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Search: All articles in the CMB digital archive with keyword trace

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1. CMB Online first

Shahrokhi-Dehkordi, M. S.
Traceless maps as the singular minimizers in the multi-dimensional calculus of variations
Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain and consider the energy functional \begin{equation*} {\mathcal F}[u, \Omega] := \int_{\Omega} {\rm F}(\nabla {\bf u}(\bf x))\, d{\bf x}, \end{equation*} over the space of $W^{1,2}(\Omega, \mathbb{R}^m)$ where the integrand ${\rm F}: \mathbb M_{m\times n}\to \mathbb{R}$ is a smooth uniformly convex function with bounded second derivatives. In this paper we address the question of regularity for solutions of the corresponding system of Euler-Lagrange equations. In particular we introduce a class of singular maps referred to as traceless and examine them as a new counterexample to the regularity of minimizers of the energy functional $\mathcal F[\cdot,\Omega]$ using a method based on null Lagrangians.

Keywords:traceless map, singular minimizer, null-Lagrangian
Category:34AXX

2. CMB Online first

Le Fourn, Samuel
Nonvanishing of central values of $L$-functions of newforms in $S_2 (\Gamma_0 (dp^2))$ twisted by quadratic characters
We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $\chi$, if a prime $p$ is large enough compared to $D$, there is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \otimes \chi,1) \neq 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions which generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$-functions $L(f \otimes \chi, \cdot)$ and a Petersson trace formula restricted to Atkin-Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

Keywords:nonvanishing of $L$-functions of modular forms, Petersson trace formula, rank zero quotients of jacobians
Categories:14J15, 11F67

3. CMB Online first

Ghaani Farashahi, Arash
Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups
This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be the normalized $G$-invariant measure on $G/H$ associated to the Weil's formula. Then, we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space $L^2(G/H,\mu)$.

Keywords:compact group, homogeneous space, dual space, Plancherel (trace) formula
Categories:20G05, 43A85, 43A32, 43A40

4. CMB 2016 (vol 59 pp. 585)

Lin, Minghua
A Determinantal Inequality Involving Partial Traces
Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes \mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved an inequality for Schatten $p$-norms: \[ 1+\|\mathbf{A}\|_p\ge \|\tr_1 \mathbf{A}\|_p+\|\tr_2 \mathbf{A}\|_p, \] where $\tr_1, \tr_2$ stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality \[ 1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n. \]

Keywords:determinantal inequality, partial trace, block matrix
Categories:47B65, 15A45, 15A60

5. CMB 2009 (vol 52 pp. 416)

Malik, Shabnam; Qureshi, Ahmad Mahmood; Zamfirescu, Tudor
Hamiltonian Properties of Generalized Halin Graphs
A Halin graph is a graph $H=T\cup C$, where $T$ is a tree with no vertex of degree two, and $C$ is a cycle connecting the end-vertices of $T$ in the cyclic order determined by a plane embedding of $T$. In this paper, we define classes of generalized Halin graphs, called $k$-Halin graphs, and investigate their Hamiltonian properties.

Keywords:$k$-Halin graph, Hamiltonian, Hamiltonian connected, traceable
Categories:05C45, 05C38

6. CMB 2008 (vol 51 pp. 140)

Rossi, Julio D.
First Variations of the Best Sobolev Trace Constant with Respect to the Domain
In this paper we study the best constant of the Sobolev trace embedding $H^{1}(\Omega)\to L^{2}(\partial\Omega)$, where $\Omega$ is a bounded smooth domain in $\RR^N$. We find a formula for the first variation of the best constant with respect to the domain. As a consequence, we prove that the ball is a critical domain when we consider deformations that preserve volume.

Keywords:nonlinear boundary conditions, Sobolev trace embedding
Categories:35J65, 35B33

7. CMB 2000 (vol 43 pp. 60)

Farkas, Daniel R.; Linnell, Peter A.
Trivial Units in Group Rings
Let $G$ be an arbitrary group and let $U$ be a subgroup of the normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$ as a subgroup of finite index, then $U = G$. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

Keywords:units, trace, finite conjugate subgroup
Categories:16S34, 16U60

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