Integral representations of risk functions for basket derivatives

Michał Barski

Displaying similar documents to “Integral representations of risk functions for basket derivatives”

Representations of a class of positively based algebras

Shiyu Lin, Shilin Yang (2023)

Czechoslovak Mathematical Journal

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We investigate the representation theory of the positively based algebra $A_{m, d}$ , which is a generalization of the noncommutative Green algebra of weak Hopf algebra corresponding to the generalized Taft algebra. It turns out that $A_{m, d}$ is of finite representative type if $d \leq 4$ , of tame type if $d = 5$ , and of wild type if $d \geq 6 .$ In the case when $d \leq 4$ , all indecomposable representations of $A_{m, d}$ are constructed. Furthermore, their right cell representations as well as left cell representations of $A_{m, d}$ are described. ...

Formal deformations and principal series representations of $SL (2, ℝ)$ and $SL (2, ℂ)$

Benjamin Cahen (2020)

Czechoslovak Mathematical Journal

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In this note, we study formal deformations of derived representations of the principal series representations of $SL (2, ℝ)$ . In particular, we recover all the representations of the derived principal series by deforming one of them. Similar results are also obtained for $SL (2, ℂ)$ .

Stacks of group representations

Paul Balmer (2015)

Journal of the European Mathematical Society

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We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$ , the derived and the stable categories of representations of a subgroup $H$ can be constructed out of the corresponding category for $G$ by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods...

Irreducibility of automorphic Galois representations of $G L (n)$ , $n$ at most $5$

Frank Calegari, Toby Gee (2013)

Annales de l’institut Fourier

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Let $π$ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of ${GL}_{n} (𝔸_{F})$ , where $F$ is a totally real field and $n$ is at most $5$ . We show that for all primes $l$ , the $l$ -adic Galois representations associated to $π$ are irreducible, and for all but finitely many primes $l$ , the mod $l$ Galois representations associated to $π$ are also irreducible. We also show that the Lie algebras of the Zariski closures of the $l$ -adic representations are independent of $l$ .

Dual Blobs and Plancherel Formulas

Ju-Lee Kim (2004)

Bulletin de la Société Mathématique de France

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Let $k$ be a $p$ -adic field. Let $G$ be the group of $k$ -rational points of a connected reductive group $𝖦$ defined over $k$ , and let $𝔤$ be its Lie algebra. Under certain hypotheses on $𝖦$ and $k$ , wethe tempered dual $\hat{G}$ of $G$ via the Plancherel formula on $𝔤$ , using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on $𝔤$ and $G$ . As a consequence, we prove that any tempered representation contains a good minimal $𝖪$ -type; we extend this result to irreducible...

Asymptotic values of modular multiplicities for ${GL}_{2}$

Sandra Rozensztajn (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the irreducible constituents of the reduction modulo $p$ of irreducible algebraic representations $V$ of the group ${Res}_{K / ℚ_{p}} {GL}_{2}$ for $K$ a finite extension of $ℚ_{p}$ . We show that asymptotically, the multiplicity of each constituent depends only on the dimension of $V$ and the central character of its reduction modulo $p$ . As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

Locally analytic vectors of unitary principal series of ${GL}_{2} (ℚ_{p})$

Ruochuan Liu, Bingyong Xie, Yuancao Zhang (2012)

Annales scientifiques de l'École Normale Supérieure

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The $p$ -adic local Langlands correspondence for ${GL}_{2} (ℚ_{p})$ attaches to any $2$ -dimensional irreducible $p$ -adic representation $V$ of $G_{ℚ_{p}}$ an admissible unitary representation $Π (V)$ of ${GL}_{2} (ℚ_{p})$ . The unitary principal series of ${GL}_{2} (ℚ_{p})$ are those $Π (V)$ corresponding to trianguline representations. In this article, for $p > 2$ , using the machinery of Colmez, we determine the space of locally analytic vectors $Π {(V)}_{an}$ for all non-exceptional unitary principal series $Π (V)$ of ${GL}_{2} (ℚ_{p})$ by proving a conjecture of Emerton.

The harmonic Cesáro and Copson operators on the spaces $L^{p} (ℝ)$ , 1 ≤ p ≤ 2

Ferenc Móricz (2002)

Studia Mathematica

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The harmonic Cesàro operator is defined for a function f in $L^{p} (ℝ)$ for some 1 ≤ p < ∞ by setting $(f) (x) : = \int_{x}^{\infty} (f (u) / u) d u$ for x > 0 and $(f) (x) : = - \int_{- \infty}^{x} (f (u) / u) d u$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L ¹_{l o c} (ℝ)$ by setting $* (f) (x) : = (1 / x) \int^{x ₀} f (u) d u$ for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If $f \in L^{p} (ℝ)$ for some 1 ≤ p ≤ 2, then ${((f))}^{\land} (t) = * (f ̂) (t)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f \in L^{p} (ℝ)$ for some 1 < p ≤ 2, then...

$L_{p, q}$ spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p, q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p, q} (B)$ ........ 235. Examples in $L_{p, q}$ theory...................................................................................

On the infinite fern of Galois representations of unitary type

Gaëtan Chenevier (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $E$ be a CM number field, $p$ an odd prime totally split in $E$ , and let $X$ be the $p$ -adic analytic space parameterizing the isomorphism classes of $3$ -dimensional semisimple $p$ -adic representations of $Gal (\bar{E} / E)$ satisfying a selfduality condition “of type $U (3)$ ”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3 [E : ℚ]$ . As important steps, and in any rank, we prove that any...

Optimal estimators in learning theory

V. N. Temlyakov (2006)

Banach Center Publications

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This paper is a survey of recent results on some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs $x_{i}$ and outputs $y_{i}$ , i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator $f_{z}$ on the base of given data $z : = ((x ₁, y ₁), . . ., (x_{m}, y_{m}))$ that approximates well the regression function...

Theoretical analysis for $ℓ_{1}$ - $ℓ_{2}$ minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

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We investigate the recovery of $k$ -sparse signals using the $ℓ_{1}$ - $ℓ_{2}$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$ -sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...

On reliability analysis of consecutive $k$ -out-of- $n$ systems with arbitrarily dependent components

Ebrahim Salehi (2016)

Applications of Mathematics

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In this paper, we consider the linear and circular consecutive $k$ -out-of- $n$ systems consisting of arbitrarily dependent components. Under the condition that at least $n - r + 1$ components ( $r \leq n$ ) of the system are working at time $t$ , we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive $k$ -out-of- $n$ systems. In the following, we investigate the inactivity time of the component with lifetime...

Representations of the general linear group over symmetry classes of polynomials

Yousef Zamani, Mahin Ranjbari (2018)

Czechoslovak Mathematical Journal

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Let $V$ be the complex vector space of homogeneous linear polynomials in the variables $x_{1}, ..., x_{m}$ . Suppose $G$ is a subgroup of $S_{m}$ , and $χ$ is an irreducible character of $G$ . Let $H_{d} (G, χ)$ be the symmetry class of polynomials of degree $d$ with respect to $G$ and $χ$ . For any linear operator $T$ acting on $V$ , there is a (unique) induced operator $K_{χ} (T) \in End (H_{d} (G, χ))$ acting on symmetrized decomposable polynomials by $K_{χ} (T) (f_{1} * f_{2} * ... * f_{d}) = T f_{1} * T f_{2} * ... * T f_{d} .$ In this paper, we show that the representation $T \mapsto K_{χ} (T)$ of the general linear group $G L (V)$ is equivalent to the direct sum of $χ (1)$ copies...

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

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We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$ , then for every subset $B$ of $G$ with $| B | > | G | / k^{1 / 3}$ we have $B^{3} = G$ . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k \geq 2$ , then $G$ has a...

Representation growth of linear groups

Michael Larsen, Alexander Lubotzky (2008)

Journal of the European Mathematical Society

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Let $Γ$ be a group and $r_{n} (Γ)$ the number of its $n$ -dimensional irreducible complex representations. We define and study the associated representation zeta function $𝒵_{Γ} (s) = \sum_{n = 1}^{\infty} r_{n} (Γ) n^{- s}$ . When $Γ$ is an arithmetic group satisfying the congruence subgroup property then $𝒵_{Γ} (s)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite...

Optimization problem under two-sided (max, +)/(min, +) inequality constraints

Karel Zimmermann (2020)

Applications of Mathematics

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$(max, +)$ -linear functions are functions which can be expressed as the maximum of a finite number of linear functions of one variable having the form $f (x_{1}, \dots, x_{h}) = {max}_{j} (a_{j} + x_{j})$ , where $a_{j}$ , $j = 1, \dots, h$ , are real numbers. Similarly $(min, +)$ -linear functions are defined. We will consider optimization problems in which the set of feasible solutions is the solution set of a finite inequality system, where the inequalities have $(max, +)$ -linear functions of variables $x$ on one side and $(min, +)$ -linear functions of variables $y$ on the other side....