Can and Cannot Do
1. What You Always Wanted to Compute
A generic solution cannot be given for this exercise. If nothing occurs to you, here are a few suggestions that, after studying the references, can be more or less easily programmed in Mathematica. Some might look complicated at first glance, but they are not so complicated after some thinking about the subject; but some are not easy either.
a) Do noninteger derivatives such as d 2 x2 expH-xL ídx 2 exist? How are they defined? Are they unique? (For details, see [875÷], [1028÷], [1236÷], [1163÷], [228÷], [1408÷], [634÷], [389÷], [695÷], and [9÷].) A similar question would be: Are there fractional iterations, like fHfH…fHxLLL (n f’s, nœ)? (See [942÷], [672÷], [810÷], [569÷], [576÷], [23÷], [1387÷], [799÷], [1129÷], [1119÷], [24÷], [22÷], and [131÷].) Another similar one would be: Are there fractional finite differences? (See [690÷], [571÷], [1257÷], and [981÷].) For fractional differentials, see [314÷] and [271÷]. For fractional summation, see [974÷].
b) Is there a multivalued analytic function fHzL, where the value of the function on another sheet is just the derivative of the function on the principal sheet? (See [980÷], [588÷], [1345÷], and [125÷].) Are there functions fHzL such that
⁄0¶ fHnL=Ÿ0¶fHzL dz? (See [162÷], [1084÷], and [1227÷].) Which differential equations have solutions that are succes-sive derivatives of some function? (See [321÷] and [322÷])
c) Is it possible to visualize the Banach-Tarski paradox? Loosely speaking, it is the creation of two oranges by slicing one into nonmeasurable pieces. (For details on the Banach-Tarski paradox, see [1334÷], [492÷], [758÷], [1336÷], and [335÷].)
d) How fast should one run in rain (if caught without an umbrella) to keep as dry as possible? (For a solution based on an idealized box-person in homogeneous rain, see [347÷], [350÷], and [1040÷]; for the properties of real rain, see [863÷], [1133÷], [1134÷], [318÷], [1064÷], [1065÷], [381÷], and [1110÷]; and for single rain drops, see [603÷].) e) How does one calculate puns (plays on words)? (See [148÷], [149÷], [1250÷].)
f) Why are falling layers of water on fountains and waterfalls often wavy in the vertical direction? (See [243÷] and [244÷].)
g) What is the position of a regular-shaped piece of wood or other symmetric object floating in water? (See [357÷], [1120÷], [1274÷], [532÷], [688÷], [446÷], [1353÷], [1354÷], [1145÷], and [93÷]; for moving floating objects see [683÷] and [1233÷].) For the not unrelated problem of hanging pictures, see [160÷].
h) Can one approximate locally a parametrically given curve better than via direct Taylor expansion in polynomials?
(See [1103÷], [1104÷], [1178÷], and [353÷].)
i) How can Newton’s equation of motion be used to describe the movement of a bicycle (for simplicity, without a rider)? The problem involves a mechanical system with anholonomic side conditions. (See [158÷], [601÷], [484÷], [211÷], [67÷], [471÷], [157÷], [1045÷], [414÷], [744÷], [903÷], [476÷], [783÷], [1044÷], [993÷], [842÷], [1295÷], [789÷], [1256÷], [1012÷], [311÷], [156÷], [1030÷], [517÷], and [841÷] and the references cited therein.) How does one express the closed form solutions of the equations of motions for the simplest nonholonomic systems— a rolling disk? (See [333÷], [785÷], [172÷], [1323÷], [171÷], [801÷], [1049÷], and [1324÷].) How does one describe the motion of a human symplectically? (See [666÷].)
j) Taking into account air resistance, does a ball thrown straight up return earlier or later than without taking into account air resistance? (See [828÷], [349÷], [854÷], and [1094÷]; for a rotating ball see [499÷], [500÷], and [501÷].) k) What is the shape of a mylar balloon made from two circular sheets? (See [953÷], [1058÷], and [952÷]; for inflating rubber balloons, see [973÷]; for larger balloons, see [64÷].)
l) What model would be used for a falling cat that always lands on its legs? (For a model cat solution, see [701÷], [961÷], [497÷], [853÷], [1091÷], [459÷], and [962÷].) At which position does a falling tower brake? (See [879÷] and [1310÷].)
m) Why does sand in shallow sea water have ridges? What determines the wavelength and height of these ridges? (For an appropriate model, see [1006÷], [878÷], and [338÷]. Concerning the outside-water behavior of sand, see [932÷].) n) How does one derive the scaling relation between the mass and the metabolic rate of an animal or plant? (See [406÷]
and [1366÷].) How many different animal species could exist? (See [1293÷].)
o) Can one calculate closed-form expressions for the gravitational potential and the moment of inertia of the regular polyhedra? (See [986÷], [1339÷], [786÷], [577÷], [596÷], [1340÷], [130÷], [847÷], [1276÷], [74÷], [1364÷], [1197÷], [816÷], [585÷], and [253÷]. For polarizabilities of Platonic solids, see [1211÷].)
p) Can one calculate how a piece of paper tears? (See [1026÷], [1161÷].) (For crumpling of paper, see [382÷], [1199÷], [909÷], [544÷], [21÷], [383÷], [397÷], [1385÷], [1162÷], [845÷], and [1074÷]; for wrinkling, see [256÷].)
q) Given the path of the front wheels of a car, what is the path of the rear wheels? (See [490÷], [1317÷], [540÷], [1275÷], [220÷], and [943÷]. For four steerable wheels, see [1402÷]. For bicycle tracks, see [469÷]. For towing, see [1121÷].)
r) Given a square with integer side lengths, is it possible to tile this square into triangles so that all triangle side lengths again are an integer? (See [594÷].) And what is the largest square that can be inscribed in a unit cube? (See [325÷].)
s) Can one model how a piece of paper (or a leaf) falls? (See [1063÷], [1263÷], [43÷], [465÷], [722÷], [882÷], [913÷], and [475÷].)
t) How high can a given kite at given wind and with given cord length fly? (See [1372÷].)
u) What is the apparent form of a train moving with relativistic velocity? (For the appearance of fast moving simple geometric bodies, see [1361÷], [10÷], [161÷], [633÷], [604÷], [646÷], [1386÷], [1270÷], [198÷], [1152÷], [464÷], [890÷], [1113÷], [791÷], [463÷], [60÷], [995÷], [1004÷], [1360÷], [141÷], [1010÷], and [685÷].) And do very fast large cars (vº0.9…9c, c the velocity of light) fit in short garages because of length contraction? (See [518÷], [1200÷], and [324÷].) For submarines, see [911÷].
v) What is the probability that a thick coin will fall on its side if dropped randomly? (See [169÷], and [719÷].)
w) How does a tippe top work? (See [183÷], [1014÷], [659÷], [1228÷], [293÷], [1082÷], [839÷], [425÷], [1036÷], [807÷], [94÷], and [956÷].) For spinning cooked eggs, see [1167÷].
x) What is the form of the closed plane curve of greatest possible area that can be moved around a right-angled corner in a hallway? (See [528÷].)
y) Does a buttered slice of toast land with the buttered side down really more often? (See [915÷], [62÷], and [432÷].) z) How is the scale of a sundial determined? (See [1176÷], [1160÷], [650÷], [1300÷], [1175÷], [1188÷], [1351÷], [1418÷], [1147÷], [504÷], [154÷], and http://www.mathsource.com/cgi-bin/MathSource/0209-001.)
a£) How can the growth of icicles be modeled? (See [1020÷], [736÷], [888÷], and [1019÷]. For similar patterns on water columns, see [1217÷]. For modeling snowflakes, see [846÷], [819÷], [1319÷], [676÷], and [677÷].)
b£) Mathematically, how does a queue of cars (on the freeway) form, and how long does one have to wait in line (as a function of the parameters traffic density, average speed, etc.)? (For mathematical models of traffic flow, see, e.g., [1369÷], [1149÷], [1172÷], [623÷], [883÷], [283÷], [503÷], [323÷], [404÷], [405÷], [1380÷], [153÷], [800÷], [984÷], [1170÷], [985÷], [493÷], [1168÷], [621÷], [1169÷], [482÷], [526÷], [1187÷], [867÷], [481÷], [684÷], and the refer-ences cited therein. For modeling the driver’s experience, see [1117÷]. For pedestrian traffic, see [218÷], [649÷], [1171÷], [219÷], [624÷], [755÷], [895÷]. For the modeling of the corrugation of roads, see [182÷].)
c£) How does one algorithmically measure k gallons given n jugs with given capacities? (See [168÷].)
d£) Which point of a hypercube in n dimensions maximizes the product of the distances to its vertices? (See [1371÷].) e£) When leaves fall from the trees in the autumn, assume that all of the ground is covered by leaves. How many leaves does one in average see inside a certain area? (See [320÷], [396÷], [558÷], and [790÷] for circular leaves.) A related, but easier problem is: What is the average height children will pile rectangular blocks while building towers before they collapse? (See [671÷].)
f£) How does one model a dripping tap? (See [1186÷], [498÷], [316÷], [712÷], [760÷], [657÷], [315÷], [31÷], [1127÷], [1126÷], [387÷], and [208÷], [761÷].)
g£) How does one calculate the shape of a water drop on a smooth surface? (See [204÷], [1015÷], [1140÷], [103÷], [798÷], [1÷], [1267÷], [817÷], and [900÷]; for moving drops, see [30÷].)
h£) How does one describe the motion of a curling rock? (See [1201÷], [680÷], [365÷], [1203÷], [1204÷], [1060÷], [364÷], [1202÷], and [457÷].) How does one model stones skimming over water? (See [165÷]). How does one model the increasing frequency of the whirring sound of a coin rotating on a table? (See [954÷], [955÷], [730÷], [1067÷], [424÷], [146÷], and [1238÷].)
i£) How does one calculate the optimal form of the teeth of gears? (See [855÷], [1392÷], [193÷], [1346÷], and [1086÷].) j£) How does one model a Levitron”? (See [135÷], [1212÷], [559÷], [415÷], [1213÷], [523÷], and [136÷]; for nonlin-ear levitation, see [946÷].) How can one model the woodpecker toy? See [1069÷] and [826÷].
k£) How does one model the shape of a human trail system on a meadow? (See [622÷].) (For modeling the flow going out of a large hall, see [1261÷], [194÷], [664÷]; for modeling standing, see [407÷]; for ski slopes, see [429÷].)
l£) Can two losing games yield a winning game? (See [606÷], [386÷], [47÷], [1265÷], [1278÷], [1316÷], [265÷], [948÷], [607÷], [936÷], [937÷], [472÷], [1112÷], [732÷], [717÷], [25÷], [945÷], and [1052÷].)
m£) How does a grooved cylinder roll down an inclined plane? (See [931÷].) How does one model the “Indian rope trick”? (See [4÷], [1038÷], [5÷], [652÷], [972÷], and [262÷].)
n£) How does one calculate the shape of the two pieces used to cover of baseball? (See [1273÷].) o£) How does one model the learning of grammar? (See [1009÷].)
p£) How does one describe the path of a single air bubble rising in water? (See [1395÷], [971÷], [376÷], [1308÷], [641÷], [1089÷], and [848÷].)
q£) Which numbers can be expressed in a closed form? (See [281÷] and [98÷].) And what numbers are computable?
(See [1359÷].)
r£) Can one use the logistic map to generate random numbers? (See [33÷], [545÷], [546÷], [1384÷], [547÷], [1299÷], and [453÷].)
s£) What are the side lengths of a rectangle with a given maximal area, such that the area/perimeter ratio is as large as possible? (See [905÷].)
t£) How does one model a continuous transition from Taylor series coefficients to Fourier series coefficients? (See [54÷], [1077÷], and [1076÷].)
u£) How does one model the waiting time for a web browser connection? (See [1374÷], [18÷], [19÷], [1262÷], [77÷], [918÷], [648÷] and [836÷], [743÷] for the cables.)
v£) Is there a multidimensional version of Simpson’s rule? (See [644÷].)
w£) What is the (continuous) symmetry of the genetic code? (See [643÷], [92÷], [486÷], [91÷], [478÷], [487÷], [39÷], [40÷], [417÷], [1193÷], [488÷], [1194÷], [686÷], [992÷], [491÷], and [727÷].)
x£) Are there functions whose reciprocal is equal to their inverse? (See [272÷].) How to calculate the Fourier coeffi-cients of the reciprocal function from the Fourier coefficoeffi-cients of a function? (See [413÷].)
y£) Is there a linkage that signs your name? (See [705÷], [749÷], [750÷], [751÷], [155÷], [531÷], [458÷], and [399÷].) z£) How does one model bird and fish swarms? (See [334÷] and [957÷].)
a≥) How does one model the various gaits of a horse? (See [298÷], [299÷], [1303÷], [1304÷], [213÷], [929÷], [529÷], and [1248÷]; for human gait modeling, see [1258÷], [419÷], [769÷], and [1318÷].) How to classify juggling patterns?
(See [210÷], [431÷], [1085÷], and [1237÷].)
b≥) How does one model the shape of a cracking whip? (See [551÷], [930÷], and [793÷].) c≥) What is the probability of going to jail in the Monopoly” game? (See [1393÷].)
d≥) How does one construct a computable bijection between the rational numbers and the integers (the classical diago-nal method is not easy to compute for reduced fractions)? (See [1047÷] and [248÷].)
e≥) How does one model collapsing bridges? (See [927÷].)
f≥) How does one model the movement of a camphor scraping on water? (See [988÷], [989÷], [982÷], and [616÷].) g≥) Given a rectangle, how many congruent rectangles can you position around it such that each one touches the given rectangle, but does not intersect with any of the others? (See [723÷] and for polyhedra [1342÷].)
h≥) What are the possible equilibrium shapes for closed elastic rods? (See [818÷], [1331÷], [914÷], [670÷], [998÷], and [49÷].)
i≥) How does one model the movement (and potential self-knotting) of a moving hanging chain? See ([106÷] and [233÷]).
j≥) How many fingers form an “optimal” hand? (See [950÷], [756÷], [901÷], and [949÷].)
k≥) How many different ancestors do humans have on average in their genealogical tree? (See [368÷], [369÷], [370÷], [359÷], [1218÷], [1024÷], [1222÷], [1266÷], and [987÷].) (And how does one model the shape of the phylogenetic tree? See [912÷], [147÷], and [1243÷]. For the related problem: the distribution of family names, see [1407÷], [891÷], [304÷], [1118÷], and [662÷].)
l≥) Are the magnetic field lines around a current-carrying wire really closed? (See [1219÷], [1279÷], [377÷], [1068÷], [1158÷], problem 18 of [1297÷], and [1073÷].)
m≥) How does one calculate polynomials orthogonal over a regular polygon? (See [1422÷].)
n≥) On which day of the week should a teacher hold an exam to maximize the surprise when it happens? (See [282÷]
and [1221÷].)
o≥) How does one model river basins? (See [1135÷], [390÷], [391÷], [224÷], and [392÷].) p≥) How does one model a ball rolling on a rough surface? (See [1311÷].)
q≥) What is the probability for a random walker in d dimensions to return to the origin? (See [108÷] and [109÷].) r≥) How does one model the expansion of a popcorn kernel? (See [640÷] and [1102÷].)
s≥) What is the explicit form of the eigenfunctions of the curl operator? (See [1406÷], [231÷], [965÷], [968÷], and [1071÷].) For the exponential of the curl operator, see [1088÷]; for the discretized version, see [285÷].
t≥) How does one model the bubbling of wine bottle labels? (See [200÷].)
u≥) How does one analytically map a polygon with a hole to an annulus? (See [780÷], [779÷], and [358÷].)
v≥) How does one construct and model a gravity-powered toy that can walk but not stand? (See [296÷] and [297÷].) w≥) How does one represent a function of several variables as a superposition of functions of one variable? (See [496÷], [15÷], and [534÷].)
x≥) Why can dolphins swim so fast? (See [852÷].)
y≥) Can a band-limited function oscillate faster than its bandwidth? (See [11÷], [725÷], [726÷], [225÷], [1100÷], [134÷], [133÷], [12÷], [13÷], and [724÷].) For the definition of an instantaneous frequency, see [1031÷].
z≥) How does one straighten out a chain of connected rods in three and four dimensions? (See [145÷] and [290÷].) a£££) How does one model the generation and sound of canary songs? (See [520÷], [1277÷], and [753÷]; for the model-ing of snormodel-ing, see [14÷].)
b£££) How does one model the sand flow in a hourglass? (See [462÷].)
c£££) How does one model the consequences of increasing information exchange on inter-personal interactions? (See [1419÷].)
d£££) How does one cut out any planar straight-line figure from one sheet of paper with a single straight cut? (See [361÷]
and [1039÷].)
e£££) If integration is the limit of a sum, what is the corresponding limit for a product? (See [395÷], [533÷], [706÷], [32÷], [602÷] and [1181÷] for matrices.)
f£££) How densely can Platonic solids be packed on a lattice? (See [140÷].)
g£££) Given a polynomial with complex roots only, what is the “nearest” polynomial with a real root? (See [635÷].) h£££) Are there nonlinear differential equations whose solutions obey a superposition principle? (See [1376÷], [1210÷], [1291÷], [214÷], [215÷], [549÷], [978÷], [237÷], [129÷], [735÷], [236÷], [1062÷], and [235÷].)
i£££) How “quadratic” are the natural numbers? (See [1146÷].)
k£££) How does one mathematically discriminate between a novel and a poem? (See [44÷], [302÷], and [303÷].)
l£££) How does one calculate the ideal steak cooking time and flipping times? (See [925÷], [1154÷], [1032÷], and [81÷].) m£££) How does one effectively fight a hydra that regrows its heads? (See [754÷], [609÷], and [869÷])
n£££) Can one eliminate all variables from a symbolic calculation? (See [1246÷], [332÷], and [1018÷].)
o£££) How does one model the noise of helicopter blades? (See [456÷], [455÷], [241÷], [605÷], [864÷], and [197÷]; for the squeal of train wheels, see [619÷]; and the sound of rubbing hands, see [1403÷].)
p£££) How does one experimentally measure and mathematically model a Riemann surface? (See [1195÷].) q£££) Given the first terms of a Taylor series, how does one recover the original function? (See [384÷] and [651÷].) r£££) What is the probability to encounter a matrix difficult to invert? (See [362÷].)
s£££) How does one model folded proteins? (See [1377÷], [1264÷], [75÷], [201÷], and [27÷].)
t£££) How frequently does a given word or phrase statistically appear as a subsequence in a text? (See [470÷].)
u£££) How does one model the creation of aeolian sand ripples? (See [331÷], [1405÷], [1008÷], [628÷], [796÷], [797÷], [418÷], [630÷], [1189÷], [1007÷], [941÷], [34÷], [35÷], [849÷], [958÷], and [629÷].)
v£££) How does one calculate all possible tie knots? (See [468÷].)
w£££) Can calculations exhibit phase transitions? (See [638÷], [665÷], [959÷], [939÷], [409÷], [260÷], [933÷], [1398÷], [1399÷], [1358÷], [757÷], [873÷], [934÷], [87÷], [195÷], [291÷], and [1357÷].) (For phase transitions in the World Wide Web, see [144÷]; for phase transitions in data compression, see [977÷]; for phase transitions in parameter-dependent wave functions, see [667÷].)
x£££) What is the expected average of the chord length of random lines intersecting a closed plane curve? (See [924÷].) y£££) How does one dissect a polygon into polygonal pieces that are connected by flexible hinges and allow to form the mirror image of the original polygon? (See [445÷].)
z£££) How does a prismatic cylinder roll down an inclined plane? (See [1254÷] and [2÷].) aiv) How many zeros does a random trigonometric polynomial have in average? (See [454÷].)
A host of other suggestions, both large and small, can be found almost daily in the newsgroup rec.puzzles http://dejanews.com, http://star.tau.ac.il/QUIZ, http://problems.math.umr.edu and related websites (http://dmoz.org/-Science/Math/Mathematical_Recreations contains a listing of such websites). We also mention the American Journal of Physics, http://www.amherst.edu/~ajp, and European Journal of Physics, and Eric Weisstein’s MathWorld http://mathworld.wolfram.com (Concise Encyclopedia of Mathematics [1363÷]), the Journal of Recreational MathematÖ ics as well as http://www.seanet.com/~ksbrown. (See also [1247÷].)
For the more theoretical physics-interested reader, we mention a few more technical possibilities.
a ) How does one construct (pseudodifferential) cube roots from a differential operator (similar to gm ∑m+m is a square root of ∑m∑m+m2)? (See [729÷], [728÷], and [1083÷].) For square roots of the heat equation, see [1314÷].
b ) How does one construct p+1 orthonormal bases in a p-dimensional vector space over , such that all possible scalar products between vectors from different bases have the same magnitude? (See [46÷], [1388÷], [1389÷], [764÷], [121÷], [1151÷], [1391÷], [763÷], [1125÷], [1079÷], [669÷], [441÷], [269÷], [45÷], [1330÷], [355÷], [76÷], [1379÷], [122÷], [53÷], and [1390÷].)
g ) In how many different orthogonal coordinate systems is the wave equation separable? (See [696÷], [697÷], [124÷], and [1337÷].)
d) Is there a potential VHxL, such that the eigenvalues of the corresponding one-dimensional Schrödinger equation are the prime numbers? (See [979÷] and [1381÷].) (For the related problem of a potential whose eigenvalues are the imaginary parts of the nontrivial zeros of the Riemann Zeta function, see [246÷], [822÷], [1394÷], [1309÷], [1150÷];
for the Jost function having the zeros of the Riemann Zeta function, see [737÷] and [738÷] and for potentials that represent the prime numbers, see [366÷].)
e) Given two hermitean matrixes K and L, what can be said about the spectrum of K+L? (See [771÷], [772÷], [773÷], [774÷], [339÷], and [506÷].) What about the spectrum of K.L? (See [1350÷].) Given two polynomials pHxL and qHxL, what can be said about the factorization of pHxL+qHxL? (See [746÷].)
¶) How fast is the “ultimate laptop”? (See [860÷], [999÷], [1000÷], [861÷], [483÷], [1056÷], [227÷], [1234÷], [792÷], and [1041÷].) (For space-time possibilities to speed up computations, see [422÷], [447÷], [1037÷], and [205÷]; for superluminal methods, see [1225÷]; for limits on the hard drive capacities, see [104÷], [292÷].)
z) How does one generate Greechie diagrams efficiently? (See [926÷].)
h) Can one model a DLA cluster deterministically? (See [610÷], [840÷], [344÷], [345÷], [83÷], [84÷], [625÷], [85÷], and [82÷].)
q) Are there (sensible) nonhermitian Hamiltonians with real spectra? (See [119÷], [1425÷], [970÷], [767÷], [969÷], [1356÷], [66÷], [401÷], [110÷], [229÷], [114÷], [115÷], [38÷], [112÷], [230÷], [113÷], [112÷], [461÷], [116÷], [117÷], [940÷], [118÷], [111÷], and [120÷].)
J) How does one model the movement of an adiabatic movable piston between two gases in equilibrium? (See [278÷], [583÷], [581÷], [731÷], [935÷], [823÷], [892÷], [222÷], [327÷], [582÷], [276÷], [328÷], [1367÷], [329÷], [277÷], [963÷], [279÷], [1070÷], [330÷], [199÷], and [994÷].)
i) Can knots be stable solutions of classical field theories? (See [96÷], [1002÷], [1023÷] and [1136÷].) And can knots be formed by the zero lines of hydrogen wave functions? (See [137÷].)
k) How does one (numerically) calculate the length and the dimension of the path of a quantum particle? (See [795÷], [543÷], and [794÷].)
¿) How does a rope or chain slide off the edge of a table? A little contemplation of the conservation of momentum law shows immediately that the standard solution from experimental physics books is wrong. (For details, see [1165÷], [363÷], [1092÷], [102÷], [186÷], and [1333÷]; for folded chains, see [1245÷].)
l) How does one “properly” discretize Maxwell’s equations? (See [1268÷], [916÷], [708÷], [579÷], [917÷], [1307÷], and [745÷].) For superconsistent discretizations in general, see [507÷]. Are there oscillating charge distributions that do not radiate? (See [514÷], [494÷], [539÷], [899÷], [637÷], [745÷], [374÷], [1005÷], and [245÷].)
m) Can bend cylinders support bound states (in a quantum-mechanical sense)? (See [653÷], [656÷], [1029÷], [862÷], [410÷], [37÷], [542÷], [411÷], [570÷], [238÷], [239÷], [240÷], [1226÷], [1087÷], [881÷], [951÷], and [508÷].) For additional linking, see [747÷]. Can neutral multipole arrangements of charges support bounds states? See [1097÷] for the general case and [1292÷], [668÷], [1072÷], [1057÷], [1130÷] for dipols.
n) Can one model any one-dimensional contact interaction with delta function potentials (in a quantum-mechanical sense)? (See [17÷], [477÷], [1207÷], [20÷], [317÷], [275÷], [682÷], [1294÷], [1050÷], [1184÷], [451÷], [1396÷],
[788÷], [1208÷], [1209÷], [808÷], [505÷], [983÷], [1287÷], [1343÷], [1288÷], [273÷], [26÷], [1289÷], and [274÷].) x) What is the efficiency of a Carnot machine that uses an ideal Bose gas or Fermi gas? (See [1214÷], [1215÷], and [1216÷].)
o) How does one construct the quantum mechanical hydrogen wave functions from classical orbits? (See [716÷], [714÷], and [715÷].)
v) How does one calculate terms of the Rayleigh–Schrödinger perturbation theory when the integrals in Xi»V» j\
diverge? (For instance VHxL~expIx4M in the harmonic oscillator basis?) (See [765÷], [373÷], [573÷], [352÷], [766÷], [608÷], [885÷], and [287÷].)
p) Can one observe the spin of a free electron in a Stern-Gerlach-type experiment? (See [95÷] and [521÷].)
r) How does one stabilize classical mechanics? (See [1325÷], [627÷], [1326÷], [97÷], and [420÷].) (For the dequantiza-tion of quantum mechanics, see [660÷]. For the fundamental constants of classical mechanics, see [928÷].)
·) What is the connection between Huygens’ principle with the wave equation? (Huygens’ principle states that from every point on a wave, a spherical basic wave emerges there.) Is it possible to model the spreading out of a wave directly from Huygens’ principle numerically? (See [69÷], [342÷], [105÷], [341÷], [289÷], [440÷], [589÷], [247÷], [1021÷], [159÷], [655÷], [827÷], [1423÷], and [1375÷].)
s) Do one-dimensional lattices show Fourier’s law in heat conduction? (See [837÷], [1095÷], [647÷], [41÷], [42÷], [428÷], [843÷], [378÷], [379÷], [522÷], and [530÷].)
V) How does one formulate classical mechanics using a Hilbert space? (See [921÷], [784÷], [560÷], [561÷], [562÷], [919÷], [8÷], [284÷], [192÷], and [191÷]. For a path integral formulation, see [920÷], [3÷], and [563÷]; for a Wigner distribution, see [512÷]; and for a unifying approach, see [759÷].)
t) What is the relation between a d-dimensional Kepler problem with a 2d-2 dimensional harmonic oscillator prob-lem? (See [1081÷], [1269÷], [1426÷], [814÷], [698÷], [343÷], [739÷], [309÷], [996÷], [305÷], [897÷], [254÷], [86÷], [699÷], [1417÷], [88÷], [234÷], [896÷], and [740÷].)
u) Are there stable atoms in d dimensions and how does the corresponding periodic table look like? (See [216÷], [894÷], [856÷], [1109÷], [631÷], [591÷], [1198÷], [58÷], [741÷], [742÷], [787÷], [809÷], and [692÷].)
f) How does one calculate the numerical value of the Boltzmann constant kB? (See [782÷], [437÷], and [825÷]. For the experimental determination, see [474÷]; for the status as a constant, see [412÷].)
j) How does one model the wave function of a photon emitted from an excited atom? (See [966÷], [967÷], [721÷], [221÷], [264÷], [527÷], [251÷], [356÷], [832÷], [143÷], [6÷], [473÷], and [898÷]. For absorbing a photon, see [61÷].
For localizing photons, see [1156÷].)
c) How does one calculate higher-order Foldy–Wouthuysen transformations? (See [964÷], [1306÷], [1122÷], [1124÷], [805÷], [421÷], [466÷], [89÷], [90÷], [734÷], [1123÷], and [1003÷].)
y) Is the charge distribution of a finite one-dimensional wire uniform? (See [674÷], [575÷], [673÷], [348÷], [1206÷], [16÷], [720÷], [1301÷], and [718÷].)
z) What are the crystal classes in 4D? (See [216÷], [1034÷], [1368÷], [202÷], [212÷], [997÷], [1383÷]; [1322÷], [1230÷], [1253÷] for 5D; and [1190÷], [1191÷], and [626÷], [1107÷] for nD.)
w) How does a light beam behave in a water vertex? (See [833÷], [1329÷], [835÷], [73÷], [1332÷], [944÷], [851÷], [80÷], and [834÷].)
¸) How does one calculate the electromagnetic field of a charge moving above a conducting surface? (See [188÷], [1174÷], [1043÷], and [1173÷]; for a corrugated surface, see [1302÷] and [597÷]; for an array of half-planes, see
[812÷]; for moving current loops, see [1042÷].) What is the magnetic field around a magnet moving through a metallic tube? See [1053÷].
º) How does one model physical systems with negative specific heat? (See [876÷].)
¹) Are there 2D potentials with two families of orthogonal trajectories? (See [1098÷].)
») What are the eigenvalues of a (grand) canonical density matrix? (See [263÷], [1027÷], [733÷], [495÷], [286÷], and [1290÷].)
a£) What is the relativistic generalization of the Ampere–Maxwell law in integral form? (See [960÷], [524÷].) b£) What is the relativistic generalization of the Fokker–Planck equation for Brownian motion? (See [416÷].) g£) Is there a fluctuation theorem for a single anharmonic oscillator? (See [1183÷].)
d£) In how many different orders can k spacelike separated events in Minkowski space be observed? (See [1239÷].) e£) What is the average shape of a random walk in many dimensions? (See [1155÷].)
¶£) What is the nature of entanglement in a free electron gas? (See [1022÷], [871÷], [872÷], and [1315÷].)
z£) How to quantify statistical properties of thermodynamic fluctuations? (See [880÷], [326÷], [1132÷], [678÷], [923÷], [448÷], [449÷], and [866÷].)
(For a set of more advanced problems, see [595÷], [1373÷], [675÷], [29÷], [325÷], [203÷], [101÷], and http://www.math.princeton.edu/~aizenman/OpenProblems.iamp . For more advanced computational geometry problems, see http://www.cs.smith.edu/~orourke/TOPP/). For the “big” problems, see [1101÷] and http://boudin.fnal.gov/NNP/-B1798866615/ .