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Skeletonized version of algebraic geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication
Tropical_geometry
Mathematics textbook
Introduction to Tropical Geometry is a book on tropical geometry, by Diane Maclagan and Bernd Sturmfels. It was published by the American Mathematical
Introduction to Tropical Geometry
Introduction_to_Tropical_Geometry
Mathematical concept
In algebraic geometry, a tropical compactification is a compactification (projective completion) of a subvariety of an algebraic torus, introduced by Jenia
Tropical_compactification
Overview of and topical guide to geometry
Absolute geometry Affine geometry Algebraic geometry Analytic geometry Birational geometry Complex geometry Computational geometry Conformal geometry Constructive
Outline_of_geometry
Russian programmer and mathematician (born 1980)
objects in a generalized algebraic geometry. Versions of a tropical geometry, of an absolute geometry over a field with one element and an algebraic analogue
Nikolai_Durov
Study of the tropical semiring
is cryptography based on the tropical semiring. Tropical geometry is an analog to algebraic geometry, using the tropical semiring. Litvinov, G. L. (2005)
Tropical_analysis
Structure in convex geometry
applications to polyhedral combinatorics, linear programming, tropical geometry, toric geometry and other areas of mathematics. Given a convex polytope P
Normal_fan
Professor of mathematics
commutative algebra and algebraic geometry, with an emphasis on toric varieties, Hilbert schemes, and tropical geometry. As a student at Burnside High School
Diane_Maclagan
Semiring with minimum and addition replacing addition and multiplication
respectively. The tropical semiring has various applications (see tropical analysis), and forms the basis of tropical geometry. The name tropical is a reference
Tropical_semiring
Branch of mathematics
mathematics are tropical geometry, commutative algebra, and optimization. Nonlinear algebra is closely related to algebraic geometry, where the main objects
Nonlinear_algebra
American mathematician
an American mathematician known for her work on cluster algebras, tropical geometry, algebraic combinatorics, amplituhedra, and the positive Grassmannian
Lauren Williams (mathematician)
Lauren_Williams_(mathematician)
Math concept
simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements. A polyhedral complex
Polyhedral_complex
Branch of mathematics
algebraic geometry. Versions of a tropical geometry, of an absolute geometry over a field of one element, and an algebraic analogue of Arakelov's geometry were
Algebraic_geometry
Cubic graph with 10 vertices and 15 edges
appearance in tropical geometry. The cone over the Petersen graph is naturally identified with the moduli space of five-pointed rational tropical curves. The
Petersen_graph
Mathematician
mathematician, who works on real algebraic geometry, singularity theory, tropical geometry, and enumerative geometry. Eugenii Shustin obtained his diploma
Eugenii_Shustin
In tropical geometry, a tropical projective space is the tropical analog of the classic projective space. Given a module M over the tropical semiring
Tropical_projective_space
Several equations of degree 1 to be solved simultaneously
more exotic structure to which linear algebra can be applied, see Tropical geometry. The system of one equation in one unknown 2 x = 4 {\displaystyle
System_of_linear_equations
Set associated with a complex-valued polynomial
complex variables. Amoebas have applications in algebraic geometry, especially tropical geometry. Consider the function Log : ( C ∖ { 0 } ) n → R n {\displaystyle
Amoeba_(mathematics)
Study of systems of inequalitites
In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations
Real_algebraic_geometry
Argentine mathematician
mathematician known for her work on algebraic geometry, particularly toric geometry, tropical geometry, and their applications to biological systems.
Alicia_Dickenstein
Mathematical fallacy
general using induction.) The freshman's dream is valid for all n in tropical geometry (where multiplication is replaced with addition, so exponentiation
Freshman's_dream
German American mathematician
mathematics, including algebraic geometry, commutative algebra, discrete geometry, Gröbner bases, toric varieties, tropical geometry, algebraic statistics, and
Bernd_Sturmfels
Concept in physics
continuous model of self-organised criticality is proposed by using tropical geometry. Key theoretical issues yet to be resolved include the calculation
Self-organized_criticality
Russian mathematician (born 1948)
algebraic geometry, most notably real algebraic geometry, tropical geometry and knot theory. Viro developed a "patchworking" technique in algebraic geometry, which
Oleg_Viro
Arithmetic operation
reasons, in tropical geometry one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication"
Addition
Cryptography using tropical algebra
In tropical analysis, tropical cryptography refers to the study of a class of cryptographic protocols built upon tropical algebras. In many cases, tropical
Tropical_cryptography
American mathematician and violinist
research involves combinatorial commutative algebra, graph theory, and tropical geometry. Chan was inspired to become a violinist as a pre-schooler, seeing
Melody_Chan
American mathematician (born 1965)
of geometry, analysis and combinatorics and has made a deep impact on fields such as tropical and non-archimedean geometry, logarithmic geometry, the
Mark_Gross_(mathematician)
German mathematician
diophantine geometry and the algebraic geometry of non-Archimedean ordered fields, including the study of buildings, Berkovich spaces, and tropical geometry. She
Annette_Werner
Italian mathematician and academic
Her research includes work in algebraic geometry, arithmetic geometry, tropical geometry and enumerative geometry. Caporaso earned a laurea from Sapienza
Lucia_Caporaso
Theoretical object in mathematics
{\displaystyle {\mathcal {G}}.} F1‑geometry has been linked to tropical geometry, via the fact that semirings (in particular, tropical semirings) arise as quotients
Field_with_one_element
Computer algebra system
areas of mathematics involving convex and algebraic geometry, such as toric and tropical geometry. Free and open-source software portal Comparison of
Singular_(software)
American mathematician
Katz is a mathematician working in combinatorial algebraic geometry and arithmetic geometry. He is currently an associate professor in the Department of
Eric_Katz
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Mathematician
Mathematical Society. Abouzaid, Mohammed (2009). "Morse homology, tropical geometry, and homological mirror symmetry for toric varieties". Selecta Mathematica
Mohammed_Abouzaid
Topics referred to by the same term
Tanzanian airline Tropical Airways, a defunct Haitian airline Tropical geometry, a mathematical area studying geometry over the tropical semiring MS Tropicale
Tropical_(disambiguation)
German mathematician (born 1972)
internationalization and diversity. Topics in her research have included tropical geometry, matroid polytopes, Chow rings, toric varieties, lattices and semilattices
Eva-Maria_Feichtner
name of Ricci calculus Absolute geometry Also called neutral geometry, a synthetic geometry similar to Euclidean geometry but without the parallel postulate
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
American expert in combinatorial commutative algebra, graph theory, and tropical geometry Sun-Yung Alice Chang (born 1948), Chinese-American mathematical analyst
List_of_women_in_mathematics
Mathematical operation on matrices
LNCS 5125, pp. 622–633, 2008. Floyd–Warshall algorithm – Algorithm in graph theory Tropical geometry – Skeletonized version of algebraic geometry v t e
Min-plus matrix multiplication
Min-plus_matrix_multiplication
Type of mathematical function
least squares is used. Linear interpolation Spline interpolation Tropical geometry Polygonal chain Apps, P., Long, N., & Rees, R. (2014). Optimal piecewise
Piecewise_linear_function
Tree graph with one central node and leaves of length 1
of some fixed length, is used as a local model of curves in tropical geometry. A tropical curve is defined to be a metric space that is locally isomorphic
Star_(graph_theory)
of Minkowski. Newton polytopes are the central object of study in tropical geometry and characterize the Gröbner bases for an ideal. Toric varieties Hilbert
Newton_polytope
Research honor for women
in number theory and algebraic geometry, particularly her contributions in arithmetic statistics and tropical geometry, as well as her work with Ravi
AWM–Microsoft Research Prize in Algebra and Number Theory
AWM–Microsoft_Research_Prize_in_Algebra_and_Number_Theory
Rapidly rotating storm system
Takagi, Hiroshi (August 18, 2021). "A new tropical cyclone surge index incorporating the effects of coastal geometry, bathymetry and storm information". Scientific
Tropical_cyclone
Spanish mathematician
power series to solve differential equations, singularity theory, and tropical geometry. She has also published research on screening for mental health, and
Fuensanta_Aroca
Morrison. Gröbner fan is a base for the nowadays active field of tropical geometry. One implementation of the Gröbner fan is called Gfan, based on an
Gröbner_fan
equidistribute to this Chambert-Loir measure and studied its properties using tropical geometry, leading to a proof of the Bogomolov conjecture when A {\displaystyle
Bogomolov_conjecture
British economist
auctions. He has also worked (with Elizabeth Baldwin) on applying tropical geometry to economics and auctions, including the Product Mix auction. In addition
Paul_Klemperer
Association of German research institutes
Planck-Humboldt Medal Robert Wood Harvard University Soft robotics Max Planck-Humboldt Medal Sam Payne University of Texas at Austin Tropical geometry
Max_Planck_Society
German mathematician
(born 19 November 1980) is a German mathematician specializing in tropical geometry. In 2010 she won both the Heinz Maier-Leibnitz Prize of the Deutsche
Hannah_Markwig
Vladimir Rokhlin. Hilbert's sixteenth problem Tropical geometry Arnold, Vladimir I. (2013). Real Algebraic Geometry. Springer. p. 95. ISBN 978-3-642-36243-9
Gudkov's_conjecture
French academic (born 1954)
(network calculus) and mathematics (tropical geometry). He is best known for contributions to stochastic geometry. His results on the Poisson-Voronoi
François_Baccelli
German mathematician
a sequence of relevant papers that relate to mirror symmetry and tropical geometry. In 2014, jointly with Mark Gross, he became an invited speaker at
Bernd_Siebert
Software for the algorithmic treatment of convex polyhedra
shelling and Hasse diagrams. Tropical: functions for exploring tropical geometry; in particular, tropical hypersurfaces and tropical cones. polymake version
Polymake
American mathematician
tight spans. His 2004 paper, "Tropical Convexity", with Sturmfels, is regarded as one of the seminal papers of tropical geometry, garnering over 300 citations
Mike_Develin
Public university in Argentina
differential equations; Alicia Dickenstein, known for her work on toric geometry, tropical geometry, and their applications to biological systems; Miguel Walsh,
University_of_Buenos_Aires
Mathematical conjecture
In mathematics, specifically in the fields of model theory and complex geometry, Existential Closedness problems aim to determine when systems of equations
Existential closedness conjecture
Existential_closedness_conjecture
Irrational system of points and lines
In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization
Perles_configuration
E. Briggs B&W 11m July 3, 1945 video [703] The Play of Imagination in Geometry (ERPI); David Eugene Smith B&W 10m February 24, 1931 Play the Shopping
List of Encyclopædia Britannica Films titles
List_of_Encyclopædia_Britannica_Films_titles
Graduate-level textbooks in mathematics
Theory Marius Overholt 2014 978-1-4704-1706-2 161 Introduction to Tropical Geometry Diane Maclagan, Bernd Sturmfels 2015 978-0-8218-5198-2 162 A Course
Graduate Studies in Mathematics
Graduate_Studies_in_Mathematics
Pacific severe tropical storm in 2026
Severe Tropical Storm Jangmi, known in the Philippines as Typhoon Domeng, was a large and moderately strong tropical cyclone that enhanced the southwest
Tropical_Storm_Jangmi_(2026)
Sums of sets of vectors are nearly convex
The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. The lemma may be intuitively
Shapley–Folkman_lemma
Violations of the convexity assumptions of elementary economics
non-convex sets using advanced mathematics, particularly differential geometry and topology, Baire category, measure and integration theory, and ergodic
Non-convexity_(economics)
Forms in Nature c-12m 1977 Geometry and You David A. Smart (producer); Harold P. Fawcett bw-11m September 24, 1948 Video Geometry: Inductive and Deductive
List_of_Coronet_Films_films
Category 5 Atlantic hurricane in 2018
Hurricane Michael was a powerful and destructive tropical cyclone that became the first Category 5 hurricane to make landfall in the contiguous United
Hurricane_Michael
Mathematical theory of the geometry of space and time
frame. Objects move along geodesics—curved paths determined by the local geometry of spacetime—rather than being influenced directly by distant bodies. This
Curved_spacetime
Granite rock in southwest England
that the kaolinization resulted from intense supergene weathering in a tropical to warm climate during the Cretaceous to Cenozoic, based on studies of
Cornubian_batholith
2026 video game
fallback in the ray tracing bounding volume hierarchy (BVH). Full strand geometry, on the other hand, is included in the BVH for path tracing. Path tracing
Pragmata
Robert L. Devaney; German mathematician Heinz-Otto Peitgen and fractal geometry; strange attractors in dynamical systems, such as weather systems; David
List_of_Equinox_episodes
Pandya, Viraj; et al. (2023), "Galaxies Going Bananas: Inferring the 3D Geometry of High-Redshift Galaxies with JWST-CEERS", The Astrophysical Journal,
2024_in_science
internal tissues of the studied specimens, as well as of diffences in the geometry of skin melanosomes of extant and fossil anurans, is presented by Falk
2025_in_paleontology
Publishing Group. p. 113. ISBN 978-0-313-28803-6. Retrieved 2011-04-18. girl geometry 1829. Teresa A. Meade; Merry E. Wiesner-Hanks (eds.). A companion to gender
Timeline_of_women's_education
Thrust belt
The latter has a limited extent at its extremities, which influences the geometry of the deformation. Thus, a shortening of 7 to 9 km is estimated at the
Geology_of_the_Jura_Massif
from his textbooks. His algebra textbook was reprinted 42 times and his geometry textbook 24 times, most recently in 1980. JPL · 8477 8479 Held 1987 HD2
Meanings of minor-planet names: 8001–9000
Meanings_of_minor-planet_names:_8001–9000
Technology development overview
Satellite orbits are chosen to match a mission's coverage, altitude, viewing geometry, communications needs, lifetime, and propulsion limits. They range from
Outline of artificial satellites
Outline_of_artificial_satellites
Manufactured wood panel made from thin sheets of wood veneer
hardness, and damage- and wear-resistance. Tropical plywood is made of mixed hardwood species of tropical timber. Originally from the Asian region, it
Plywood
Spanish philosopher and essayist (1941–2021)
and Order (Espasa, 1999), when he discovered that Mandelbrot's fractal geometry was an alternative to Euclid's idealization, and Prigogine's dissipative
Antonio_Escohotado
Division of the year
2018. Khavrus, V.; Shelevytsky, I. (2010). "Introduction to solar motion geometry on the basis of a simple model". Physics Education. 45 (6): 641–653. Bibcode:2010PhyEd
Season
American treasure hunter
Colombia. The emeralds of Muzo are renowned for their color, fire, and geometry. Large as it was, this was only roughly half of the treasure that went
Mel_Fisher
Tropical pitcher plants
(/nɪˈpɛnθiːz/ nih-PEN-theez) is a genus of carnivorous plants, also known as tropical pitcher plants, or monkey cups, in the monotypic family Nepenthaceae. The
Nepenthes
introduce an algebra for atomic constraints. This algebra is similar to the tropical semiring, with two modifications: An arbitrary ordered monoid may be used
Difference_bound_matrix
Surgical procedure on the shoulder
fixed-fulcrum shoulder replacement in which he reversed the ball and socket geometry. His design resulted in several early failures, leading him to abandon
Reverse_shoulder_replacement
(chemical kinetics) (chemical reactions) (collision theory) (molecular geometry) (stereochemistry) Subadditivity effect (cognitive biases) Subject-expectancy
List_of_effects
Significant topic in economics
economics depends upon the following definitions and results from convex geometry. A real vector space of two dimensions may be given a Cartesian coordinate
Convexity_in_economics
East Asian ethnic group
Shiing-Shen Chern has been regarded as the "father of modern differential geometry" and has also been recognized as one of the greatest mathematicians of
Han_Chinese
Genus of viruses
Phala bat lyssavirus Lyssavirions are enveloped, with bullet shaped geometries. These virions are about 75 nm wide and 180 nm long. Lyssavirions have
Lyssavirus
Branch of mathematics
method of indivisibles, and Evangelista Torricelli extended such methods in geometry and mechanics. These works did not yet constitute mathematical analysis
Mathematical_analysis
L.; Cartolano, M.; Spinapolice, E. E.; Ferrara, S. (2026). "Earliest geometries: A cognitive investigation of Howiesons Poort engraved ostrich eggshells"
2026_in_paleomammalogy
geometry and its relation to hair coloration in extant therian mammals and Vilevolodon diplomylos, indicative of limited range of melanosome geometry
2025_in_paleomammalogy
Fluid flow revolving around an axis of rotation
rings, whirlpools in the wake of a boat, and in the winds surrounding a tropical cyclone, tornado, or dust devil. Vortices are a major component of turbulent
Vortex
Imaginary line halfway between Earth's North and South poles
for any other celestial body that is roughly spherical. In spatial (3D) geometry, as applied in astronomy, the equator of a rotating spheroid (such as a
Equator
a similar vein, live on. Stillwell, John (1994). Elements of algebra: geometry, numbers, equations. Springer. p. 42. Bunch, Bryan H. (1982). Mathematical
List of common misconceptions about science, technology, and mathematics
List_of_common_misconceptions_about_science,_technology,_and_mathematics
1111/j.1365-246X.2006.03236.x. Maharaj, R.J. (1994). "The morphology, geometry and kinematics of Judgement Cliff rock avalanche, Blue Mountains, Jamaica
List_of_landslides
Sedimentary body with topographic relief composed of autochthonous calcareous deposits
amount of nutrients than tropical factories. Carbonate platforms built by the "cool-water factory" show two types of geometry or depositional profile,
Carbonate_platform
Public park in Abuja, Nigeria
forest, rainforest and brushwood as well as greenhouses for butterflies and tropical birds. The other side, corresponding to the main entrance from road, is
Millennium_Park_(Abuja)
Country in East Asia
contexts. The spiritual concept of feng shui influences the layout and geometry of houses and settlements, seeking to orient constructions around the flow
China
Rotation of Earth around its axis
except for an instance not relevant to this value. In astronomy, unlike geometry, 360° means returning to the same point in some cyclical time scale, either
Earth's_rotation
Tennant, C. A.; La, D.; Torres-Chiriboga, F.; Cooke, S. B. (2024). "Incisor Geometry, Relief, and Diet in Anthropoid Primates With Implications for Antillothrix"
2024_in_paleomammalogy
Family of mammals
Functional Integration and Mechanical Performance as Revealed by Mandible Geometry". Systematic Biology. 62 (6): 878–900. doi:10.1093/sysbio/syt053. hdl:11590/132981
Felidae
TROPICAL GEOMETRY
TROPICAL GEOMETRY
Surname or Lastname
English
English : habitational name from a place in Wootton Fitzpaine, Dorset, Gupehegh in Middle English. This is named with the Old English personal name Guppa (a short form of Gūðbeorht ‘battle bright’) + (ge)hæg ‘enclosure’. The tropical fish denoted by this word was named in the 19th century in honor of R.J.L. Guppy, a clergyman in Trinidad who first presented specimens to the British Museum.The earliest known bearer of the name is Nicholas de Gupehegh (Somerset, 1253/4). Most if not all present-day bearers of the name are thought to descend from a certain William Guppy of Chardstock, Devon, who in 1497 was fined forty shillings for his alleged part in the rebellion of Perkin Warbeck.
Female
English
English name derived from the vocabulary word, MISTY means "dim, cloudy, vague." It is often chosen as a name because of the mental images it evokes, such as a misty autumn morning or a hazy tropical island.
Surname or Lastname
English
English : variant spelling of Rich.French : nickname for a rich man or perhaps an ironical name for a pauper, from Old French riche ‘rich’.
Surname or Lastname
English
English : nickname for an amiable person, also perhaps sometimes given in an ironical sense, from Middle English luvelich, loveli (Old English luflic). During the main period of surname formation the word was used in an active sense, ‘loving’, ‘kind’, ‘affectionate’, as well as the passive ‘lovable’, ‘worthy of love’. The meaning ‘attractive’, ‘beautiful’ is not clearly attested before the 14th century, and remained rare throughout the Middle Ages.New England Americanized form of French Lavallée (see Lavallee) or a similar name.
Surname or Lastname
English
English : variant of Dixon.Possibly a German topographic name from a reduced form (typical of the Lower Rhine) of Middle Low German dīk ‘dike’ + hūs ‘house’.
Boy/Male
Greek
Greek surname. Euclid was an early developer of geometry theories.
Surname or Lastname
English
English : topographic name for someone who lived on a patch of land where woodruff grew, Old English wudurofe (a compound of wudu ‘wood’ with a second element of unknown origin). The leaves of the plant have a sweet smell and the surname may also have been a nickname for one who used it as a perfume, or perhaps an ironical nickname for a malodorous person.Two English families brought the name Woodruff to the American colonies: those of Matthew Woodruff and of John and Ann Woodruffe. The latter migrated to Lynn, MA, from Kent, and moved to Southampton, Long Island, NY, before 1640. John and Ann’s many descendants were established in NJ, NC, and SC by 1790. The city of Woodruff, SC, is named for this family. The name is variously spelled Woodrove, Woodroffe, Woodruffe, Woodrough, and Woodruff in colonial records.
Surname or Lastname
Italian
Italian : from the personal name Gentile, a continuation of Late Latin Gentilis meaning ‘of the same stock (Latin gens)’ and then ‘non-Christian’, ‘pagan’; as a medieval name it was an omen name with the sense ‘noble’, ‘courteous’, also ‘delicate’, ‘charming’, ‘graceful’ (Italian gentile). In some cases the surname may have arisen from a nickname, sometimes possibly ironical, from the same word.English : variant of Gentle.
TROPICAL GEOMETRY
TROPICAL GEOMETRY
Boy/Male
Muslim
The protected one, The protector
Girl/Female
Muslim
Night
Girl/Female
Hindu, Indian, Traditional
Eye Lid
Surname or Lastname
English
English : variant spelling of Petit.
Girl/Female
English, Hindu, Indian, Kerala, Telugu
Cool; Winter; Adorable; Loyal to First Love
Boy/Male
Hindu, Indian
God is Beautiful
Boy/Male
Muslim
Gold flower
Female
English
English variant spelling of French Sophie, SOPHY means "wisdom."Â
Surname or Lastname
English and Scottish
English and Scottish : variant spelling of Steele.Americanization of German Stahl.
Female
English
Pet form of English Augusta, GUSSIE means "venerable."
TROPICAL GEOMETRY
TROPICAL GEOMETRY
TROPICAL GEOMETRY
TROPICAL GEOMETRY
TROPICAL GEOMETRY
n.
A white tropical tern (Cygis candida).
a.
Topical.
n.
Rhetorically changed from its exact original sense; being of the nature of a trope; figurative; metaphorical.
a.
Pertaining to irony; containing, expressing, or characterized by, irony; as, an ironical remark.
a.
Situated beyond, or outside of, the tropics; extratropical; also, having an excessively tropical temperature; warmer than the tropics.
n.
One of the two small circles of the celestial sphere, situated on each side of the equator, at a distance of 23¡ 28/, and parallel to it, which the sun just reaches at its greatest declination north or south, and from which it turns again toward the equator, the northern circle being called the Tropic of Cancer, and the southern the Tropic of Capricorn, from the names of the two signs at which they touch the ecliptic.
n.
Same as Troupial.
adv.
In a tropical manner; figuratively; metaphorically.
a.
Diseased with dropsy; hydropical; tending to dropsy; as, a dropsical patient.
n.
Of or pertaining to a place; limited; logical application; as, a topical remedy; a topical claim or privilege.
a.
Of or pertaining to the tropics; tropical.
a.
Combining or exhibiting the essential characteristics of a group; as, a typical genus.
n.
One of the two parallels of terrestrial latitude corresponding to the celestial tropics, and called by the same names.
a.
Ironical.
a.
Nearly tropical.
a.
tropical; figurative; as, a translative sense.
n.
Pertaining to, or consisting of, a topic or topics; according to topics.
n.
Of or pertaining to the tropics; characteristic of, or incident to, the tropics; being within the tropics; as, tropical climate; tropical latitudes; tropical heat; tropical diseases.
n.
A tropical fruit. See Sapodilla.
n.
Resembling a topic, or general maxim; hence, not demonstrative, but merely probable, as an argument.