{ "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