiment analysis. The comparison of Naive bayes classifier and word2vec classifier used for identifying intent to the question, is made. Asim Roy, Shiban Qureshi, Kartikeya Pande, Divitha Nair, Kartik Gairola, Pooja Jain, Suraj Singh, Kirti Sharma, Akshay Jagadale, Yi Yang Lin, Shashank Sharma, Ramya. NN and SVM) in classiﬁcation and regression. N-gram Model. Decision trees are easy to use for small amounts of classes. However, Weka is a GPL-licensed Java library, and was not written with scalability in mind, so we feel there is a need for alternatives to Auto-Weka. Breiman, L. Firstly to make predictions with SVM for sparse data, it must have been fit on the dataset. An Empirical Bayes Approach to Optimizing Machine Learning Algorithms James McInerney Spotify Research 45 W 18th St, 7th Floor New York, NY 10011 [email protected] Naive bayes classifiers are commonly used for text classification, and are a traditional solution for spam detection. 이 논문은 hyperparameter tuning 문제를 Bayesian optimization을 사용해여 해결하는 방법을 제안한다. Naive Bayes; Support-Vector machines; Gradient Boosting; Random Regression Forests and Random Classification Forests; Kriging; The base class of learners is Learner, specialized for regression as LearnerRegr and for classification as LearnerClassif. But hyperparameter tuning requires a number of training jobs on different subsets of the training data. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Smaller loss is better. Thus, the second contribution of this work is a comparison of Bayesian optimization and random search in hyperparameter tuning, namely in support vector regression. Fold 5 (160 reviews) is used to estimate the performance of the classi ers. You will now practice this yourself, but by using logistic regression on the diabetes dataset instead!. Roadmap 1 Tuning hyperparameters Motivation Machine learning without data Assessing the quality of a trained SVM Model selection log of the bandwith log of C 1. Despite its success, standard BO focuses on a single ta. Perform support vector machine (SVM) and Naive Bayes classification, create bags of decision trees, and fit lasso regression on out-of-memory data. Hyperparameters are learned during training and allow the algorithm to generalize beyond the training set. We now use the Sonar dataset from the mlbench package to explore a new regularization method, regularized discriminant analysis (RDA), which combines the LDA and QDA. , 2017], Google’s internal hyperparameter tuning service. The caret package contains train() function which is helpful in setting up a grid of tuning parameters for a number of classification and regression routines, fits each model and calculates a resampling based performance measure. class sklearn. An ensemble-learning meta-classifier for stacking. We take a step by step approach to understand Bayes and implementing the different options in Scikitlearn. There's an endless supply of industries and applications machine learning can be applied to to make them more efficient and intelligent. One of the advantages of Cloud ML Engine is that it provides out-of-the-box support for hyperparameter tuning using a simple YAML configuration without any changes required in the training code. Specifically, the hyperparameter tuning service in ML Engine allows users to evaluate different types of hyperparameter combinations, while also benefiting from the managed hyper-parameter tuning service using Bayesian optimization that speeds up optimization process compared to a naive grid search. Gain expertise in ML techniques with AWS to create interactive apps using SageMaker, Apache Spark, and TensorFlow. Enterprises Training Courses. See the complete profile on LinkedIn and discover Pradip’s connections and jobs at similar companies. 1BestCsharp blog 6,288,604 views. The Naive Bayes model showed this initially, and was proven out even further in other succeeding models used. Naive Bayes Classifier and Black Box Machine Learning | Day 15. People who are familiar with Machine Learning might want to fast forward to Section 3 for details. • Arjun Mukherjee and Bing Liu. This paper investigates the effects of the hyperparameter tuning on the predictive performance of DT induction algorithms, as well as the impact hyperparameters have on the ﬁnal predictive performance of the induced models. In this tutorial, you covered a lot of ground about Support vector machine algorithm, its working, kernels, hyperparameter tuning, model building and evaluation on breast cancer dataset using the Scikit-learn package. Smaller loss is better. Discover how machine learning algorithms work including kNN, decision trees, naive bayes, SVM, ensembles and much more in my new book, with 22 tutorials and examples in excel. Naive Bayes. Model evaluation, model selection, and algorithm selection in machine learning - Part III - Cross-validation and hyperparameter tuning Google developers: Machine Learning Recipes Rules of Machine Learning: Best Practices for ML Engineering. I am an NTU Singapore and BITS Goa alumnus with a background in Signal Processing, Electronics, and Instrumentation. [18] propose the Decision Tree classiﬁer on the benchmark Heart UCI (University of California, Irvine, CA, USA) dataset by applying several tuning techniques to Decision Trees like different combinations of discretization,. import optunity import optunity. Find something interesting to watch in seconds. Bayes Theorem and Naive Bayes 09:30 We'll build a real spam classifier using Naive Bayes, and see how it well it works on our problem of classifying vehicle speeds based on upcoming obstacles in the road. Understanding Bayes theorem with conditional probability. Gaussian processes for regression without hyperparameter-tuning: unary, binary, nominal, numeric Class for generating a decision tree with naive Bayes classifiers. Naive Bayes SMS spam classification example. Laplace estimator. CaseStudy1 Predicting Income Status¶The objective of this case study is to fit and compare three different binary classifiers to predict whether an individual earns more than USD 50,000 (50K) or less in a year using the 1994 US Census Data sourced from the UCI Machine Learning Repository (Lichman, 2013). These baselines provide an understanding of the possible predictive power of a dataset. As the search progresses, the algorithm switches from exploration — trying new hyperparameter values — to exploitation — using hyperparameter values that resulted in the lowest objective function loss. Because hyperparameter optimization can lead to an overfitted model, the recommended approach is to create a separate test set before importing your data into the Classification Learner app. The comparison of Naive bayes classifier and word2vec classifier used for identifying intent to the question, is made. For naive bayes we fine-tuned the Laplace parameter testing values between 0 and 30. Flexible Data Ingestion. In multi-label classification, this is the subset accuracy which is a harsh metric since you require for each sample that each label set be correctly predicted. They demonstrated that a random search across a large hyperparameter space is more effective than a manual approach, as we did for Multinomial Naive Bayes, and often as effective—or more so—than GridSearch. Hyperparameter Optimization on Spark MLLib using Monte Carlo methods Some time back I wrote a post titled Hyperparameter Optimization using Monte Carlo Methods , which described an experiment to find optimal hyperparameters for a Scikit-Learn Random Forest classifier. In order to carry this out, we segmented our training data into a training set and a validation set. This makes them less useful for large scale or online learning models. The final task of this chapter will be to apply our newly gained skills to a real spam filter! This task deals with solving a binary-class (spam/ham) classification problem using the Naive Bayes algorithm. 927$, and the AUC has increased from$0. Evaluation for Naive Bayes Classifier after grid-search and cross-validation. For many machine learning algorithms, predictive performance is critically affected by the hyperparameter values used to train them. Live! จากงาน Workshop: Python Data Science for Developer 🐍 by Sorratat Sirirattanajakarin & 3Digits Academy Agenda #Day2: 9. API Reference. This also helped us to avoid overfitting issues. This comprehensive 2-in-1 course is a comprehensive, practical guide to master the basics and learn from real-life applications of machine learning. This course will introduce the learner to applied data analytics with Python, focusing more on the techniques and methods than on the statistics behind these methods. Bayesian optimization (BO) is a popular methodology to tune the hyperparameters of expensive black-box functions. My thesis based on approaches and algorithms of deep learning and applications in various domains. Scikit has CalibratedClassifierCV, which allows us to calibrate our models on a particular X, y pair. In our previous articles, we have introduced you to Random Forest and compared it against a CART model. , 2017], Google’s internal hyperparameter tuning service. shape [0] positive_digit = 3 negative_digit = 9 positive_idx = [i for i in range (n) if digits. Decision trees are supervised learning models used for problems involving classification and regression. (2001), Random Forests, Machine Learning 45(1), 5-32. SVM's are pretty great at text classification tasks; Models based on simple averaging of word-vectors can be surprisingly good too (given how much information is lost in taking the average). Choosing a right value of K is a process called Hyperparameter Tuning. algorithms, namely Naïve Bayes, Logistic Regression and Random Forest. The parameter test_size is given value 0. Leveraging parallel and distributed computational resources presents a solution to the increasingly challenging problem of hyperparameter optimization. naive_bayes. For many machine learning algorithms, predictive performance is critically affected by the hyperparameter values used to train them. individual words as tokens). Flexible Data Ingestion. The comparison of Naive bayes classifier and word2vec classifier used for identifying intent to the question, is made. In my last blog post I showed how to create a multi class classification ensemble using scikit-learn's VotingClassifier and finished mentioning that I didn't know which classifiers should be part of the ensemble. Furthermore, we include a Naïve Bayes Classifier in our tests, because of its high accuracy in binary classification. Kallirroi has 4 jobs listed on their profile. Machine Learning. Quick Start. a parameter that controls the form of the model itself. This is how important tuning these machine learning algorithms are. Applied Machine Learning - Beginner to Professional course by Analytics Vidhya aims to provide you with everything you need to know to become a machine learning expert. In Bayesian statistics, a hyperparameter is a parameter of a prior distribution; the term is used to distinguish them from parameters of the model for the underlying system under analysis. The Naive Bayes model showed this initially, and was proven out even further in other succeeding models used. This course will introduce the learner to applied data analytics with Python, focusing more on the techniques and methods than on the statistics behind these methods. Hyperparameter gradients might also not be available. One study by Tomana et al. See the complete profile on LinkedIn and discover Ye’s connections and jobs at similar companies. SVM Parameter Tuning in Scikit Learn using GridSearchCV. Source: https://github. To reduce this, we performed the parameter tuning to get the optimal value of "laplace" parameter of X and Y coordinates. Si No No No No Si * HNB Contructs Hidden Naive Bayes classification model with high classification accuracy and AUC. For details on algorithm used to update feature means and variance online, see Stanford CS tech report STAN-CS-79-773 by Chan, Golub, and LeVeque:. Includes an example with, - brief definition of what is svm? - svm classification model - svm classification plot - interpretation - tuning or hyperparameter optimization - best model selection. SK3 SK Part 3: Cross-Validation and Hyperparameter Tuning¶ In SK Part 1, we learn how to evaluate a machine learning model using the train_test_split function to split the full set into disjoint training and test sets based on a specified test size ratio. Naive Bayes classification. The tutorial provides an example for doing this while also doing additional hyperparameter tuning in a nested CV-setting. Naive Bayes methods are a set of supervised learning algorithms based on applying Bayes’ theorem with the “naive” assumption of conditional independence between every pair of features given the value of the class variable. Naive Bayes. At a high level, there are four kinds of machine learning: supervised learning, unsupervised learning, reinforcement learning, and active machine learning. In this Video I will show you how you can easily tune the crap out of your model… using python and scikit-learn. How to use for loops for hyperparameter tuning using fitcnb. This yields the benefit of integrating hyperparameter tuning with model-based optimization into your machine learning experiments without any overhead. Tree models present a high flexibility that comes at a price: on one hand, trees are able to capture complex non-linear relationships; on the other hand, they are prone to memorizing the noise present in a dataset. Helping 3M+ developers be better through coding contests, data science competitions, and hackathons. The AUTOTUNE statement in the NNET, TREESPLIT, FOREST, and GRADBOOST procedures defines tunable parameters, default ranges, user overrides, and. hyperparameter tuning) An important task in ML is model selection, or using data to find the best model or parameters for a given task. Next problem is tuning hyperparameters of one of the basic machine learning models, Support Vector Machine. Hyperparameter Optimization on Spark MLLib using Monte Carlo methods Some time back I wrote a post titled Hyperparameter Optimization using Monte Carlo Methods , which described an experiment to find optimal hyperparameters for a Scikit-Learn Random Forest classifier. Decision trees are easy to use for small amounts of classes. Through hyperparameter optimization, a practitioner identifies free parameters in the model that can be tuned to achieve better model performance. Manipulate and analyze data that is too big to fit in memory. An ensemble-learning meta-classifier for stacking. Vector Machine, Naive Bayes and Decision Tree are also. This is the class and function reference of scikit-learn. The classification template uses the Naive Bayes algorithm by default. In today's big data world, many companies have gathered huge amounts of customer data about marketing success, use of financial services, online usage, and even fraud behavior. The following subsection will go over our Naive Bayes implementation in NBModel. This process is called hyperparameter tuning. We now turn to the challenge of tuning our GBM’s hyperparameters. In a cartesian grid search, users specify a set of values for each hyperparameter that they want to search over, and H2O will train a model for every combination of the hyperparameter values. View Teja Krishna Talluri’s profile on LinkedIn, the world's largest professional community. python,syntax,machine-learning,scikit-learn. I am a data scientist mainly focusing on business side and have a background on machine learning and deep learning. Naive Bayes; Support-Vector machines; Gradient Boosting; Random Regression Forests and Random Classification Forests; Kriging; The base class of learners is Learner, specialized for regression as LearnerRegr and for classification as LearnerClassif. Por Erick Almaraz. They used the multinomial Naive Bayes classifier in their tests. Quite the same Wikipedia. [email protected] Answer Wiki. You can test a large number of potential smoothing parameters, evaluating the accuracy of the classifier using each. 92$, the recall for class 0 has increased from$0. Naïve Bayes classification with caret package. naive_bayes. You will find tutorials to implement machine learning algorithms, understand the purpose and get clear and in-depth knowledge. Flexible Data Ingestion. This time pretrained embeddings do better than Word2Vec and Naive Bayes does really well, otherwise same as before. This algorithm exhibited 99 percent accuracy on test data, and a Matthews correlation coefficient of 0. com Abstract There is rapidly growing interest in using Bayesian optimization to tune model and inference hyperparameters for machine learning algorithms that take a long time to run. pt Abstract— pThe starting point in every Machine Learning model application is the input features. The classification template uses a naive bayesian algorithm that has a smoothing parameter. Stemmerb, Alceu de S. Random Forest is one of the easiest machine learning tool used in the industry. Machine Learning tools are known for their performance. This tutorial will focus on the model building process, including how to tune hyperparameters. Hyperparameter Optimization. Run and report on experiments for tuning the a hyperparameter. One study by Tomana et al. In the [next tutorial], we will create weekly predictions based on the model we have created here. 4 - a Python package on PyPI - Libraries. However, recall that the predicted results required in the specifications listed in the overview are of the form:. I would like to tune the threshold (and only the threshold) for the classification. TPOT is a Python library that automatically creates and optimizes full machine learning pipelines using genetic programming. It means that your hyperparameter space is tree-like: the value chosen for one hyperparameter determines what hyperparameter will be chosen next and what values are available for it. Parameter ranges. Naive Bayes and K-NN, are both examples of supervised learning (where the data comes already labeled). Catboost is a gradient boosting library that was released by Yandex. The performance of these classification algorithms is evaluated based on accuracy. Since the curve is not known, a naive approach would be the pick a few values of x and try to observe the corresponding values f(x). Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Seeing as XGBoost is used by many Kaggle competition winners, it is worth having a look at CatBoost! Contents. I am training a Naive Bayes model using the mlr package. Grid object is ready to do 10-fold cross validation on a KNN model using classification accuracy as the evaluation metric. This is the best case scenario. Specifically, this tutorial will cover a. Distance Metrics, K Nearest Neighbors, Clustering, Decision Trees, Ensemble Methods, Dimensionality Reduction, Pipeline Building, Hyperparameter Tuning, Grid Search, Scikit-Learn In the final module, you'll learn how to use regular expressions in Python and how to manage string values, analyze text, and perform sentiment analysis. The number of peaks. The algorithms were subsequently trained with optimized hyperparameter settings on the full training set and evaluated on the hold-out test set, which has not been used for preprocessing and hyperparameter tuning in any form. However, recent evidence on a benchmark of over a hundred hyperparameter optimization datasets suggests that such enthusiasm may call for increased scrutiny. Are Naive Bayes algorithms affected by outliers in the data? Suppose there is a data set, does one need to. These baselines provide an understanding of the possible predictive power of a dataset. Smaller loss is better. • creating custom- model docker containers and using amazon Sagemaker for hyperparameter tuning and deployment Naive Bayes, Logistic Regression. The most popular member of the family is probably Multinomial Naive Bayes (MNB), and it's one of the algorithms that we use here at MonkeyLearn. Conclusions. It was conceived by the Reverend Thomas Bayes, an 18th-century British statistician who sought to explain how humans make predictions based on their changing beliefs. The hyperparameter $$\rho$$ is the length-scale, and corresponds to the frequency of the functions represented by the Gaussian process prior with respect to the domain. shape [0] positive_digit = 3 negative_digit = 9 positive_idx = [i for i in range (n) if digits. Support Vector Machine (SVM) or Naive Bayes. What exactly is a hyperparameter? 1 answer Can anyone give me full details about what we mean by hyperparameters, and what in the Dirichlet distribution are called hyperparameters? A practice example for the estimation of those parameters would also be useful. For details on algorithm used to update feature means and variance online, see Stanford CS tech report STAN-CS-79-773 by. classifier import StackingClassifier. number of units, learning rate, L 2 weight cost, dropout probability You can evaluate them using a validation set, but there’s still the problem of which values to try. Live! จากงาน Workshop: Python Data Science for Developer by Sorratat Sirirattanajakarin & 3Digits Academy Agenda #Day2: 9. You will find tutorials to implement machine learning algorithms, understand the purpose and get clear and in-depth knowledge. Kaggle competitors spend considerable time on tuning. 927$, and the AUC has increased from$0. Distance Metrics, K Nearest Neighbors, Clustering, Decision Trees, Ensemble Methods, Dimensionality Reduction, Pipeline Building, Hyperparameter Tuning, Grid Search, Scikit-Learn In the final module, you'll learn how to use regular expressions in Python and how to manage string values, analyze text, and perform sentiment analysis. , [1] focused on lemmatization and stemming algorithms as a means to normalize text. Naive Bayes SMS spam classification example. Machine Learning tools are known for their performance. Model Tuning. Module Identifier Overfitting - Naive Bayes and Bayesian Hyperparameter tuning Other topics and in machine learning Reinforcement. This course will introduce the learner to applied data analytics with Python, focusing more on the techniques and methods than on the statistics behind these methods. Laplace estimator. In today's big data world, many companies have gathered huge amounts of customer data about marketing success, use of financial services, online usage, and even fraud behavior. The examples in this post will demonstrate how you can use the caret R package to tune a machine learning algorithm. Consultez le profil complet sur LinkedIn et découvrez les relations de Guillaume, ainsi que des emplois dans des entreprises similaires. Specifically, the hyperparameter tuning service in ML Engine allows users to evaluate different types of hyperparameter combinations, while also benefiting from the managed hyper-parameter tuning service using Bayesian optimization that speeds up optimization process compared to a naive grid search. Proceedings of the 2010 Conference on. The naive Bayes algorithm leverages Bayes theorem and makes the assumption that predictors are conditionally independent, given the class. Por Erick Almaraz. Tuning the learning rate. View PRASHANT BANSOD’S profile on LinkedIn, the world's largest professional community. Feature engineering is a hard problem to automate, however, and not all AutoML systems handle it. Seeing as XGBoost is used by many Kaggle competition winners, it is worth having a look at CatBoost! Contents. Explore Popular Topics Like Government, Sports, Medicine, Fintech, Food, More. The model we will be using in this video is again the model from the Video about. Used Logistic Regression, KNN, Naïve Bayes, Decision Tree, Random Forest, finally selected Ranodm forest with hyperparameter tuning Classification model, predicting churning behavior of customer for a telecom company. Also try practice problems to test & improve your skill level. There are actually further under the hood features implemented by Google for their AI Platform hyperparameter tuning service that further improves the quality of life during parameter searching. Hyperparameter tuning may be one of the most tricky, yet interesting, topics in Machine Learning. CaseStudy1 Predicting Income Status¶The objective of this case study is to fit and compare three different binary classifiers to predict whether an individual earns more than USD 50,000 (50K) or less in a year using the 1994 US Census Data sourced from the UCI Machine Learning Repository (Lichman, 2013). This Microsoft Data Science Online Training Course includes the necessary skillset required for Data Scientists with Microsoft Platform. Bayes SMBO is probably the best candidate as long as resources are not a constraint for you or your team, but you should also consider establishing a baseline with Random Search. Quite the same Wikipedia. Hyperparameter Optimization. Bayesian Optimization) that attempt to improve upon grid/random search. At a high level, there are four kinds of machine learning: supervised learning, unsupervised learning, reinforcement learning, and active machine learning. The performance of these classification algorithms is evaluated based on accuracy. Hyperparameters are learned during training and allow the algorithm to generalize beyond the training set. I set up my Dask cluster using Kubernetes. Package ‘rminer’ April 19, 2011 Type Package Title Simpler use of data mining methods (e. Statistical Data Mining and Machine Learning Hilary Term 2016 Supervised Learning Naïve Bayes Naïve Bayes is a tuning parameter (or hyperparameter ) and. Course Production Assistant. Naive Bayes can be trained very efficiently. See the complete profile on LinkedIn and discover PRASHANT’S connections and jobs at similar companies. As a system architect, I explore, design, and implement pilot projects in object detection, emotion recognition, and human speech source classification. They demonstrated that a random search across a large hyperparameter space is more effective than a manual approach, as we did for Multinomial Naive Bayes, and often as effective—or more so—than GridSearch. Use Machine Learning (Naive Bayes, Random Forest and Logistic Regression) to process and transform Pima Indian Diabetes data to create a prediction model. Quick Start. To know more about SVM, Support Vector Machine; GridSearchCV; Secondly, tuning or hyperparameter optimization is a task to choose the right set of optimal hyperparameters. Naive Bayes baseline. A Naive Bayes classifier assumes that all attributes are conditionally independent, thereby, computing the likelihood is simplified to the product of the conditional. number of units, learning rate, L 2 weight cost, dropout probability You can evaluate them using a validation set, but there’s still the problem of which values to try. Model Building & Hyperparameter Tuning¶ Welcome to the third part of this Machine Learning Walkthrough. In Bayesian statistics, a hyperparameter is a parameter of a prior distribution; the term is used to distinguish them from parameters of the model for the underlying system under analysis. For example: random forests theoretically use feature selection but effectively may not, support vector machines use L2 regularization etc. Using the hyperparameter settings we provided in Table 2, we tuned the ML models, and Table 3 provides the best parameters from our hyperparameter tuning and also the performance improvement compared with corresponding baseline ML models that used default hyperparameter settings. Bayes’ theorem states the following relationship, given class. Training time: Naive Bayes algorithm only requires one pass on the entire dataset to calculate the posterior probabilities for each value of the feature in the dataset. Manipulate and analyze data that is too big to fit in memory. The number of peaks used to classify the structure (see Fig. Choosing the right parameters for a machine learning model is almost more of an art than a science. In this talk we will explain the details of Bayesian optimization for hyperparameter tuning and contrast it with more traditional but naive methods. Shown below, we’ve built a parameter grid totaling 1050 required training jobs. Documentation for the TensorFlow for R interface. Possible appli- cations of this classifier are detecting online harassment, trolling, and other negative behavior on forums, groups, and other forms of social media. Quite the same Wikipedia. (SVM), Naive Bayes and Decision Tree. svm import SVC # Naive Bayes from sklearn. Naïve Bayes for Digits Naïve Bayes: assume all features are independent effects of the label Simple version for digits: One feature F ij for each grid position Possible feature values are on / off, based on whether intensity is more or less than 0. Gain expertise in ML techniques with AWS to create interactive apps using SageMaker, Apache Spark, and TensorFlow. Certain Statistics and Machine Learning Toolbox™ classification functions offer automatic hyperparameter tuning through Bayesian optimization, grid search, or random search. Ravi má na svém profilu 2 pracovní příležitosti. It really starts to pay off when you get into hyperparameter tuning, but I'll save that for another post. In his engaging and informal style, author and R expert Hefin Ioan Rhys lays a firm foundation of ML basics and introduces you to the tidyverse, a powerful set of R tools designed specifically for practical data science. Gradient Descent is not always the best method to calculate the weights, nevertheless it is a relatively fast and easy method. Prajakta has 4 jobs listed on their profile. Hyperparameter tuning, Regularization and Optimization Coursera. If the dataset at time T is similar to some previous datasets, their optimal hyperparameter configuration will be similar, or the corresponding black-box functions will be correlated. There are many approaches that allow for predicting the class of an unknown object, from simple algorithms like Naive Bayes to more complex ones like XGBoost. This feature is intended to prevent users from trying several hyperparameter values on their own and selecting the best results a posteriori, a strategy which would obviously lead to severe bias [ 11 ]. 01, regularization parameter of 0. Pipelines unfortunately do not support the fit_partial API for out-of-core training. Tuning of k-value in KNN classifier. The approach that is used in this research succeeded to increase the classification accuracy for all feature. pdf from CS 101 at Vidya Bharti Senior Secondary School. Découvrez le profil de Guillaume Cazenave sur LinkedIn, la plus grande communauté professionnelle au monde. You will now practice this yourself, but by using logistic regression on the diabetes dataset instead!. metrics import numpy as np # k nearest neighbours from sklearn. A hyperparameter is a parameter that measures the process of learning using its value. This tutorial will focus on the model building process, including how to tune hyperparameters. Stacking is an ensemble learning technique to combine multiple classification models via a meta-classifier. Certain Statistics and Machine Learning Toolbox™ classification functions offer automatic hyperparameter tuning through Bayesian optimization, grid search, or random search. View Denis Gavrielov’s profile on LinkedIn, the world's largest professional community. Created and evaluated model using machine learning multiclassifications alghoritms (Naive Bayes, kNN, SVM, Random Forest). Instead of doing a single training/testing split, we can systematise this process, produce multiple, different out-of-sample train/test splits, that will lead to a better estimate of the out-of-sample RMSE. This paper presents an automatic tuning implementation that uses SAS/OR® local search optimization for tuning hyperparameters of modeling algorithms in SAS® Visual Data Mining and Machine Learning. See the complete profile on LinkedIn and discover Denis’ connections and jobs at similar companies. Naive Bayes classifiers are easy to interpret and useful for multiclass classification. Enterprises Training Courses. Neural networks can be difficult to tune. This Helm chart sets up a Dask scheduler + web UI, Dask worker(s), and a Jupyter Notebook instance. Feature engineering is a hard problem to automate, however, and not all AutoML systems handle it. 문제는 이런 hyperparameter들을 어떻게 설정하느냐에 따라 그 결과가 크게 바뀌기 때문에 소위 말하는 ‘튜닝’에 시간을 매우 많이 쏟아야한다는 점이다. API Reference. There are a number of machine learning blogs and books that describe how to use hyperparameters to achieve better text classification results. However, text normalization is an important step that occurs prior to hyperparameter tuning. An hands-on introduction to machine learning with R. Name Publication Description Attribute Class Papers; Bayes: AODE: 2005: Improved Naive Bayes: unary, binary, nominal: binary, nominal : AODEsr: 2006: Improved AODE. The tutorial provides an example for doing this while also doing additional hyperparameter tuning in a nested CV-setting. Membandingkan tingkat akurasi antar metode dan memilih tiga metode yang memiliki tingkat akurasi paling baik 8. pdf from CS 101 at Vidya Bharti Senior Secondary School. Bayes SMBO is probably the best candidate as long as resources are not a constraint for you or your team, but you should also consider establishing a baseline with Random Search. Machine Learning. Depending on the use case, 10% could be a massive improvement, but it came at a significant time investment! We can also time how long it takes to train the two models using the %timeit magic command in Jupyter Notebooks. This is also called tuning. individual words as tokens). Saishruthi has 7 jobs listed on their profile. More about Naïve Bayes (hyperparameter) Hyperparameters are parameters that cannot - Dev (used for tuning hyperparameters). Grid (Hyperparameter) Search¶. A model can have many hyperparameters and the process of finding the best possible combination of hyperparameters is referred to as hyperparameter tuning. naive_bayes. It is sometimes called Hyperparameter optimization where the algorithm parameters are referred to as hyperparameters whereas the coefficients found by the machine learning algorithm itself are referred to as parameters. Discover how machine learning algorithms work including kNN, decision trees, naive bayes, SVM, ensembles and much more in my new book, with 22 tutorials and examples in excel. You can find the complete modified source code here. Hyperparameter tuning for performance optimization is an art in itself, and there are no hard-and-fast rules that guarantee best performance on a given dataset. Certain Statistics and Machine Learning Toolbox™ classification functions offer automatic hyperparameter tuning through Bayesian optimization, grid search, or random search. Also try practice problems to test & improve your skill level. In Part I and Part II, we saw different holdout and bootstrap techniques for estimating the generalization performance of a model. It was conceived by the Reverend Thomas Bayes, an 18th-century British statistician who sought to explain how humans make predictions based on their changing beliefs. Shown below, we’ve built a parameter grid totaling 1050 required training jobs. 92$, the recall for class 0 has increased from$0. If you're trying to decide between the three, your best option is to take all three for a test drive on your data, and see which produces the best results. Fold 5 (160 reviews) is used to estimate the performance of the classi ers. When you have a numeric parameter to optimize, the first question is: discrete or continuous? For example, a number of units in a neural network layer is an integer, while amount of L2 regularization is a real number. Model Tuning. Hyperparameter Tuning Model Performance • CNNs were more effective than Naive Bayes and SVM, as expected. TPOT is a Python library that automatically creates and optimizes full machine learning pipelines using genetic programming. This course examines a variety of machine learning models including popular machine learning algorithms such as k-nearest neighbors, logistic regression, naive Bayes, k-means, decision trees, and artificial neural networks. In today's big data world, many companies have gathered huge amounts of customer data about marketing success, use of financial services, online usage, and even fraud behavior. Manipulate and analyze data that is too big to fit in memory. Why do i get different accuracy value when i use different values for random_state?. Download Open Datasets on 1000s of Projects + Share Projects on One Platform. Vector Machine, Naive Bayes and Decision Tree are also. a parameter that controls the form of the model itself. To get more detailed information, visit our website now. We can use Pandas to conduct Bayes Theorem and Scikitlearn to implement the Naive Bayes Algorithm. python,syntax,machine-learning,scikit-learn. understand the importance of hyperparameter tuning, they are K-nearest neighbors, Gradient Boosting. However, recent evidence on a benchmark of over a hundred hyperparameter optimization datasets suggests that such enthusiasm may call for increased scrutiny. To reduce this, we performed the parameter tuning to get the optimal value of "laplace" parameter of X and Y coordinates. Melakukan klasifikasi dengan metode KNN, Naïve Bayes, Decision Tree, Random Forest, Adadptive Boosting, SVC, MLP, dan Logistic Regression 7.


text:
## shaqadry Group Title integrate -x(1+2x)^5 dx one year ago one year ago

1. ash2326 Group Title

$\int -x(1+2x)^5dx$ We can expand (1+2x)^5, but that will be tiring Do you know u substitution method ?

2. shaqadry Group Title

how do i solve it by parts?

3. ash2326 Group Title

Not by parts, do u know u-substitution? It'll make it easy

4. ash2326 Group Title

@shaqadry Put 2x+1=t so $2dx=dt$ $dx=\frac{dt}{2}$ Do you get this?

5. shaqadry Group Title

yes!

6. shaqadry Group Title

and then?

7. shaqadry Group Title

wait but the question requires me to solve it by parts :-/

8. ash2326 Group Title

Then, we will use this $\int uv=u\int v-\int(du \int v)$

9. ash2326 Group Title

Can you try, apply it here?

10. shaqadry Group Title

@ash2326 which one should i equate to 'u' ? which is the better choice?

11. ash2326 Group Title

Equate x to u, it'll become 1 in the second part of by parts and then we find the integral easily @shaqadry

12. shaqadry Group Title

okay thanks i will try to solve it!


text:
# Thread: Mechanics Question

2. ## Re: Mechanics Question

Originally Posted by 1729
$\noindent Here are some hints.$

 Spoiler (rollover to view): $\noindent 1. Show that \ddot{\theta} = \frac{g}{L}\sin \theta (where dots refer to time derivatives).$

 Spoiler (rollover to view): $\noindent 2. Recall that \ddot{\theta} = \frac{d}{d\theta}\left(\frac{1}{2}\omega^{2}\right ) (same as the formula \ddot{x} = \frac{d}{dx}\left(\frac{1}{2}v^{2}\right) that you would have seen in 3U probably).$

 Spoiler (rollover to view): $\noindent 3. Observe that it follows from the previous two hints that \frac{d}{d\theta}\left( \frac{1}{2}\omega^{2}\right) = \frac{g}{L}\sin \theta.$

 Spoiler (rollover to view): $\noindent 4. Integrate both sides from \theta_{0} to \theta. (Given that the bob is released from rest, what is the value of \omega_{0}?)$

 Spoiler (rollover to view): $\noindent 5. What is the formula for \cos \theta and \cos \theta_{0} in terms of tension, m and g (it follows from resolving of forces)? Use this to finish off.$

3. ## Re: Mechanics Question

Originally Posted by InteGrand
$\noindent Here are some hints.$

 Spoiler (rollover to view): $\noindent 1. Show that \ddot{\theta} = \frac{g}{L}\sin \theta (where dots refer to time derivatives).$

 Spoiler (rollover to view): $\noindent 2. Recall that \ddot{\theta} = \frac{d}{d\theta}\left(\frac{1}{2}\omega^{2}\right ) (same as the formula \ddot{x} = \frac{d}{dx}\left(\frac{1}{2}v^{2}\right) that you would have seen in 3U probably).$

 Spoiler (rollover to view): $\noindent 3. Observe that it follows from the previous two hints that \frac{d}{d\theta}\left( \frac{1}{2}\omega^{2}\right) = \frac{g}{L}\sin \theta.$

 Spoiler (rollover to view): $\noindent 4. Integrate both sides from \theta_{0} to \theta. (Given that the bob is released from rest, what is the value of \omega_{0}?)$

 Spoiler (rollover to view): $\noindent 5. What is the formula for \cos \theta and \cos \theta_{0} in terms of tension, m and g (it follows from resolving of forces)? Use this to finish off.$
Do you use Newton's Second Law tangentially to show $\ddot{\theta} = \frac{g}{L}\sin \theta$

With $\sum F = -mg\sin \theta and a = L\frac{d^2\theta}{dt^2}$

Wouldn't it then be $\ddot{\theta} = -\frac{g}{L}\sin \theta$ and how come $a = L\frac{d^2\theta}{dt^2}$ instead of $a = \frac{d^2\theta}{dt^2}$

4. ## Re: Mechanics Question

Originally Posted by 1729
Do you use Newton's Second Law tangentially to show $\ddot{\theta} = \frac{g}{L}\sin \theta$

With $\sum F = -mg\sin \theta and a = L\frac{d^2\theta}{dt^2}$

Wouldn't it then be $\ddot{\theta} = -\frac{g}{L}\sin \theta$ and how come $a = L\frac{d^2\theta}{dt^2}$ instead of $a = \frac{d^2\theta}{dt^2}$

$\noindent Ah yes, I meant -g/L, not g/L, sorry.$

5. ## Re: Mechanics Question

Originally Posted by 1729
how come $a = L\frac{d^2\theta}{dt^2}$ instead of $a = \frac{d^2\theta}{dt^2}$

$\noindent It's because the displacement along the arc is given by s = L\theta, so \ddot{s} = a = L\ddot{\theta}. (Acceleration being the second derivative wrt time of displacement.)$

6. ## Re: Mechanics Question

Originally Posted by InteGrand
$\noindent It's because the displacement along the arc is given by s = L\theta, so \ddot{s} = a = L\ddot{\theta}. (Acceleration being the second derivative wrt time of displacement.)$
Why do we only consider half of the entire arc? Is it because the net force would then be 0?

Also what is the physical significance of integrating from $\theta _0 to \theta$

7. ## Re: Mechanics Question

Originally Posted by 1729
Why do we only consider half of the entire arc? Is it because the net force would then be 0?

Also what is the physical significance of integrating from $\theta _0 to \theta$
$\noindent Well basically the acceleration of a particle at any moment in time is independent of its initial location''. So it suffices to only consider displacement from the middle'' of the arc when deducing the acceleration expression.$

$\noindent The reason I said to integrate from \theta_{0} to \theta is because we have given data at \theta_{0} (i.e. we know T_{0} then and we know the value \omega_{0} at that point in time (since it was released \underline{from rest})).$

$\noindent Integrating from \theta_{0} to \theta is like integrating over the path taken so far''; i.e. view a start of the pendular cycle as being at angle \theta_{0} (the right extreme'' of the motion) and integrating up to its current location, \theta. If you don't like this, you can just integrate using indefinite integrals over \theta, and then you'll have to introduce a +C to one side. Then use the initial condition (at \theta_{0}) to find the C. But it is quicker to just integrate from \theta_{0} to \theta straightaway.$

8. ## Re: Mechanics Question

Originally Posted by InteGrand
$\noindent Well basically the acceleration of a particle at any moment in time is independent of its initial location''. So it suffices to only consider displacement from the middle'' of the arc when deducing the acceleration expression.$

$\noindent The reason I said to integrate from \theta_{0} to \theta is because we have given data at \theta_{0} (i.e. we know T_{0} then and we know the value \omega_{0} at that point in time (since it was released \underline{from rest})).$

$\noindent Integrating from \theta_{0} to \theta is like integrating over the path taken so far''; i.e. view a start of the pendular cycle as being at angle \theta_{0} (the right extreme'' of the motion) and integrating up to its current location, \theta. If you don't like this, you can just integrate using indefinite integrals over \theta, and then you'll have to introduce a +C to one side. Then use the initial condition (at \theta_{0}) to find the C. But it is quicker to just integrate from \theta_{0} to \theta straightaway.$
Thank you

9. ## Re: Mechanics Question

Another mechanics question (sry trying to self-learn)

I let x = 0 be the wharf and t = 0 be at 1:10 am which would mean the endpoints of the tidal motion are x = -1 and x = 0.4. Since we are told it is SHM, we can deduce $x = a\cos{(nt+\alpha)} + b$

Can we immediately deduce b = -0.3 since it is the centre of motion and a = 0.7? And if t is in hours, the period would be 12 hours and 25 minutes or T = 149/12 which would make n = 24pi/149.

And using $\frac{dx}{dt} = -na\sin{(nt+\alpha)}$ and dx/dt = 0 when t = 0 I got alpha = 0.

Meaning I got $x = 0.7\cos{\left(\frac{24\pi}{149}t\right)} - 0.3$ which is clearly wrong since at t = 0, x = 0.4 and not x = -1. What have I done wrong?

Screen Shot 2018-04-17 at 2.30.37 am.png

10. ## Re: Mechanics Question

Originally Posted by 1729
Another mechanics question (sry trying to self-learn)

I let x = 0 be the wharf and t = 0 be at 1:10 am which would mean the endpoints of the tidal motion are x = -1 and x = 0.4. Since we are told it is SHM, we can deduce $x = a\cos{(nt+\alpha)} + b$

Can we immediately deduce b = -0.3 since it is the centre of motion and a = 0.7? And if t is in hours, the period would be 12 hours and 25 minutes or T = 149/12 which would make n = 24pi/149.

And using $\frac{dx}{dt} = -na\sin{(nt+\alpha)}$ and dx/dt = 0 when t = 0 I got alpha = 0.

Meaning I got $x = 0.7\cos{\left(\frac{24\pi}{149}t\right)} - 0.3$ which is clearly wrong since at t = 0, x = 0.4 and not x = -1. What have I done wrong?

Screen Shot 2018-04-17 at 2.30.37 am.png
$\noindent The mistake is in this line:$

Originally Posted by 1729
And using $\frac{dx}{dt} = -na\sin{(nt+\alpha)}$ and dx/dt = 0 when t = 0 I got alpha = 0.
$\noindent Putting t = 0 there, we have \sin \alpha = 0. This does not imply that \alpha = 0, but only that \alpha = 0 or \alpha = \pi (note it suffices to take \alpha between 0 and 2\pi due to periodicity). The correct one to choose turns out to be \alpha = \pi.$

$\noindent Think of an easier question: fit a cosine wave of the form y = \cos(x + \alpha) with period 2\pi that goes through (0,-1) with slope 0 at x = 0. (By periodicity of \cos, it suffices to only consider \alpha \in [0,2\pi).) By doing your slope = 0 at x = 0'' method, we are basically not narrowing down the possibilities enough, since we admit both y = \cos x and y= -\cos x (these both have slope 0 at the origin). So the slope isn't really the right thing to look at, the right thing to look at first is the \textbf{value} at x= 0. Doing this, we have y = -1 at x = 0, so \cos(0 + \alpha) = -1\Rightarrow \cos \alpha = -1. Hence \alpha = \pi.$

$\noindent So in your question, try looking at the value at t = 0 first rather than the slope. Also you can do these sorts of questions more intuitively quickly. Since we're starting at low tide and moving towards high tide, this is like using a cosine curve that is starting at its minimum point, and moving up. This looks like a -\cos t curve. So the form of our function will be y = -A\cos (nt) + b, where A is the amplitude (note that A > 0 because we took care of sign by putting the minus sign out front). Now you can just inspect the values of A, n, b as you did before to get the answer. Upshot: \color{blue}{We don't to worry about \alpha by using this method}\color{black}; it's accounted for already since we know the curve must look like a negative cosine, due to it being a sinusoid that starts off at its minimum.$

11. ## Re: Mechanics Question

Originally Posted by InteGrand
$\noindent The mistake is in this line:$

$\noindent Putting t = 0 there, we have \sin \alpha = 0. This does not imply that \alpha = 0, but only that \alpha = 0 or \alpha = \pi (note it suffices to take \alpha between 0 and 2\pi/n due to periodicity). The correct one to choose turns out to be \alpha = \pi.$

$\noindent Think of an easier question: fit a cosine wave of the form y = \cos(x + \alpha) with period 2\pi that goes through (0,-1) with slope 0 at x = 0. (By periodicity of \cos, it suffices to only consider \alpha \in [0,2\pi).) By doing your slope = 0 at x = 0'' method, we are basically not narrowing down the possibilities enough, since we admit both y = \cos x and y= -\cos x (these both have slope 0 at the origin). So the slope isn't really the right thing to look at, the right thing to look at first is the \textbf{value} at x= 0. Doing this, we have y = -1 at x = 0, so \cos(0 + \alpha) = -1\Rightarrow \cos \alpha = -1. Hence \alpha = \pi.$

$\noindent So in your question, try looking at the value at t = 0 first rather than the slope. Also you can do these sorts of questions more intuitively quickly. Since we're starting at low tide and moving towards high tide, this is like using a cosine curve that is starting at its minimum point, and moving up. This looks like a -\cos t curve. So the form of our function will be y = -A\cos (nt) + b, where A is the amplitude (note that A > 0 because we took care of sign by putting the minus sign out front). Now you can just inspect the values of A, n, b as you did before to get the answer. Upshot: \color{blue}{We don't to worry about \alpha by using this method}\color{black}; it's accounted for already since we know the curve must look like a negative cosine, due to it being a sinusoid that starts off at its minimum.$
Thank you

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text:
Test: Transformation Geometry II - Normal

Double click on maths expressions to zoom
Question 1:   Find the reflection of the triangle $ABC$ across the $y=-x$ line

${A}^{\prime }\left(1,-1\right),{B}^{\prime }\left(1,-3\right),{C}^{\prime }\left(3,-1\right)$
${A}^{\prime }\left(-1,1\right),{B}^{\prime }\left(-1,3\right),{C}^{\prime }\left(-3,1\right)$
${A}^{\prime }\left(1,1\right),{B}^{\prime }\left(1,3\right),{C}^{\prime }\left(3,1\right)$
${A}^{\prime }\left(-1,-1\right),{B}^{\prime }\left(-1,-3\right),{C}^{\prime }\left(-3,-1\right)$
Question 2:   Find the reflection of the triangle $ABC$ across the $y=-x$ line

${A}^{\prime }\left(-2,-1\right),{B}^{\prime }\left(0,4\right),{C}^{\prime }\left(-4,2\right)$
${A}^{\prime }\left(2,1\right),{B}^{\prime }\left(0,-4\right),{C}^{\prime }\left(4,-2\right)$
${A}^{\prime }\left(2,-1\right),{B}^{\prime }\left(0,4\right),{C}^{\prime }\left(4,2\right)$
${A}^{\prime }\left(-2,1\right),{B}^{\prime }\left(0,-4\right),{C}^{\prime }\left(-4,-2\right)$
Question 3:   Find the coordinates of the triangle $ABC$ enlarged by a scale factor of $3$ and centered about the origin

${A}^{\prime }\left(-3,0\right),{B}^{\prime }\left(3,-1\frac{1}{2}\right),{C}^{\prime }\left(1\frac{1}{2},\frac{1}{3}\right)$
${A}^{\prime }\left(-6,0\right),{B}^{\prime }\left(6,-3\right),{C}^{\prime }\left(3,9\right)$
${A}^{\prime }\left(-2,3\right),{B}^{\prime }\left(2,2\right),{C}^{\prime }\left(1,5\right)$
${A}^{\prime }\left(1,3\right),{B}^{\prime }\left(5,2\right),{C}^{\prime }\left(4,6\right)$
Question 4:   Find the coordinates of the triangle $ABC$ enlarged by the scale factor of $\frac{1}{2}$ and centered about the origin

${A}^{\prime }\left(-8,-4\right),{B}^{\prime }\left(8,2\right),{C}^{\prime }\left(0,8\right)$
${A}^{\prime }\left(-2,-1\right),{B}^{\prime }\left(2,1\right),{C}^{\prime }\left(0,2\right)$
${A}^{\prime }\left(-2,-2\right),{B}^{\prime }\left(6,2\right),{C}^{\prime }\left(2,4\right)$
${A}^{\prime }\left(-4,0\right),{B}^{\prime }\left(4,4\right),{C}^{\prime }\left(0,6\right)$
Question 5:   Find the coordinates of the triangle $ABC$ rotated by ${90}^{\circ }$ about the origin

${A}^{\prime }\left(0,0\right),{B}^{\prime }\left(3,0\right),{C}^{\prime }\left(0,-3\right)$
${A}^{\prime }\left(0,0\right),{B}^{\prime }\left(0,3\right),{C}^{\prime }\left(-3,0\right)$
${A}^{\prime }\left(0,0\right),{B}^{\prime }\left(-3,0\right),{C}^{\prime }\left(0,-3\right)$
${A}^{\prime }\left(0,0\right),{B}^{\prime }\left(0,-3\right),{C}^{\prime }\left(3,0\right)$
Please note, you have solved only half of the test. For the complete test get a Must Have account. Get started


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# Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell?

The question is in the title:

Question: Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the 120-cell?

I consider only convex polytopes (convex hull of finitely many points) that are full-dimensional (not contained in a proper subspace). And I consider a polytope to be distinct from the 120-cell if it has a non-isomorphic face-lattice.

It is known that any 4-polytope must have a 3-gonal, 4-gonal or 5-gonal face of dimension two. The 120-cell has only 5-gonal faces of dimension two.

• "other than" means: not isomorphic as polyhedral complex? (this is a reasonable isomorphism notion; an a priori stronger one would be being isotopic, i.e., have a continuous deformation from one to another)
– YCor
May 23, 2020 at 16:38
• @YCor Yes, thanks. I edited that into the question. May 23, 2020 at 16:41
• Did you find who proved that the only possible faces are 3-, 4-, or 5-gonal? (An earlier version requested a citation for that result.) May 23, 2020 at 16:54
• @BrianHopkins I still don't have a source, but I realized the following: one can show (via a standard double counting arguments) that a planar graph has a vertex of degree 5 or smaller, or equivalently (considering its dual), a 3-gonal, 4-gonal or 5-gonal face. Since the edge-graph of a (3-dimensional) polyhedron is a planar graph, this proves it in dimension three. This then carries over to higher dimensions by considering the 3-faces of the polytopes. May 23, 2020 at 16:57
• I found this highly relevant question with an equivalently relevant answer. May 23, 2020 at 18:17

There are other polytopes. To construct one let's do the following. Remember first that in the hyperbolic $$4$$-space there exists a regular compact right-angled 120-cell. Here, right-angled means that any two adjacent faces intersect under angle $$\frac{\pi}{2}$$. Regular means, that all the faces are isomeric, and the polytope has the same group of self-isometries as the Euclidean 120-cell. This polytope is discussed, for example, in

https://pdfs.semanticscholar.org/a0eb/ccbed0687d966a9aaaac2f370bc930a556be.pdf

at the bottom of page 65. The references to more classical articles are given there.

Now, if we double it in one face then we get a new convex polytope, and it is not hard to see, that it doesn't have 2-faces that are triangles and quadrilaterals. But any convex hyperbolic polytope is also combinatorially equivalent to a Euclidean one.

More generally, you can take any compact right-angled hyperbolic polytope in $$\mathbb H^4$$. Since it is hyperbolic and right-angled, it can not have $$2$$-faces that are triangles of quadrilaterals. And there is a infinite number of such polytopes in dimension 4. Each of them gives a Euclidean one as well.

• Thank you for your answer. I am not familiar with some terminology: 1. what is this "regular compact right-angled 120-cell" in hyperbolic 4-space. Is it like a tiling of 4-space? I wasn't able to find out what a "convex hyperbolic polytope" is via google. 2. What does it mean to "double it on one face"? May 23, 2020 at 17:06
• I added a reference. Yes, starting with such a polytope you get a tiling of $\mathbb H^4$, in the same way as cubes tile $\mathbb R^n$. And what I say is that you need to take the union of two tiles that share a common hyperface to get a new polytope as you want. This can be continued as far as you want. May 23, 2020 at 17:13

Dmitri's answer is definitely correct. I just want to add my geometric intuition, and a generalization, which, in hindsight, is quite obvious.

All in all, we can have the following:

If $$P\subset\Bbb R^d$$ is a polytope with $$n$$ facets, each of which is combinatorially (or projectively) equivalent to $$Q\subset\smash{\Bbb R^{d-1}}\!$$, then for each $$k\ge 1$$ there also exists a polytope $$P_k\subset\Bbb R^d$$ with $$k(n-2)+2$$ facets, all of which are combinatorially (or projectively) equivalent to $$Q$$.

With this, it should be clear that there are many 4-polytopes with only 5-gonal 2-faces.

The main idea is visualized below.

Construction:

1. Fix a face $$\sigma\subset P$$.
2. Let $$P'$$ be the polytope obtained from $$P$$ by applying a certain projective transformation that a) fixes $$\sigma$$, and b) moves all vertices of $$P$$ "beyond" $$\sigma$$ (see the image). This construction is related to the idea behind the Schlegel diagram, in particular, this transformations always exists.
3. Glue $$P'$$ and $$P$$ on their common face isomorphic to $$\sigma$$ (if we have chosen the correct transformation in 2., then this is a convex polytope).

Repeat this to obtain as many $$Q$$-facets as you like.

Still, it might be interesting to determine the atomic $$Q$$-facetted polytopes, i.e. those, which are not "stacked" in the sense above.


text:
Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01ns0648369
 Title: Calculating Mass Balance Per Degree for the Aleutian Crust Using a Novel Method Authors: Klee, Trevor Advisors: Schoene, Blair Contributors: Higgins, John Department: Geosciences Class Year: 2015 Abstract: Several global volumetric balances have been carried out for intraoceanic arcs and continental arcs in an attempt to tabulate the global sources and sinks for continental and island-arc crust. However, these balances are flawed in that they cannot recognize geochemical differences in the crust produced by different arcs or different parts of arcs, and that they cannot properly account for sedimentary erosion. In this paper, I calculate a longitudinal degree-by-degree mass balance for the Aleutian arc, emphasizing the shift from basaltic magma production in the east/central Aleutians to andesitic juvenile continental crust production in the west Aleutians. I also calculate for the first time the sedimentary erosion of the Aleutians. I then carry out a more general volumetric balance for the Aleutians historically, which makes clear that there could not have been a steady state for magmatic production throughout the lifetime of the Aleutian arc, or else not enough crust would have been produced. Finally, I calculate that the Aleutians produced approximately 2,500,000 km$$^{3}$$ of juvenile continental crust since inception, but lost around 1,000,000 km$$^{3}$$ of continental crust through sedimentary erosion, leaving the Aleutians as a small source for continental crust. Extent: 30 pages URI: http://arks.princeton.edu/ark:/88435/dsp01ns0648369 Type of Material: Princeton University Senior Theses Language: en_US Appears in Collections: Geosciences, 1929-2016


text:
03 360 933

# Station  7BShould we believe infinite sums?

The geometric series formula is $1+x+x{2}+x{3}+... = \dfrac {1} {1-x}$.

We saw this by taking the polynomial $1$ and dividing it by the polynomial $1-x$. The infinite sum of the powers of $x$ naturally appear.

But can this formula be true?

### Question 1

What if $x$ is, say, the number $2$?

Then the geometric series formula says that $1+2+4+8+16+...$ equals $\dfrac {1} {1-2}$, which is $-1$. That’s just absurd!

On the other hand, if $x = 0.1$, then the geometric series formula says that

$1+\left(0.1\right)+\left(0.1\right){2}+\left(0.1\right){3}+... = 1+0.1+0.01+0.001+...=1.111...$

equals

$\dfrac {1} {1-0.1}=\dfrac {1} {0.9}=\dfrac {10} {9}$.

This is one and one-ninth. From primary school we learn that $\dfrac {1} {9}$ as a decimal is $0.111...$. (Also see the Decimalia Island!)
So one-and-one-ninth equals $1.111...$. The geometric series formula is correct in this case!

$1+\left(0.1\right)+\left(0.1\right){2}+\left(0.1\right){3}+... = \dfrac {1} {1-0.1}$

What is going on? When can we believe the formula and when can we not?

# ALGEBRA VERSUS ARITHMETIC

The $1\leftarrow x$ machine is a machine that displays mechanical algebra. Everything it does and all we conclude from it is true in terms of mechanical algebra.

For example, the geometric series formula says that $1+2+4+8+...$ equals $\dfrac{1}{1-2}$, which is absurd as a statement of arithmetic. However, in a purely mechanical sense, without regard to arithmetic, there is some version of truth to this claim. Here’s how we can see some truth.

If $1+2+4+8+...$ equals $\dfrac{1}{1-2}$, then multiplying $1+2+4+8+...$ by $1-2$ should give $1$. Does it?
Yes it does!

\begin{aligned}
\left(1-2\right) \times \left(1+2+4+8+...\right) &= \left(1-2\right) + \left(1-2\right) \times 2 + \left(1-2\right) \times 4 + \left(1-2\right) \times 8 + ...\
&= 1-2+2-4+4-8+8-16+...\
&= 1+0+0+0+...\
&= 1
\end{aligned}

So $1+2+4+8+...$ really does behave like $\dfrac{1}{1-2}$.
But still, this is not very satisfying. We want to know when $1+x+x{2}+x{3}+... = \dfrac {1} {1-x}$ is actually true as a statement of arithmetic.

# THE CALCULUS ANSWER (FOR THE BOLD)

One studies infinite sums in a calculus class. You learn there, for example, that $1+x+x{2}+x{3}+... = \dfrac {1} {1-x}$ is true as a statement of arithmetic for small values of $x$, (specifically, for all values of $x$ strictly between $-1$ and $1$.) The formula is valid for $x=0.1$, as we saw, and not for $x=2$.

For those who are game, here is how the calculus argument goes.

Regular polynomial division shows that $\dfrac {1-x^2} {1-x}=1+x$ and $\dfrac {1-x^3} {1-x} = 1+x+x^{2}$ and $\dfrac {1-x^4} {1-x} = 1+x+x{2}+x{3}$, and so on. (Try these!)
In general, we see that
$1+x+x{2}+...+x{n-1}=\dfrac {1-x^n} {1-x}$.

Now as we let $n$ get bigger and bigger it looks like we’re getting the infinite geometric sum.
\begin{aligned}
1+x &= \dfrac{1-x^2}{1-x}\
1+x+x^2 &= \dfrac{1-x^3}{1-x}\
1+x+x2+x3 &= \dfrac{1-x^4}{1-x}\
\vdots\
1+x+x2+x3+... &= ?
\end{aligned}

So the question is: What does $\dfrac {1-x^n} {1-x}$ go to as $n$ gets bigger and bigger? If there is an answer to this question, then the answer will be the value of $1+x+x{2}+x{3}+...$.

So does $\dfrac {1-x^n} {1-x}$ have a limit value? Well, this depends on whether or not $x^{n}$ has a limit value as $n$ grows. So for which values of $x$ do the powers of it approach a value?

We know the that powers of $0.1$, for example, and of $0.83$ and of $- \dfrac {1}{2}$, all approach zero for bigger and bigger powers. In fact, $x^{n}$ gets closer and closer to zero as $n$ grows for any value between $-1$ and $1$.

So for $-1, we have
\begin{aligned}
1+x &= \dfrac{1-x^2}{1-x}\
1+x+x^2 &= \dfrac{1-x^3}{1-x}\
1+x+x2+x3 &= \dfrac{1-x^4}{1-x}\
\vdots\
1+x+x2+x3+... &= \dfrac{1-0}{1-x} = \dfrac{1}{1-x}
\end{aligned}

The geometric series formula can be believed, as a statement of arithmetic, for $-1, at least.

# THE HONEST APPROACH

Another approach to examining the geometric series formula is to make an explicit leap of faith. Let’s assume that the infinite sum $1+x+x{2}+x{3}+...$ is meaningful and has an answer. Call that answer $S$.

Then

\begin{aligned}
S &= 1+x+x{2}+x{3}+...\
&= 1+x \left(1+x+x^{2}+...\right) \
&= 1+xS
\end{aligned}

from which we get $S= \dfrac{1}{1-x}$.

This is an honest approach. It proves:
IF the infinite sum $1+x+x^{2}+...$ has an answer, then that answer must be $\dfrac{1}{1-x}$. It makes no assertion as to whether or not the infinite sum is meaningful in the first place.

The same is true for the dots-and-boxes approach as it too proves: IF $1+x+x{2}+x{3}+...$ is meaningful to you, then it has the answer $\dfrac{1}{1-x}$. It is up to you to decide if this infinite sum is meaningful.
(Calculus likes to say it is if $-1.)

# OTHER SYSTEMS OF ARITHMETIC OFFER OTHER MEANINGS

The statement $1+2+4+8+...=-1$ is meaningless in ordinary arithmetic. But who says we have to stay with ordinary arithmetic? Is there an extraordinary way to view matters?

We tend to view numbers as spaced apart on the number line additively. Walk one step to the right of $0$ and we end up at position $1$. Now add to that two steps and we end up position $3$. Now add four steps, position $7$. And so on. The sum $1+2+4+8+...$, in this viewpoint, takes infinitely far to the right of $0$ on the number line.

$1+2+4+8+...=\infty$ in ordinary arithmetic. It does not equal $-1$.

But let’s think of numbers multiplicatively. In particular, since we are focusing on the sum $1+2+4+8+...$, let’s think of factors and multiples of powers of two.

Now $0$ is a highly divisible number. It is the most divisible number of all. With regard to just two-ness it can be divided by $2$ once, in fact twice, in fact thrice. In fact, you can divide $0$ by two as many times as you like—and still keep dividing.

With regard to two-ness, the number $8$ is somewhat zero-like: you can divide it by two three times. But $32$ is even more zero-like: you can divide it by two five times. And $2^{100}$ is even more zero-like still.

So in this sense, $2^{100}$ is a number very close to $0$. The number $32$ is somewhat close to $0$. The number $8$ is less close. The number $1$ is not very close to zero at all: it cannot be evenly divided by two even once.

So, in this context, could $1+2+4+8+...$ possibly be $-1$?

Well

\begin{aligned}
1+2 = 3 &= 4-1\
1+2+4 = 7 &= 8-1\
1+2+4+8 = 15 &= 16-1\
\vdots &= \vdots \
1+2+\ldots + 2^{99} &= 2^{100}-1
\end{aligned}

These finite sums grow to become “a number very close to zero, minus one.” In the limit, the infinite sum thus has value $0-1 = -1$, just as our formal arguments said it would be.

So in this multiplicative view of arithmetic, $1+2+4+8+...$ is a meaningful quantity, and it does indeed have value $-1$. The geometric series formula is meaningful and correct for $x = 2$ in this context.

The point is that our dots-and-boxes work shows what the answers to many infinite sums have to be, IF the infinite sum has meaning to you. It is thus up to you to decide what arithmetic context you want to play in and whether or not the infinite sum you are examining is meaningful in that context. If it is, then dots and boxes tells you its answer!

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