Leq 2n Research Articles

An Adaptive BIST Design for Detecting Multiple Stuck-Open Faults in a CMOS Complex Cell

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper describes a new adaptive built-in self-test (BIST) technique for detecting stuck-open faults in a CMOS complex cell. A test pattern generator (TPG) that adaptively generates a sequence of single-input-change test pairs based on the past responses of the circuit under test (CUT) is designed. Conventional TPGs produce a predefined sequence of test vectors. The novelty of the proposed approach lies in the fact that the test sequence (TS) generated by the TPG depends on the actual error produced by the CUT during testing. The BIST design is universal, i.e., independent of the structure or functionality of the CUT, and depends only on the number of inputs to the CUT. The length of the TS <formula formulatype="inline"><tex>$(\vert \hbox{TS}\vert)$</tex></formula> also depends on the error behavior; hence, it significantly reduces the average test application time. For an <formula formulatype="inline"><tex>$n$</tex></formula>-input CUT, it is shown that <formula formulatype="inline"> <tex>$4 \leq \vert \hbox{TS} \vert \leq 2n \cdot 2^{n}$</tex></formula>. The design of the response analyzer is also simple. Barring a few exceptions, any irredundant multiple stuck-open faults, including those simultaneously occurring in both the <formula formulatype="inline"><tex>$n$</tex></formula>- and <formula formulatype="inline"><tex> $p$</tex></formula>- parts of a complex cell, are guaranteed to be detected using the proposed BIST scheme. </para>

An Algorithmic Friedman&amp;ndash;Pippenger Theorem on Tree Embeddings and Applications

An $(n, d)$-expander is a graph $G = (V, E)$ such that for every $X \subseteq V$ with $|X| \leq 2n - 2$ we have $|\Gamma_G(X)| \geq (d+1)|X|$. A tree $T$ is small if it has at most $n$ vertices and has maximum degree at most $d$. Friedman and Pippenger (1987) proved that any $(n, d)$-expander contains every small tree. However, their elegant proof does not seem to yield an efficient algorithm for obtaining the tree. In this paper, we give an alternative result that does admit a polynomial time algorithm for finding the immersion of any small tree in subgraphs $G$ of $(N,D,\lambda)$-graphs $\Lambda$, as long as $G$ contains a positive fraction of the edges of $\Lambda$ and $\lambda/D$ is small enough. In several applications of the Friedman–Pippenger theorem, including the ones in the original paper of those authors, the $(n,d)$-expander $G$ is a subgraph of an $(N,D,\lambda)$-graph as above. Therefore, our result suffices to provide efficient algorithms for such previously non-constructive applications. As an example, we discuss a recent result of Alon, Krivelevich, and Sudakov (2007) concerning embedding nearly spanning bounded degree trees, the proof of which makes use of the Friedman–Pippenger theorem. We shall also show a construction inspired on Wigderson–Zuckerman expander graphs for which any sufficiently dense subgraph contains all trees of sizes and maximum degrees achieving essentially optimal parameters. Our algorithmic approach is based on a reduction of the tree embedding problem to a certain on-line matching problem for bipartite graphs, solved by Aggarwal et al. (1996).

Cycle Systems in the Complete Bipartite Graph Plus a One-Factor

Let $K_{n,n}$ denote the complete bipartite graph with n vertices in each partite set and $K_{n,n}+I$ denote $K_{n,n}$ with a one-factor added. It is proved in this paper that there exists an m-cycle system of $K_{n,n}+I$ if and only if $n \equiv 1 (\rm{mod} 2)$, $m \equiv 0 (\rm{mod} 2)$, $4 \leq m \leq 2n$, and $n(n+1) \equiv$ 0 (mod m).

Cycle Decompositions of $K_{\lowercase{n,n}}-I$

Let $K_{n,n}$ denote the complete bipartite graph with $n$ vertices in each bipartition set and $K_{n,n}-I$ denote $K_{n,n}$ with a 1-factor removed. An $m$-cycle system of $K_{n,n}-I$ is a collection $T$ of $m$-cycles such that each edge of $K_{n,n}-I$ is contained in a unique $m$-cycle of $T$. In this paper, it is proved that the necessary and sufficient conditions for the existence of an $m$-cycle system of $K_{n,n}-I$ are $n\equiv 1 \({\rm 2)$, $m\equiv ({\rm 2)$, $4\leq m \leq 2n$, and $n(n-1)\equiv 0 ({\rm mod} m)$.

The singular Embry quartic moment problem

Given a collection of complex numbers $\gamma \equiv \{\gamma _{ij}\}\,\,\,(0\leq i+j\leq 2n,\,\,\,|i-j|\leq n)$ with $\gamma _{00}>0$ and $\gamma _{ji}=\bar{\gamma}_{ij},$ we consider the moment problem for $\gamma$ in the case of $n=2$, which is refered to Embry quartic moment problem. In this note we give a solution for the singular case.

On the Schatten class membership of Hankel operators on the unit ball

A well-known theorem of K. Zhu \cite{6} asserts that, for $2 \leq p <\infty $, the Hankel operators $H_f$ and $H_{\bar f}$ on the Bergman space $L^2_a(B_n,dV)$ of the unit ball belong to the Schatten class ${\mathcal{C}}_p$ if and only if the mean oscillation $\MO(f)(z) = \{\widetilde{|f|^2}(z) - |\tilde f(z)|^2\}^{1/2}$ belongs to $L^p(B_n,(1-|z|^2)^{-n-1}dV(z))$. It is well known that, for trivial reasons, this theorem cannot be extended to the case $p \leq 2n/(n+1)$. This paper fills the gap between $2n/(n+1)$ and 2. More precisely, we prove that, when $2n/(n+1) < p < 2$, the same theorem holds true.

Koszul property for points in projective spaces

A graded $K$-algebra $R$ is said to be Koszul if the minimal $R$-free graded resolution of $K$ is linear. In this paper we study the Koszul property of the homogeneous coordinate ring $R$ of a set of $s$ points in the complex projective space $\boldsymbol P^n$. Kempf proved that $R$ is Koszul if $s\leq 2n$ and the points are in general linear position. If the coordinates of the points are algebraically independent over $\boldsymbol Q$, then we prove that $R$ is Koszul if and only if $s\le 1 +n + n^2/4$. If $s\le 2n$ and the points are in linear general position, then we show that there exists a system of coordinates $x_0,\dots,x_n$ of $\boldsymbol P^n$ such that all the ideals $(x_0,x_1,\dots,x_i)$ with $0\le i \le n$ have a linear $R$-free resolution.

Low-dimensional linear representations of 𝐴𝑢𝑡𝐹_{𝑛},𝑛≥3

We classify all complex representations of A u t F n , \mathrm {Aut} \: F_n, the automorphism group of the free group F n F_n ( n ≥ 3 ) , (n \geq 3), of dimension ≤ 2 n − 2. \leq 2n - 2. Among those representations is a new representation of dimension n + 1 n + 1 which does not vanish on the group of inner automorphisms.

Milnor link invariants and quantum 3-manifold invariants

Let $ {\cal Z}(M) $ be the 3-manifold invariant of Le, Murakami and Ohtsuki. We show that $ {\cal Z}(M) = 1 + o(n) $ , where $ o(n) $ denotes terms of degree $ \geq n $ , if M is a homology 3-sphere obtained from $ S^3 $ by surgery on an n-component Brunnian link whose Milnor $ \overline\mu $ -invariants of length $ \leq 2n $ vanish.¶We prove a realization theorem which is a partial converse to the above theorem.¶Using the Milnor filtration on links, we define a new bifiltration on the $ \Bbb Q $ vector space with basis the set of oriented diffeomorphism classes of homology 3-spheres. This includes the Milnor level 2 filtration defined by Ohtsuki. We show that the Milnor level 2 and level 3 filtrations coincide after reindexing.

MEDIANS OF GRAPHS AND KINGS OF TOURNAMENTS*

We first prove that for any graph $G$ with a positive vertex weight function $w$, there exists a graph $H$ with a positive weight function $w'$ such that $w(v)=w'(v)$ for all vertices $v$ in $G$ and whose $w'$-median is $G$. This is a generalization of a previous result for the case in which all weights are 1. The second result is that for any $n$-tournament $T$ without transmitters, there exists an integer $m\leq 2n-1$ and an $m$-tournament $T'$ whose kings are exactly the vertices of $T$. This improves upon a previous result for $m\leq 2n$.

On the $\lambda$-Number of $Q_n $ and Related Graphs

An $L(2, l)$-labeling of graph $G$ is an integer labeling of $V(G)$ such that adjacent vertices have labels that differ by at least 2 and such that vertices distance 2 apart have labels that differ by at least 1. The $\lambda$-number of $G, \lambda(G)$, is the minimum range over all $L(2, 1)$-labelings. We examine the properties of $\lambda$-labelings of the $n$-cube $Q_n$. Griggs and Yeh have determined $\lambda(Q_n)$ for $n \leq 5$ and have established $n + 3 \leq \lambda(Q_n) \leq 2n + 1$ for $n \geq 6$. We modify a technique used in coding theory to improve the upper bound. We also examine the $\lambda$-labelings of related graphs, such as the subdivision of the $n$-cube and the Cartesian products of paths.

Complex interpolating polynomials

Let I n , m ( f , z ) {I_{n,m}}\left ( {f,z} \right ) be the unique interpolatory polynomial of degree ≤ 2 n − 1 \leq 2n - 1 satisfying the conditions given by (1.1) where the z k n {z_{kn}} ’s are the zeros of the polynomial z n − 1 {z^n} - 1 . The object of this paper is to consider the rate of convergence of I m , n ( f , z ) {I_{m,n}}\left ( {f,z} \right ) to f ( z ) f\left ( z \right ) in the L p {L_p} norm where f ∈ C [ | z | ≤ 1 ] f \in C\left [ {|z| \leq 1} \right ] . This problem was initially raised by P. Turán in the case p = 2 p = 2 and in this case the solution was obtained by J. Szabados and A. K. Varma in [7].

Essential Dimension Lowering Mappings having Dense Deficiency Set

Two classes of surjective maps $f:{S^m} \to {S^n}$ that are one-to-one over the image of a dense set are constructed. We show that for $m,n \geq 3$ there is a monotone surjection $f:{S^m} \to {S^n}$ that is one-to-one over the image of a dense set; and for $3 \leq n \leq m \leq 2n - 3$, each element of ${\pi _m}({S^n})$ can be represented as a monotone surjection $f:{S^m} \to {S^n}$ that is one-to-one over the image of a dense set.

An analytic stability test for a class of time-delay systems

Stability properties of linear dynamic systems with characteristic equations of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Sigma\min{i=0}\max{n} P_{i}(s)e^{-shi}=0</tex> , where all the polynomials P <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> are of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\leq r</tex> in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> , can be determined from a polynomial equation in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\leq 2n + r</tex> .

Deligne–Beilinson cycle maps for Lichtenbaum cohomology

We define Deligne-Beilinson cycle maps for Lichtenbaum cohomology $H_L^m(X, \mathbb Z(n))$ and that with compact supports $H_{c,L}^m(X, \mathbb Z(n))$ of an arbitrary complex algebraic variety $X.$ When $(m,n)=(2,1),$ the homological part of our cycle map with compact supports gives a generalization of the Abel-Jacobi theorem and its projection to the Betti cohomology yields that of the Lefschetz theorem on $(1,1)$-cycles for arbitrary complex algebraic varieties. In general degrees $(m,n),$ we show that the Deligne-Beilinson cycle maps are always surjective on torsion and have torsion-free cokernels. If $m \leq 2n,$ the version with compact supports induces an isomorphism on torsion, and so does the one without compact supports if $min \{2m-1, 2 \dim X+1 \} \leq 2n.$ We also characterize the algebraic part of Griffiths's intermediate Jacobians with a universal property.