Polynomials that Sign Represent Parity and Descartes' Rule of Signs

A real polynomial $P(X_1,..., X_n)$ sign represents $f: A^n \to \{0,1\}$ if for every $(a_1, ..., a_n) \in A^n$, the sign of $P(a_1,...,a_n)$ equals $(-1)^{f(a_1,...,a_n)}$. Such sign representations are well-studied in computer science and have applications to computational complexity and computational learning theory. In this work, we present a systematic study of tradeoffs between degree and sparsity of sign representations through the lens of the parity function. We attempt to prove bounds that hold for any choice of set $A$. We show that sign representing parity over $\{0,...,m-1\}^n$ with the degree in each variable at most $m-1$ requires sparsity at least $m^n$. We show that a tradeoff exists between sparsity and degree, by exhibiting a sign representation that has higher degree but lower sparsity. We show a lower bound of $n(m -2) + 1$ on the sparsity of polynomials of any degree representing parity over $\{0,..., m-1\}^n$. We prove exact bounds on the sparsity of such polynomials for any two element subset $A$. The main tool used is Descartes' Rule of Signs, a classical result in algebra, relating the sparsity of a polynomial to its number of real roots. As an application, we use bounds on sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with a Threshold Gate at the top. We use this to give a simple proof that such circuits need size $1.5^n$ to compute parity, which improves the previous bound of ${4/3}^{n/2}$ due to Goldmann (1997). We show a tight lower bound of $2^n$ for the inner product function over $\{0,1\}^n \times \{0, 1\}^n$.Comment: To appear in Computational Complexit

On the importance of eliminating errors in cryptographic computations

Abstract. We present a model for attacking various cryptographic schemes by taking advantage of random hardware faults. The model consists of a black-box containing some cryptographic secret. The box interacts with the outside world by following a cryptographic protocol. The model supposes that from time to time the box is affected by a random hardware fault causing it to output incorrect values. For example, the hardware fault flips an internal register bit at some point during the computation. We show that for many digital signature and identification schemes these incorrect outputs completely expose the secrets stored in the box. We present the following results: (1) The secret signing key used in an implementation of RSA based on the Chinese Remainder Theorem (CRT) is completely exposed from a single erroneous RSA signature, (2) for non-CRT implementations of RSA the secret key is exposed given a large number (e.g. 1000) of erroneous signatures, (3) the secret key used in Fiat-Shamir identification is exposed after a small number (e.g. 10) of faulty executions of the protocol, and (4) the secret key used in Schnorr's identification protocol is exposed after a much larger number (e.g. 10,000) of faulty executions. Our estimates for the number of necessary faults are based on standard security parameters such as a 1024-bit modulus, and a 2 −40 identification error probability. Our results demonstrate the importance of preventing * This is an expanded version of an earlier paper that appeared in Proc. of Eurocrypt '97. 101 102 D. Boneh, R. A. DeMillo, and R. J. Lipton errors in cryptographic computations. We conclude the paper with various methods for preventing these attacks

The migraine postdrome:An electronic diary study

OBJECTIVE: To report migraine postdrome symptoms in patients who report nonheadache symptoms as part of their attacks. METHODS: A prospective daily electronic diary study was conducted over 3 months in 120 patients with migraine. Nonheadache symptoms before, during, and after headache were collected on a daily basis. Visual analogue scales were used to capture the overall level of functioning and the severity of the headache. The postdrome was defined as the time from resolution of troublesome headache to return to normal. RESULTS: Of 120 evaluable patients, 97 (81%) reported at least one nonheadache symptom in the postdrome. Postdrome symptoms, in order of frequency, included feeling tired/weary and having difficulty concentrating and stiff neck. Many patients also reported a mild residual head discomfort. In most attacks (93%), there was return to normal within 24 hours after spontaneous pain resolved. There was no relationship between medication taken for the headache and the duration of the postdrome. The severity of the migraine was not associated with the duration of the postdrome. Overall state of health scores remained low during the postdrome. CONCLUSION: Nonheadache symptoms in the postdrome were common and may contribute to the distress and disability in the patients studied. Postdrome symptoms merit larger observational studies and careful recording in clinical trials of acute and preventive migraine treatments

Erenumab in chronic migraine: Patient-reported outcomes in a randomized double-blind study.

OBJECTIVE: To determine the effect of erenumab, a human monoclonal antibody targeting the calcitonin gene-related peptide receptor, on health-related quality of life (HRQoL), headache impact, and disability in patients with chronic migraine (CM). METHODS: In this double-blind, placebo-controlled study, 667 adults with CM were randomized (3:2:2) to placebo or erenumab (70 or 140 mg monthly). Exploratory endpoints included migraine-specific HRQoL (Migraine-Specific Quality-of-Life Questionnaire [MSQ]), headache impact (Headache Impact Test-6 [HIT-6]), migraine-related disability (Migraine Disability Assessment [MIDAS] test), and pain interference (Patient-Reported Outcomes Measurement Information System [PROMIS] Pain Interference Scale short form 6b). RESULTS: Improvements were observed for all endpoints in both erenumab groups at month 3, with greater changes relative to placebo observed at month 1 for many outcomes. All 3 MSQ domains were improved from baseline with treatment differences for both doses exceeding minimally important differences established for MSQ-role function-restrictive (≥3.2) and MSQ-emotional functioning (≥7.5) and for MSQ-role function-preventive (≥4.5) for erenumab 140 mg. Changes from baseline in HIT-6 scores at month 3 were -5.6 for both doses vs -3.1 for placebo. MIDAS scores at month 3 improved by -19.4 days for 70 mg and -19.8 days for 140 mg vs -7.5 days for placebo. Individual-level minimally important difference was achieved by larger proportions of erenumab-treated participants than placebo for all MSQ domains and HIT-6. Lower proportions of erenumab-treated participants had MIDAS scores of severe (≥21) or very severe (≥41) or PROMIS scores ≥60 at month 3. CONCLUSIONS: Erenumab-treated patients with CM experienced clinically relevant improvements across a broad range of patient-reported outcomes. CLINICALTRIALSGOV IDENTIFIER: NCT02066415. CLASSIFICATION OF EVIDENCE: This study provides Class II evidence that for patients with CM, erenumab treatment improves HRQoL, headache impact, and disability

Provable Virus Detection: Using the Uncertainty Principle to Protect Against Malware

Protecting software from malware injection is the holy grail of modern computer security. Despite intensive efforts by the scientific and engineering community, the number of successful attacks continues to increase. We have a breakthrough novel approach to provably detect malware injection. The key idea is to use the very insertion of the malware itself to allow for the systems to detect it. This is, in our opinion, close in spirit to the famous Heisenberg Uncertainty Principle. The attackers, no matter how clever, no matter when or how they insert their malware, change the state of the system they are attacking. This fundamental idea is a game changer. And our system does not rely on heuristics; instead, our scheme enjoys the unique property that it is proved secure in a formal and precise mathematical sense and with minimal and realistic CPU modification achieves strong provable security guarantees. Thus, we anticipate our system and formal mathematical security treatment to open new directions in software protection

Natural history of malignant bone disease in breast cancer and the use of cumulative mean functions to measure skeletal morbidity

BACKGROUND: Bone metastases are a common cause of skeletal morbidity in patients with advanced cancer. The pattern of skeletal morbidity is complex, and the number of skeletal complications is influenced by the duration of survival. Because many patients with cancer die before trial completion, there is a need for survival-adjusted methods to accurately assess the effects of treatment on skeletal morbidity. METHODS: Recently, a survival-adjusted cumulative mean function model has been generated that can provide an intuitive graphic representation of skeletal morbidity throughout a study. This model was applied to the placebo-control arm of a pamidronate study in patients with malignant bone disease from breast cancer. RESULTS: Analysis by bone lesion location showed that spinal metastases were associated with the highest cumulative mean incidence of skeletal-related events (SREs), followed by chest and pelvic metastases. Metastases located in the extremities were associated with an intermediate incidence of SREs, and those in the skull were associated with the lowest incidence of SREs. CONCLUSION: Application of this model to data from the placebo arm of this trial revealed important insight into the natural history of skeletal morbidity in patients with bone metastases. Based on these observations, treatment for the prevention of SREs is warranted regardless of lesion location except for metastases on the skull

A close association of freedom from pain, migraine-related functional disability, and other outcomes: results of a post hoc analysis of randomized lasmiditan studies SAMURAI and SPARTAN

Background: While pain freedom at 2 h is a key primary outcome for current trials for acute treatment of migraine, the relationship between the degree of head pain and other efficacy measures at 2 h has rarely been explored. Following lasmiditan treatment of a migraine attack with moderate or severe head pain, we contrast those who achieve pain freedom with those who achieve mild pain but not pain freedom 2 h post dosing. Methods: Patient-level data were pooled across studies and treatment arms from two Phase 3 trials comparing lasmiditan and placebo, SAMURAI and SPARTAN. This post hoc analysis assessed freedom from the most bothersome symptom (MBS), freedom from migraine-related functional disability (disability), and improved patient global impression of change (PGIC) in patients who achieved 2 h pain freedom compared to those who experienced 2 h mild pain. Mild pain differs from pain relief which is defined as either mild pain or pain freedom. Results: Patients who achieved 2 h pain freedom (N = 913), in comparison with those with 2 h mild pain (N = 864), were significantly more likely to experience MBS freedom (91.9% vs. 44.9%), disability freedom (87.1% and 13.4%), and improved PGIC (86.5% and 31.5%) (p \u3c 0.001 for all combinations). In addition, more patients who were pain free experienced both 2 h MBS freedom and 2 h functional disability freedom (83.6%) compared to those with mild pain (10.8%; p \u3c 0.001). The proportion of patients with pain freedom who did not achieve either MBS or disability freedom (4.6%) was lower than in patients with mild pain (52.4%). Lastly, 55.2% of patients experienced mild pain before disability freedom compared to 72.1% who experienced pain freedom and disability freedom at the same time. Conclusions: This study demonstrated that, at 2 h post treatment, patients who were pain free were more likely to achieve other outcomes including freedom from their MBS, freedom from migraine-related functional disability, and improved PGIC compared to those with mild pain, confirming that 2 h pain freedom is more robustly associated with other clinical outcomes than the 2 h mild pain endpoint. Trial Registration: SAMURAI (NCT02439320); SPARTAN (NCT02605174)

Workshop on Environmental Research Needs in Support of Potential Virginia Offshore Oil and Gas Activities

The MMS, a bureau within the Department of the Interior, sponsored a workshop on the environmental research needs in support of potential Virginia offshore oil and gas activities 3 and 4 December 2008, in Williamsburg, Virginia. The focus of the workshop was to assess the existing scientific knowledgebase along the Virginia Coast and the information gaps that need to 2 be addressed should a lease sale for oil and gas activities be held for the Virginia outer continental shelf. This report summarizes the outcome of the workshop

Defining Refractory Migraine: Results of the RHSIS Survey of American Headache Society Members

To gauge consensus regarding a proposed definition for refractory migraine proposed by Refractory Headache Special Interest Section, and where its use would be most appropriate. Background.— Headache experts have long recognized that a subgroup of headache sufferers remains refractory to treatment. Although different groups have proposed criteria to define refractory migraine, the definition remains controversial. The Refractory Headache Special Interest Section of the American Headache Society developed a definition through a consensus process, assisted by a literature review and initial membership survey. Design.— A 12-item questionnaire was distributed at the American Headache Society meeting in 2007 during a platform session and at the Refractory Headache Special Interest Section symposium. The same questionnaire was subsequently sent to all American Headache Society members via e-mail. A total of 151 responses from AHS members form the basis of this report. The survey instrument was designed using Survey Monkey. Frequencies and percentages of the survey were used to describe survey responses. Results.— American Headache Society members agreed that a definition for refractory migraine is needed (91%) that it should be added to the International Classification of Headache Disorders-2 (86%), and that refractory forms of non-migraine headache disorders should be defined (87%). Responders believed a refractory migraine definition would be of greatest value in selecting patients for clinical drug trials. The current refractory migraine definition requires a diagnosis of migraine, interference with function or quality of life despite modification of lifestyle factors, and adequate trials of acute and preventive medicines with established efficacy. The proposed criteria for the refractory migraine definition require failing 2 preventive medications to meet the threshold for failure. Although 42% of respondents agreed with the working definition of refractory migraine, 43% favored increasing the number to 3 (50%) or 4 (26%) preventive treatment failures. When respondents were asked if they felt that the proposed definition was appropriate to select patients for invasive procedures (patent foramen ovale repair or stimulators) only 44% agreed. Conclusions.— There is a consensus for a need for a definition for refractory migraine and that it should be added to the International Classification of Headache Disorder-2. There was also general agreement by the responders that refractory forms of non-migraine headache disorders should be defined.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/72860/1/j.1526-4610.2009.01370.x.pd