A proof of the Grünbaum conjecture

Bruce L. Chalmers; Grzegorz Lewicki

Displaying similar documents to “A proof of the Grünbaum conjecture”

On the generalized vanishing conjecture

Zhenzhen Feng, Xiaosong Sun (2019)

Czechoslovak Mathematical Journal

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We show that the GVC (generalized vanishing conjecture) holds for the differential operator $Λ = (\partial_{x} - Φ (\partial_{y})) \partial_{y}$ and all polynomials $P (x, y)$ , where $Φ (t)$ is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.

Dynamically defined Cantor sets under the conditions of McDuff's conjecture

Jorge Iglesias, Aldo Portela (2010)

Colloquium Mathematicae

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We prove that if the Cantor set K, dynamically defined by a function $S \in C^{1 + α}$ , satisfies the conditions of McDuff’s conjecture then it cannot be C¹-minimal.

A note on Nakai’s conjecture for the ring $K [X ₁, . . ., X ₙ] / (a ₁ X ₁^{m} + \dots + a ₙ X ₙ^{m})$

Paulo Roberto Brumatti, Marcelo Oliveira Veloso (2011)

Colloquium Mathematicae

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A new proof of the $L^{p}$ -conjecture for locally compact abelian groups

David L. Johnson (1982)

Colloquium Mathematicae

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Order of the smallest counterexample to Gallai's conjecture

Fuyuan Chen (2018)

Czechoslovak Mathematical Journal

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In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Zamfirescu conjectured that the smallest counterexample to Gallai’s conjecture is a graph on 12 vertices. We prove that Gallai’s conjecture is true for every connected graph $G$ with $α^{'} (G) \leq 5$ , which implies that Zamfirescu’s conjecture is true.

On the Brocard-Ramanujan problem and generalizations

Andrzej Dąbrowski (2012)

Colloquium Mathematicae

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Let $p_{i}$ denote the ith prime. We conjecture that there are precisely 28 solutions to the equation $n ² - 1 = p ₁^{α ₁} \dots p_{k}^{α_{k}}$ in positive integers n and α₁,..., $α_{k}$ . This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).

The strength of the projective Martin conjecture

C. T. Chong, Wei Wang, Liang Yu (2010)

Fundamenta Mathematicae

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We show that Martin’s conjecture on Π¹₁ functions uniformly $\leq_{T}$ -order preserving on a cone implies Π¹₁ Turing Determinacy over ZF + DC. In addition, it is also proved that for n ≥ 0, this conjecture for uniformly degree invariant $Π ¹_{2 n + 1}$ functions is equivalent over ZFC to $Σ ¹_{2 n + 2}$ -Axiom of Determinacy. As a corollary, the consistency of the conjecture for uniformly degree invariant Π¹₁ functions implies the consistency of the existence of a Woodin cardinal.

Greenberg’s conjecture for $ℤ_{p}^{d}$ -extensions

Andrea Bandini (2003)

Acta Arithmetica

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A geometric construction for spectrally arbitrary sign pattern matrices and the $2 n$ -conjecture

Dipak Jadhav, Rajendra Deore (2023)

Czechoslovak Mathematical Journal

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We develop a geometric method for studying the spectral arbitrariness of a given sign pattern matrix. The method also provides a computational way of computing matrix realizations for a given characteristic polynomial. We also provide a partial answer to $2 n$ -conjecture. We determine that the $2 n$ -conjecture holds for the class of spectrally arbitrary patterns that have a column or row with at least $n - 1$ nonzero entries.

Two remarks on the Suita conjecture

Nikolai Nikolov (2015)

Annales Polonici Mathematici

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It is shown that the weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $C^{1 + ε}$ -smooth boundary. On the other hand, it is proved that the weak converse to the Suita conjecture holds for any finitely connected planar domain.

The Cohen-Lenstra heuristics, moments and $p^{j}$ -ranks of some groups

Christophe Delaunay, Frédéric Jouhet (2014)

Acta Arithmetica

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This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result due to É. Fouvry and J. Klüners which proves that a conjecture provided by the Cohen-Lenstra philosophy implies another such conjecture. As a consequence of our work, we can deduce, for example, a conjecture for the probability laws of $p^{j}$ -ranks of...

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

Ke-Pao Lin, Xue Luo, Stephen S.-T. Yau, Huaiqing Zuo (2014)

Journal of the European Mathematical Society

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It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function $ψ (x, y)$ which is the number of positive integers $\leq x$ and free of prime factors $> y$ . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ( $n \geq 3$ ) real right-angled simplices. In this...

A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions

Karl-Joachim Wirths (2004)

Annales Polonici Mathematici

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Let D denote the open unit disc and f:D → ℂ̅ be meromorphic and injective in D. We further assume that f has a simple pole at the point p ∈ (0,1) and an expansion $f (z) = z + \sum_{n = 2}^{\infty} a ₙ (f) z ⁿ$ , |z| < p. In particular, we consider f that map D onto a domain whose complement with respect to ℂ̅ is convex. Because of the shape of f(D) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by Co(p). It is proved that for p ∈ (0,1) the domain of variability...

Results related to Huppert’s $ρ$ - $σ$ conjecture

Xia Xu, Yong Yang (2023)

Czechoslovak Mathematical Journal

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We improve a few results related to Huppert’s $ρ$ - $σ$ conjecture. We also generalize a result about the covering number of character degrees to arbitrary finite groups.

Homotopy invariance of higher signatures and $3$ -manifold groups

Michel Matthey, Hervé Oyono-Oyono, Wolfgang Pitsch (2008)

Bulletin de la Société Mathématique de France

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For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable $3$ -manifolds, including the “piecewise geometric” ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable $3$ -manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients...

Recent progress on the Jacobian Conjecture

Michiel de Bondt, Arno van den Essen (2005)

Annales Polonici Mathematici

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We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski’s result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form $x + {(A x)}^{* 3}$ with A² = 0. Then we describe the authors’ result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form x - ∇f, with $f \in k^{[n]}$ homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the...

On coarse embeddability into $ℓ_{p}$ -spaces and a conjecture of Dranishnikov

Piotr W. Nowak (2006)

Fundamenta Mathematicae

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We show that the Hilbert space is coarsely embeddable into any $ℓ_{p}$ for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into $ℓ_{p}$ are equivalent for 1 ≤ p < 2.

Semifields and a theorem of Abhyankar

Vítězslav Kala (2017)

Commentationes Mathematicae Universitatis Carolinae

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Abhyankar proved that every field of finite transcendence degree over $ℚ$ or over a finite field is a homomorphic image of a subring of the ring of polynomials $ℤ [T_{1}, \dots, T_{n}]$ (for some $n$ depending on the field). We conjecture that his result cannot be substantially strengthened and show that our conjecture implies a well-known conjecture on the additive idempotence of semifields that are finitely generated as semirings.

Selection principles and upper semicontinuous functions

Masami Sakai (2009)

Colloquium Mathematicae

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In connection with a conjecture of Scheepers, Bukovský introduced properties wQN* and SSP* and asked whether wQN* implies SSP*. We prove it in this paper. We also give characterizations of properties S₁(Γ,Ω) and $S_{f i n} (Γ, Ω)$ in terms of upper semicontinuous functions

Diagonalization in proof complexity

Jan Krajíček (2004)

Fundamenta Mathematicae

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We study diagonalization in the context of implicit proofs of [10]. We prove that at least one of the following three conjectures is true: ∙ There is a function f: 0,1* → 0,1 computable in that has circuit complexity $2^{Ω (n)}$ . ∙ ≠ co . ∙ There is no p-optimal propositional proof system. We note that a variant of the statement (either ≠ co or ∩ co contains a function $2^{Ω (n)}$ hard on average) seems to have a bearing on the existence of good proof complexity generators. In particular, we prove that...

A counterexample to a conjecture of Bass, Connell and Wright

Piotr Ossowski (1998)

Colloquium Mathematicae

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Let F=X-H: $k^{n}$ → $k^{n}$ be a polynomial map with H homogeneous of degree 3 and nilpotent Jacobian matrix J(H). Let G=(G1,...,Gn) be the formal inverse of F. Bass, Connell and Wright proved in [1] that the homogeneous component of $G_{i}$ of degree 2d+1 can be expressed as $G_{i}^{(d)} = \sum_{T} α {(T)}^{- 1} σ_{i} (T)$ , where T varies over rooted trees with d vertices, α(T)=CardAut(T) and $σ_{i} (T)$ is a polynomial defined by (1) below. The Jacobian Conjecture states that, in our situation, $F$ is an automorphism or, equivalently, $G_{i}^{(d)}$ is zero for sufficiently...

On Lee's conjecture and some results

Lixia Fan, Zhihe Liang (2009)

Discussiones Mathematicae Graph Theory

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S.M. Lee proposed the conjecture: for any n > 1 and any permutation f in S(n), the permutation graph P(Pₙ,f) is graceful. For any integer n > 1 and permutation f in S(n), we discuss the gracefulness of the permutation graph P(Pₙ,f) if $f = \prod_{k = 0}^{l - 1} (m + 2 k, m + 2 k + 1)$ , and $\prod_{k = 0}^{l - 1} (m + 4 k, m + 4 k + 2) (m + 4 k + 1, m + 4 k + 3)$ for any positive integers m and l.

On Grosswald's conjecture on primitive roots

Stephen D. Cohen, Tomás Oliveira e Silva, Tim Trudgian (2016)

Acta Arithmetica

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Grosswald’s conjecture is that g(p), the least primitive root modulo p, satisfies g(p) ≤ √p - 2 for all p > 409. We make progress towards this conjecture by proving that g(p) ≤ √p -2 for all $409 < p < 2.5 \times 10^{15}$ and for all $p > 3.38 \times 10^{71}$ .

A basis of ℤₘ, II

Min Tang, Yong-Gao Chen (2007)

Colloquium Mathematicae

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Given a set A ⊂ ℕ let $σ_{A} (n)$ denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, $σ_{A} (n)$ is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and $σ_{A} (n ̅) \leq 5120$ for all n̅ ∈ ℤₘ.