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from article Circle
A circle, with radius, diameter, and circumference indicated. (Credit: McGraw-Hill Education)
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from article Circle
Nine-point circle and Euler line PMH.
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from article Trihedron
Trihedron and trihedral angles.
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from article Trihedron
Spherical triangle ABC formed by intersection of sphere with a trihedron. The center of the sphere is at the vertex of the trihedron.
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from article Gamma function
Depiction of gamma function Γ(z) for real values of z.
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from article Hypergeometric functions
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from article Slope
Slope of a line.
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from article Slope
Slope of a curve.
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from article Bioinformatics
Bioinformatics (indicated by the star in the diagram) is a scientific discipline that combines biology, mathematics, and computer science. It applies the power of computer processing to the study of biological systems. (Copyright © McGraw Hill)
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from article Bioinformatics
The central dogma of molecular biology explains how genetic material is transcribed into RNA, which is translated into proteins; then, proteins function to create metabolites, which affect the phenotypes of cells, tissues, and organisms. Scientists…
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from article Finite element method
Illustration of a Von Mises stress plot for the bending of an I-beam. The finite element method can be used to investigate these stresses. (Credit: Shutterstock / Mathew Alexander)
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from article Finite element method
Structures modeled by fitting together structural elements: (a) truss structure; (b) two-dimensional planar structure.
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from article Finite element method
Element types: (a) bar; (b) beam; (c) membrane elements; (d) solid elements; (e) axisymmetric solid; (f) plate-bending elements; (g) curved-shell elements; (h) axisymmetric thin-shell element.
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from article Finite element method
Finite element method of (a) fuselage section and (b) wing of a business jet.
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from article Euclidean geometry
The Greek mathematician Euclid. (Credit: ZU_09/E+/Getty Images)
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from article Euclidean geometry
Basic objects in mathematics from left to right: a point, a line, and a ray. (Credit: Christopher Thomas)
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from article Euclidean geometry
A sketch of two parallel lines, l and m. (Credit: Christopher Thomas)
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from article Euclidean geometry
Two similar triangles, which have different sizes but the same angles. (Credit: Christopher Thomas)
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from article Euclidean geometry
Representation of statements that are true in Euclidean geometry. (5a) A circle with radius r and a circumference (C) of 2πr and area (A) of πr2. (5b) One line through point p when p is not on line l will form a parallel line to line l. (Credit:…
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from article Euclidean geometry
A right triangle with sides A, B, and C. Knowing any two of the lengths of the three sides can be used to find the length of the third side using the Pythagorean theorem. (Credit: Christopher Thomas)
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from article Wavelets
Members of a wavelet family: an original and two shifted and dilated copies. (a) Original wavelet ψ with width w. (b) Large-scale wavelet ψa,b, with width aw that is larger than the original width w. (c) Fine-scale wavelet ψa,b with narrow width.
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from article Wavelets
Two examples of wavelets which generate an orthonormal basis, and which are associated with fast algorithms.
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from article Logic
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from article Logic
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from article Graphic methods
Percentages of distribution of grades in a class, depicted by (a) bar graph and (b) circle graph.
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from article Graphic methods
Line graph of the functional relation y = 5 − x.
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from article Fractal
Koch curve, (a) first and (b) second stages.
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from article Decision theory
Decision tree for the problem given in the text. Numbers at nodes represent time past alarm (7:00 A.M.), and numbers at branches refer to the decision mode or state of nature.
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from article Decision theory
From of a decision matrix. Oij = outcome corresponding to the interaction of a strategy (Si) and a state of nature (Nj).
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from article Decision theory
Decision tree structure to determine the subjective utility of lottery outcomes.
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from article Decision theory
Hypothetical subjective utility curves.
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from article Catastrophe theory
Cusp catastrophe surface for the response function Vab as a function of (a,b) at a critical point. Paths A, B, and C are labeled.
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from article Catastrophe theory
Paths A, B, and C from Fig. 1, with points 1–10 marked. The response function Vab has the same values at (1,3,5) and at (2,4,10).
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from article Catastrophe theory
Paths that exhibit features of catastrophe theory. (a) Path on curve C showing a jump in the response function. (b) Path using curves A and B which shows hysteresis. No change in potential energy Vab is required to go from 1 to 3 or from 2 to 4. The…
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from article Progression (mathematics)
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from article Tangent
Line L tangent to curve C at P.
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from article Hyperbola
Hyperbola as a conic section.
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from article Hyperbola
Hyperbola as a locus of points.
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from article Geometrization theorem
Making a 2-D torus.
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from article Geometrization theorem
Life in Seifert-Weber dodecahedral space. (Courtesy of Geometry Center, University of Minnesota)
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from article Geometrization theorem
Hyperbolic dodecahedron of infinite diameter. The infinite spikes are rounded off “chimneys,” which are here triangular rather than rectangular. (Courtesy of Paul Nylander from http://bugman123.com/)
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from article Geometrization theorem
Neck pinching, resulting in infinite curvature.
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from article Geometrization theorem
Ricci flow with surgery.
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from article Matrix theory
Vector rotation that rotates any vector v in the plane counterclockwise through 45° to a new vector Tv.
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from article Trigonometry
Point P on the unit circle corresponding to θ = t radians. (a) t ≥ 0: length of arc from (1,0) to P is t units. (b) t < 0: length of arc from (1,0) to P is |t| units.
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from article Trigonometry
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from article Trigonometry
Angles and coordinates of corresponding points on the unit circle. (a) Angles that are integral multiples of π/4 (45°). (b) Angles that are integral multiples of π/6 (30°) and π/3 (60°).
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from article Trigonometry
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from article Trigonometry
Point P on a circle of radius r corresponding to angle θ, and corresponding point P* on the unit circle.
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from article Trigonometry
Relationship between angles θ and −θ, and coordinates of corresponding points on the unit circle.
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from article Trigonometry
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from article Trigonometry
Geometric construction used to prove the difference formula for cosines. (a) Distance between P1 and P2 equals (b) distance between A and P3.
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from article Trigonometry
Graphs of trigonometric functions. (a) y = sin x. (b) y = cos x. (c) y = tan x, x not equal to odd multiples of π/2. (d) y = csc x, x not equal to integral multiples of π. (e) y = sec x, x not equal to odd multiples of π/2. (f) y = cot x, x not…
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from article Trigonometry
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from article Trigonometry
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from article Trigonometry
Right triangle. (a) Labeling of sides and angles. (b) Relationship of sides to coordinates defining trigonometric functions.
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from article Trigonometry
Labeling of sides, angles, and vertices of oblique triangle, with altitude h used in proving the law of sines. (a) Angle α is acute. (b) Angle α is obtuse.
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from article Trigonometry
Triangle and rectangular coordinates used in proving the laws of cosines. (a) Angle γ is acute. (b) Angle γ is obtuse.
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from article Trigonometry
Relationship between rectangular coordinates (x, y) and polar coordinates (r, θ).
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from article Boolean algebra
A portrait of George Boole.
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from article Calculus of variations
Weierstrass condition. Symbols explained in text.
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from article Calculus of variations
Example of an area-minimizing surface that cannot be written as the graph of a single function. (a) Nonconvex domain C. (b) Minimal graph over C that is not area-minimizing. (c) Second surface over C that has less area. (After F. Morgan, Geometric…
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from article Calculus of variations
Comparison of surfaces. (a) Surface with no handles. (b) Surface with one handle and less area. (c) Surface with two handles and still less area. (d, e, f) These three surfaces are deformed to show that they do have zero, one, and two handles,…
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from article Calculus of variations
Examples of surfaces minimizing the integral of a nonelliptic integrand among all bodies having the same volume (the surface at upper left) and among all surfaces having the same boundary (the saddle-shaped surface). Surfaces were computed as part…
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from article Linear algebra
Geometric representation of a vector OP.
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from article Radian measure
Diagrams showing radian measurement. Symbols are explained in the text.
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from article Z transform
Extraction of discrete-time sample values from a continuous signal.
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from article Pythagorean theorem
Proof of Pythagorean theorem based on comparison of areas of squares.
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from article Pythagorean theorem
Three-dimensional Pythagorean theorem.
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from article Asymptote
Asymptotes of a hyperbola.
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from article Tractrix
Tractrix (upper half).
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from article Decision analysis
A methodology of decision analysis.
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from article Decision analysis
Model for revising the state of information or knowledge.
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from article Factor analysis
Diagram of the total variance of a variable in standard score units subdivided into types of variances.
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from article Factor analysis
Vectorial representation of correlation coefficients.
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from article Factor analysis
Representation demonstrating that the three-variable correlation matrix cannot be represented correctly in two dimensions.
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from article Factor analysis
Vectorial representation of the intercorrelations among four variables in a two-dimensional space.
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from article Factor analysis
A simple structure configuration for 10 variables in three dimensions, shown on the surface of a sphere, with reference axes I, II, and III at the intersections, and the corresponding factor-pattern matrix.
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from article Factor analysis
A simple structure for 12 variables in three dimensions with bipolar factors, and the corresponding factor-pattern matrix.
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from article Conic section
Four conic sections. (a) Circle. (b) Ellipse. (c) Parabola. (d) Hyperbola.
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from article Catenary
Catenary and force triangle which keeps a portion of the string in equilibrium.
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from article Catenary
Relation of catenary to the tractrix which is its involute.
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from article Legendre functions
Legendre polynomials Pn(x), n = 2, 3, 4, 5.
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from article Legendre functions
Legendre functions of the second kind, Qn(x). (After E. Jahnke and F. Emde, Tables of Functions with Formulae and Curves, 4th ed., Dover Publications, 1945)
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from article Mathematical biology
Illustration of the computationally derived structure of the Nsp 15 protein in the SARS-CoV-2 virus, generated as a potential drug target. Calculations of protein folding are an example of the application of mathematics to biology. (Credit:…
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from article Mathematical biology
The fish (a) Diodon and (b) Mola, showing the basic relatedness of the two genera by the superimposed grids.
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from article Mathematical biology
Crab carapaces from five different species (b–f) as deformations of (a). By transforming the grid coordinates, the variation in the shapes of the carapaces is minimized.
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from article Statistics
An example of statistics collection and presentation regarding median age of the populations within each United States county as of 2018. (Credit: U.S. Census Bureau)
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from article Statistics
A probability distribution of annual incomes expressed graphically.
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from article Statistics
Normal distribution, also known as a "bell curve" because of its bell-like shape when plotted.
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from article Set theory
Sets of points on a plane.
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from article Set theory
Vertical projection of set X onto segment Y.
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from article Polygon
Triangle ABC with nine-point circle. Symbols are explained in text.
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from article Polygon
Regular pentagon, showing approximate construction without a protractor. Symbols are explained in text.
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from article Polygon
Polar triangles A, B, C and A′, B′, C′. The center of the sphere is at O. AQ and PB are 90° arcs, so angle C′ = arc PQ is the supplement of AB.
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from article Epicycloid
Diagram of an epicycloid.
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from article Difference equation
Numerical solution of Eq. (32) for a = 0.05, b = 0.1, c = 0.01, and a step input. The solution uses the state-variable representation of Eq. (33) and the difference-equation approximation of Eq. (36) with h = 0.1.
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from article Cycloid
Diagram of cycloid; symbols explained in text.
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from article Reference value and reference interval
The reference values measured on the reference sample group are arranged as a histogram, the distribution of which is imitated by a suitable mathematical function where the reference limits are easy to calculate (parametric method). Here, the usual…
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from article Torus
Diagram of a torus.
