Determining when the absolute state complexity of a Hermitian code achieves its DLP bound

Let g be the genus of the Hermitian function field H/F(q)2 and let C-L(D,mQ(infinity)) be a typical Hermitian code of length n. In [Des. Codes Cryptogr., to appear], we determined the dimension/length profile (DLP) lower bound on the state complexity of C-L(D,mQ(infinity)). Here we determine when this lower bound is tight and when it is not. For m less than or equal to n-2/2 or m greater than or equal to n-2/2 + 2g, the DLP lower bounds reach Wolf's upper bound on state complexity and thus are trivially tight. We begin by showing that for about half of the remaining values of m the DLP bounds cannot be tight. In these cases, we give a lower bound on the absolute state complexity of C-L(D,mQ(infinity)), which improves the DLP lower bound. Next we give a good coordinate order for C-L(D,mQ(infinity)). With this good order, the state complexity of C-L(D,mQ(infinity)) achieves its DLP bound (whenever this is possible). This coordinate order also provides an upper bound on the absolute state complexity of C-L(D,mQ(infinity)) (for those values of m for which the DLP bounds cannot be tight). Our bounds on absolute state complexity do not meet for some of these values of m, and this leaves open the question whether our coordinate order is best possible in these cases. A straightforward application of these results is that if C-L(D,mQ(infinity)) is self-dual, then its state complexity (with respect to the lexicographic coordinate order) achieves its DLP bound of n /2 - q(2)/4, and, in particular, so does its absolute state complexity.

Lower Bounds on the State Complexity of Geometric Goppa Codes

We reinterpret the state space dimension equations for geometric Goppa codes. An easy consequence is that if deg G less than or equal to n-2/2 or deg G greater than or equal to n-2/2 + 2g then the state complexity of C-L(D, G) is equal to the Wolf bound. For deg G is an element of [n-1/2, n-3/2 + 2g], we use Clifford's theorem to give a simple lower bound on the state complexity of C-L(D, G). We then derive two further lower bounds on the state space dimensions of C-L(D, G) in terms of the gonality sequence of F/F-q. (The gonality sequence is known for many of the function fields of interest for defining geometric Goppa codes.) One of the gonality bounds uses previous results on the generalised weight hierarchy of C-L(D, G) and one follows in a straightforward way from first principles; often they are equal. For Hermitian codes both gonality bounds are equal to the DLP lower bound on state space dimensions. We conclude by using these results to calculate the DLP lower bound on state complexity for Hermitian codes.

Local complexity of Delone sets and crystallinity

This paper characterizes when a Delone set X in R-n is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set X, let N-X (T) count the number of translation-inequivalent patches of radius T in X and let M-X (T) be the minimum radius such that every closed ball of radius M-X(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a gap in the spectrum of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to X being an ideal crystal. Explicitly, for N-X (T), if R is the covering radius of X then either N-X (T) is bounded or N-X (T) greater than or equal to T/2R for all T > 0. The constant 1/2R in this bound is best possible in all dimensions. For M-X(T), either M-X(T) is bounded or M-X(T) greater than or equal to T/3 for all T > 0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has M-X(T) greater than or equal to c(n)T for all T > 0, for a certain constant c(n) which depends on the dimension n of X and is > 1/3 when n > 1.

Lower bounds on the complexity of simulating quantum gates

We give a simple proof of a formula for the minimal time required to simulate a two-qubit unitary operation using a fixed two-qubit Hamiltonian together with fast local unitaries. We also note that a related lower bound holds for arbitrary n-qubit gates.

Capturing the complexity of women's experiences: A mixed-method approach to studying incontinence in older women

Capturing the voices of women when the issue is of a sensitive nature has been a major concern of feminist researchers. It has often been argued that interpretive methods are the most appropriate way to collect such information, but there are other appropriate ways to approach the design of research. This article explores the use of a mixed-method approach to collect data on incontinence in older women and argues for the use of a variety of creative approaches to collect and analyze data.

Unravelling the complexity of muscle impairment in chronic neck pain

Exercise interventions are deemed essential for the effective management of patients with neck pain. However, there has been a lack of consensus on optimal exercise prescription, which has resulted from a paucity of studies to quantify the precise nature of muscle impairment, in people with neck pain. This masterclass will present recent research from our laboratory, which has utilized surface electromyography to investigate cervical flexor muscle impairment in patients with chronic neck pain. This research has identified deficits in the motor control of the deep and superficial cervical flexor muscles in people with chronic neck pain, characterized by a delay in onset of neck muscle contraction associated with movement of the upper limb. In addition, people with neck pain demonstrate an altered pattern of muscle activation, which is characterized by reduced deep cervical flexor muscle activity during a low load cognitive task and increased activity of the superficial cervical flexor muscles during both cognitive tasks and functional activities. The results have demonstrated the complex, multifaceted nature of cervical muscle impairment, which exists in people with a history of neck pain. In turn, this has considerable implications for the rehabilitation of muscle function in people with neck pain disorders. (C) 2004 Elsevier Ltd. All rights reserved.

The complexity of drinking: Interactions between the cognitive and behavioural determinants of alcohol consumption

This study expanded the earlier work conducted by this laboratory ( Hasking, P.A. and Oei, T.P.S. (2002a) . The differential role of alcohol expectancies, drinking refusal self-efficacy and coping resources in predicting alcohol consumption in community and clinical samples. Addiction Research and Theory , 10 , 465-494), by examining the independent and interactive effects of avoidant coping strategies, positive and negative expectancies and self-efficacy, in predicting volume and frequency of alcohol consumption in a sample of community drinkers. Differential relationships were found between the variables when predicting the two consumption measures. Specifically, while self-efficacy, seeking social support for emotional reasons and using drugs or alcohol to cope were independently related to both volume and frequency of drinking, complex interactions with positive and negative alcohol expectancies were also found. These interactions are discussed in terms of the cognitive and behavioural mechanisms thought to underlie drinking behaviour.

The complexity of p53 stabilization and activation

A number of proteins are activated by stress stimuli but none so spectacularly or with the degree of complexity as the tumour suppressor p53 (human p53 gene or protein). Once stabilized, p53 is responsible for the transcriptional activation of a series of proteins involved in cell cycle control, apoptosis and senescence. This protein is present at low levels in resting cells but after exposure to DNA-damaging agents and other stress stimuli it is stabilized and activated by a series of post-translational modifications that free it from MDM2 (mouse double minute 2 but used interchangeably to denote human also), a ubiquination ligase that ubiquitinates it prior to proteasome degradation. The stability of p53 is also influenced by a series of other interacting proteins. In this review, we discuss the post-translational modifications to p53 in response to different stresses and the consequences of these changes.

Quantifying the complexity of relative clause and cleft sentences

Explanations of the difficulty of relative-clause sentences implicate complexity but the measurement of complexity remains controversial. Four experiments investigated how far relational complexity (RC) theory, that has been found valid for cognitive development and human reasoning, accounts for the difficulty of 16 types of English, object- and subject-extracted relative-clause constructions. RC corresponds to the number of nouns assigned to thematic roles in the same decision. Complexity estimates based on RC and those based on maximal integration cost (MIC) were strongly correlated and accounted for similar variance in sentence difficulty (subjective ratings, comprehension accuracy, reading times). Consistent with RC theory, sentences that required more than 4 role assignments in the same decision were extremely difficult for many participants. Performance on nonlinguistic relational tasks predicted comprehension of object-extracted sentences, before and after controlling for subject-extractions. Working memory tasks predicted comprehension of object-extractions before controlling for subjectextractions. The studies extend the RC approach to a linguistic domain.