MATVB_query_nl.json

{
  "q_1": "unit root vs. trend stationary diagram",
  "q_2": "Sombrero function",
  "q_3": "How is vector field divergence visualized?",
  "q_4": "Vector addition diagram",
  "q_5": "Venn diagram of symmetric difference of three sets",
  "q_6": "How does cutting plane method work with unit cube?",
  "q_7": "What does the Vamos matroid structure look like?",
  "q_8": "What is the fundamental polygon for Bring's curve?",
  "q_9": "What does the bundle theorem show about cycles?",
  "q_10": "What are extreme points that aren't exposed points?",
  "q_11": "Regular decagon symmetries vertices mirrors gyration orders",
  "q_12": "How can 36 be both square and triangular?",
  "q_13": "betweenness centrality graph theory",
  "q_14": "What is spherical triangle geometric shape?",
  "q_15": "horocycle in Poincaré disk model with normals",
  "q_16": "quaternion spatial rotation axis angle sphere",
  "q_17": "How does logarithm function grow?",
  "q_18": "Rule of succession circle diagram",
  "q_19": "3-adic integers visualization",
  "q_20": "Why does joining two truncated icosahedra produce a rhombic enneacontahedron?",
  "q_21": "What is the absolutely convex hull of a cross?",
  "q_22": "Bessel Processes realizations",
  "q_23": "Fibonacci polynomials from Pascal's triangle",
  "q_24": "Venn diagram for converse nonimplication $P$ is not implied by $Q$",
  "q_25": "matching distance between two lines",
  "q_26": "What's the difference between convex and affine combinations?",
  "q_27": "What is the Tutte fragment and how does it lead to the full Tutte graph?",
  "q_28": "hypercube graph 1-skeleton of tesseract",
  "q_29": "Why are some numbers called untouchable?",
  "q_30": "How does isoelastic utility vary with different parameters?",
  "q_31": "What are cospectral polyhedral graphs?",
  "q_32": "hazard function with different $\beta$ when $\\alpha =1$ in a log-logistic",
  "q_33": "What is complete coloring of the Clebsch graph?",
  "q_34": "Tschirnhausen cubic graph for $a=1$",
  "q_35": "Show an illustration of the combinatorial proof of Pascal's rule",
  "q_36": "What properties make the regular dodecahedron a convex body?",
  "q_37": "Whewell equation quantities",
  "q_38": "How are stationary points distinguished from inflection points on a curve?",
  "q_39": "view from inside a 3-torus",
  "q_40": "bijective function diagram",
  "q_41": "How is $K_{3,3}$ edge-colored with parallel segments?",
  "q_42": "What does a uniform triangular prism look like?",
  "q_43": "Cohomological spectral sequence pages",
  "q_44": "What is a minimum cut in graph theory?",
  "q_45": "Why isn't  $f(x) = \\sin(1/x)$ of bounded variation?",
  "q_46": "How do stacked rectangles demonstrate the divergence of the harmonic series?",
  "q_47": "How does line intersection create a circle locus?",
  "q_48": "What does the fourth iteration of a Gosper curve look like?",
  "q_49": "What is planar subdivision in computational geometry?",
  "q_50": "What are limiting parallels in hyperbolic geometry?",
  "q_51": "What are the key features of the logarithmic integral function?",
  "q_52": "What simplexes can be represented in 3D space?",
  "q_53": "hyperbola inscribed angle theorem",
  "q_54": "What distinguishes convex kites from concave kites in geometry?",
  "q_55": "How many distinct necklaces with 3 red and 3 green beads?",
  "q_56": "What are different types of polygons?",
  "q_57": "What is incidence coloring in graph theory?",
  "q_58": "finite projective planes order 2 and 3 constructions",
  "q_59": "Why is $[0,1]/\\{0,1\\}$ homeomorphic to circle $S^1$?",
  "q_60": "ladder graphs $L_1$ to $L_5$",
  "q_61": "What does the intersection of two planes look like?",
  "q_62": "weight representation theory positive roots",
  "q_63": "How do conic sections vary with eccentricity?",
  "q_64": "Thales's theorem diagram",
  "q_65": "How can an equilateral triangle be constructed using only compass and straightedge?",
  "q_66": "amoeba of a polynomial",
  "q_67": "How are eigenfaces used as eigenvector?",
  "q_68": "How do you compute the image and preimage of subsets under a finite mapping?",
  "q_69": "What are demicubes from alternating hypercubes?",
  "q_70": "Euler diagram with a set of polygons",
  "q_71": "How does Playfair’s axiom underlie translations in affine geometry?",
  "q_72": "How does Huber loss compare to squared error?",
  "q_73": "What is an Eisenstein triple?",
  "q_74": "What does a cone that has been truncated by an inclined plane look like?",
  "q_75": "Ceva's theorem concurrent case",
  "q_76": "What are osculating circles of an Archimedean spiral?",
  "q_77": "Six permutations of three distinct balls",
  "q_78": "divisor summatory function with leading terms removed",
  "q_79": "Constructing regular heptadecagon with Gaussian",
  "q_80": "Venn diagram for union of two sets",
  "q_81": "Hyperbolic trajectories with the same velocity but different impact parameters",
  "q_82": "angle on Cartesian coordinate system",
  "q_83": "What is the graphic matroid of a cycle graph?",
  "q_84": "How do implicit surfaces transform between shapes?",
  "q_85": "How do Cartesian coordinates work on the plane?",
  "q_86": "How can stopped Brownian motion be visualized as a martingale?",
  "q_87": "Compare the shapes of logit and appropriately scaled probit curves at the origin.",
  "q_88": "What is a power diagram of circles?",
  "q_89": "How is a circular paraboloid represented as a polygon mesh?",
  "q_90": "What is a 2-dimensional cyclohedron?",
  "q_91": "Simple polygon shape",
  "q_92": "Explain why the octahedron is dual to the cube.",
  "q_93": "Gamma function plot in real domain",
  "q_94": "How is a torus obtained by revolving a square around a skew axis?",
  "q_95": "Why is the $\\{4,4\\}$ square tiling self-dual?",
  "q_96": "A supergolden rectangle contains three scaled copies of itself",
  "q_97": "Projective geometry illustration of Simson line",
  "q_98": "Simplex algorithm on a polytope",
  "q_99": "one-one vs. one-to-many vs. many-to-many relations",
  "q_100": "What conditions define a eutactic star?",
  "q_101": "How is the slope $m$ of a line defined in terms of $\\Delta y$ and $\\Delta x$?",
  "q_102": "How does trilinear interpolation work with cube corner points?",
  "q_103": "What does a free abelian group look like as a lattice?",
  "q_104": "What is the definition and structure of a cantellated cubic tiling?",
  "q_105": "quasiconvex vs pseudoconvex function plot",
  "q_106": "Whitney embedding double-point cancellation",
  "q_107": "First Pythagorean triples plot with legs $a$ and $b$",
  "q_108": "topology set and its boundary",
  "q_109": "monkey saddle surface",
  "q_110": "Why isn't $e^{-1/x^2}$ analytic at $x=0$?",
  "q_111": "What does the leading Lyapunov vector represent?",
  "q_112": "Normal distribution grading methods chart",
  "q_113": "Dawson integral function ($F(z)$) complex plot",
  "q_114": "What is phase space in Melnikov distance theory?",
  "q_115": "How are weak orderings represented in different ways?",
  "q_116": "unit circles for different p-norms",
  "q_117": "interpolation with polyharmonic splines",
  "q_118": "basis for two-dimensional subspace of $R^3$",
  "q_119": "What is the Perles configuration in matroid theory?",
  "q_120": "geometric intuition for integral of $\frac{1}{x}$",
  "q_121": "Hasse diagram subgroups of $Dih_4$",
  "q_122": "What is a breather pseudospherical surface?",
  "q_123": "Complete elliptic integral of the third kind $\\Pi(n, k)$",
  "q_124": "Csaszar polyhedron model",
  "q_125": "counit as a universal morphism",
  "q_126": "illustration of sandwich theorem",
  "q_127": "Bessel function of the first kind with $\\alpha =0.5$",
  "q_128": "analytic tableaux a or not b generates a and not b",
  "q_129": "universal property of the initial topology",
  "q_130": "How does Playfair's axiom work with interior angles?",
  "q_131": "How is a surface of genus 3 triangulated?",
  "q_132": "correction factor  $c_4$ vs. sample size $n$",
  "q_133": "Why does a hanging chain form a catenary curve?",
  "q_134": "Massey product for the complement of the Borromean rings",
  "q_135": "How does the scaled bump function vary with epsilon?",
  "q_136": "Relative error in truncated Stirling series in regards to number of terms",
  "q_137": "What are the symmetries of a regular icositetragon?",
  "q_138": "Circular right cylinder",
  "q_139": "What is a p-box in probability theory?",
  "q_140": "Levi graph of Fano plane",
  "q_141": "How do Cassini ovals vary with parameter $e$?",
  "q_142": "What problems occur with normal approximation intervals?",
  "q_143": "example of a tree decomposition of a graph",
  "q_144": "area of a parallelogram as a determinant",
  "q_145": "Shrikhande graph embedded on a torus",
  "q_146": "How can three squares be rearranged into a dodecagon?",
  "q_147": "How do rational points correspond under stereographic projection?",
  "q_148": "three-leaved clover quartic plane curve",
  "q_149": "odd graph $O_4$",
  "q_150": "What is the Cayley diagram for binary tetrahedral group?",
  "q_151": "Examples of basic three-dimensional cell shapes",
  "q_152": "What is an ungula of a circular cylinder?",
  "q_153": "Chernoff bound for chi-squared CDF",
  "q_154": "What does Cayley's nodal cubic surface look like?",
  "q_155": "Why is a star domain simply connected?",
  "q_156": "How is a Möbius band an I-bundle?",
  "q_157": "How do you calculate distance to the horizon?",
  "q_158": "Apollonius problem solution circle",
  "q_159": "witch of Agnesi curve",
  "q_160": "How does the golden rectangle demonstrate the golden ratio?",
  "q_161": "monomorphism scenarios illustration",
  "q_162": "How is the involute of a cycloid generated?",
  "q_163": "What does the formula $m={\\frac {\\Delta y}{\\Delta x}}$ represent?",
  "q_164": "What truncations of the cube exist beyond rectification?",
  "q_165": "What are parabolic cylinder functions in coordinate systems?",
  "q_166": "Lambert's problem ellipse with foci through points",
  "q_167": "How many squares are in different grid configurations?",
  "q_168": "How many ways can you triangulate a convex heptagon?",
  "q_169": "What does the Heighway dragon curve look like?",
  "q_170": "How does compressing a circle produce an ellipse?",
  "q_171": "How does chaotic mixing evolve for an advected contour?",
  "q_172": "Walk-on-spheres algorithm illustration",
  "q_173": "How are Pell numbers used to construct silver spirals?",
  "q_174": "cobweb diagram Gauss iterated map example",
  "q_175": "Candido identity geometric interpretation",
  "q_176": "probability of not observing $1/n$ event in $n$ trials",
  "q_177": "What is the Heilbronn triangle problem with 8 points?",
  "q_178": "branch decomposition of a grid graph",
  "q_179": "unknotting a trefoil knot",
  "q_180": "($10_3$) configuration not Desargues",
  "q_181": "How are curves used to construct the cissoid?",
  "q_182": "How is drunkard's walk an absorbing Markov chain?",
  "q_183": "Graph of a sine wave's voltage vs. times",
  "q_184": "What is a 3-dimensional foliated chart?",
  "q_185": "Bhargava cube for binary quadratic forms",
  "q_186": "How does erf(z) behave over the complex plane?",
  "q_187": "Cayley's formula for spanning trees of complete graphs",
  "q_188": "semiorder Hasse diagram",
  "q_189": "Open neighborhood of the Moore plane",
  "q_190": "How is an equilateral pentagon constructed from four equal circles?",
  "q_191": "skew lines on a rectangular parallelepiped",
  "q_192": "What does a geodesic triangle look like on a sphere?",
  "q_193": "probability mass function graph",
  "q_194": "illustrate convex metric space",
  "q_195": "Fuhrmann circle and triangle",
  "q_196": "Sierpinski triangle fractal",
  "q_197": "path-connected subspace of $R^2$",
  "q_198": "What is a section of a fiber bundle?",
  "q_199": "How do gradient and directional derivatives appear in contour plots?",
  "q_200": "Young's lattice Hasse diagram",
  "q_201": "Dini's Surface",
  "q_202": "What are all possible nets of a cube?",
  "q_203": "example of injective partial function",
  "q_204": "Building a forest with trees",
  "q_205": "What are the asymptotes for the function $f(x)=x+1/x$?",
  "q_206": "What defines an inellipse, and how is it constructed?",
  "q_207": "How does the Euler method approximate a curve?",
  "q_208": "foci of a lemniscate of Bernoulli",
  "q_209": "spiral of equilateral triangles following Padovan sequence",
  "q_210": "linear interpolation between two points",
  "q_211": "How does the generalized Pythagorean theorem hold for Bregman divergence?",
  "q_212": "How can a parabola be used to trisect an angle?",
  "q_213": "What do line graphs of linear equations look like when plotted on a Cartesian plane?",
  "q_214": "Prime knots with seven or fewer crossings",
  "q_215": "cube and octahedron Hanner polytopes",
  "q_216": "ordered field property $a>0 \\land x<y \\Rightarrow ax<ay$",
  "q_217": "What is a 5-cell and why is it considered self-dual?",
  "q_218": "How does a Pochhammer contour wind around two points in complex analysis?",
  "q_219": "barycentric subdivision 2-simplices iterations",
  "q_220": "jacobi elliptic functions fundamental rectangle",
  "q_221": "How does the regularized incomplete gamma function behave in the complex plane?",
  "q_222": "cartesian tensor coordinate rotation angles",
  "q_223": "What is a group extension and how does it form a short exact sequence?",
  "q_224": "creating a projective plane from a hemisphere",
  "q_225": "What is the envelope of a family of lines?",
  "q_226": "Riemann zeta function plot",
  "q_227": "intersection of sphere and cylinder",
  "q_228": "How to prove that two lines are parallel using corresponding angles?",
  "q_229": "truncated triangular prism",
  "q_230": "What are inversions in permutations and how are they counted?",
  "q_231": "What is Jordan's inequality and how does it bound the sine function?",
  "q_232": "Minkowski sum of two sets",
  "q_233": "taxicab geometry circle grid",
  "q_234": "What is a tangential quadrilateral and how does it relate to its incircle?",
  "q_235": "Thales's theorem with rays",
  "q_236": "How do you use the dot product to find the angle between two vectors?",
  "q_237": "Hamiltonian path in the Herschel graph",
  "q_238": "Walsh matrix LDU decomposition",
  "q_239": "How do the DFT and DTFT of a Hann window compare?",
  "q_240": "moser circle problem regions",
  "q_241": "What is a Bethe lattice and how does it relate to apeirogonal tiling?",
  "q_242": "What characterizes a mixed graph, and can you give an example?",
  "q_243": "Cantor pairing function bijection",
  "q_244": "permutations of the square $D_4$",
  "q_245": "Tharnack curve theorem M-curves",
  "q_246": "What is the golden angle and how is it related to the golden ratio?",
  "q_247": "Eulerian path in Königsberg Bridges and Five room puzzles multigraphs",
  "q_248": "What are the intermediate stages when cantellating a cube?",
  "q_249": "What are lemniscate elliptic functions and how do they relate to Fermat curves?",
  "q_250": "How are points and lines arranged in the Fano plane?",
  "q_251": "Cayley graph of the free group on two generators",
  "q_252": "comparison of mean median and mode of two log-normal distributions",
  "q_253": "Explain the concept of ramification in geometry",
  "q_254": "Graphs of $\text{csch}(x)$, $\text{sech}(x)$, and $\text{coth}(x)$",
  "q_255": "What are the fundamental properties and parametrization of the unit circle by an angle $t$?",
  "q_256": "How is the Kostant partition function defined for the $A_2​$ root system",
  "q_257": "How do you construct a harmonic conjugate using a complete quadrangle in projective geometry?",
  "q_258": "Tutte polynomial of the bull graph",
  "q_259": "What conditions characterize Poncelet’s porism for bicentric quadrilaterals?",
  "q_260": "How are sample paths of a Wiener process defined?",
  "q_261": "Why does Morley’s trisector theorem guarantee an equilateral triangle from angle trisectors of any triangle?",
  "q_262": "How does applying the escape-time algorithm to the Perrin function’s Newton map reveal its real and complex roots?",
  "q_263": "Archimedean spiral $r = \frac{1}{2} \theta + 2$",
  "q_264": "What relationships hold among the bases, diagonals, and their intersection in an isosceles trapezoid?",
  "q_265": "Keplerian orbital elements used to describe an orbit",
  "q_266": "Illustration of Miquel's Pentagon Theorem",
  "q_267": "rotation represented by an Euler axis and angle",
  "q_268": "What is the axiom schema of replacement in set theory?",
  "q_269": "Under what conditions do the main diagonals of a tangential polygon concur?",
  "q_270": "What is the figure-eight knot in knot theory?",
  "q_271": "power graph analysis primitive motifs",
  "q_272": "What do spiric sections look like when cutting a torus?",
  "q_273": "Pappus's hexagon theorem",
  "q_274": "rose curve $r = \\cos(4\\theta)$",
  "q_275": "How does the Pythagorean theorem derive the hypotenuse length $c$ from sides $a$ and $b$?",
  "q_276": "What are the positive roots of the $G_2​$ root system?",
  "q_277": "What does transversality mean for curves on the surface of a sphere?",
  "q_278": "T-square fractal order 8",
  "q_279": "Define a Misiurewicz point and its preperiodic orbit.",
  "q_280": "chromatic polynomials of all non-isomorphic graphs on 3 vertices",
  "q_281": "What is a Pappus chain?",
  "q_282": "What is the Cairo pentagonal tiling?",
  "q_283": "finite affine plane of order 2",
  "q_284": "How is numerical differentiation performed using finite differences?",
  "q_285": "height of an equilateral triangle with side length 2",
  "q_286": "What does the fundamental domain of a three-generator Schottky group look like?",
  "q_287": "Fractional Fourier transform, and it generalization to the classical Fourier transform",
  "q_288": "What are the defining properties of a regular square pyramid?",
  "q_289": "Why do perpendicular bisectors of chords between concyclic points concur?",
  "q_290": "What mathematical principles underlie the geometric construction of a spirograph curve?",
  "q_291": "What is the duality between poles and polars in projective geometry?",
  "q_292": "congruent vs similar triangles",
  "q_293": "How can the law of cosines be proved trigonometrically for an acute angle?",
  "q_294": "What are the properties of a uniform pentagonal prism?",
  "q_295": "Lyapunov fractal swallow",
  "q_296": "triangle-free graphs vs. bipartite graphs",
  "q_297": "How is the dimension of a graph defined in graph theory?",
  "q_298": "What is an oval in a projective plane?",
  "q_299": "What is the Kiepert hyperbola of triangle $ABC$?",
  "q_300": "What are the properties of gamma distribution density functions?",
  "q_301": "angle bisection construction",
  "q_302": "What does a negative Spearman’s rank correlation coefficient indicate about the monotonic trend between two variables?",
  "q_303": "What conditions characterize a two-dimensional figure that is invariant under point reflection?",
  "q_304": "What is an orthocentric system?",
  "q_305": "How do the shapes of Student’s t-distribution density functions change with different degrees of freedom?",
  "q_306": "single-wide vs multi-wide block-stacking problem",
  "q_307": "What is the pole–polar relation for an ellipse?",
  "q_308": "How do curvilinear, affine, and Cartesian coordinate systems differ in two-dimensional space?",
  "q_309": "circles on a ring torus",
  "q_310": "How does increasing kurtosis affect the shape and intersections of symmetric probability density functions?",
  "q_311": "bijective function diagram $f: X \\rightarrow Y$",
  "q_312": "What are the basic properties and symmetries of a regular dodecahedron?",
  "q_313": "What is the connected sum operation in topology?",
  "q_314": "unit circle with trigonometric functions",
  "q_315": "What are the fundamental domains of convex uniform honeycombs in a cubic cell?",
  "q_316": "measuring the width of a Reuleaux triangle",
  "q_317": "branches of the unit hyperbola",
  "q_318": "bounded non-convergent sequence",
  "q_319": "first six triangular numbers",
  "q_320": "equidiagonal quadrilateral properties",
  "q_321": "What is the epigraph of a real-valued function?",
  "q_322": "parallelogram law visualization",
  "q_323": "Petersen graph drawing slope number 3",
  "q_324": "triangle centers medial triangle",
  "q_325": "triangle orthocenter centroid and orthocentroidal disk",
  "q_326": "Fiedler vector of a graph",
  "q_327": "What characterizes umbilical points on an ellipsoid, and how are they located via lines of curvature?",
  "q_328": "What is the standard proof of the Butterfly Theorem?",
  "q_329": "How does the four-color theorem apply to map coloring?",
  "q_330": "What is the structure of the 4-dimensional hypercube graph $Q_4​$?",
  "q_331": "packing of equal-sized regular pentagons on a plane",
  "q_332": "limit of perimeters of circumscribed and inscribed polygons",
  "q_333": "Villarceau circles intersection of torus and plane",
  "q_334": "visual representation of well-separated pair decomposition",
  "q_335": "How do the hexagonal faces of a truncated tetrahedron correspond to the original tetrahedron’s faces and vertices?",
  "q_336": "How is a two-dimensional rotation about a point $O$ defined and applied?",
  "q_337": "What is a Pythagorean Hugieia Pentagram and it properties?",
  "q_338": "What characterizes an injective function that is not surjective?",
  "q_339": "How is the Klein quartic decomposed into 24 heptagons in the Klein map?",
  "q_340": "How is a Young diagram of shape $(5,4,1)$ constructed in English notation?",
  "q_341": "How are data points mapped into feature space before applying kernel PCA?",
  "q_342": "What is a Morin surface and how does it facilitate sphere eversion?",
  "q_343": "What is the behavior of the cardinal sine function $\\sinc(z)$ in the complex plane?",
  "q_344": "What are the dimension rules for multiplying two matrices?",
  "q_345": "How is a saddle point defined in the context of Morse theory?",
  "q_346": "What is exact graph coloring?",
  "q_347": "How is addition of two complex numbers represented geometrically?",
  "q_348": "How to compute the gradient of $f(x,y) = -\\left(\\cos^2 x + \\cos^2 y\\right)^2$?",
  "q_349": "Riemann sum approximation methods",
  "q_350": "hexagonal close-packed vs. face-centered cubic lattices",
  "q_351": "How is the Weierstrass sigma function defined?",
  "q_352": "Explain Pascal’s theorem for a self-crossing hexagon inscribed in an ellipse",
  "q_353": "how does intersecting line $CB$ with a circle produce a rational solution $(x_0,y_0)$?",
  "q_354": "Regular tetrahedron shape",
  "q_355": "Which four Hermite basis functions are used in cubic Hermite spline interpolation?",
  "q_356": "Why does adding the line at infinity to a 20-line simplicial arrangement from a regular decagon produce a 21-line arrangement?",
  "q_357": "Describe the Lorenz oscillator’s equations and how they give rise to the Lorenz attractor.",
  "q_358": "How do the four gluings of a square correspond to the four closed surfaces?",
  "q_359": "What construction rules generate quasiregular polyhedra from Schwarz triangle fundamental domains without right angles?",
  "q_360": "Flexible graph vs. rigid graph",
  "q_361": "How do sections of the Mandelbrot set follow logarithmic spirals?",
  "q_362": "Dihedral and gyration symmetries of a regular enneagon",
  "q_363": "Circles of Apollonius and isodynamic points",
  "q_364": "great-circle distance vs straight-line distance on a sphere",
  "q_365": "How does the trapezoidal rule approximate $\\int _{a}^{b}f(x)\\,dx$ using linear interpolants over each interval?",
  "q_366": "What example demonstrates a convex cone that cannot be generated by finitely many extreme rays?",
  "q_367": "Why is a cusp point continuous yet non-differentiable?",
  "q_368": "Describe the stereographic projection that defines a Laguerre plane via the $x–z$ plane onto a cylinder.",
  "q_369": "Intersection graph of disks",
  "q_370": "Geometric construction of regular triacontagon",
  "q_371": "How are the four corner values used in bilinear interpolation on a unit square to estimate interior $z$ values?",
  "q_372": "Torricelli's geometrical solution of the Fermat triangle problem",
  "q_373": "What makes an edge singular when gluing manifolds to form a pseudomanifold?",
  "q_374": "How does an Apollonian gasket arise as the limit set of a Kleinian group?",
  "q_375": "Diagram of Aristotle's wheel",
  "q_376": "In Pascal’s tetrahedron, how do the first four layers extend Pascal’s triangle into three dimensions?",
  "q_377": "Why does subdividing a volume cause the ratio $\\Phi(V_i)$",
  "q_378": "Kissing number in two dimensions",
  "q_379": "How do mean, median, and mode shift when skewness varies between two log-normal distributions?",
  "q_380": "How a trefoil knot can be linked with a simple circle in knot theory.",
  "q_381": "What is the connection between the j-invariant and modular functions on the upper half-plane?",
  "q_382": "In a Randolph diagram, how is the logical disjunction $P \\lor Q$ represented?",
  "q_383": "Can Brownian bridge be represented as union of excursions?",
  "q_384": "How is the nephroid described as the envelope of a pencil of circles?",
  "q_385": "High school exterior angle theorem proof illustration",
  "q_386": "What structure and symmetries emerge in a hand-threaded 2D projection of the $E_8$ root lattice?",
  "q_387": "What is a lattice in the Euclidean plane?",
  "q_388": "Explain the Hadamard product of two matrices and its algebraic properties",
  "q_389": "How does the differential proof of the Pythagorean theorem proceed using infinitesimal triangles?",
  "q_390": "How to measure a mountain's height with trigonometry?",
  "q_391": "Why is the Fermat point also known as the first isogonic center in triangle geometry?",
  "q_392": "Comparison of p-value of chi-squared distribution for different degrees of freedom",
  "q_393": "What vertex and edge identifications produce a regular skew octagon as a square antiprism’s edge graph?",
  "q_394": "How do variations in slope and intercept affect regression‐analysis confidence and prediction bands?",
  "q_395": "Why might one fit a quadratic rather than a line to data in least-squares regression?",
  "q_396": "What distinguishes stationary points from inflection points on a graph?",
  "q_397": "How is an oloid constructed from two circular sectors?",
  "q_398": "Explain how the altitude in an isosceles triangle splits it into two right triangles and preserves the triangle inequality.",
  "q_399": "Compute all 28 bitangent lines of the Trott quartic curve",
  "q_400": "Explain the relationship between the logistic curve’s parameters and its inflection point."
}