399 results on '"Classical theorem"'
Search Results
2. Ratios of Sensed Segments (RS)
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Specht, Edward John, Jones, Harold Trainer, Calkins, Keith G., Rhoads, Donald H., Specht, Edward John, Jones, Harold Trainer, Calkins, Keith G., and Rhoads, Donald H.
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- 2015
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3. Lagrange’s Theorem
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Karpenkov, Oleg, Cohen, Arjeh M., Series editor, Cohen, Henri, Series editor, Eisenbud, David, Series editor, Singer, Michael F., Series editor, Sturmfels, Bernd, Series editor, and Karpenkov, Oleg
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- 2013
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4. More on a Problem of Zarankiewicz
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Dutta, Chinmoy, Radhakrishnan, Jaikumar, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Chao, Kun-Mao, editor, Hsu, Tsan-sheng, editor, and Lee, Der-Tsai, editor
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- 2012
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5. On the Fourier transform of functions from the classes $$H_p^\alpha ({{\mathbb {R}}})$$
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S. S. Platonov
- Subjects
Physics ,Combinatorics ,symbols.namesake ,Fourier transform ,Lipschitz class ,General Mathematics ,010102 general mathematics ,symbols ,0101 mathematics ,Classical theorem ,Lebesgue integration ,01 natural sciences - Abstract
Let a function f belongs to the Lebesgue class $$L^p({{\mathbb {R}}})$$ , $$1\le p\le 2$$ , and let $${\widehat{f}}$$ be the Fourier transform of f. The classical theorem of E. Titchmarsh states that if the function f belongs to the Lipschitz class $$Lip(\alpha ,p; {{\mathbb {R}}})$$ , $$00$$ .
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- 2021
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6. On the pair correlations of powers of real numbers
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Christoph Aistleitner and Simon Baker
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11K06, 11K60 ,General Mathematics ,Modulo ,FOS: Physical sciences ,0102 computer and information sciences ,Lebesgue integration ,01 natural sciences ,Combinatorics ,symbols.namesake ,Pair correlation ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Algebra over a field ,Classical theorem ,Mathematical Physics ,Real number ,Mathematics ,Sequence ,Mathematics - Number Theory ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,010201 computation theory & mathematics ,symbols ,Martingale (probability theory) ,Mathematics - Probability - Abstract
A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method., Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics
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- 2021
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7. Classical Theorems
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Lin, Pei-Kee and Lin, Pei-Kee
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- 2004
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8. The effect on the adjacency and signless Laplacian spectral radii of uniform hypergraphs by grafting edges
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Peng Xiao and Ligong Wang
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Numerical Analysis ,Hypergraph ,Algebra and Number Theory ,Spectral radius ,Grafting (decision trees) ,Mathematics::Spectral Theory ,Signless laplacian ,Combinatorics ,Discrete Mathematics and Combinatorics ,Order (group theory) ,Adjacency list ,Geometry and Topology ,Classical theorem ,Laplace operator ,Mathematics - Abstract
In this paper, we investigate how the adjacency spectral radius and signless Laplacian spectral radius behave when a connected uniform hypergraph is perturbed by grafting edges. We extend the classical theorem of Li and Feng (1979) [10] about spectral radius from connected graphs to connected uniform hypergraphs by using a constructive method. This result also generalizes the results of Cvetkovic and Simic (2009) [2] , and Su et al. (2018) [22] . As applications, we determine the k-uniform supertrees of order n with the first two smallest adjacency spectral radii (signless Laplacian spectral radii, respectively). Also, we determine the k-uniform supertrees of order n with the first two smallest Laplacian spectral radii, in the case when k is even.
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- 2021
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9. On the Hankel transform of functions from Nikol'ski type classes
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Sergey S. Platonov
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Class (set theory) ,Pure mathematics ,Hankel transform ,Lipschitz class ,Applied Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,Type (model theory) ,Lebesgue integration ,01 natural sciences ,symbols.namesake ,Fourier transform ,symbols ,0101 mathematics ,Classical theorem ,Analysis ,Mathematics - Abstract
Let a function f belong to the Lebesgue class , , and let be the Fourier transform of f. The classical theorem of E. Titchmarsh states that if the function f belongs to the Lipschitz class , , then...
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- 2020
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10. A strongly irreducible affine iterated function system with two invariant measures of maximal dimension
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Cagri Sert and Ian Morris
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Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Invariant subspace ,Open set ,self-affine set ,iterated function system ,equilibrium state ,non-conformal repeller ,subadditive thermodynamic formalism ,01 natural sciences ,Linear subspace ,Iterated function system ,0103 physical sciences ,Attractor ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,Invariant (mathematics) ,Classical theorem ,Mathematics - Abstract
A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb {R}^{d}$ . In this paper we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.
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- 2020
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11. Non-solvable groups each of whose vanishing class sizes has at most two prime divisors
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Sajjad Mahmood Robati
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Finite group ,Class (set theory) ,Pure mathematics ,Algebra and Number Theory ,Group (mathematics) ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Prime (order theory) ,Solvable group ,Order (group theory) ,0101 mathematics ,Classical theorem ,Mathematics - Abstract
Let G be a finite group. A classical theorem of Burnside shows that a group with order divisible by at most two primes is solvable. The aim of this article is studying finite groups each of whose v...
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- 2020
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12. Fourier Transform of Dini-Lipschitz Functions on Locally Compact Vilenkin Groups
- Author
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Sergey S. Platonov
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General Mathematics ,010102 general mathematics ,Dual group ,Lebesgue integration ,Lipschitz continuity ,01 natural sciences ,Combinatorics ,symbols.namesake ,Fourier transform ,Bounded function ,0103 physical sciences ,symbols ,010307 mathematical physics ,Locally compact space ,0101 mathematics ,Classical theorem ,Mathematics - Abstract
Let $$G$$ be a locally compact bounded Vilenkin group, $$\Gamma$$ be the dual group of $$G$$ . Suppose that a function $$f(x)$$ belongs to the the Lebesgue class $$L^p(G)$$ , $$10$$ , $$\beta\in{\mathbb R}$$ , then for which values of $$r$$ we can guarantee that $$\widehat{f}\in L^r(\Gamma)$$ ? The result is an analogue of one classical theorem of E. Titchmarsh about the Fourier transform of functions from the Lipschitz classes on $${\mathbb R}$$ .
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- 2020
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13. Geometric properties of normal submanifolds
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Mario Alfredo Hernández and Josué Meléndez
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Pure mathematics ,Overline ,Geodesic ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,Extension (predicate logic) ,Riemannian manifold ,Space (mathematics) ,Curvature ,Submanifold ,01 natural sciences ,010101 applied mathematics ,Mathematics::Differential Geometry ,0101 mathematics ,Classical theorem ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
This paper deals with normal submanifolds immersed in a Riemannian manifold $${\overline{M}}$$ . We generalized some recent results of surfaces in space forms obtained by Hernandez-Lamoneda and Ruiz-Hernandez (Bull Braz Marh Soc (NS) 49:447–462, 2018) to arbitrary submanifolds. More precisely, given a submanifold M in $${\overline{M}}$$ , we study the submanifolds formed by orthogonal geodesics to M, and call it a ruled normal submanifold to M. In the first part of this paper, we analyze these submanifolds and establish some geometric properties of them. Furthermore, we extend some properties about the lines of curvature and using the ideas of [3] also give an extension of the classical Theorem of Bonnet to hypersurfaces of $${\overline{M}}$$ .
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- 2020
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14. On the non-tangential convergence of Poisson and modified Poisson semigroups at the smoothness points of $$L_{p}$$-functions
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Simten Bayrakci, Ilham A. Aliev, and M. F. Shafiev
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Pure mathematics ,Smoothness (probability theory) ,Semigroup ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,Function (mathematics) ,Poisson distribution ,01 natural sciences ,symbols.namesake ,Convergence (routing) ,symbols ,0101 mathematics ,Classical theorem ,Mathematics - Abstract
The high-dimensional version of Fatou’s classical theorem asserts that the Poisson semigroup of a function $$f\in L_{p}(\mathbb {R}^{n}), \ 1\le p \le \infty $$, converges to f non-tangentially at Lebesque points. In this paper we investigate the rate of non-tangential convergence of Poisson and metaharmonic semigroups at $$\mu $$-smoothness points of f.
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- 2020
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15. On Some Topological Properties of Fourier Transforms of Regular Holonomic -Modules
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Yohei Ito and Kiyoshi Takeuchi
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Pure mathematics ,Holonomic ,General Mathematics ,010102 general mathematics ,01 natural sciences ,symbols.namesake ,Fourier transform ,0103 physical sciences ,Converse ,D-module ,symbols ,010307 mathematical physics ,0101 mathematics ,Classical theorem ,Mathematics - Abstract
We study Fourier transforms of regular holonomic ${\mathcal{D}}$-modules. In particular, we show that their solution complexes are monodromic. An application to direct images of some irregular holonomic ${\mathcal{D}}$-modules will be given. Moreover, we give a new proof of the classical theorem of Brylinski and improve it by showing its converse.
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- 2020
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16. Further generalisations of a classical theorem of Daboussi
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Jean-Marie De Koninck and Imre Kátai
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Pure mathematics ,General Mathematics ,Classical theorem ,Mathematics - Published
- 2020
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17. The Fundamental Theorems
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Bridges, Douglas S., Mehta, Ghanshyam B., Fandel, G., editor, Trockel, W., editor, Bridges, Douglas S., and Mehta, Ghanshyam B.
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- 1995
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18. Monotone Flows with Dense Periodic Orbits
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Morris W. Hirsch
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Statistics and Probability ,Numerical Analysis ,Pure mathematics ,Algebra and Number Theory ,Applied Mathematics ,Open set ,Dynamical Systems (math.DS) ,Theoretical Computer Science ,Monotone polygon ,Flow (mathematics) ,FOS: Mathematics ,Order (group theory) ,Periodic orbits ,Interval (graph theory) ,Convex cone ,Geometry and Topology ,Mathematics - Dynamical Systems ,Classical theorem ,math.DS ,Mathematics - Abstract
The main result is Theorem 1: A flow on a connected open set X ⊂ R d is globally periodic provided (i) periodic points are dense in X, and (ii) at all positive times the flow preserves the partial order defined by a closed convex cone that has nonempty interior and contains no straight line. The proof uses the analog for homeomorphisms due to B. Lemmens et al. [27], a classical theorem of D. Montgomery [31, 32], and a sufficient condition for the nonstationary periodic points in a closed order interval to have rationally related periods (Theorem 2).
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- 2019
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19. Extensions of the Erdős–Gallai theorem and Luo’s theorem
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Bo Ning and Xing Peng
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Statistics and Probability ,Erdős–Gallai theorem ,Applied Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Circumference ,01 natural sciences ,Upper and lower bounds ,Graph ,Theoretical Computer Science ,Turán number ,Extremal graph theory ,Combinatorics ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,0101 mathematics ,Classical theorem ,Mathematics - Abstract
The famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least $${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$ edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G). We also construct a family of graphs which shows our extension improves the estimate given by the Erdős–Gallai theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with ℓ vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.
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- 2019
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20. Fourier Transform of Dini-Lipschitz Functions on the Field of p-Adic Numbers
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Sergey S. Platonov
- Subjects
Class (set theory) ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,Function (mathematics) ,Lebesgue integration ,Lipschitz continuity ,01 natural sciences ,Combinatorics ,symbols.namesake ,Fourier transform ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Classical theorem ,Mathematics ,p-adic number - Abstract
Let ℚp be the field of p-adic numbers, a function f(x) belongs to the the Lebesgue class Lρ(ℚp), 1 ρ ≤ 2, and let $$\hat{f}(\xi)$$ be the Fourier transform of f. In this paper we give an answer to the next problem: if the function f belongs to the Dini-Lipschitz class DLip(α, β, ρ; ℚp), α > 0, β ∈ ℝ, then for which values of r we can guarantee that $$\hat{f} \in {L^r}(\mathbb{Q}_p)$$? The result is an analogue of one classical theorem of E. Titchmarsh about the Fourier transform of functions from the Lipschitz classes on ℝ.
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- 2019
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21. Envy-free cake division without assuming the players prefer nonempty pieces
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Shira Zerbib and Frédéric Meunier
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Computer Science::Computer Science and Game Theory ,Lemma (mathematics) ,Simplex ,Conjecture ,General Mathematics ,010102 general mathematics ,Prime number ,0102 computer and information sciences ,01 natural sciences ,Envy-free ,Combinatorics ,n-connected ,Computer Science - Computer Science and Game Theory ,010201 computation theory & mathematics ,05E45, 54H25, 91B32 ,Mathematics - Combinatorics ,0101 mathematics ,Classical theorem ,Mathematics - General Topology ,Mathematics - Abstract
Consider $n$ players having preferences over the connected pieces of a cake, identified with the interval $[0,1]$. A classical theorem, found independently by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it is possible to divide the cake into $n$ connected pieces and assign these pieces to the players in an envy-free manner, i.e, such that no player strictly prefers a piece that has not been assigned to her. One of these conditions, considered as crucial, is that no player is happy with an empty piece. We prove that, even if this condition is not satisfied, it is still possible to get such a division when $n$ is a prime number or is equal to $4$. When $n$ is at most $3$, this has been previously proved by Erel Segal-Halevi, who conjectured that the result holds for any $n$. The main step in our proof is a new combinatorial lemma in topology, close to a conjecture by Segal-Halevi and which is reminiscent of the celebrated Sperner lemma: instead of restricting the labels that can appear on each face of the simplex, the lemma considers labelings that enjoy a certain symmetry on the boundary.
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- 2019
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22. Triangles in diophantine approximation
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Daniele Mundici
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Combinatorics ,Infinite set ,Sequence ,Algebra and Number Theory ,Dense set ,Rank (differential topology) ,Diophantine approximation ,Continued fraction ,Classical theorem ,Mathematics ,Additive group - Abstract
For any point x = ( x 1 , x 2 ) ∈ R 2 we let G x = Z x 1 + Z x 2 + Z be the subgroup of the additive group R generated by x 1 , x 2 , 1 . When rank ( G x ) = 3 we say that x is a rank 3 point. We prove the existence of an infinite set I ⊆ R 2 of rank 3 points having the following property: For every two-dimensional continued fraction expansion μ and x ∈ I , letting μ ( x ) = T 0 ⊇ T 1 ⊇ ⋯ , it follows that infinitely many triangles T n have some angle ≤ arcsin ( 23 1 / 2 / 6 ) ≈ π / ( 3.3921424 ) ≈ 53 ∘ . Thus lim inf n → ∞ area ( T n ) / diam ( T n ) 2 ≤ 23 1 / 2 / 12 . At the opposite extreme, we construct a two-dimensional continued fraction expansion μ and a dense set D ⊆ R 2 of rank 3 points such that for each x ∈ D the sequence T 0 ⊇ T 1 ⊇ ⋯ of triangles of μ ( x ) has the following property: Letting ω n denote the smallest angle of T n , it follows that ω 0 ω 1 ⋯ and lim n → ∞ ω n = π / 3 . Further, the other two angles of T n are > π / 3 . Thus lim n → ∞ area ( T n ) / diam ( T n ) 2 = 3 1 / 2 / 4 , and the vertices of the triangles T n strongly converge to x. Our proofs combine a classical theorem of Davenport and Mahler with binary stellar operations of regular fans.
- Published
- 2019
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23. Baker–Pixley theorem for algebras in relatively congruence distributive quasivarieties
- Author
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Diego Vaggione
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,0102 computer and information sciences ,Term (logic) ,01 natural sciences ,Distributive property ,010201 computation theory & mathematics ,Clone (algebra) ,Computer Science::General Literature ,Congruence (manifolds) ,0101 mathematics ,Algebra over a field ,Classical theorem ,Mathematics - Abstract
A classical theorem of Baker and Pixley states that if [Formula: see text] is a finite algebra with a majority term and [Formula: see text] is an [Formula: see text]-ary operation on [Formula: see text] which preserves every subuniverse of [Formula: see text], then [Formula: see text] is representable by a term in [Formula: see text]. We give a generalizacion of this theorem for the case in which [Formula: see text] is a finite algebra belonging to some relatively congruence distributive quasivariety.
- Published
- 2019
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24. An Erdős-Gallai type theorem for vertex colored graphs
- Author
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Nika Salia, Casey Tompkins, and Oscar Zamora
- Subjects
Mathematics::Combinatorics ,Conjecture ,Colored graph ,0211 other engineering and technologies ,021107 urban & regional planning ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Upper and lower bounds ,Graph ,Theoretical Computer Science ,Vertex (geometry) ,Combinatorics ,Colored ,Computer Science::Discrete Mathematics ,010201 computation theory & mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Classical theorem ,Mathematics - Abstract
While investigating odd-cycle free hypergraphs, Győri and Lemons introduced a colored version of the classical theorem of Erdős and Gallai on $$P_k$$ -free graphs. They proved that any graph G with a proper vertex coloring and no path of length $$2k+1$$ with end vertices of different colors has at most 2kn edges. We show that Erdős and Gallai’s original sharp upper bound of kn holds for their problem as well. We also introduce a version of this problem for trees and present a generalization of the Erdős-Sos conjecture.
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- 2019
- Full Text
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25. Extended nilHecke algebras and symmetric functions in type B
- Author
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Michael Reeks
- Subjects
Pure mathematics ,Ring (mathematics) ,Algebra and Number Theory ,010102 general mathematics ,Structure (category theory) ,Type (model theory) ,01 natural sciences ,Action (physics) ,Symmetric function ,Symmetric polynomial ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Classical theorem ,Mathematics - Abstract
We formulate a type B extended nilHecke algebra, following the type A construction of Naisse and Vaz. We describe an action of this algebra on extended polynomials and describe some results on the structure on the extended symmetric polynomials. Finally, following Appel, Egilmez, Hogancamp, and Lauda, we prove a result analogous to a classical theorem of Solomon connecting the extended symmetric polynomial ring to a ring of usual symmetric polynomials and their differentials.
- Published
- 2019
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26. Spectral Decompositions Arising from Atzmon’s Hyperinvariant Subspace Theorem
- Author
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Miguel Monsalve-López and Eva A. Gallardo-Gutiérrez
- Subjects
Pure mathematics ,Algebra and Number Theory ,Subspace theorem ,010102 general mathematics ,01 natural sciences ,Linear subspace ,Matrix decomposition ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Classical theorem ,Analysis ,Mathematics - Abstract
By means of a weaker functional model, we prove the existence of non-trivial closed hyperinvariant subspaces for linear bounded operators generalizing, in particular, a classical theorem of Atzmon and revealing the spectral nature of the hyperinvariant subspaces involved. As an application, we show non-trivial spectral subspaces for Bishop operators on $$L^p[0,1)$$ , $$1\le p
- Published
- 2021
- Full Text
- View/download PDF
27. Continued Fractions, Hurwitz and Stieltjes
- Author
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Tomas Sauer
- Subjects
Pure mathematics ,Signal processing ,Unit circle ,Simple (abstract algebra) ,Stability (learning theory) ,Riemann–Stieltjes integral ,Classical theorem ,Mathematics - Abstract
In the chapter on Signal Processing we found out that the stability of filters or difference equations is equivalent to the fact that certain polynomials have all their zeros inside the unit circle. In this chapter we consider polynomials with such restrictions. With a simple change of parameters, we see that polynomials with zeros inside the unit circle are equivalent to polynomials that have their zeros located in the left half-plane. This is what is known as Hurwitz polynomials, and they have once again close relations to continued fractions. Indeed a classical theorem by Stieltjes will characterize the Hurwitz polynomials using continued fractions which will also have a description using certain Hankel matrices and operators.
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- 2021
- Full Text
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28. Commutators of singular integrals on generalized Lp spaces with variable exponent
- Author
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Andrei K. Lerner and A. Yu. Karlovich
- Subjects
Pure mathematics ,Variable exponent ,General Mathematics ,Mathematical analysis ,Mathematics::Classical Analysis and ODEs ,Commutator (electric) ,Singular integral ,Generalized Lp space with variable exponent ,law.invention ,46E30 ,law ,Mathematics - Classical Analysis and ODEs ,Commutator ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,42B20 ,Classical theorem ,Local sharp maximal function ,Mathematics ,Calderón-Zygmund singular integral ,BMO - Abstract
A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized $L^p$ spaces with variable exponent., 13 pages
- Published
- 2021
29. A class of summing operators acting in spaces of operators
- Author
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Enrique A. Sánchez-Pérez and Jorge Tomás Rodríguez
- Subjects
Physics ,Mathematics::Functional Analysis ,Banach space ,Summing operator ,¿-product of Banach spaces ,Negative - answer ,Combinatorics ,Operator (computer programming) ,Strong operator topology ,Dominated operator ,Classical theorem ,MATEMATICA APLICADA ,Universally measurable function - Abstract
Let \(X\), \(Y\) and \(Z\) be Banach spaces and let \(U\) be a subspace of \(\mathcal{L}(X^*,Y)\), the Banach space of all operators from \(X^*\) to \(Y\). An operator \(S\colon U \to Z\) is said to be \((\ell^s_p,\ell_p)\)-summing (where \(1\leq p
- Published
- 2021
30. The Frucht property in the quantum group setting
- Author
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Teodor Banica and J.P. McCarthy
- Subjects
Automorphism group ,Finite group ,Pure mathematics ,Property (philosophy) ,Quantum group ,General Mathematics ,46L65 (46L53, 81R50) ,Structure (category theory) ,Mathematics - Operator Algebras ,Automorphism ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Mathematics - Combinatorics ,Quantum Algebra (math.QA) ,Combinatorics (math.CO) ,Classical theorem ,Operator Algebras (math.OA) ,Quantum ,Mathematics - Abstract
A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results., Comment: 41 pages; v3 further revisions, to appear in Glasg. Math. J
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- 2021
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31. Refinements of the Bohr and Rogosinski phenomena
- Author
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Nilanjan Das
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,Applied Mathematics ,Holomorphic function ,Unit disk ,Bohr model ,symbols.namesake ,Bounded function ,Simply connected space ,symbols ,Convex domain ,Classical theorem ,Analysis ,Mathematics - Abstract
We obtain improved versions of a classical theorem of Rogosinski concerning the partial sums of a bounded holomorphic function defined on the open unit disk D . Further, we establish refined versions of a generalized Bohr inequality for holomorphic functions mapping D inside a simply connected or convex domain Ω ⊊ C . In addition, we improve on the classical Bohr inequality for the family of holomorphic self mappings of D and for its subfamily consisting of functions that fix the origin.
- Published
- 2022
- Full Text
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32. Dirichlet series of integers with missing digits
- Author
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Melvyn B. Nathanson
- Subjects
Sequence ,Algebra and Number Theory ,Mathematics - Number Theory ,010102 general mathematics ,Abscissa ,010103 numerical & computational mathematics ,01 natural sciences ,Decimal ,Combinatorics ,symbols.namesake ,Convergence (routing) ,FOS: Mathematics ,symbols ,11A63, 11B05, 11B75, 11K16 ,Number Theory (math.NT) ,0101 mathematics ,Classical theorem ,Dirichlet series ,Mathematics - Abstract
For certain sequences $A$ of positive integers with missing $g$-adic digits, the Dirichlet series $F_A(s) = \sum_{a\in A} a^{-s}$ has abscissa of convergence $\sigma_c < 1$. The number $\sigma_c$ is computed. This generalizes and strengthens a classical theorem of Kempner on the convergence of the sum of the reciprocals of a sequence of integers with missing decimal digits., Comment: Minor improvements and corrected typos; 7 pages
- Published
- 2020
33. Convergent series of integers with missing digits
- Author
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Melvyn B. Nathanson
- Subjects
Discrete mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Decimal ,symbols.namesake ,Number theory ,010201 computation theory & mathematics ,Fourier analysis ,FOS: Mathematics ,symbols ,11A63, 11B05, 11B75, 11K16 ,Number Theory (math.NT) ,0101 mathematics ,Classical theorem ,Harmonic series (mathematics) ,Convergent series ,Mathematics - Abstract
A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with convergent harmonic series., Minor changes, 8 pages
- Published
- 2020
34. Equidecomposition in cardinal algebras
- Author
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Forte Shinko
- Subjects
Algebra and Number Theory ,Generalization ,Group (mathematics) ,Astrophysics::High Energy Astrophysical Phenomena ,010102 general mathematics ,Mathematics - Operator Algebras ,Mathematics::General Topology ,Mathematics - Logic ,01 natural sciences ,Combinatorics ,Mathematics::Logic ,FOS: Mathematics ,Countable set ,0101 mathematics ,Logic (math.LO) ,Operator Algebras (math.OA) ,Classical theorem ,Mathematics ,Probability measure - Abstract
Let $\Gamma$ be a countable group. A classical theorem of Thorisson states that if $X$ is a standard Borel $\Gamma$-space and $\mu$ and $\nu$ are Borel probability measures on $X$ which agree on every $\Gamma$-invariant subset, then $\mu$ and $\nu$ are equidecomposable, i.e. there are Borel measures $(\mu_\gamma)_{\gamma\in\Gamma}$ on $X$ such that $\mu = \sum_\gamma \mu_\gamma$ and $\nu = \sum_\gamma \gamma\mu_\gamma$. We establish a generalization of this result to cardinal algebras.
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- 2020
35. Admissible Banach Function Spaces and Nonuniform Stabilities
- Author
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Liviu Popescu and Nicolae Lupa
- Subjects
010101 applied mathematics ,Class (set theory) ,Pure mathematics ,Function space ,General Mathematics ,Bounded function ,010102 general mathematics ,Stability (learning theory) ,Banach space ,0101 mathematics ,Classical theorem ,01 natural sciences ,Mathematics - Abstract
For nonuniform exponentially bounded evolution families defined on Banach spaces, we introduce a class of Banach function spaces, whose norms are completely determined by the nonuniform behaviour of the corresponding evolution family. We generalize a classical theorem of Datko on these spaces. In addition, we obtain new criteria for the existence of nonuniform stability.
- Published
- 2020
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36. On commuting billiards in higher-dimensional spaces of constant curvature
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Alexey Glutsyuk, UMPA, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Dynamical Systems (math.DS) ,Space (mathematics) ,01 natural sciences ,Ellipsoid ,Constant curvature ,Nonlinear Sciences::Chaotic Dynamics ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,37C25, 70H99 ,Mathematics - Dynamical Systems ,0101 mathematics ,Dynamical billiards ,Classical theorem ,Convex function ,Mathematics - Abstract
We consider two nested billiards in $\mathbb R^d$, $d\geq3$, with $C^2$-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due to Sergei Tabachnikov. The main result is deduced from the classical theorem due to Marcel Berger saying that in higher dimensions only quadrics may have caustics. We also prove versions of Berger's theorem and the main result for billiards in spaces of constant curvature: space forms., Comment: 21 pages. The main result on commuting billiards and Berger's result on caustics are extended to billiards in spaces of constant curvature
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- 2020
- Full Text
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37. Range-kernel characterizations of operators which are adjoint of each other
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Zoltán Sebestyén and Zsigmond Tarcsay
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Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Hilbert space ,010103 numerical & computational mathematics ,Operator theory ,Mathematics::Spectral Theory ,01 natural sciences ,Symmetry (physics) ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Range (mathematics) ,symbols.namesake ,Kernel (algebra) ,FOS: Mathematics ,symbols ,Order (group theory) ,47A5, 47B25 ,0101 mathematics ,Classical theorem ,Analysis ,Mathematics ,Von Neumann architecture - Abstract
We provide necessary and sufficient conditions for a pair $S,T$ of Hilbert space operators in order that they satisfy $S^*=T$ and $T^*=S$. As a main result we establish an improvement of von Neumann's classical theorem on the positive self-adjointness of $S^*S$ for two variables. We also give some new characterizations of self-adjointness and skew-adjointness of operators, not requiring their symmetry or skew-symmetry, respectively., 10 pages
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- 2020
38. Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results
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Salvador Romaguera and Pedro Tirado
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General Mathematics ,Fixed-point theorem ,fuzzy metric space ,complete ,fixed point ,hicks contraction ,Fixed point ,01 natural sciences ,Fuzzy logic ,Computer Science (miscellaneous) ,0101 mathematics ,Classical theorem ,Engineering (miscellaneous) ,Contraction (operator theory) ,Mathematics ,Discrete mathematics ,Hicks contraction ,Fuzzy metric space ,lcsh:Mathematics ,010102 general mathematics ,lcsh:QA1-939 ,010101 applied mathematics ,Metric space ,Complete ,If and only if ,MATEMATICA APLICADA - Abstract
[EN] With the help of C-contractions having a fixed point, we obtain a characterization of complete fuzzy metric spaces, in the sense of Kramosil and Michalek, that extends the classical theorem of H. Hu (see "Am. Math. Month. 1967, 74, 436-437") that a metric space is complete if and only if any Banach contraction on any of its closed subsets has a fixed point. We apply our main result to deduce that a well-known fixed point theorem due to D. Mihet (see "Fixed Point Theory 2005, 6, 71-78") also allows us to characterize the fuzzy metric completeness., This research was partially funded by Ministerio de Ciencia, Innovacion y Universidades, under grant PGC2018-095709-B-C21 and AEI/FEDER, UE funds.
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- 2020
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39. Computing on Lattice-Ordered Abelian Groups
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Daniele Mundici
- Subjects
Word problem (mathematics education) ,Pure mathematics ,Lattice (order) ,MV-algebra ,Abelian group ,Decision problem ,Classical theorem ,Mathematics - Abstract
Starting from a classical theorem of Gurevich and Kokorin we survey recent diverging developments of the theories of lattice-ordered abelian groups and their counterparts equipped with a distinguished order unit. We will focus on decision and recognition problems. As an application of Elliott’s classification, we will touch on word problems of AF C*-algebras.
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- 2020
- Full Text
- View/download PDF
40. The Alon–Milman Theorem for Non-symmetric Bodies
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Márton Naszódi
- Subjects
Combinatorics ,Section (fiber bundle) ,Projection (relational algebra) ,Euclidean ball ,Non symmetric ,Dimension (graph theory) ,Regular polygon ,Mathematics::Metric Geometry ,Convex body ,Classical theorem ,Mathematics - Abstract
A classical theorem of Alon and Milman states that any d dimensional centrally symmetric convex body has a projection of dimension \(m\geq e^{c\sqrt {\ln {d}}}\) which is either close to the m-dimensional Euclidean ball or to the m-dimensional cross-polytope. We extended this result to non-symmetric convex bodies.
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- 2020
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- View/download PDF
41. Effective Erdős-Wintner theorems
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Gérald Tenenbaum, Johann Verwee, Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
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effective averages ,Work (thermodynamics) ,number of prime factors ,010102 general mathematics ,Asymptotic distribution ,0102 computer and information sciences ,Conditional probability distribution ,Function (mathematics) ,mean values of complex multiplicative function ,16. Peace & justice ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Term (time) ,distribution of real additive functions ,Mathematics (miscellaneous) ,2010 Mathematics Subject Classification:11N25, 11N37, 11N60 ,010201 computation theory & mathematics ,Applied mathematics ,Arithmetic function ,0101 mathematics ,Remainder ,Classical theorem ,Mathematics ,Erdős-Wintner theorem - Abstract
International audience; The classical theorem of Erdős & Wintner furnishes a criterion for the existence of a limiting distribution for a real, additive arithmetical function. This work is devoted to providing an effective estimate for the remainder term under the assumption that the conditions in the criterion are fulfilled. We also investigate the case of a conditional distribution.
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- 2020
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- View/download PDF
42. On a quaternionic Picard theorem
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Jörg Winkelmann and Cinzia Bisi
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Holomorphic function ,Socio-culturale ,theorem of Picard ,01 natural sciences ,30G35 ,Mathematics - Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Number Theory (math.NT) ,0101 mathematics ,Complex Variables (math.CV) ,PE1_5 ,Classical theorem ,Algebraic Geometry (math.AG) ,Variable (mathematics) ,Mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics - Complex Variables ,010102 general mathematics ,Function (mathematics) ,Mathematics - Rings and Algebras ,Differential Geometry (math.DG) ,Rings and Algebras (math.RA) ,010307 mathematical physics ,Geometry and Topology ,Value (mathematics) ,Analysis ,Picard theorem - Abstract
The classical theorem of Picard states that a non-constant holomorphic function $f:\mathbb{C}\to\mathbb{C}$ can avoid at most one value. We investigate how many values a non-constant slice regular function of a quaternionic variable $f:\mathbb{H}\to\mathbb{H}$ may avoid., Comment: 15 pages. To appear on Proc. Americ. Math. Soc. (2020)
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- 2020
- Full Text
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43. Inclusion modulo nonstationary
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Miguel Moreno, Gabriel Fernandes, and Assaf Rinot
- Subjects
010505 oceanography ,General Mathematics ,Modulo ,010102 general mathematics ,Mathematics::General Topology ,Mathematics - Logic ,03E35 (Primary) 03E45, 54H05 (Secondary) ,Lipschitz continuity ,01 natural sciences ,Omega ,Combinatorics ,Mathematics::Logic ,Cofinal ,FOS: Mathematics ,Uncountable set ,0101 mathematics ,Logic (math.LO) ,Partially ordered set ,Classical theorem ,Maximal element ,0105 earth and related environmental sciences ,Mathematics - Abstract
A classical theorem of Hechler asserts that the structure $\left(\omega^\omega,\le^*\right)$ is universal in the sense that for any $\sigma$-directed poset P with no maximal element, there is a ccc forcing extension in which $\left(\omega^\omega,\le^*\right)$ contains a cofinal order-isomorphic copy of P. In this paper, we prove a consistency result concerning the universality of the higher analogue $\left(\kappa^\kappa,\le^S\right)$: Theorem. Assume GCH. For every regular uncountable cardinal $\kappa$, there is a cofinality-preserving GCH-preserving forcing extension in which for every analytic quasi-order Q over $\kappa^\kappa$ and every stationary subset S of $\kappa$, there is a Lipschitz map reducing Q to $(\kappa^\kappa,\le^S)$., Comment: Slow filtrations made explicit in the LCC derivation
- Published
- 2020
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44. Linking over cones for the Neumann Fractional $p-$Laplacian
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Edoardo Proietti Lippi and Dimitri Mugnai
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Pure mathematics ,35A15, 47J30, 35S15, 47G10, 45G05 ,Applied Mathematics ,010102 general mathematics ,Structure (category theory) ,Eigenfunction ,01 natural sciences ,Term (time) ,010101 applied mathematics ,Nonlinear system ,Mathematics - Analysis of PDEs ,Face (geometry) ,Neumann boundary condition ,p-Laplacian ,FOS: Mathematics ,0101 mathematics ,Classical theorem ,Analysis ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We consider nonlinear problems governed by the fractional p-Laplacian in presence of nonlocal Neumann boundary conditions and we show three different existence results: the first two theorems deal with a p-superlinear term, the last one with a source having p-linear growth. For the p-superlinear case we face two main difficulties. First: the p-superlinear term may not satisfy the Ambrosetti-Rabinowitz condition. Second, and more important: although the topological structure of the underlying functional reminds the one of the linking theorem, the nonlocal nature of the associated eigenfunctions prevents the use of such a classical theorem. For these reasons, we are led to adopt another approach, relying on the notion of linking over cones.
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- 2020
- Full Text
- View/download PDF
45. The Weyl problem of isometric immersions revisited
- Author
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Siran Li
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Isometric exercise ,01 natural sciences ,symbols.namesake ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,35J60, 53C23, 53C42, 53C21, 53C20 ,Gaussian curvature ,symbols ,FOS: Mathematics ,Mathematics::Differential Geometry ,0101 mathematics ,GEOM ,Classical theorem ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We revisit the classical problem due to Weyl, as well as its generalisations, concerning the isometric immersions of $\mathbb{S}^2$ into simply-connected $3$-dimensional Riemannian manifolds with non-negative Gauss curvature. A sufficient condition is exhibited for the existence of global $C^{1,1}$-isometric immersions. Our developments are based on the framework \`{a} la Labourie (Immersions isom\'{e}triques elliptiques et courbes pseudo-holomorphes, J. Diff. Geom. 30 (1989), 395--424) of studying isometric immersions using $J$-holomorphic curves. We obtain along the way a generalisation of a classical theorem due to Heinz and Pogorelov., Comment: 11 pages
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- 2020
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46. The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed
- Author
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Marco Mazzucchelli, Daniel Cristofaro-Gardiner, University of California [Santa Cruz] (UCSC), University of California, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS), University of California [Santa Cruz] (UC Santa Cruz), University of California (UC), and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Dynamical Systems (math.DS) ,Rank (differential topology) ,01 natural sciences ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics - Dynamical Systems ,[MATH]Mathematics [math] ,Classical theorem ,53C22, 58E10 ,Mathematics::Symplectic Geometry ,Mathematics ,010102 general mathematics ,Manifold ,[MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG] ,Flow (mathematics) ,Differential Geometry (math.DG) ,Mathematics - Symplectic Geometry ,Orbit (dynamics) ,Symplectic Geometry (math.SG) ,010307 mathematical physics ,Mathematics::Differential Geometry - Abstract
A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed connected 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods of its closed Reeb orbits., Comment: 18 pages; version 3: we specified that the contact manifolds are required to be connected. To appear in Commentarii Mathematici Helvetici
- Published
- 2020
- Full Text
- View/download PDF
47. Solvability of Fractional Differential Inclusion with a Generalized Caputo Derivative
- Author
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Tamer Nabil
- Subjects
Article Subject ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Derivative ,01 natural sciences ,010101 applied mathematics ,QA1-939 ,Applied mathematics ,0101 mathematics ,Fractional differential ,Inclusion (mineral) ,Classical theorem ,Analysis ,Mathematics - Abstract
This paper is devoted to the investigation of a kind of generalized Caputo semilinear fractional differential inclusions with deviated-advanced nonlocal conditions. Solvability of the problem is established by means of the Leray-Schauder’s alternative approach with the help of the Lagrange mean-value classical theorem. Finally, some examples are given to delineate the efficient of theoretical results.
- Published
- 2020
48. Pure strictly uniform models of non-ergodic measure automorphisms
- Author
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Tomasz Downarowicz and Benjamin Weiss
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Applied Mathematics ,Dynamical Systems (math.DS) ,Extension (predicate logic) ,Automorphism ,Measure (mathematics) ,Set (abstract data type) ,Primary 37B05, 37B20, Secondary 37A25 ,Compact space ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Ergodic theory ,Mathematics - Dynamical Systems ,Classical theorem ,Analysis ,Mathematics - Abstract
The classical theorem of Jewett and Krieger gives a strictly ergodic model for any ergodic measure preserving system. An extension of this result for non-ergodic systems was given many years ago by George Hansel. He constructed, for any measure preserving system, a strictly uniform model, i.e. a compact space which admits an upper semicontinuous decomposition into strictly ergodic models of the ergodic components of the measure. In this note we give a new proof of a stronger result by adding the condition of purity, which controls the set of ergodic measures that appear in the strictly uniform model., 5 figures
- Published
- 2022
- Full Text
- View/download PDF
49. On a refinement of a theorem of Landau on Koebe domains
- Author
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Manabu Ito
- Subjects
010101 applied mathematics ,Pure mathematics ,Mathematics::Complex Variables ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Holomorphic function ,State (functional analysis) ,0101 mathematics ,Classical theorem ,01 natural sciences ,Mathematics - Abstract
We state and prove a refinement of a classical theorem due to Landau on the Koebe domains for certain families of holomorphic functions introduced by A. W. Goodman. Our geometric approach in this article enables us to derive several statements of interest, which would not be produced via the methods in Goodman's paper, as immediate corollaries of the proof of the main theorem.
- Published
- 2018
- Full Text
- View/download PDF
50. Erratum to 'On a Classical Theorem on the Diameter and Minimum Degree of a Graph'
- Author
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Verónica Hernández, José M. Rodríguez, and Domingo Pestana
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,Mistake ,Classical theorem ,Graph ,Mathematics - Abstract
The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake: in Theorem 6.2 appears that β(n, Δ) = (n−Δ+5)/4 but the correct statement is β(n, Δ) = (n − Δ + 4)/4. In this erratum we correct the theorem and give the correct proof.
- Published
- 2018
- Full Text
- View/download PDF
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