Geometric approaches to establish the fundamentals of Lorentz spaces $\mathbb {R}_2^3$ and $\mathbb {R}_1^2$

Sevilay Çoruh Şenocak; Salim Yüce

Displaying similar documents to “Geometric approaches to establish the fundamentals of Lorentz spaces $\mathbb {R}_2^3$ and $\mathbb {R}_1^2$”

Fundamental solution $E_{n}$ of the operator $\partial^{2} / \partial t^{2} - Δ_{n}$ for n>3

Zofia Szmydt, Bogdan Ziemian (1979)

Annales Polonici Mathematici

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How many clouds cover the plane?

James H. Schmerl (2003)

Fundamenta Mathematicae

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The plane can be covered by n + 2 clouds iff $2^{ℵ ₀} \leq ℵ ₙ$ .

Hyperideal polyhedra in hyperbolic 3-space

Xiliang Bao, Francis Bonahon (2002)

Bulletin de la Société Mathématique de France

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A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic $3$ -space $ℍ^{3}$ which, in the projective model for $ℍ^{3} \subset {ℝℙ}^{3}$ , is just the intersection of $ℍ^{3}$ with a projective polyhedron whose vertices are all outside $ℍ^{3}$ and whose edges all meet $ℍ^{3}$ . We classify hyperideal polyhedra, up to isometries of $ℍ^{3}$ , in terms of their combinatorial type and of their dihedral angles.

The existence of Carathéodory solutions of hyperbolic functional differential equations

Adrian Karpowicz (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We consider the following Darboux problem for the functional differential equation $\partial ² u / \partial x \partial y (x, y) = f (x, y, u_{(x, y)}, \partial u / \partial x (x, y), \partial u / \partial y (x, y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b] $0, a] \times (0, b],$ where the function $u_{(x, y)} : [- a ₀, 0] \times [- b ₀, 0] \to ℝ^{k}$ is defined by $u_{(x, y)} (s, t) = u (s + x, t + y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.

A multiparameter variant of the Salem-Zygmund central limit theorem on lacunary trigonometric series

Mordechay B. Levin (2013)

Colloquium Mathematicae

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We prove the central limit theorem for the multisequence $\sum_{1 \leq n ₁ \leq N ₁} \dots \sum_{1 \leq n_{d} \leq N_{d}} a_{n ₁, . . ., n_{d}} c o s (⟨ 2 π m, A ₁^{n ₁} . . . A_{d}^{n_{d}} x ⟩)$ where $m \in ℤ^{s}$ , $a_{n ₁, . . ., n_{d}}$ are reals, $A ₁, . . ., A_{d}$ are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in ${[0, 1]}^{s}$ . The main tool is the S-unit theorem.

The Niemytzki plane is $ϰ$ -metrizable

Wojciech Bielas, Andrzej Kucharski, Szymon Plewik (2021)

Mathematica Bohemica

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We prove that the Niemytzki plane is $ϰ$ -metrizable and we try to explain the differences between the concepts of a stratifiable space and a $ϰ$ -metrizable space. Also, we give a characterisation of $ϰ$ -metrizable spaces which is modelled on the version described by Chigogidze.

G-matrices, $J$ -orthogonal matrices, and their sign patterns

Frank J. Hall, Miroslav Rozložník (2016)

Czechoslovak Mathematical Journal

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A real matrix $A$ is a G-matrix if $A$ is nonsingular and there exist nonsingular diagonal matrices $D_{1}$ and $D_{2}$ such that $A^{- T} = D_{1} A D_{2}$ , where $A^{- T}$ denotes the transpose of the inverse of $A$ . Denote by $J = diag (\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+ 1$ or $- 1$ . A nonsingular real matrix $Q$ is called $J$ -orthogonal if $Q^{T} J Q = J$ . Many connections are established between these matrices. In particular, a matrix $A$ is a G-matrix if and only if $A$ is diagonally (with positive diagonals) equivalent to a column permutation...

Sharp bounds for the intersection of nodal lines with certain curves

Junehyuk Jung (2014)

Journal of the European Mathematical Society

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Let $Y$ be a hyperbolic surface and let $φ$ be a Laplacian eigenfunction having eigenvalue $- 1 / 4 - τ^{2}$ with $τ > 0$ . Let $N (φ)$ be the set of nodal lines of $φ$ . For a fixed analytic curve $γ$ of finite length, we study the number of intersections between $N (φ)$ and $γ$ in terms of $τ$ . When $Y$ is compact and $γ$ a geodesic circle, or when $Y$ has finite volume and $γ$ is a closed horocycle, we prove that $γ$ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between $N (φ)$ and $γ$ is $O (τ)$ . This bound is...

Hyperbolic spaces in Teichmüller spaces

Christopher J. Leininger, Saul Schleimer (2014)

Journal of the European Mathematical Society

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We prove, for any $n$ , that there is a closed connected orientable surface $S$ so that the hyperbolic space $ℍ^{n}$ almost-isometrically embeds into the Teichmüller space of $S$ , with quasi-convex image lying in the thick part. As a consequence, $ℍ^{n}$ quasi-isometrically embeds in the curve complex of $S$ .

Linear preservers of rc-majorization on matrices

Mohammad Soleymani (2024)

Czechoslovak Mathematical Journal

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Let $A$ , $B$ be $n \times m$ matrices. The concept of matrix majorization means the $j$ th column of $A$ is majorized by the $j$ th column of $B$ and this is done for all $j$ by a doubly stochastic matrix $D$ . We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

Nonexistence results for the Cauchy problem of some systems of hyperbolic equations

Mokhtar Kirane, Salim Messaoudi (2002)

Annales Polonici Mathematici

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We consider the systems of hyperbolic equations ⎧ $u ₜ ₜ = Δ (a (t, x) u) + Δ (b (t, x) v) + {h (t, x) | v |}^{p}$ , t > 0, $x \in ℝ^{N}$ , (S1) ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {k (t, x) | u |}^{q}$ , t > 0, $x \in ℝ^{N}$ ⎧ $u ₜ ₜ = Δ (a (t, x) u) + {h (t, x) | v |}^{p}$ , t > 0, $x \in ℝ^{N}$ , (S2) ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {l (t, x) | v |}^{m} + k (t, x) {| u |}^{q}$ , t > 0, $x \in ℝ^{N}$ , (S3) ⎧ $u ₜ ₜ = Δ (a (t, x) u) + Δ (b (t, x) v) + {h (t, x) | u |}^{p}$ , t > 0, $x \in ℝ^{N}$ , ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {k (t, x) | v |}^{q}$ , t > 0, $x \in ℝ^{N}$ , in $(0, \infty) \times ℝ^{N}$ with u(0,x) = u₀(x), v(0,x) = v₀(x), uₜ(0,x) = u₁(x), vₜ(0,x) = v₁(x). We show that, in each case, there exists a bound B on N such that for 1 ≤ N ≤ B solutions to the systems blow up in finite time.

Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$

Daniel Uzcátegui Contreras, Dardo Goyeneche, Ondřej Turek, Zuzana Václavíková (2021)

Communications in Mathematics

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It is known that a real symmetric circulant matrix with diagonal entries $d \geq 0$ , off-diagonal entries $\pm 1$ and orthogonal rows exists only of order $2 d + 2$ (and trivially of order $1$ ) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries $d \geq 0$ and any complex entries of absolute value $1$ off the diagonal. As a particular case, we consider...

Effective bounds for Faltings’s delta function

Jay Jorgenson, Jürg Kramer (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces $X$ , nowadays called Faltings’s delta function and here denoted by $δ_{Fal} (X)$ . For a given compact Riemann surface $X$ of genus $g_{X} = g$ , the invariant $δ_{Fal} (X)$ is roughly given as minus the logarithm of the distance with respect to the Weil-Petersson metric of the point in the moduli space $ℳ_{g}$ of genus $g$ curves determined by $X$ to its boundary $\partial ℳ_{g}$ . In this paper we begin by revisiting a formula derived...

$(m, r)$ -central Riordan arrays and their applications

Sheng-Liang Yang, Yan-Xue Xu, Tian-Xiao He (2017)

Czechoslovak Mathematical Journal

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For integers $m > r \geq 0$ , Brietzke (2008) defined the $(m, r)$ -central coefficients of an infinite lower triangular matrix $G = (d, h) = {(d_{n, k})}_{n, k \in ℕ}$ as $d_{m n + r, (m - 1) n + r}$ , with $n = 0, 1, 2, \dots$ , and the $(m, r)$ -central coefficient triangle of $G$ as $G^{(m, r)} = {(d_{m n + r, (m - 1) n + k + r})}_{n, k \in ℕ} .$ It is known that the $(m, r)$ -central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array $G = (d, h)$ with $h (0) = 0$ and $d (0), h^{'} (0) \neq 0$ , we obtain the generating function of its $(m, r)$ -central coefficients and give an explicit representation for the $(m, r)$ -central Riordan array $G^{(m, r)}$ in terms of the Riordan array $G$ ....

On the structural theory of ${II}_{1}$ factors of negatively curved groups

Ionut Chifan, Thomas Sinclair (2013)

Annales scientifiques de l'École Normale Supérieure

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Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor $L Γ$ is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that $L Γ$ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in $Sp (n, 1)$ , $n \geq 2$ , are virtually $W^{*}$ -superrigid.

Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps

Martine Babillot, Marc Peigné (2006)

Bulletin de la Société Mathématique de France

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We consider a large class of non compact hyperbolic manifolds $M = ℍ^{n} / Γ$ with cusps and we prove that the winding process $(Y_{t})$ generated by a closed $1$ -form supported on a neighborhood of a cusp $𝒞$ , satisfies a limit theorem, with an asymptotic stable law and a renormalising factor depending only on the rank of the cusp $𝒞$ and the Poincaré exponent $δ$ of $Γ$ . No assumption on the value of $δ$ is required and this theorem generalises previous results due to Y. Guivarc’h, Y. Le Jan, J. Franchi and N. Enriquez. ...

On the combinatorial structure of $0 / 1$ -matrices representing nonobtuse simplices

Jan Brandts, Abdullah Cihangir (2019)

Applications of Mathematics

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A $0 / 1$ -simplex is the convex hull of $n + 1$ affinely independent vertices of the unit $n$ -cube $I^{n}$ . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute $0 / 1$ -simplices in $I^{n}$ can be represented by $0 / 1$ -matrices $P$ of size $n \times n$ whose Gramians $G = P^{⊤} P$ have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part $D$ of the transposed inverse $P^{- ⊤}$ of $P$ is doubly stochastic and has the...

Nonconventional limit theorems in averaging

Yuri Kifer (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider “nonconventional” averaging setup in the form $\frac{d X^{ε} (t)}{d t} = ε B (X^{ε} (t)$ , $𝛯 (q_{1} (t)), 𝛯 (q_{2} (t)), ..., 𝛯 (q_{ℓ} (t)))$ where $𝛯 (t)$ , $t \geq 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_{j} (t) = α_{j} t$ , $α_{1} l t; α_{2} l t; \dots l t; α_{k}$ and $q_{j}$ , $j = k + 1, ..., ℓ$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

Hyperbolic measure of maximal entropy for generic rational maps of $ℙ^{k}$

Gabriel Vigny (2014)

Annales de l’institut Fourier

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Let $f$ be a dominant rational map of $ℙ^{k}$ such that there exists $s < k$ with $λ_{s} (f) > λ_{l} (f)$ for all $l$ . Under mild hypotheses, we show that, for $A$ outside a pluripolar set of $Aut (ℙ^{k})$ , the map $f \circ A$ admits a hyperbolic measure of maximal entropy $log λ_{s} (f)$ with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of $f$ to $ℙ^{k + 1}$ . This provides many examples where non uniform hyperbolic dynamics is established. One of the key tools is to approximate...