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In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered...

the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan–Hadamard manifolds. The conjecture...

omitted). The number 0 is the smallest nonnegative integer, and the largest nonpositive integer. The natural number following 0 is 1 and no natural number precedes...

1080/0025570X.1998.11996652. Ballmann, Werner (1995). Lectures on spaces of nonpositive curvature. DMV Seminar 25. Basel: Birkhäuser Verlag. pp. viii+112. ISBN 3-7643-5242-6...

nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions...

is an integral of a function involving the logarithm, and so on. For nonpositive integer orders s, the polylogarithm is a rational function. In the case...

metrics exist on any closed Kähler manifold whose first Chern class is nonpositive. Yau's method adapted earlier work of Calabi, Jürgen Moser, and Aleksei...

derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for F j {\displaystyle F_{j}} appears in...

derivatives of all orders of f {\displaystyle f} are nonnegative or all nonpositive at all points on the interval. All strictly monotonic functions are invertible...

and their automorphisms, groups acting on trees, various notions of nonpositive curvature for groups (CAT(0) groups, Dehn functions, automaticity...)...

Square matrix whose off-diagonal entries are nonpositive...

defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities: ε1x1 ≥ 0     ...

theorem states that a complete simply connected Riemannian manifold M with nonpositive sectional curvature is diffeomorphic to the Euclidean space Rn with n...

Joannes Stieltjes, is a real symmetric positive definite matrix with nonpositive off-diagonal entries. A Stieltjes matrix is necessarily an M-matrix....

show that any smooth map from a closed manifold to a closed manifold of nonpositive curvature can be deformed to a harmonic map. In 1975, Hamilton considered...

with Brad Rodgers, Tao showed that the de Bruijn–Newman constant, the nonpositivity of which is equivalent to the Riemann hypothesis, is nonnegative. In...

Faber–Krahn inequality should also hold in all simply connected manifolds of nonpositive curvature. Hearing the shape of a drum Benguria, Rafael D. (2001) [1994]...

{\displaystyle \mathbb {C} \backslash \mathbb {Z} _{\leq 0}} . At all the nonpositive integers these polygamma functions have a pole of order m + 1. The function...

(real) critical points. In the two latter cases, that is, if b2 − 3ac is nonpositive, the cubic function is strictly monotonic. See the figure for an example...

{1}{3^{3}}}+\cdots =1.202056903159594285399...} is Apéry's constant. For nonpositive integers, the series does not converge, but via analytic continuation...

function that extends the usual nth root to the complex plane without the nonpositive real numbers. If the complex number is moved around zero by increasing...

three-digit number abc, where a, b, and c are the digits, each nonpositive if abc is nonpositive; we have (abc) mod 7 = 9*a + 3*b + c. Repeat the formula down...

point of u, only says that a weighted average of manifestly nonpositive quantities is nonpositive. This is trivially true, and so one cannot draw any nontrivial...

i {\displaystyle x_{i}} , the function f i {\displaystyle f_{i}} is nonpositive when x i = − 1 {\displaystyle x_{i}=-1} and nonnegative when x i = 1...

Alexandrov spaces), specifically polyhedral spaces of nonnegative and nonpositive curvature. Nonnegative curvature on singularities of codimension 2 implies...

Hadamard manifolds (simply connected complete Riemannian manifolds of nonpositive curvature). They are named after Herbert Busemann, who introduced them;...

An example is a typical aquarium air pump. "Figure 4-11.—Mechanical nonpositive fuel pump". www.tpub.com. Archived from the original on 2004-01-15. Retrieved...

Ruth Charney, he established the "hyperbolization" method for using nonpositive curvature to construct aspherical manifolds. His contributions also include...

Glaser, that energy densities in local quantum field theories are always nonpositive. A large amount of Jaffe's work deals with the mathematical construction...

negative real part. It is Lyapunov stable if all Floquet exponents have nonpositive real part and, in addition, the Floquet multipliers of modulus 1 are...

functions are closely related to notions of sectional curvature and nonpositive curvature. Important examples of geodesic metric spaces include complete...

system that removes flue gases entirely by buoyancy (stack effect) under nonpositive static pressure, without mechanical fans. Natural-draft venting is typical...

isoperimetric inequality for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as Cartan–Hadamard manifolds? Chern's conjecture (affine...

nonnegative on 0 ≤ y < π 2 , {\textstyle 0\leq y<{\frac {\pi }{2}},} but nonpositive on π 2 < y ≤ π . {\textstyle {\frac {\pi }{2}}<y\leq \pi .} For a similar...

quadratic form must therefore be nonpositive. (The fact that the real part of x ∗ A x , {\displaystyle x^{*}Ax,} must be nonpositive implies that the real parts...

Riemannian manifolds, and suppose that the sectional curvature of (N, h) is nonpositive. Then for any continuously differentiable map f from M to N, the maximal...

"Turán Memorial Lectures". Heintze, Ernst (1987). "Review: Manifolds of nonpositive curvature, by W. Ballmann, M. Gromov & V. Schroeder". Bull. Amer. Math...

bisectional curvature decreases on submanifolds, X has a Kähler metric of nonpositive holomorphic bisectional curvature. The reciprocal is known to be false:...

91 Stat. 1265 (Nov. 8, 1977), and is codified at 48 U.S.C. § 1821, a nonpositive law title. As of April 16, 2024[update]: List of current United States...

differential geometry and geometric group theory. If the manifold is nonpositively curved then its volume entropy coincides with the topological entropy...

be used to prove identities for the multiset coefficients. If α is a nonpositive integer n, then all terms with k > −n are zero, and the infinite series...

Riemannian manifolds of nonpositive sectional curvature. Important examples of Hadamard manifolds are simply connected nonpositively curved symmetric spaces...

{\displaystyle \mu } if μ {\displaystyle \mu } is nonnegative (resp. nonpositive) everywhere on A . {\displaystyle A.} Of course, if μ {\displaystyle...

Bergman metric of X {\displaystyle X} is a complete Kähler metric with nonpositive sectional curvature. Every K3 surface is Kähler (by Siu). A smooth real-valued...

213–247. doi:10.2307/2007062. JSTOR 2007062. MR 0672836. —— (1985). "Nonpositively Curved Manifolds of Higher Rank". The Annals of Mathematics. 122 (3):...

components with respect to this basis that are either all nonnegative or all nonpositive. Given the E8 Cartan matrix (above) and a Dynkin diagram node ordering...

sometimes used to refer to a number that is either positive or zero, while nonpositive is used to refer to a number that is either negative or zero. Zero is...

This may be confusing, as sometimes nonnegative matrices (respectively, nonpositive matrices) are also denoted in this way. It follows from the above definitions...

Sturm–Liouville theory. Cartan–Hadamard theorem Berger 2004; Jost, Jürgen (1997), Nonpositive curvature: geometric and analytic aspects, Lectures in Mathematics, ETH...

^{n})} . Hadamard manifolds are complete simply connected manifolds with nonpositive curvature. Thus they generalize the Euclidean space R n {\displaystyle...

curvature implies there are no nontrivial (nonzero) Killing fields; nonpositive Ricci curvature implies that any Killing field is parallel. i.e. covariant...

some users of bubble charts resort to graphic symbology to express nonpositive data values. As an example, a negative value v < 0 {\displaystyle v<0}...

Z-matrix (mathematics) – Square matrix whose off-diagonal entries are nonpositive M-matrix – Matrix in mathematics P-matrix – Complex square matrix for...

equivalent to G containing a directed spanning tree, a(G) can still be nonpositive as the exploding star and Theorem 1 indicate. Fiedler, Miroslav (1973)...

Chern and Kuiper even says that any closed m-dimensional manifold of nonpositive sectional curvature cannot be smoothly isometrically immersed in R2m–1...

theory. Margulis lemma Ballmann, Werner (1995). Lectures on spaces of nonpositive curvature. DMV Seminar 25. Basel: Birkhäuser Verlag. pp. viii+112. ISBN 3-7643-5242-6...

(q+n-1)&n>0\end{cases}}} The series terminates if either a or b is a nonpositive integer, in which case the function reduces to a polynomial: 2 F 1 (...

Lawson, H. Blaine Jr.; Yau, Shing Tung (1972). "Compact manifolds of nonpositive curvature". Journal of Differential Geometry. 7 (1–2): 211–228. doi:10...

Calabi conjecture, had settled the case of closed Kähler manifolds with nonpositive first Chern class. His work in applying the method of continuity showed...

diagonally dominant. L is an M-matrix (its off-diagonal entries are nonpositive, yet the real parts of its eigenvalues are nonnegative). Every row sum...

positive y. Only positive yL are permitted in the formula, with any nonpositive terms being ignored (and yR are always positive). This formula involves...

b s {\displaystyle b=b_{s}} , where b s {\displaystyle b_{s}} is a (nonpositive) constant. For b s = 0 {\displaystyle b_{s}=0} the highest waves occur...

(v){\pmod {p^{b}}},} where u = 1 − m and v = 1 − n, so that u and v are nonpositive and not congruent to 1 modulo p − 1. This tells us that the Riemann zeta...

distance convexity. A book by Papadopoulos studies metric spaces of nonpositive curvature in the sense of Busemann, that is, metric spaces whose distance...

that any complete and noncompact smooth Einstein manifold must have nonpositive Einstein constant. Since M {\displaystyle M} is connected, there exists...

is a linear combination of elements of Δ {\displaystyle \Delta } with nonpositive coefficients. 6.  long root 7.  short root 8.  inverse of a root system:...

Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor (1985). Manifolds of nonpositive curvature. Progress in Mathematics. Vol. 61. Birkhäuser Boston, Inc....

Attribution/Share-Alike License. Ballmann, Werner; Brin, Michael (1995), "Orbihedra of nonpositive curvature", Publications Mathématiques de l'Institut des Hautes Études...

S3 is aspherical, by the sphere theorem Complete metric spaces with nonpositive curvature in the sense of Aleksandr D. Aleksandrov (locally CAT(0) spaces)...

generalization of the classical isoperimetric inequality to spaces of nonpositive sectional curvature Cauchy–Hadamard theorem, a statement in complex analysis...

group theory, geometric representation theory in Lie groups, spaces of nonpositive curvature [de], and configuration spaces of arrangements and mechanical...

Retrieved 2024-11-02. Brady, Thomas (1995-05-26). "Complexes of nonpositive curvature for extensions of F2 by Z". Topology and Its Applications....

independently by Blaine Lawson and Shing-Tung Yau. This establishes that, under nonpositivity of the sectional curvature, abelian subgroups of the fundamental group...

Ballmann, Werner; Gromov, Mikhail; Schroeder, Viktor (1985). Manifolds of Nonpositive Curvature. Birkhâuser. Raghunathan, M. S. (1972). Discrete subgroups...

applications to construction of Kac–Moody groups in algebra, and to nonpositively curved manifolds and hyperbolic groups in topology and geometric group...

standard among authors, for example some sources force non-pivots to be nonpositive or place no sign restriction on them. However, these definitions are...

by implementations. In the BOUNDS section, if a variable is given a nonpositive upper bound but no lower bound, its lower bound may default to zero or...

∈ t* such that λ-ρ does not pair with any positive root α to give a nonpositive integer (it is "regular dominant"): ( λ − ρ , α )   ∈   C − Z ≤ 0 . {\displaystyle...

curvature. As a corollary, a closed manifold with a Riemannian metric of nonpositive curvature, such as a torus, has no metric with positive scalar curvature...

(but not identically zero) sectional curvature. Non-compact type: M has nonpositive (but not identically zero) sectional curvature. A more refined invariant...

metrics on the projective plane, every function except for those which are nonpositive arises as a scalar curvature. Moser's prior analysis of the Moser–Trudinger...

Assume that all the off-diagonal entries of a real symmetric matrix A are nonpositive. Then A is copositive if and only if it is positive semidefinite. The...

function would be convex in income, and the Atkinson index would be nonpositive. Shorrocks, AF (1980). The class of additively decomposable inequality...

space of Kähler metrics with the structure of a Alexandrov space of nonpositive curvature. The technical difficulty of their work is that geodesics in...

list (link) Papadopoulos, Athanase (2005). Metric Spaces, Convexity and Nonpositive Curvature. European Mathematical Society. p. 77. ISBN 9783037190104....

∂ f ∂ y {\displaystyle y{\frac {\partial f}{\partial y}}} is always nonpositive. He also studied the existence of periodic solutions to the third-order...

where G ( t ) {\displaystyle G(t)} is a q × q {\displaystyle q\times q} nonpositive definite matrix which is a unique solution of the matrix-valued Riccati...

Leary, I. J.; Minasyan, Ashot (2021). "Commensurating HNN extensions: nonpositive curvature and biautomaticity". Geom. Topol. 25: 1819–1860. arXiv:1907...

1007/978-1-4684-9167-8_10. Ballmann, Werner (1995), Lectures on spaces of nonpositive curvature, DMV Seminar 25, Basel: Birkhäuser Verlag, pp. viii+112, ISBN 3-7643-5242-6...

Synonym for Hurwitz matrix. Stieltjes matrix A real symmetric positive definite matrix with nonpositive off-diagonal entries. Special case of an M-matrix....

{Ric} (X,X).} In the case of a Riemannian metric, the left-hand side is nonpositive at any local maximum of the length of X. However, on a Riemannian metric...

These are central examples in Riemannian geometry of manifolds with nonpositive sectional curvature. For example, SL(2,R)/SO(2) is the hyperbolic plane...

geodesics, and curvature; the sectional curvature of Mabuchi's metric is nonpositive. Along geodesics in the Kähler class, the Mabuchi energy is convex. So...

2 2 = − P 2 2 {\displaystyle mE-{P^{2} \over 2}={-P^{2} \over 2}} is nonpositive. Suppose it is zero. Here, it is also the boosts as well as the rotations...

doi:10.1142/S0218196705002669, S2CID 2786607. Davis, M. W. (2002), "Nonpositive curvature and reflection groups", in Daverman, R. J.; Sher, R. B. (eds...

"Maximum entropy analytic continuation for spectral functions with nonpositive spectral weight". Physical Review B. 92 (6) 060509. arXiv:1507.01956...

L, Lovell MA, Wei Q, Jamieson BA, Williams SA, et al. (June 2004). "Nonpositive terminal deoxynucleotidyl transferase in pediatric precursor B-lymphoblastic...

)^{2}\leq \gamma _{2}\,\mathrm {area} } is satisfied by all surfaces of nonpositive Euler characteristic is unknown. For orientable surfaces of genus 2 and...

Academy of Arts and Sciences. H. Epstein; V. Glaser; A. Jaffe (1965). "Nonpositivity of the Energy Density in Quantized Field Theories". Nuovo Cimento. 36...

constant-speed biharmonic curve in a three-dimensional space form of nonpositive curvature must be geodesic. Any constant-speed biharmonic curves in the...

Taylor, L.R.; Williams, E.B. (July 1995). "Rigidity of fibrations over nonpositively curved manifolds". Topology. 34 (3): 565–574. doi:10.1016/0040-9383(94)00035-J...

University Press. ISBN 978-0-521-05712-7. OCLC 39658898. Jost, Jürgen (1997). Nonpositive curvature : geometric and analytic aspects. Basel Boston: Birkhäuser...

minimizing geodesic. Hadamard space is a complete simply connected space with nonpositive curvature. Hausdorff dimension Hausdorff distance Hausdorff measure Hilbert...

empty. Scores are read from left to right and are processed as follows. Nonpositive scores require no special processing, so the next score is read. A positive...

Margulis, Grigori (1975). "Discrete groups of motions of manifolds of nonpositive curvature". Proceedings of the International Congress of Mathematicians...

thus all eigenvalues of ( A / 2 − B / 2 ) {\textstyle (A/2-B/2)} are nonpositive, thus A ⪯ B {\textstyle A\preceq B} . The inequality has been generalized...

MR 2132405. Papadopoulos, Athanase (2005). Metric Spaces, Convexity and Nonpositive Curvature. IRMA lectures in mathematics and theoretical physics. Vol...

if a compact group acts by isometries on a complete simply connected nonpositively curved Riemannian manifold then it has a fixed point. Maximal compact...

must be positive: if it were not, then every term in the sum would be nonpositive, and therefore so would the sum itself. Thus, 0 < k + 1 1 + 1 − m + (...

(\mathbf {a_{p}} )=\prod _{j=1}^{p}\Gamma (a_{j}).} This holds unless a nonpositive integer value of at least one of its parameters ap reduces the hypergeometric...

verified the conjecture for various cases, most notably flat manifolds, nonpositively curved manifolds. In his thesis, Farrell solved the problem of determining...

and reciprocity laws on algebraic varieties Bruce Kleiner, Spaces of nonpositive curvature Grigory Margulis, Quantitative Oppenheim Conjecture Sergei...

Exponential Laplace Approximations to Expectations and Variances of Nonpositive Functions". Journal of the American Statistical Association. 84 (407):...

which is a positive weak solution on the complement of U and which is nonpositive on U; moreover such a function is uniquely determined on the complement...