Search references for LEVI GRAPH. Phrases containing LEVI GRAPH
See searches and references containing LEVI GRAPH!LEVI GRAPH
Graph representing incident points and lines
In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure. From a collection of points
Levi_graph
Abstract mathematical system of two types of objects and a relation between them
to a bipartite graph called the Levi graph or incidence graph of the structure. As any bipartite graph is two-colorable, the Levi graph can be given a
Incidence_structure
Geometry with 7 points and 7 lines
has order 168. As with any incidence structure, the Levi graph of the Fano plane is a bipartite graph, the vertices of one part representing the points
Fano_plane
Area of discrete mathematics
computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context
Graph_theory
Undirected graph with 14 vertices
mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood. The graph is cubic, and
Heawood_graph
Generalization of graph theory
"incidence graph" or "Levi graph" corresponding to every hypergraph, and conversely, every bipartite graph can be regarded as the incidence graph of a hypergraph
Hypergraph
3-regular graph with 30 vertices and 45 edges
unique smallest cubic graph of girth 8, it is a cage and a Moore graph. It is bipartite, and can be constructed as the Levi graph of the generalized quadrangle
Tutte–Coxeter_graph
Study of graphs defined by geometric means
2009) states that every planar graph can be represented as the intersection graph of line segments in the plane. A Levi graph of a family of points and lines
Geometric_graph_theory
Graphs formed by a hypercube's edges and vertices
graph Q 4 {\displaystyle Q_{4}} is the Levi graph of the Möbius configuration. It is also the knight's graph for a toroidal 4 × 4 {\displaystyle 4\times
Hypercube_graph
Graph divided into two independent sets
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets
Bipartite_graph
Symbolic representation of information using visualization techniques
collective term standing for the whole class of technical genres, including graphs, technical drawings and tables. specific kind of visual display : This is
Diagram
Matrix that shows the relationship between two classes of objects
matrix is also a biadjacency matrix of the Levi graph of the structure. As there is a hypergraph for every Levi graph, and vice versa, the incidence matrix
Incidence_matrix
per circle and three circles through each point. Its Levi graph is the rhombic dodecahedral graph, the skeleton of the rhombic dodecahedron. The configuration
Miquel_configuration
Geometry theorem
as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges
Pappus's_hexagon_theorem
Distance-transitive cubic graph with 20 nodes and 30 edges
points of one decagon to the corresponding points of the other. It is the Levi graph of the Desargues configuration. This configuration consists of ten points
Desargues_graph
Points and lines with equal incidences
most one point. That is, the girth of the corresponding bipartite graph (the Levi graph of the configuration) must be at least six. A configuration in the
Configuration_(geometry)
Catalan solid with 12 faces
dodecahedron is called a rhombic dodecahedral graph, with 14 vertices and 24 edges. It is the Levi graph of the Miquel configuration (83 64). For edge
Rhombic_dodecahedron
7-regular undirected graph with 50 nodes and 175 edges
of graph theory, the Hoffman–Singleton graph is a 7-regular undirected graph with 50 vertices and 175 edges. It is the unique strongly regular graph with
Hoffman–Singleton_graph
edge-transitive graph is either regular or biregular. The Levi graphs of geometric configurations are biregular; a biregular graph is the Levi graph of an (abstract)
Biregular_graph
Geometric structure of 8 points and 8 lines
projective configuration of type (8383). The Möbius–Kantor graph derives its name from being the Levi graph of the Möbius–Kantor configuration. It has one vertex
Möbius–Kantor_configuration
Bipartite, 3-regular undirected graph
field of graph theory, the Pappus graph is a bipartite, 3-regular, undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus
Pappus_graph
through it, and each line has exactly three points on it. The Gray graph is the Levi graph of this configuration; it has a vertex for every point and every
Gray_graph
Type of pseudorotation in chemistry
through successive pseudorotations to reach every possible arrangement. The Levi graph of trigonal bipyramidal isomers via Berry pseudorotations has D3d symmetry
Berry_mechanism
Undirected bipartite graph with 112 vertices and 168 edges
51, −47, −33, 19, 51, −21, 29, 21, −31, −39]2. The Ljubljana graph is the Levi graph of the Ljubljana configuration, a quadrangle-free configuration
Ljubljana_graph
Fewest graph edges whose removal breaks all cycles
of a hypergraph can be derived by its Levi graph, with the same cyclomatic number but reduced to a simple graph. It is r = g − ( v + e ) + c , {\displaystyle
Cyclomatic_number
Symmetric bipartite cubic graph with 16 vertices and 24 edges
the Möbius–Kantor configuration. The Möbius–Kantor graph derives its name from being the Levi graph of the Möbius–Kantor configuration. It has one vertex
Möbius–Kantor_graph
Type of incidence structure
Cage. Incidence graphs of configurations are today generally called Levi graphs, but the original Levi graph was the incidence graph of the GQ(2,2). If
Generalized_quadrangle
Geometric configuration of ten points and lines
The Levi graph of the Desargues configuration, a graph having one vertex for each point or line in the configuration, is known as the Desargues graph. Because
Desargues_configuration
Field of mathematics which studies incidence structures
corresponding incidence graph (Levi graph), namely the length of the shortest path between two vertices in this bipartite graph. The distance between two
Incidence_geometry
Unsolved problem in extremal graph theory
2 , 2 {\displaystyle K_{2,2}} may be obtained as the Levi graph, or point-line incidence graph, of a projective plane of order q {\displaystyle q} ,
Zarankiewicz_problem
Geometric system of two mutually inscribed tetrahedra
surfaces of three-dimensional space than the latter configuration. The Levi graph of the Möbius configuration has 16 vertices, one for each point or plane
Möbius_configuration
McKay–Miller–Širáň graphs by modifying the Levi graph of an affine plane over the field of order q {\displaystyle q} . The spectrum of a McKay–Miller–Širáň graph has
McKay–Miller–Širáň_graph
28 lines which touch a general quartic plane curve in two places
correspondence with the 28 triangles of the Fano plane. The Levi graph of the Fano plane is the Heawood graph, in which the triangles of the Fano plane are represented
Bitangents_of_a_quartic
is a generalized quadrangle with parameters (2,2). Its Levi graph is the Tutte–Coxeter graph. The points of the Cremona–Richmond configuration may be
Cremona–Richmond configuration
Cremona–Richmond_configuration
24-vertex symmetric bipartite cubic graph
In the mathematical field of graph theory, the Nauru graph is a symmetric, bipartite, cubic graph with 24 vertices and 36 edges. It was named by David
Nauru_graph
Undirected graph derived from a hypercube graph
of dimension five is the Clebsch graph. The folded cube graph of dimension six is the Kummer graph, i.e. the Levi graph of the Kummer point-plane configuration
Folded_cube_graph
Structure in combinatorial mathematics
configuration has a corresponding biregular bipartite graph known as its incidence or Levi graph. Given a finite set X (of elements called points) and
Block_design
Geometric configuration of 9 points and 9 lines
pairs of points. The Levi graph of the Pappus configuration is known as the Pappus graph. It is a bipartite symmetric cubic graph with 18 vertices and
Pappus_configuration
Geometric configuration of 9 points and 12 lines
geometries, vol. Second edition, Cambridge University, p. 41-42 On the Levi graph of point-line configurations, Jessica Hauschild, Jazmin Ortiz and Oscar
Hesse_configuration
Grünbaum. The Levi graph of the configuration is the Kronecker cover of the odd graph O4, and is isomorphic to the middle layer graph of the seven-dimensional
Danzer's_configuration
German mathematician
Mathematics Department at the University of Calcutta. He introduced the Levi graph in 1940 at a series of lectures on finite geometry. He contributed to
Friedrich_Wilhelm_Levi
Geometric configuration of 12 points and 6 lines
The Levi graph of the Reye configuration
Reye_configuration
Incidence matrix Incidence algebra Incidence structure Incidence geometry Levi graph Hilbert's axioms Joel G. Broida & S. Gill Williamson (1998) A Comprehensive
Incidence_(geometry)
Arrangement of 30 points and 12 lines
color. The Levi graph of the double six may be obtained by replacing each edge of the crown graph by a two-edge path. The intersection graph of the entire
Schläfli_double_six
determine that Alexander I Land is an island. Friedrich Wilhelm Levi introduces the Levi graph in a series of lectures on finite geometry at the University
1940_in_science
8, 2014. Levi Gundert (December 11, 2013). "Big Data in Security – Part III: Graph Analytics". Readwrite. Retrieved September 8, 2014. "Graph Database
InfiniteGraph
thus to tα = β. This approach results in a graph of substitutions generated by repeatedly applying Levi's lemma. If each unknown appears at most twice
Levi's_lemma
Number of cops needed to catch a robber on a graph
{\displaystyle n} -vertex graph has cop number O ( n ) {\displaystyle O({\sqrt {n}})} . The Levi graphs (or incidence graphs) of finite projective planes
Cop_number
City and county in California, US
companies—many in the technology sector—including Salesforce, Uber, Airbnb, Levi's, Gap, OpenAI, Chime, Anthropic, Databricks, Dropbox, and Lyft. In 2022,
San_Francisco
Binary operation in graph theory
In graph theory, the modular product of graphs G and H is a graph formed by combining G and H that has applications to subgraph isomorphism. It is one
Modular_product_of_graphs
Curve formed by a hanging chain
problem involving a hanging rope. Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary
Catenary
cryptographer Beppo Levi (1875–1961), mathematician Eugenio Levi (1883–1917), mathematician Friedrich Levi (1888–1966), algebra Leone Levi (1821–1888), statistician
List_of_Jewish_mathematicians
Theorem that every subgroup of a free group is itself free
and Friedrich Levi (1936). The original proof by Schreier forms the Schreier graph in a different way as a quotient of the Cayley graph of G modulo the
Nielsen–Schreier_theorem
Searching, pursuing, and killing wild animals
are hunting, even as population rises. National Public Radio reported, a graph shows 2016 statistics, that only about 5 per cent of Americans, 16 years
Hunting
City in Sicily, Italy
updated to 2006, except for those of Tiche and Ortigia, updated to 2016. Cf. graph in Ruggiero, Scrofani. Tourism and Urban Competitiveness. p. 215.. "Syracuse
Syracuse,_Sicily
Israeli communities built on land occupied in the 1967 Six-Day War
Machine: "Foundation for Middle East Peace -- Settler Population 1972-2005, Graph". Archived from the original on 1 December 2007. Retrieved 1 December 2007
Israeli_settlement
Washington, D.C. The San Jose Sharks hosted the 2015 NHL Stadium Series at Levi's Stadium in Santa Clara, California, against the Los Angeles Kings. Jobing
List of National Hockey League attendance figures
List_of_National_Hockey_League_attendance_figures
1980 video game
cover art designer at Atari to create the illustrations of the invaders on graph paper to use in the game. Unable to find a designer, Maurer created his
Space Invaders (Atari 2600 video game)
Space_Invaders_(Atari_2600_video_game)
Coordinates comprising a distance and an angle
then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r. Note that, in contrast to Cartesian coordinates
Polar_coordinate_system
U.S. presidential administration from 1974 to 1977
the Ford administration. Ford also appointed Edward H. Levi as Attorney General, charging Levi with cleaning up a Justice Department that had been politicized
Presidency_of_Gerald_Ford
South African-born mathematician
Lubotzky and R. Phillips) applied results in number theory to Ramanujan graphs, with connections to combinatorics and computer science. Sarnak has made
Peter_Sarnak
Diffusion process with a non-linear relationship to time
F. (1 April 1926). "Atmospheric Diffusion Shown on a Distance-Neighbour Graph". Proceedings of the Royal Society A: Mathematical, Physical and Engineering
Anomalous_diffusion
Dominican baseball player (born 1998)
2019 season by Baseball Prospectus (#47), ESPN's Keith Law (#62), and FanGraphs (#94). He returned to Down East to open the 2019 season, and hit .294/.368/
Leody_Taveras
German polymath (1646–1716)
employed integral calculus for the first time to find the area under the graph of a function y = f(x). He introduced several notations used to this day
Gottfried_Wilhelm_Leibniz
Social structure made up of a set of social actors
field which emerged from social psychology, sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing
Social_network
American singer (born 1941)
Frankie Valli Four Tops Renaldo Benson, Abdul "Duke" Fakir, Lawrence Payton, Levi Stubbs The Kinks Mick Avory, Dave Davies, Ray Davies, Pete Quaife The Platters
Art_Garfunkel
Origin and evolution of the symbols used to write equations and formulas
(1976), "A canonical representation of trivalent Hamiltonian graphs", Journal of Graph Theory, 1 (1): 45–60, doi:10.1002/jgt.3190010111 Fraleigh 2002:89;
History of mathematical notation
History_of_mathematical_notation
Branch of discrete mathematics
right. One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas
Combinatorics
American baseball player (born 1995)
Levi Jordan Hitting His Way Back to Iowa". northsidebound.com. June 2, 2022. Retrieved September 17, 2024. "Levi Jordan - Stats - Batting". FanGraphs
Levi_Jordan
studied by Levi (1955) and independently, Gohberg & Markus (1960). Additionally, there is a different Hadwiger conjecture concerning graph coloring, and
Hadwiger conjecture (combinatorial geometry)
Hadwiger_conjecture_(combinatorial_geometry)
This graph was using the legacy Graph extension, which is no longer supported. It needs to be converted to the new Chart extension.
2022 United States House of Representatives elections in Oregon
2022_United_States_House_of_Representatives_elections_in_Oregon
Straight path on a curved surface or a Riemannian manifold
mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold
Geodesic
Longenhagen, Eric (July 9, 2023). "Texas Rangers Top 43 Prospects". FanGraphs.com. Retrieved July 20, 2023. Parker, Mark (July 12, 2023). "PROVIDING A
Washington Nationals minor league players
Washington_Nationals_minor_league_players
Knowledge Graph which when clicked, makes confetti explode. "panipuri( see it )" will show three types of panipuris in the Knowledge Graph, which when
List_of_Google_Easter_eggs
American musical duo
and Garfunkel assumed the name Tom & Jerry; Garfunkel named himself Tom Graph, a reference to his interest in mathematics, and Simon Jerry Landis, after
Simon_&_Garfunkel
Five-pointed star polygon
essentially "good". However, the influential but controversial writer Éliphas Lévi, known for believing that magic was a real science, had called it evil whenever
Pentagram
Canadian local elections
in campaign items (signs, literature, etc.) or colours used in polling graphs and are used as a visual differentiation between candidates. Sources: Office
2021 Quebec municipal elections
2021_Quebec_municipal_elections
Japanese singer, dancer, model, and actress (born 2005)
Japanese). 講談社. January 23, 2024. p. 143. ASIN B0CQMFRQYM. blt graph. vol.77 [blt graph. vol.77] (in Japanese). 東京ニュース通信社. April 4, 2022. p. 21. ISBN 9784867014042
Ten_Yamasaki
Israeli mathematician and computer scientist
won the Levi L. Conant Prize of the American Mathematical Society for best mathematical exposition for this article, a survey on expander graphs. "2008
Nati_Linial
Swiss mathematician (1707–1783)
logician, geographer, music theorist and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches
Leonhard_Euler
Four-dimensional number system
Charles F.F. (January 2007). "Quaternions in molecular modeling". J. Mol. Graph. Mod. 25 (5): 595–604. arXiv:physics/0506177. Bibcode:2007JMGM...25..595K
Quaternion
Array of numbers
The adjacency matrix of a finite graph is a basic notion of graph theory. It records which vertices of the graph are connected by an edge. Matrices
Matrix_(mathematics)
Neo-avant-garde literary and art movement
by introducing non-verbal symbols into the text (drawings, photographs, graphs.) for the purpose of creating a collage of verbal and visual elements. This
Signalism
Heuristic pathfinding algorithm
Iterative deepening A* (IDA*) is a graph traversal and path search algorithm that can find the shortest path between a designated start node and any member
Iterative_deepening_A*
8th-century Abbasid provincial governor
revolution. Jerusalem: Graph Press Ltd. ISBN 965-223-388-9. Zetterstéen, K.V. (1960). "Abū ʿAwn". In Gibb, H. A. R.; Kramers, J. H.; Lévi-Provençal, E.; Schacht
Abu Awn Abd al-Malik ibn Yazid
Abu_Awn_Abd_al-Malik_ibn_Yazid
Mathematician at Courant Institute of Mathematical Sciences
Mathematical Sciences at New York University working in combinatorics and graph theory. She and Huy Tuan Pham proved the Kahn–Kalai conjecture on estimating
Jinyoung_Park_(mathematician)
Two-dimensional geometrical space
Euclidean metric is used instead of the Euclidean plane. For example, in a graph, which can be drawn on paper, and in which the position of a particle is
Affine_plane
Property of a mathematical space
0 dimension Point Zero-dimensional space Integer 1 dimension Line Curve Graph (combinatorics) Real number Length 2 dimensions Plane Surface Polygon Net
Dimension
1993 Japanese book by Wataru Tsurumi
Drowning Self-immolation Freezing Miscellaneous Each chapter begins with a graph assessing the method in question in terms of painfulness, time/effort required
The Complete Manual of Suicide
The_Complete_Manual_of_Suicide
Branch of mathematics
astronomers calculated Jupiter's position from the area under a time-velocity graph". Science. 351 (6272): 482–484. Bibcode:2016Sci...351..482O. doi:10.1126/science
Geometry
varieties. Algebraic graph theory a branch of graph theory in which methods are taken from algebra and employed to problems about graphs. The methods are
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Symbols for constants, special functions
investment the α-conversion in lambda calculus the independence number of a graph a placeholder for ordinal numbers in mathematical logic a type of receptor
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
German mathematician (1871–1928)
thesis also contains the proof of Kőnig's theorem for regular bipartite graphs, phrased in the language of configurations. In 1910 Steinitz published the
Ernst_Steinitz
Smooth manifold with an inner product on each tangent space
Cartan connection, one of the first concepts of a connection. Levi-Civita defined the Levi-Civita connection, a special connection on a Riemannian manifold
Riemannian_manifold
Inversion (discrete mathematics) Major index Ménage problem Permutation graph Permutation pattern Permutation polynomial Permutohedron Rencontres numbers
List_of_permutation_topics
Mathematics prize
The Levi L. Conant Prize is a mathematics prize of the American Mathematical Society, which has been awarded every year since 2001 for outstanding expository
Levi_L._Conant_Prize
goat-headed Baphomet image seen here is a 19th-century drawing made by Eliphas Levi as a metaphorical symbol from Dogme et Rituel de la Haute Magie. It was not
List_of_occult_symbols
Numerological practice of reading a word or phrase as a number
ha-Malchut" by Rabbi David ha-Levi of the Draa Valley, a Spanish-Moroccan Kabbalist of the 15th–16th century. Rabbi David ha-Levi's methods also consider the
Gematria
Function of a matrix
hafnian counts perfect matchings in a graph from its adjacency matrix, the permanent counts matchings in a bipartite graph from its biadjacency matrix. The
Hafnian
(combinatorics) Graph structure theorem (graph theory) Grinberg's theorem (graph theory) Grötzsch's theorem (graph theory) Hajnal–Szemerédi theorem (graph theory)
List_of_theorems
LEVI GRAPH
LEVI GRAPH
Boy/Male
Biblical American Hebrew
Associated with him.
Boy/Male
American, Australian, British, Chinese, Christian, Danish, English, Finnish, French, German, Hawaiian, Hebrew, Jamaican, Portuguese
United; Combined; Attached; Possibly Governor; Guide; Joined; Governor; Adhesion; Joined to; Crown; Garland
Male
Russian
 Yiddish name LEV means "lion." In use by the Russians. Compare with other forms of Lev.
Biblical
associated with himjoined, to adhere,adhesion
Female
Hindi/Indian
(देवी) Hindi name derived from Sanskrit devi DEVI means "goddess."
Male
Hebrew
(לֵב) Hebrew name LEV means "heart." Compare with other forms of Lev.
Girl/Female
English, Swedish
Life
Male
Hungarian
 Short form of Hungarian Levente, possibly LEVI means "governor, guide." Compare with other forms of Levi.
Surname or Lastname
Jewish (Ashkenazic)
Jewish (Ashkenazic) : from German Levit ‘Levite’ (see Levy).English : variant spelling of Leavitt.
Female
German
Pet form of German Helene, possibly LENI means "torch."
Male
Greek
(Λευί) Greek form of Hebrew Leviy, LEUI means "adhesion, joined to" or "crown, garland." In the bible, this is the name of several characters, including the third son of Jacob.
Male
Finnish
Finnish form of Greek Leui, LEEVI means "adhesion, joined to" or "crown, garland."
Male
Yiddish
(לֵב) Yiddish name LEV means "lion." In use by the Russians. Compare with other forms of Lev.
Girl/Female
Hindu
Lakshmi Devi
Male
Hebrew
(לֵוִי) Hebrew name derived from the word lewi, LEVIY means "adhesion, joined to" or "crown, garland." In the bible, this is the name of several characters, including the third son of Jacob.
Female
English
Variant spelling of English Lexy, LEXI means "defender of mankind."
Female
Hebrew
(לָבִי×) Hebrew unisex name LAVI means "lion."
Female
Finnish
Finnish form of Greek Eva, EEVI means "life."
Male
English
 English short form of English Levi, LEV means "adhesion, joined to" or "crown, garland." Compare with other forms of Lev.
Male
Hebrew
Variant spelling of Hebrew Tzevi, ZEVI means "deer, gazelle."
LEVI GRAPH
LEVI GRAPH
Boy/Male
Indian, Sanskrit
Tamable
Girl/Female
Assamese, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu, Traditional
Goddess of Victory
Boy/Male
Tamil
Maheshwaram | மஹேஷà¯à®µà®¾à®°à®¾à®®
Lord of the universe
Boy/Male
Muslim
Similar. Comparable.
Boy/Male
British, English
Town by a Clay Bed
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Radiant
Girl/Female
Hindu
Coming from Utkal
Boy/Male
Indian, Punjabi, Sikh
Life Imbued with the Love of God
Girl/Female
English American
Path; roadway.Lane and Laine.
Boy/Male
American, British, English
From the Barley Ford
LEVI GRAPH
LEVI GRAPH
LEVI GRAPH
LEVI GRAPH
LEVI GRAPH
n.
A name formerly given in Pennsylvania, Maryland, and Virginia to the Spanish real of one eighth of a dollar (or 12/ cents), valued at eleven pence when the dollar was rated at 7s. 6d.
pl.
of Levy
v. t.
To raise; to collect; said of troops, to form into an army by enrollment, conscription, etc.
v. t.
To gather or exact; as, to levy money.
v. t.
To take or seize on execution; to collect by execution.
v. t.
To raise, as a siege.
v. t.
To erect, build, or set up; to make or construct; to raise or cast up; as, to levy a mill, dike, ditch, a nuisance, etc.
v. i.
To seize property, real or personal, or subject it to the operation of an execution; to make a levy; as, to levy on property; the usual mode of levying, in England, is by seizing the goods.
n.
The taking or seizure of property on executions to satisfy judgments, or on warrants for the collection of taxes; a collecting by execution.
imp. & p. p.
of Levy
v. t.
To raise or collect by assessment; to exact by authority; as, to levy taxes, toll, tribute, or contributions.
v. i.
To levy a distress.
v. t.
To raise; to levy.
p. pr. & vb. n.
of Levy
n.
That which is levied, as an army, force, tribute, etc.
n.
The act of levying or collecting by authority; as, the levy of troops, taxes, etc.