1,691 results
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2. A weighted uniform $L^{p}$--estimate of Bessel functions: A note on a paper of Guo
- Author
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Krzysztof Stempak
- Subjects
symbols.namesake ,Cylindrical harmonics ,Bessel process ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Struve function ,Bessel polynomials ,symbols ,Calculus ,Bessel function ,Lommel function ,Mathematics - Published
- 2000
3. Remarks on DiPerna’s paper 'Convergence of the viscosity method for isentropic gas dynamics'
- Author
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Gui-Qiang Chen
- Subjects
Discrete mathematics ,Isentropic process ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Vacuum state ,Finite difference method ,Euler equations ,Binary entropy function ,symbols.namesake ,Riemann hypothesis ,Compact space ,Mathematics Subject Classification ,symbols ,Mathematics - Abstract
Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.
- Published
- 1997
4. An operator valued function space integral: A sequel to Cameron and Storvick’s paper
- Author
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D. L. Skoug and G. W. Johnson
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Multiple integral ,Integral representation theorem for classical Wiener space ,Mathematical analysis ,Riemann integral ,Riemann–Stieltjes integral ,Singular integral ,Fourier integral operator ,Volume integral ,symbols.namesake ,symbols ,Daniell integral ,Mathematics - Abstract
Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.
- Published
- 1971
5. Book Review: The lost notebook and other unpublished papers
- Author
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Richard Askey
- Subjects
symbols.namesake ,Applied Mathematics ,General Mathematics ,symbols ,Ramanujan's sum ,Mathematics - Published
- 1988
6. Two papers on similarity of certain Volterra integral operators
- Author
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Stanley Osher
- Subjects
Algebra ,symbols.namesake ,Similarity (network science) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,symbols ,Microlocal analysis ,Volterra integral equation ,Fourier integral operator ,Mathematics - Published
- 1967
7. Observations on a paper by Rosenblum
- Author
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S. Cater
- Subjects
Complex conjugate ,Applied Mathematics ,General Mathematics ,Hilbert space ,Uniform limit theorem ,Combinatorics ,symbols.namesake ,Operator (computer programming) ,Skew-Hermitian matrix ,Bounded function ,symbols ,Normal operator ,Complex number ,Mathematics - Abstract
M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.
- Published
- 1961
8. A remark on Neuwirth and Newman’s paper: 'Positive 𝐻^{1/2} functions are constants'
- Author
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Shinji Yamashita
- Subjects
Combinatorics ,Lemma (mathematics) ,symbols.namesake ,Applied Mathematics ,General Mathematics ,Blaschke product ,symbols ,Function (mathematics) ,Absolute value (algebra) ,Boundary values ,Decomposition theorem ,Mathematical physics ,Mathematics - Abstract
PROOF. By a theorem of Rudin a function gEH' in U whose boundary values are real a.e. on I can be analytically continued to D [3, p. 59]. The lemma follows on applying Rudin's result to gi= (1/2) (fl+f2) and g2=(i/2) (fi-f2). PROOF OF THEOREM 1. By a well-known decomposition theorem [2, p. 87], f(z)=B(z)F2(Z), where B(z) is a Blaschke product and F(z) EH1. Since the boundary values of B (z) have absolute value one a.e. on K, we have a.e. on I, f(ei0)= |f(eio)I, or B(ei0)F2(ei0) = F2(ei0) |, and hence
- Published
- 1969
9. Integers represented as the sum of one prime, two squares of primes and powers of 2
- Author
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Haiwei Sun and Guangshi Lü
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Short paper ,MathematicsofComputing_GENERAL ,Prime number ,Prime (order theory) ,Algebra ,symbols.namesake ,Integer ,symbols ,Idoneal number ,Prime power ,Sphenic number ,Mathematics - Abstract
In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .
- Published
- 2008
10. Higher order Turán inequalities for the Riemann $\xi$-function
- Author
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Dimitar K. Dimitrov, Fábio Rodrigues Lucas, Universidade Estadual Paulista (Unesp), and Universidade Estadual de Campinas (UNICAMP)
- Subjects
Degree (graph theory) ,Applied Mathematics ,General Mathematics ,Entire function ,Mathematical analysis ,Short paper ,Function (mathematics) ,Maclaurin coefficients ,Riemann ξ function ,Combinatorics ,Riemann hypothesis ,symbols.namesake ,Jensen polynomials ,symbols ,Order (group theory) ,Shape function ,Laguerre-Pólya class ,Turán inequalities ,Mathematics - Abstract
Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160
- Published
- 2011
11. The Lane-Emden equation with variable double-phase and multiple regime
- Author
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Vicenţiu D. Rădulescu and Claudianor O. Alves
- Subjects
Variable exponent ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Mathematical proof ,Supercritical fluid ,symbols.namesake ,Mathematics - Analysis of PDEs ,Criticality ,Feature (computer vision) ,Dirichlet boundary condition ,FOS: Mathematics ,symbols ,Lane–Emden equation ,Analysis of PDEs (math.AP) ,Variable (mathematics) ,Mathematics - Abstract
We are concerned with the study of the Lane-Emden equation with variable exponent and Dirichlet boundary condition. The feature of this paper is that the analysis that we develop does not assume any subcritical hypotheses and the reaction can fulfill a mixed regime (subcritical, critical and supercritical). We consider the radial and the nonradial cases, as well as a singular setting. The proofs combine variational and analytic methods with a version of the Palais principle of symmetric criticality., The final version this paper will be published in Proc. AMS
- Published
- 2020
12. Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials
- Author
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Daniel Erman, Steven V Sam, and Andrew Snowden
- Subjects
Pure mathematics ,General Mathematics ,media_common.quotation_subject ,MathematicsofComputing_GENERAL ,Hilbert's basis theorem ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematics - Algebraic Geometry ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,Ideal (ring theory) ,0101 mathematics ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics ,media_common ,Conjecture ,Hilbert's syzygy theorem ,Mathematics::Commutative Algebra ,Degree (graph theory) ,Applied Mathematics ,010102 general mathematics ,13A02, 13D02 ,Mathematics - Commutative Algebra ,Infinity ,Bounded function ,symbols ,010307 mathematical physics - Abstract
Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert's results are not uniform in n: unsurprisingly, polynomials in n variables will exhibit greater complexity as n increases. However, an array of recent work has shown that in a certain regime---namely, that where the number of polynomials and their degrees are fixed---the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's Conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including: the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables., This expository paper was written in conjunction with Craig Huneke's talk on Stillman's Conjecture at the 2018 JMM Current Events Bulletin
- Published
- 2018
13. On symmetric linear diffusions
- Author
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Liping Li and Jiangang Ying
- Subjects
Discrete mathematics ,Representation theorem ,Dirichlet form ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Disjoint sets ,01 natural sciences ,Dirichlet distribution ,010104 statistics & probability ,symbols.namesake ,Closure (mathematics) ,symbols ,Interval (graph theory) ,Countable set ,0101 mathematics ,Mathematics - Abstract
The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let ( E , F ) (\mathcal {E},\mathcal {F}) be a regular and local Dirichlet form on L 2 ( I , m ) L^2(I,m) , where I I is an interval and m m is a fully supported Radon measure on I I . We shall first present a complete representation for ( E , F ) (\mathcal {E},\mathcal {F}) , which shows that ( E , F ) (\mathcal {E},\mathcal {F}) lives on at most countable disjoint “effective" intervals with an “adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C c ∞ ( I ) C_c^\infty (I) being a special standard core of ( E , F ) (\mathcal {E},\mathcal {F}) and shall identify the closure of C c ∞ ( I ) C_c^\infty (I) in ( E , F ) (\mathcal {E},\mathcal {F}) when C c ∞ ( I ) C_c^\infty (I) is contained but not necessarily dense in F \mathcal {F} relative to the E 1 1 / 2 \mathcal {E}_1^{1/2} -norm. This paper is partly motivated by a result of Hamza’s that was stated in a theorem of Fukushima, Oshima, and Takeda’s and that provides a different point of view to this theorem. To illustrate our results, many examples are provided.
- Published
- 2018
14. Asymptotics of Racah polynomials with fixed parameters
- Author
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Xiang-Sheng Wang and Roderick Wong
- Subjects
Classical orthogonal polynomials ,symbols.namesake ,Pure mathematics ,Difference polynomials ,Applied Mathematics ,General Mathematics ,Discrete orthogonal polynomials ,Orthogonal polynomials ,Wilson polynomials ,symbols ,Racah W-coefficient ,Mathematics - Abstract
In this paper, we investigate asymptotic behaviors of Racah polynomials with fixed parameters and scaled variable as the polynomial degree tends to infinity. We start from the difference equation satisfied by the polynomials and derive an asymptotic formula in the outer region via ratio asymptotics. Next, we find the asymptotic formulas in the oscillatory region via a simple matching principle. Unlike the varying parameter case considered in a previous paper, the zeros of Racah polynomials with fixed parameters may not always be real. For this unusual case, we also provide a standard method to determine the oscillatory curve which attracts the zeros of Racah polynomials when the degree becomes large.
- Published
- 2017
15. On Bohr sets of integer-valued traceless matrices
- Author
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Alexander Fish
- Subjects
Applied Mathematics ,General Mathematics ,Torus ,Ergodic Ramsey theory ,Random walk ,Bohr model ,Combinatorics ,Matrix (mathematics) ,symbols.namesake ,Conjugacy class ,Integer ,symbols ,Analytic number theory ,Mathematics - Abstract
In this paper we show that any Bohr-zero non-periodic set B B of traceless integer-valued matrices, denoted by Λ \Lambda , intersects non-trivially the conjugacy class of any matrix from Λ \Lambda . As a corollary, we obtain that the family of characteristic polynomials of B B contains all characteristic polynomials of matrices from Λ \Lambda . The main ingredient used in this paper is an equidistribution result for an S L d ( Z ) SL_d(\mathbb {Z}) random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work [J. Amer. Math. Soc. 24 (2011), 231–280].
- Published
- 2017
16. On embedding certain partial orders into the P-points under Rudin-Keisler and Tukey reducibility
- Author
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Dilip Raghavan and Saharon Shelah
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Boolean algebra (structure) ,010102 general mathematics ,Ultrafilter ,Natural number ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,symbols ,Embedding ,Continuum (set theory) ,0101 mathematics ,Partially ordered set ,Continuum hypothesis ,Axiom ,Mathematics - Abstract
The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.
- Published
- 2017
17. What are Lyapunov exponents, and why are they interesting?
- Author
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Amie Wilkinson
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Spectral theory ,General Mathematics ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,02 engineering and technology ,Lyapunov exponent ,Barycentric subdivision ,Computer Science::Computational Geometry ,Equilateral triangle ,Translation (geometry) ,01 natural sciences ,Midpoint ,Mathematics - Spectral Theory ,Mathematics - Geometric Topology ,symbols.namesake ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Ergodic theory ,Mathematics - Dynamical Systems ,0101 mathematics ,Spectral Theory (math.SP) ,Mathematical Physics ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Geometric Topology (math.GT) ,Mathematical Physics (math-ph) ,37C40, 37D25, 37H15, 34D08, 37C60, 47B36, 32G15 ,Computer Science::Graphics ,symbols ,020201 artificial intelligence & image processing ,Schrödinger's cat - Abstract
This expository paper, based on a Current Events Bulletin talk at the January, 2016 Joint Meetings, introduces the concept of Lyapunov exponents and discusses the role they play in three areas: smooth ergodic theory, Teichm\"uller theory, and the spectral theory of one-frequency Schr\"odinger operators. The inspiration for this paper is the work of 2014 Fields Medalist Artur Avila, and his work in these areas is given special attention., Comment: 27 pages. To appear in the Bulletin of the AMS
- Published
- 2016
18. Newton polyhedra and weighted oscillatory integrals with smooth phases
- Author
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Toshihiro Nose and Joe Kamimoto
- Subjects
Weight function ,Explicit formulae ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Resolution of singularities ,Critical point (mathematics) ,Polyhedron ,symbols.namesake ,Newton fractal ,symbols ,Oscillatory integral ,Asymptotic expansion ,Mathematics - Abstract
In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals whose phases and weights are contained in a certain class of smooth functions, including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho.
- Published
- 2015
19. Coisotropic subalgebras of complex semisimple Lie bialgebras
- Author
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Nicole Rae Kroeger
- Subjects
Pure mathematics ,Mathematics::Operator Algebras ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Subalgebra ,Torus ,Fixed point ,symbols.namesake ,17B62 ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,symbols ,Quantum Algebra (math.QA) ,Representation Theory (math.RT) ,Variety (universal algebra) ,Mathematics::Representation Theory ,Mathematics::Symplectic Geometry ,Mathematics - Representation Theory ,Lagrangian ,Mathematics - Abstract
In his paper "A Construction for Coisotropic Subalgebras of Lie Bialgebras", Marco Zambon gave a way to use a long root of a complex semisimple Lie biaglebra $\mathfrak{g}$ to construct a coisotropic subalgebra of $\mathfrak{g}$. In this paper, we generalize Zambon's construction. Our construction is based on the theory of Lagrangian subalgebras of the double $\mathfrak{g}\oplus\mathfrak{g}$ of $\mathfrak{g}$, and our coisotropic subalgebras correspond to torus fixed points in the variety $\mathcal{L}(\mathfrak{g}\oplus\mathfrak{g})$ of Lagrangian subalgebras of $\mathfrak{g}\oplus\mathfrak{g}$.
- Published
- 2015
20. Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities
- Author
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Song Sun, Xiuxiong Chen, and Simon Donaldson
- Subjects
Mathematics - Differential Geometry ,Series (mathematics) ,Computer Science::Information Retrieval ,Applied Mathematics ,General Mathematics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Fano plane ,Mathematical proof ,53C55 ,Mathematics - Algebraic Geometry ,symbols.namesake ,Differential Geometry (math.DG) ,Cone (topology) ,FOS: Mathematics ,symbols ,Gravitational singularity ,Einstein ,Algebraic Geometry (math.AG) ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
This is the first of a series of three papers which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. The main result of this paper is that a Kähler-Einstein metric with cone singularities along a divisor can be approximated by a sequence of smooth Kähler metrics with controlled geometry in the Gromov-Hausdorff sense.
- Published
- 2014
21. On uniqueness in the extended Selberg class of Dirichlet series
- Author
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Bao Qin Li and Haseo Ki
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Dirichlet eta function ,Class number formula ,Riemann zeta function ,Combinatorics ,Dirichlet kernel ,symbols.namesake ,Riemann hypothesis ,Selberg trace formula ,symbols ,Selberg class ,Dirichlet series ,Mathematics - Abstract
We will show that two functions in the extended Selberg class satisfying the same functional equation must be identically equal if they have sufficiently many common zeros. This paper concerns the question of how L-functions are determined by their zeros. L-functions are Dirichlet series with the Riemann zeta function ζ(s) = ∑∞ n=1 1 ns as the prototype and are important objects in number theory. The Selberg class S of L-functions is the set of all Dirichlet series L(s) = ∑∞ n=1 a(n) ns of a complex variable s = σ + it with a(1) = 1, satisfying the following axioms (see [7]): (i) (Dirichlet series) For σ > 1, L(s) is an absolutely convergent Dirichlet series. (ii) (Analytic continuation) There is a nonnegative integer k such that (s − 1)L(s) is an entire function of finite order. (iii) (Functional equation) L satisfies a functional equation of type ΛL(s) = ωΛL(1− s), where ΛL(s) = L(s)Q s ∏K j=1 Γ(λjs+μj) with positive real numbers Q, λj and with complex numbers μj , ω with Reμj ≥ 0 and |ω| = 1. (iv) (Ramanujan hypothesis) a(n) n for every e > 0; (v) (Euler product) logL(s) = ∑∞ n=1 b(n) ns , where b(n) = 0 unless n is a positive power of a prime and b(n) n for some θ < 12 . The degree dL of an L-function L is defined to be dL = 2 ∑K j=1 λj , where K, λj are the numbers in axiom (iii). The Selberg class includes the Riemann zeta-function ζ and essentially those Dirichlet series where one might expect the analogue of the Riemann hypothesis. At the same time, there are a whole host of interesting Dirichlet series not possessing a Euler product (see e.g. [3], [8]). Throughout the paper, all L-functions are assumed to be functions from the extended Selberg class of those only satisfying the axioms (i)-(iii) (see [3]). Thus, the results obtained in the present paper particularly apply to L-functions in the Selberg class. Received by the editors October 5, 2011 and, in revised form, February 12, 2012. 2010 Mathematics Subject Classification. Primary 11M36, 30D30.
- Published
- 2013
22. Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems
- Author
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Sibei Yang, Der-Chen Chang, Jun Cao, and Dachun Yang
- Subjects
Discrete mathematics ,Semigroup ,Applied Mathematics ,General Mathematics ,Order (ring theory) ,Muckenhoupt weights ,Type (model theory) ,Hardy space ,Omega ,Dirichlet distribution ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,Mathematics - Classical Analysis and ODEs ,42B35 (Primary) 42B30, 42B20, 42B25, 35J25, 42B37, 47B38, 46E30 (Secondary) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Maximal function ,Mathematics - Abstract
Let $\Omega$ be either $\mathbb{R}^n$ or a strongly Lipschitz domain of $\mathbb{R}^n$, and $\omega\in A_{\infty}(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second order divergence form elliptic operator on $L^2 (\Omega)$ with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by $L$ has the Gaussian property $(G_1)$ with the regularity of their kernels measured by $\mu\in(0,1]$. Let $\Phi$ be a continuous, strictly increasing, subadditive, positive and concave function on $(0,\infty)$ of critical lower type index $p_{\Phi}^-\in(0,1]$. In this paper, the authors introduce the "geometrical" weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega,\,r}(\Omega)$ and $h^{\Phi}_{\omega,\,z}(\Omega)$ via the weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega}(\mathbb{R}^n)$, and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$ when $p_{\Phi}^-\in(n/(n+\mu),1]$. As applications, the authors prove that the operators $\nabla^2{\mathbb G}_D$ are bounded from $h^{\Phi}_{\omega,\,r}(\Omega)$ to the weighted Orlicz space $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,r}(\Omega)$ to itself when $\Omega$ is a bounded semiconvex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, and the operators $\nabla^2{\mathbb G}_N$ are bounded from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $h^{\Phi}_{\omega,\,r}(\Omega)$ when $\Omega$ is a bounded convex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, where ${\mathbb G}_D$ and ${\mathbb G}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator., Comment: This paper has been withdrawn by the authors
- Published
- 2013
23. A note on numerators of Bernoulli numbers
- Author
-
Dinesh S. Thakur
- Subjects
Fermat's Last Theorem ,Conjecture ,Divisor ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Prime (order theory) ,Ramanujan's sum ,Combinatorics ,symbols.namesake ,Eisenstein series ,symbols ,Algorithm ,Bernoulli number ,Mathematics ,Counterexample - Abstract
The object of this short note is to give some observations on Bernoulli numbers and their function field analogs and point out ‘known’ counter-examples to a conjecture of Chowla. Bernoulli numbers Bn defined (for integer n > 1) by z/(e z − 1) = ∑ Bnz /n!, and their important cousins Bn/n, play interesting roles in many areas of mathematics. (Below we only restrict to these for n even, precisely the case when they are non-zero.) We mention some key words by which the reader can search: Power sums, Zeta special values, Eisenstein series, measures, p-adic L-functions, finite differences, combinatorics, Euler-Maclaurin formula, Todd classes in topology, Grothendieck-Hirzebruch-Riemann-Roch formula, K-theory of integers, Stable homotopy, Bhargava factorial associated to the set of primes, Kummer-HerbrandRibet theorems in cyclotomic theory, Kervaire-Milnor formula for diffeomorphism classes of exotic spheres. Their factorization is of interest, the denominators (which show up explicitly in the third-fourth items from the end) are well understood via theorems of von-Staudt, but the numerators (which show up explicitly in the last two items above) are mysterious and connected to many interesting phenomena. In one of the rare lapses, Ramanujan, in his very first paper [R1911, (14), (18) and Sec. 12], claimed to have proved (editors downgrade it to a conjecture) that the numerator Nn of Bn/n is always a prime, when it was already known since Kummer (in Fermat’s last theorem connection) that ‘irregular’ prime 37 is a proper divisor of N32, and even N20 is composite. In [C1930], Chowla showed that Ramanujan’s claim had infinity of counter-examples. Note that this also follows from one counterexample and the Kummer congruences (recalled below) for that prime! Interestingly, in his last paper [CC1986], Chowla (jointly with his daughter) asks as unsolved problem whether the numerator is always square-free. (This is also mentioned in the nice survey article by Murtys and Williams on Chowla’s work in Vol. 1 of [C1999], where the author learned about it.) Theorem 1. Chowla’s conjecture stated above has infinity of counter-examples. In fact, for any given irregular prime p less than 163 million, and given arbitrarily large k, there is n such that p divides Nn. Proof. Using the tables (or the reader can try to check directly!) giving factorizations of Bn/n, for example the table by Wagstaff at the Bernoulli web page www.bernoulli.org, we see that 37 divides N284. Now recall the well-known Kummer congruences that the value of (1 − p)Bn/n modulo p depends only on (even) n modulo pk−1(p − 1), for n not divisible by p − 1. The first claim follows by taking p = 37. Supported in part by NSA grant H98230-10-1-0200.
- Published
- 2012
24. Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type
- Author
-
Guozhen Lu, Yongsheng Han, and Ji Li
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Duality (mathematics) ,Mathematical analysis ,Banach space ,Singular integral ,Hardy space ,Space (mathematics) ,symbols.namesake ,Product (mathematics) ,symbols ,Interpolation space ,Lp space ,Mathematics - Abstract
This paper is inspired by the work of Nagel and Stein in which the L p L^p ( 1 > p > ∞ ) (1>p>\infty ) theory has been developed in the setting of the product Carnot-Carathéodory spaces M ~ = M 1 × ⋯ × M n \widetilde {M}=M_1\times \cdots \times M_n formed by vector fields satisfying Hörmander’s finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product B M O BMO space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderón identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journé-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.
- Published
- 2012
25. The Shi arrangement and the Ish arrangement
- Author
-
Drew Armstrong and Brendon Rhoades
- Subjects
Weyl group ,Mathematics::Combinatorics ,Applied Mathematics ,General Mathematics ,Complete graph ,Type (model theory) ,Interpretation (model theory) ,Combinatorics ,symbols.namesake ,Hyperplane ,Bounded function ,symbols ,Symmetry (geometry) ,Characteristic polynomial ,Mathematics - Abstract
This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$ "ceilings" and $d$ "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.
- Published
- 2012
26. On Lyapunov exponents of continuous Schrödinger cocycles over irrational rotations
- Author
-
Yingfei Yi and Wen Huang
- Subjects
Algebra ,Pure mathematics ,symbols.namesake ,Applied Mathematics ,General Mathematics ,Irrational number ,symbols ,Lyapunov exponent ,Schrödinger's cat ,Mathematics - Abstract
In this paper we consider continuous, SL ( 2 , R ) \text {SL}(2,\mathbb {R}) -valued, Schrödinger cocycles over irrational rotations. We prove two generic results on the Lyapunov exponents which improve the corresponding ones contained in a paper by Bjerklöv, Damanik and Johnson.
- Published
- 2011
27. Inequalities of Chernoff type for finite and infinite sequences of classical orthogonal polynomials
- Author
-
Przemysław Rutka and Ryszard Smarzewski
- Subjects
Pure mathematics ,Gegenbauer polynomials ,Applied Mathematics ,General Mathematics ,Discrete orthogonal polynomials ,Mathematical analysis ,MathematicsofComputing_GENERAL ,Classical orthogonal polynomials ,symbols.namesake ,Difference polynomials ,Wilson polynomials ,Orthogonal polynomials ,Hahn polynomials ,symbols ,Jacobi polynomials ,Mathematics - Abstract
In this paper we present two-sided estimates of Chernoff type for the weighted L w 2 L_{w}^{2} -distance of a smooth function to the k k -dimensional space of all polynomials of degree less than k k , whenever the weight function w w solves the Pearson differential equation and generates a finite or infinite sequence of classical orthogonal polynomials. These inequalities are simple corollaries of a unified general theorem, which is the main result of the paper.
- Published
- 2009
28. The ergodicity of weak Hilbert spaces
- Author
-
Razvan Anisca
- Subjects
Discrete mathematics ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Ergodicity ,Banach space ,Hilbert space ,State (functional analysis) ,Space (mathematics) ,Linear subspace ,symbols.namesake ,symbols ,Ergodic theory ,Isomorphism ,Mathematics - Abstract
This paper complements a recent result of Dilworth, Ferenczi, Kutzarova and Odell regarding the ergodicity of strongly asymptotic ℓ p \ell _p spaces. We state this result in a more general form, involving domination relations, and we show that every asymptotically Hilbertian space which is not isomorphic to ℓ 2 \ell _2 is ergodic. In particular, every weak Hilbert space which is not isomorphic to ℓ 2 \ell _2 must be ergodic. Throughout the paper we construct explicitly the maps which establish the fact that the relation E 0 E_0 is Borel reducible to isomorphism between subspaces of the Banach spaces involved.
- Published
- 2009
29. Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part I
- Author
-
Wilhelm Schlag, Avy Soffer, and Wolfgang Staubach
- Subjects
Wronskian ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Inverse ,Zero-point energy ,Riemannian manifold ,Eigenfunction ,Manifold ,symbols.namesake ,symbols ,Asymptotic expansion ,Hamiltonian (quantum mechanics) ,Mathematics ,Mathematical physics - Abstract
Let Ω C ℝ N be a compact imbedded Riemannian manifold of dimension d > 1 and define the (d + 1)-dimensional Riemannian manifold M := {(x, r(x)w): x ∈ ℝ, ω ∈ Ω} with r > 0 and smooth, and the natural metric ds 2 = (1 + r'(x) 2 )dx 2 + r 2 (x)ds 2 Ω . We require that M has conical ends: r(x) = |x| + O(x -1 ) as x → ±∞. The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schrodinger evolution e itΔ M and the wave evolution e it √ -Δ M are obtained for data of the form f (x, ω) = y n (ω)u(x), where Y n are eigenfunctions of -Δ Ω with eigenvalues u 2 n . In this paper we discuss all cases d + n > 1. If n ≠ 0 there is the following accelerated local decay estimate: with 0 1, ∥ω σ e itΔM Y n f ∥ L ∞ (M) ≤ C (n, M, σ) t - d+1 / 2 -σ ∥w -1 σ f ∥ L 1 (M) , where w σ (x) = 〈x〉 -σ , and similarly for the wave evolution. Our method combines two main ingredients: (A) A detailed scattering analysis of Schrodinger operators of the form -∂ 2 ξ + (v 2 - 1 / 4 )〈ξ〉 -2 + U(ξ) on the line where U is real-valued and smooth with U (l) (ξ) = 0( ξ-3-l ) for all l ≥ 0 as ξ → ±∞ and v > 0. In particular, we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the Wronskian between the outgoing Jost solutions as the energy tends to zero. In particular, the division into Part I and Part II can be explained by the former being resonant at zero energy, where the present paper deals with the nonresonant case. (B) Estimation of oscillatory integrals by (non)stationary phase.
- Published
- 2009
30. Operator-valued frames
- Author
-
David R. Larson, Victor Kaftal, and Shuang Zhang
- Subjects
Applied Mathematics ,General Mathematics ,Hilbert space ,Operator theory ,Algebra ,symbols.namesake ,Von Neumann's theorem ,Operator (computer programming) ,Operator algebra ,Von Neumann algebra ,symbols ,Affiliated operator ,Strong operator topology ,Mathematics - Abstract
We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the mul- tiplicity one case and extends to higher multiplicity their dilation approach. We prove several results for operator-valued frames concerning duality, dis- jointeness, complementarity , and composition of operator valued frames and the relationship between the two types of similarity (left and right) of such frames. A key technical tool is the parametrization of Parseval operator val- ued frames in terms of a class of partial isometries in the Hilbert space of the analysis operator. We apply these notions to an analysis of multiframe gener- ators for the action of a discrete group G on a Hilbert space. One of the main results of the Han-Larson work was the parametrization of the Parseval frame generators in terms of the unitary operators in the von Neumann algebra gen- erated by the group representation, and the resulting norm path-connectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von Neumann algebra. In this paper we general- ize this multiplicity one result to operator-valued frames. However, both the parameterization and the proof of norm path-connectedness turn out to be necessarily more complicated, and this is at least in part the rationale for this paper. Our parameterization involves a class of partial isometries of a different von Neumann algebra. These partial isometries are not path-connected in the norm topology, but only in the strong operator topology. We prove that the set of operator frame generators is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. As in the multiplicity one theory there are analogous results for general (non-Parseval) frames.
- Published
- 2009
31. Transverse LS category for Riemannian foliations
- Author
-
Steven Hurder and Dirk Töben
- Subjects
Pure mathematics ,Closed manifold ,Riemannian submersion ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Lie group ,01 natural sciences ,Upper and lower bounds ,symbols.namesake ,Compact group ,Mathematics::Category Theory ,0103 physical sciences ,symbols ,Foliation (geology) ,Lusternik–Schnirelmann category ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Category theory ,Mathematics - Abstract
We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation F on a closed manifold M. The essential transverse category cat e (M, F) is introduced in this paper, and we prove that cat e (M, F) is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category cat (M, F) is finite, and thus cat e (M, F) = cat(M, F) holds. A fundamental point of this paper is to use properties of Riemannian submersions and the Molino Structure Theory for Riemannian foliations to transform the calculation of cat e (M, F) into a standard problem about O(q)-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated O(q)-manifold W, we have that cat e (M, F) = cat O(q) (Ŵ). Hence, the traditional techniques developed for the study of smooth compact Lie group actions can be effectively employed for the study of the LS category of Riemannian foliations. A generalization of the Lusternik-Schnirelmann theorem is derived: given a C 1 -function f: M → R which is constant along the leaves of a Riemannian foliation F, the essential transverse category cat e (M, F) is a lower bound for the number of critical leaf closures of f.
- Published
- 2009
32. A Weierstrass-type theorem for homogeneous polynomials
- Author
-
David Benko and András Kroó
- Subjects
Pure mathematics ,Gegenbauer polynomials ,Applied Mathematics ,General Mathematics ,Discrete orthogonal polynomials ,Mathematical analysis ,Classical orthogonal polynomials ,symbols.namesake ,Macdonald polynomials ,Difference polynomials ,Orthogonal polynomials ,symbols ,Jacobi polynomials ,Stone–Weierstrass theorem ,Mathematics - Abstract
By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in R d . In this paper we study the following question: does the density hold if we approximate only by homogeneous polynomials? Since the set of homogeneous polynomials is nonlinear this leads to a nontrivial problem. It is easy to see that: 1) density may hold only on star-like 0-symmetric surfaces; 2) at least 2 homogeneous polynomials are needed for approximation. The most interesting special case of a star-like surface is a convex surface. It has been conjectured by the second author that functions continuous on 0-symmetric convex surfaces in R d can be approximated by a pair of homogeneous polynomials. This conjecture is not resolved yet but we make substantial progress towards its positive settlement. In particular, it is shown in the present paper that the above conjecture holds for 1) d = 2, 2) convex surfaces in R d with C 1+ǫ boundary.
- Published
- 2008
33. A summability criterion for stochastic integration
- Author
-
Nicolae Dinculeanu and Peter Gray
- Subjects
Discrete mathematics ,Integrable system ,Stochastic process ,Applied Mathematics ,General Mathematics ,Banach space ,Hilbert space ,Stochastic integral ,Stochastic integration ,symbols.namesake ,Bounded function ,symbols ,Martingale (probability theory) ,Mathematics - Abstract
In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes X with values in a Banach space E; namely, X is of class (LD), and the stochastic measure I X is bounded and strongly additive in L p E (in particular, if I X is bounded in L p E and c 0 ⊈ E) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with p-integrable semivariation or p-integrable variation, the conditions of I X to be bounded and have bounded semivariation are superfluous. For processes with 1-integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.
- Published
- 2008
34. Maps on the 𝑛-dimensional subspaces of a Hilbert space preserving principal angles
- Author
-
Lajos Molnár
- Subjects
Set (abstract data type) ,symbols.namesake ,Pure mathematics ,N dimensional ,Applied Mathematics ,General Mathematics ,Principal angles ,Mathematical analysis ,Hilbert space ,symbols ,Linear subspace ,Mathematics - Abstract
In a former paper we studied transformations on the set of all n n -dimensional subspaces of a Hilbert space H H which preserve the principal angles. In the case when dim H ≠ 2 n \dim H\neq 2n , we could determine the general form of all such maps. The aim of this paper is to complete our result by considering the problem in the remaining case dim H = 2 n \dim H=2n .
- Published
- 2008
35. Classes of Hardy spaces associated with operators, duality theorem and applications
- Author
-
Lixin Yan
- Subjects
Combinatorics ,Analytic semigroup ,Class (set theory) ,symbols.namesake ,Operator (computer programming) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,symbols ,Infinitesimal generator ,Hardy space ,Space (mathematics) ,Mathematics - Abstract
Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space H1 L(R n ) and a BMO L (R n ) space associated with the operator L were introduced and studied. In this paper we define a class of H p L (R n ) spaces associated with the operator L for a range of p < 1 acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical H p (R n ) spaces. We then establish a duality theorem between the H p L (R n ) spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on H p L (R n ) and give the inclusion between the classical H p (R n ) spaces and the H p L (R n ) spaces associated with operators.
- Published
- 2008
36. The Noether map II
- Author
-
Mara D. Neusel and Müfit Sezer
- Subjects
Finite group ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Image (category theory) ,Permutation group ,Group representation ,Surjective function ,Combinatorics ,Faithful representation ,symbols.namesake ,symbols ,Noether's theorem ,Mathematics - Abstract
Let ρ: G → GL(n, F) be a faithful representation of a finite group G. In this paper we proceed with the study of the image of the associated Noether map η G G : F[V(G)] G → F[V] G . In our 2005 paper it has been shown that the Noether map is surjective if V is a projective FG-module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for p-groups (where p is the characteristic of the ground field F) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of V.
- Published
- 2007
37. About the cover: Euler and Königsberg's Bridges: A historical view
- Author
-
Gerald L. Alexanderson
- Subjects
Class (set theory) ,Applied Mathematics ,General Mathematics ,Graph theory ,Variety (linguistics) ,Seven Bridges of Königsberg ,symbols.namesake ,Number theory ,Path (graph theory) ,Calculus ,Euler's formula ,symbols ,Title page ,Mathematics - Abstract
Graph theory almost certainly began when, in 1735, Leonhard Euler solved a popular puzzle about bridges. The East Prussian city of Konigsberg (now Kalin- ingrad) occupies both banks of the River Pregel and an island, Kneiphof, which lies in the river at a point where it branches into two parts. There were seven bridges that spanned the various sections of the river, and the problem posed was this: could a person devise a path through Konigsberg so that one could cross each of the seven bridges only once and return home? Long thought to be impossible, the first mathematical demonstration of this was presented by Euler to the members of the Petersburg Academy on August 26, 1735, and written up the following year under the title "Solutio Problematis ad Geometriam Situs Pertinentis (The solution to a problem relating to the geometry of position)" (2) in the proceedings of the Petersburg Academy (the Commentarii). The title page of this volume appears on the cover of this issue of the Bulletin. This story is well-known, and the illustrations in Euler's paper are often repro- duced in popular books on mathematics and in textbooks. Sandifer in (6) claims flatly that "The Konigsberg Bridge Problem is Euler's most famous work," though scholars in other specialties (differential equations, complex analysis, calculus of variations, combinatorics, number theory, physics, naval architecture, music, . . . ) might disagree. N. L. Biggs, E. K. Lloyd, and R. J. Wilson in their history of graph theory (1) clearly view this paper of Euler's as seminal and remark: "The origins of graph theory are humble, even frivolous. Whereas many branches of mathematics were motivated by fundamental problems of calculation, motion, and measurement, the problems which led to the development of graph theory were often little more than puzzles, designed to test the ingenuity rather than to stimulate the imagina- tion. But despite the apparent triviality of such puzzles, they captured the interest of mathematicians, with the result that graph theory has become a subject rich in theoretical results of a surprising variety and depth." Euler provides only a neces- sary condition, not a sufficient condition, for solving the problem. But he does treat more than the original problem by beginning a generalization to two islands and four rivers, as is illustrated in the plate accompanying the original paper (Figure 1). In this renowned paper Euler does not get around to stating the problem until the second page. On the first he states the reason for being interested in the problem— it was an example of a class of problems he attributes to Leibniz as belonging to something Leibniz called "geometry of position." Euler says that "this branch is concerned only with the determination of position and its properties; it does not
- Published
- 2006
38. On the restriction of Deligne-Lusztig characters
- Author
-
Mark Reeder
- Subjects
Finite group ,Weyl group ,Applied Mathematics ,General Mathematics ,Centralizer and normalizer ,Algebra ,Combinatorics ,symbols.namesake ,Conjugacy class ,Borel subgroup ,symbols ,Coset ,Maximal torus ,Identity component ,Mathematics - Abstract
results of Hagedorn gave me the courage to attempt such calculations for general groups and to obtain closed multiplicity formulas for orthogonal groups. It is a pleasure to thank Dick Gross for initiating the work in [12] which led to this paper, for helpful remarks on an earlier version, and for aquainting me with Hagedorn's thesis. The referee read the original version of this paper with care and insight, made valuable comments and simplified some of the arguments. In particular, the proof of Lemma 3.1 given below is due to the referee and is much shorter than the original one. Some general notation: The cardinality of a finite set X is denoted by \X\. Equivalence classes are generally denoted by [ ], sometimes with ornamentation. If g is an element of a group G, we write Ad(g) for the conjugation map Ad(g) : x i? gxg~x, and also write 9T := gTg~x for a subgroup T C G. The center of G is denoted Z(G) and the centralizer of g G G is denoted CG(g). We write ( , )# for the pairing on the space of class functions on a finite group H, for which the irreducible characters of H are an orthonormal basis. l?G,Gf D H are finite overgroups of H and i?, ip' are class functions on G, G' respectively, then (^^')h is understood to mean (^\h^'\h)h^ where \h denotes restriction to H. 2. Remarks on maximal tori Let G be a connected reductive algebraic J-group. We assume G is defined over f and has Frobenius F. If T is a maximal torus in G we denote its normalizer in G by NG(T) and write WG(T) = NG(T)/T for the Weyl group of T in G. If T is F-stable, we have W(T)F = NG(T)F/TF, by the Lang-Steinberg theorem. The reduction formula for Deligne-Lusztig characters (recalled in section 4 below) involves a sum over the following kind of subset of GF. Fix an F-stable maximal torus T C G, and let s be a semisimple element in GF. We must sum over the set NG(s,T)F:={j GF: *" T}. Note that NG(s, T)F, if nonempty, is a union of GF x NG(T)F double cosets, where Gs := CG(s)? is the identity component of the centralizer CG(s) of s in G. To say that s^ G T is to say that 7TcGs, so determining the GF x NG(T)F double cosets in NG(s,T)F amounts to determining the Gf-conjugacy classes of F-stable maximal tori in Gs which are contained in a given GF-conjugacy class. Such classes of tori are parameterized by twisted conjugacy classes in Weyl groups of Gs and G. The aim of this section is to parameterize the GF x NG(T)F double cosets in NG(s, T)F in terms of the fiber of a natural map between twisted conjugacy classes in the Weyl groups of Gs and G. This parameterization will be fundamental to our later calculations with Deligne-Lusztig characters. We begin by recalling the classification of F-stable maximal tori in G. See [5, chap. 3] for more details in what follows. Fix an F-stable maximal torus To in G contained in an F-stable Borel subgroup of G, and abbreviate NG = Ng(Tq), Wg = WG(T0). This content downloaded from 207.46.13.149 on Mon, 03 Oct 2016 06:07:19 UTC All use subject to http://about.jstor.org/terms ON THE RESTRICTION OF DELIGNE-LUSZTIG CHARACTERS 577 Let T(G) denote the set of all F-stable maximal tori in G. Then T(G) is a finite union of GF-orbits. For any T G T(G), let [T\g := {7T : 7 G GF} denote the GF-orbit of T. There is g G G such that T = 9T0. Since T is F-stable, we have g~1F(g) G AfcThis gives an element w := g~1F(g)T0 G WG. The map Ad(#)t =
- Published
- 2006
39. Large character sums: Pretentious characters and the Pólya-Vinogradov theorem
- Author
-
Kannan Soundararajan and Andrew Granville
- Subjects
Brauer's theorem on induced characters ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Upper and lower bounds ,Combinatorics ,Riemann hypothesis ,symbols.namesake ,Character sum ,Character (mathematics) ,Simple (abstract algebra) ,Bounded function ,symbols ,Character group ,Mathematics - Abstract
In 1918 Pólya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Pólya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Pólya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large, and it finds a hidden structure among these characters.
- Published
- 2006
40. Geometric characterization of strongly normal extensions
- Author
-
Jerald J. Kovacic
- Subjects
Non-abelian class field theory ,Galois cohomology ,Applied Mathematics ,General Mathematics ,Fundamental theorem of Galois theory ,Galois group ,Abelian extension ,Galois module ,Embedding problem ,Algebra ,symbols.namesake ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,symbols ,Galois extension ,Mathematics - Abstract
This paper continues previous work in which we developed the Galois theory of strongly normal extensions using differential schemes. In the present paper we derive two main results. First, we show that an extension is strongly normal if and only if a certain differential scheme splits, i.e. is obtained by base extension of a scheme over constants. This gives a geometric characterization to the notion of strongly normal. Second, we show that Picard-Vessiot extensions are characterized by their Galois group being affine. Our proofs are elementary and do not use "group chunks" or cohomology. We end by recalling some important results about strongly normal extensions with the hope of spurring future research.
- Published
- 2006
41. Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension
- Author
-
Joachim Krieger and Wilhelm Schlag
- Subjects
Blow-Up Rate ,Spectral theory ,General Mathematics ,Instability ,Nonintegrable Equations ,Schrödinger equation ,Standing wave ,symbols.namesake ,Klein-Gordon Equations ,Stability theory ,Invariant-Manifolds ,modulation theory ,Mathematics ,Applied Mathematics ,Non-Linear Schrodinger ,Mathematical analysis ,spectral theory ,Standing Waves ,Stability Theory ,Ground-States ,stable manifolds ,Nonlinear system ,critical Schrodinger equation ,symbols ,Solitary Waves ,Monic polynomial ,Schrödinger's cat - Abstract
Standing wave solutions of the one-dimensional nonlinear Schrödinger equation \[ i ∂ t ψ + ∂ x 2 ψ = − | ψ | 2 σ ψ i\partial _t \psi + \partial _{x}^2 \psi = -|\psi |^{2\sigma } \psi \] with σ > 2 \sigma >2 are well known to be unstable. In this paper we show that asymptotic stability can be achieved provided the perturbations of these standing waves are small and chosen to belong to a codimension one Lipschitz surface. Thus, we construct codimension one asymptotically stable manifolds for all supercritical NLS in one dimension. The considerably more difficult L 2 L^2 -critical case, for which one wishes to understand the conditional stability of the pseudo-conformal blow-up solutions, is studied in the authors’ companion paper Non-generic blow-up solutions for the critical focusing NLS in 1-d, preprint, 2005.
- Published
- 2006
42. Taylor series for the Askey-Wilson operator and classical summation formulas
- Author
-
Bernardo Lopez, José Marco, and Javier Parcet
- Subjects
Binomial type ,Basic hypergeometric series ,Applied Mathematics ,General Mathematics ,Entire function ,Function (mathematics) ,Methods of contour integration ,Binomial theorem ,Algebra ,symbols.namesake ,symbols ,Taylor series ,Jacobi polynomials ,Mathematics - Abstract
An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studying the different forms of the binomial theorem in this context. 1. Introduction and definitions The problem of expanding a function with respect to a given polynomial basis has many implications in analysis. The simplest example of this kind is the Taylor's expansion theorem. In this paper, we replace the classical derivative by a difference operator of Askey-Wilson type. Our results complement the paper (3) of Ismail and Stanton and are a natural continuation of the point of view presented in (5), where a new approach to the theory of classical hypergeometric polynomials is given. In contrast with (3), our aim is to find sufficient conditions for t he Taylor series to converge, but not necessarily to the original function. In this more general setting, we may consider non-necessarily entire functions and we give an explicit expression for the limit of the remainders in terms of a contour integral. Using this and a new estimate for the q-shifted factorials, which might be of independent interest, we obtain a summation formula which is new as far as we know. As we explain below, it can be regarded as a non-symmetrized version of the non-terminating q-Saalschutz sum. As applications, we also provide a new proof of the q-Gauss summation formula and a list of binomial type summation formulas in the same line than Ismail's paper (2). Now we give some definitions which will be used in what follows. The notions we are presenting were already introduced in (5) with the aim of studying some aspects of the theory of hypergeometric polynomials. The relevance of this approach is justified in (5), where a more detailed exposition is given.
- Published
- 2006
43. On the power series coefficients of certain quotients of Eisenstein series
- Author
-
Bruce C. Berndt and Paul R. Bialek
- Subjects
Power series ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Ramanujan summation ,Modular form ,Ramanujan's congruences ,Ramanujan's sum ,symbols.namesake ,Eisenstein series ,symbols ,Ramanujan tau function ,Reciprocal ,Mathematics - Abstract
In their last joint paper, Hardy and Ramanujan examined the coefficients of modular forms with a simple pole in a fundamental region. In particular, they focused on the reciprocal of the Eisenstein series E 6 ( τ ) E_6(\tau ) . In letters written to Hardy from nursing homes, Ramanujan stated without proof several more results of this sort. The purpose of this paper is to prove most of these claims.
- Published
- 2005
44. On the Andrews-Stanley refinement of Ramanujan’s partition congruence modulo 5 and generalizations
- Author
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Alexander Berkovich and Frank G. Garvan
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Combinatorial proof ,Congruence relation ,Ramanujan's congruences ,Ramanujan's sum ,Combinatorics ,symbols.namesake ,Rank of a partition ,symbols ,Partition (number theory) ,Partially ordered set ,Quotient ,Mathematics - Abstract
In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic s r a n k ( π ) = O ( π ) − O ( π ′ ) , \begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*} where O ( π ) {\mathcal O}(\pi ) denotes the number of odd parts of the partition π \pi and π ′ \pi ’ is the conjugate of π \pi . In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod 5 5 : p 0 ( 5 n + 4 ) a m p ; ≡ p 2 ( 5 n + 4 ) ≡ 0 ( mod 5 ) , p ( n ) a m p ; = p 0 ( n ) + p 2 ( n ) , \begin{align*} p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod {5}, p(n) &= p_0(n) + p_2(n), \end{align*} where p i ( n ) p_i(n) ( i = 0 , 2 i=0,2 ) denotes the number of partitions of n n with s r a n k ≡ i ( mod 4 ) \mathrm {srank}\equiv i\pmod {4} and p ( n ) p(n) is the number of unrestricted partitions of n n . Andrews asked for a partition statistic that would divide the partitions enumerated by p i ( 5 n + 4 ) p_i(5n+4) ( i = 0 , 2 i=0,2 ) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the 2 2 -quotient-rank and the 5 5 -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the 2 2 -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan’s congruence mod 5 5 . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5 5 . Finally, we discuss some new formulas for partitions that are 5 5 -cores and discuss an intriguing relation between 3 3 -cores and the Andrews-Garvan crank.
- Published
- 2005
45. A new approach to the theory of classical hypergeometric polynomials
- Author
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Javier Parcet and José Marco
- Subjects
Applied Mathematics ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,Type (model theory) ,Rodrigues' rotation formula ,Hypergeometric distribution ,Rodrigues' formula ,Algebra ,symbols.namesake ,Operator (computer programming) ,Functional equation ,Orthogonal polynomials ,symbols ,Jacobi polynomials ,Mathematics - Abstract
JOSE MANUEL MARCO AND JAVIER PARCET´Abstract. In this paper we present a unified approach to the spectral analysisof an hypergeometric type operator whose eigenfunctions include the classicalorthogonal polynomials. We write the eigenfunctions of this operator by meansof a new Taylor formula for operators of Askey-Wilson type. This gives rise tosome expressions for the eigenfunctions, which are unknown in such a generalsetting. Our methods also give a general Rodrigues formula from which severalwell known formulas of Rodrigues type can be obtained directly. Moreover,other new Rodrigues type formulas come out when seeking for regular solutionsof the associated functional equations. The main difference here is that, incontrast with the formulas appearing in the literature, we get non-ramifiedsolutions which are useful for applications in combinatorics. Another fact,that becomes clear in this paper, is the role played by the theory of ellipticfunctions in the connection between ramified and non-ramified solutions.
- Published
- 2004
46. Remark on well-posedness for the fourth order nonlinear Schrödinger type equation
- Author
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Jun Ichi Segata
- Subjects
Well-posed problem ,Partial differential equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Schrödinger equation ,Split-step method ,Sobolev space ,Nonlinear system ,symbols.namesake ,symbols ,Initial value problem ,Nonlinear Schrödinger equation ,Mathematics - Abstract
We consider the initial value problem for the fourth order nonlinear Schrodinger type equation (4NLS) related to the theory of vortex filament. In this paper we prove the time local well-posedness for (4NLS) in the Sobolev space, which is an improvement of our previous paper.
- Published
- 2004
47. Newton polyhedra, unstable faces and the poles of Igusa’s local zeta function
- Author
-
Kathleen Hoornaert
- Subjects
Monomial ,Pure mathematics ,Polynomial ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Congruence relation ,Igusa zeta-function ,Riemann zeta function ,symbols.namesake ,Polyhedron ,symbols ,Order (group theory) ,Local zeta-function ,Mathematics - Abstract
In this paper we examine when the order of a pole of Igusa's local zeta function associated to a polynomial f is smaller than expected. We carry out this study in the case that f is sufficiently non-degenerate with respect to its Newton polyhedron Γ(f), and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial f to the same question about polynomials f μ , where μ are faces of Γ(f) depending on the examined pole and f μ is obtained from f by throwing away all monomials of f whose exponents do not belong to μ. Secondly, we obtain a formula for Igusa's local zeta function associated to a polynomial f μ , with μ unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than expected.
- Published
- 2003
48. Reduction of Opial-type inequalities to norm inequalities
- Author
-
Gord Sinnamon
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Matrix norm ,Hilbert space ,Bilinear form ,symbols.namesake ,Quadratic form ,symbols ,Schatten norm ,Condition number ,Operator norm ,Dual norm ,Mathematics - Abstract
Weighted Opial-type inequalities are shown to be equivalent to weighted norm inequalities for sublinear operators and for nearly positive operators. Examples involving the Hardy-Littlewood maximal function and the non-increasing rearrange- ment are presented. Opial-type inequalities are related to norm inequalities much as quadratic forms are related to bilinear forms. A linear operator T on Hilbert space gives rise to the bilinear form (f,g) 7! hTf,gi and the quadratic form f 7! hTf,fi. Duality shows that the norm of T and the norm of the bilinear form coincide and a standard polarization argument shows that this norm is equivalent to but not necessarily equal to the norm of the quadratic form, called the numerical radius of T. In this paper, far from the luxuries of Hilbert spaces and linear operators, we show that the equivalence of operator norm and numerical radius persists. The work is in response to Richard Brown's suggestion that Steven Bloom's result (2, The- orem 1) which gives the equivalence for positive operators should apply in greater generality. Opial-type inequalities have been much studied since Opial's original paper in 1960 and the papers (2), (3) and (4) include many references. After the main theorem showing equivalence of Opial-type and norm inequali- ties, an example involving the Hardy-Littlewood maximal function is included to illustrate that the equivalence cannot be taken in a pointwise sense. To show that the method can be readily applied to generate non-trivial inequal- ities from known norm inequalities we give a simple weight characterization of an Opial-type inequality for the non-increasing rearrangement.
- Published
- 2003
49. Integration by parts formulas involving generalized Fourier-Feynman transforms on function space
- Author
-
Seung Jun Chang, David Skoug, and Jae Gil Choi
- Subjects
Pure mathematics ,Function space ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Generalized linear array model ,First variation ,symbols.namesake ,Fourier transform ,Probability theory ,symbols ,Feynman diagram ,Integration by parts ,Brownian motion ,Mathematics - Abstract
In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form F ( x ) = f ( ⟨ α 1 , x ⟩ , … , ⟨ α n , x ⟩ ) F(x)=f(\langle \alpha _{1} , x\rangle , \dots , \langle \alpha _{n} , x\rangle ) where ⟨ α , x ⟩ \langle {\alpha ,x}\rangle denotes the Paley-Wiener-Zygmund stochastic integral ∫ 0 T α ( t ) d x ( t ) \int _{0}^{T} \alpha (t) d x(t) .
- Published
- 2003
50. Gaussian curvature in the negative case
- Author
-
Wenxiong Chen and Congming Li
- Subjects
symbols.namesake ,Pure mathematics ,Partial differential equation ,Negative case ,Applied Mathematics ,General Mathematics ,Gaussian curvature ,symbols ,Geometry ,Manifold ,Mathematics - Abstract
In this paper, we reinvestigate an old problem of prescribing Gaussian curvature in the negative case. In 1974, Kazdan and Warner considered the equation -Δu+α=R(x)e u , x ∈ M, on any compact two dimensional manifold M with a α > α o and it is not solvable for α < α o . Then one may naturally ask: Is the equation solvable for a = α o ? In this paper, we answer the question affirmatively. We show that there exists at least one solution for α = α o .
- Published
- 2002
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