Unitary designs and codes

A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code - a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct values - and give an upper bound for the size of a code of degree s in U(d) for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary t-design in U(d), and we catalogue some t-designs that arise from finite groups.Comment: 25 pages, no figure

Eine Test- und Ansteuerschaltung für eine neuartige 3D Verbindungstechnologie

In der vorliegenden Arbeit wird eine Built-In Self-Test Schaltung (BIST) vorgestellt, welche die vertikalen Inter-Chip-Verbindungen in einer neuartigen 3D Schaltungstechnologie auf ihre Funktionalität zur Datenübertragung überprüft. Die 3D Technologie beruht auf der Stapelung mehrerer aktiver Silizium-CMOS-ICs, welche durch das Siliziumsubstrat hindurch vertikal miteinander elektrisch verbunden sind. Bei diesen Vias sind die zu erwartenden Defekte hochohmige Verbindungen und Kurzschlüsse. </p><p style=&quot;line-height: 20px;&quot;> Die entwickelte Testschaltung ermöglicht es, beliebige Konstellationen von vertikalen Verbindungen auf Fehler zu untersuchen, und das Ergebnis entweder zur Analyse der 3D Technologie auszulesen oder innerhalb des Chipstapels zu verwenden, um defekte Vias zu umgehen. Die Schaltung wurde in einer 0,13μm Technologie entworfen und simuliert. Ein Testchip ist momentan in Produktion

Yield-improving test and routing circuits for a novel 3-D interconnect technology

This work presents a system to increase the yield of a novel 3-D chip integration technology. A built-in self-test and a routing system have been developed to identify and avoid faults on vertical connections between different stacked chips. The 3-D technology is based on stacking several active CMOS-ICs, which have through-substrate electrical contacts to communicate with each other. The expected defects of these vias are shorts and resistances that are too high. <P> The test and routing system is designed to analyze an arbitrary number of connections. The result ist used to gain information about the reliability of the new 3-D processing and to increase its yield. The circuits have been developed in 0.13 μm technology, one chip has been fabricated and tested, another one is in production

Commutator Leavitt path algebras

For any field K and directed graph E, we completely describe the elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E),L_K(E)]. We then use this result to classify all Leavitt path algebras L_K(E) that satisfy L_K(E)=[L_K(E),L_K(E)]. We also show that these Leavitt path algebras have the additional (unusual) property that all their Lie ideals are (ring-theoretic) ideals, and construct examples of such rings with various ideal structures.Comment: 24 page

On the automorphisms of quantum Weyl algebras

Motivated by Weyl algebra analogues of the Jacobian conjecture and the Tame Generators problem, we prove quantum versions of these problems for a family of analogues to the Weyl algebras. In particular, our results cover the Weyl-Hayashi algebras and tensor powers of a quantization of the first Weyl algebra which arises as a primitive factor algebra of U+q(so5)

Two-parameter Quantum Affine Algebra Ur,s(sln^)U_{r,s}(\widehat{\frak {sl}_n}), Drinfel'd Realization and Quantum Affine Lyndon Basis

We further define two-parameter quantum affine algebra $U_{r,s}(\widehat{\frak {sl}_n})$ $(n>2)$ after the work on the finite cases (see [BW1], [BGH1], [HS] & [BH]), which turns out to be a Drinfel'd double. Of importance for the quantum {\it affine} cases is that we can work out the compatible two-parameter version of the Drinfel'd realization as a quantum affinization of $U_{r,s}(\frak{sl}_n)$ and establish the Drinfel'd isomorphism Theorem in the two-parameter setting, via developing a new combinatorial approach (quantum calculation) to the quantum {\it affine} Lyndon basis we present (with an explicit valid algorithm based on the use of Drinfel'd generators).Comment: 31 page

Tensor product Markov chains

We analyze families of Markov chains that arise from decomposing ten-sor products of irreducible representations. This illuminates the Burnside-Brauer Theorem for building irreducible representations, the McKay Corre-spondence, and Pitman’s2M−XTheorem. The chains are explicitly di-agonalizable, and we use the eigenvalues/eigenvectors to give sharp rates ofconvergence for the associated random walks. For modular representations,the chains are not reversible, and the analytical details are surprisingly intri-cate. In the quantum group case, the chains fail to be diagonalizable, but anovel analysis using generalized eigenvectors proves successful

Analytical and numerical modelling of AlGaNnext term/previous termGaNnext term/previous termAlNnext term heterostructure based cantilevers for mechanical sensing in harsh environments

International audienceSome industrial areas as oil, automotive and aerospace industries, require electromechanical systems working in harsh environments. An elegant solution is to use III-V materials alloys having semiconductor, piezoelectric and pyroelectric properties. These materials, particularly nitrides such as previous termGaNnext term or previous termAlNnext term, enable design of advanced devices suitable for harsh environment. A cantilever structure based on previous termAlGaNnext term/previous termGaNnext term/previous termAlN heterostructures coupled with a High Electron Mobility Transistor (HEMT) can act as an electromechanical device suited for sensing applications. In this article, we present the mechanical modelling of such a structure. An analytical and a numerical model have been developed to obtain the electrical charge distribution in the structure in response to mechanical stress. A theoretical electromechanical sensitivity of 3.5 μC m−2 was achieved for the cantilever free end displacement of several hundreds of nanometres. Both models show good agreement, presenting less than 5% deviation in almost the whole structure. The differences between the two models that are pronounced near the clamped area can be explained by particular boundary conditions of the numerical model. The topological characterization and numerical modelling allowed the estimation of the equivalent intrinsic residual stress in the structure and the stress distribution within each layer. Finally, the dynamic mechanical characterization of fabricated cantilevers using laser interferometry is presented and compared to numerical modal analysis with less than 10% deviation between theoretical and experimental resonant frequencies. The obtained results enable the use of the analytical model for further study of the electromechanical coupling with the HEMT structure