Discovering the roots: Uniform closure results for algebraic classes under factoring

Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots $r$ is small but the multiplicities are exponentially large. Our method sets up a linear system in $r$ unknowns and iteratively builds the roots as formal power series. For an algebraic circuit $f(x_1,\ldots,x_n)$ of size $s$ we prove that each factor has size at most a polynomial in: $s$ and the degree of the squarefree part of $f$. Consequently, if $f_1$ is a $2^{\Omega(n)}$-hard polynomial then any nonzero multiple $\prod_{i} f_i^{e_i}$ is equally hard for arbitrary positive $e_i$'s, assuming that $\sum_i \text{deg}(f_i)$ is at most $2^{O(n)}$. It is an old open question whether the class of poly($n$)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial $f$ of degree $n^{O(1)}$ and formula (resp. ABP) size $n^{O(\log n)}$ we can find a similar size formula (resp. ABP) factor in randomized poly($n^{\log n}$)-time. Consequently, if determinant requires $n^{\Omega(\log n)}$ size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation $\tau$, $f(\tau\overline{x})$ completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \& Burgisser, FOCS 2001).Comment: 33 Pages, No figure

Interactions between cardiac activity and conscious somatosensory perception

Fluctuations in the heart's activity can modulate the access of external stimuli to consciousness. The link between perceptual awareness and cardiac signals has been investigated mainly in the visual and auditory domain. Here, we investigated whether the phase of the cardiac cycle and the prestimulus heart rate influence conscious somatosensory perception. We also tested how conscious detection of somatosensory stimuli affects the heart rate. Electrocardiograms (ECG) of 33 healthy volunteers were recorded while applying near‐threshold electrical pulses at a fixed intensity to the left index finger. Conscious detection was not uniformly distributed across the cardiac cycle but significantly higher in diastole than in systole. We found no evidence that the heart rate before a stimulus influenced its detection, but hits (correctly detected somatosensory stimuli) led to a more pronounced cardiac deceleration than misses. Our findings demonstrate interactions between cardiac activity and conscious somatosensory perception, which highlights the importance of internal bodily states for sensory processing beyond the auditory and visual domain

How Long It Takes for an Ordinary Node with an Ordinary ID to Output?

In the context of distributed synchronous computing, processors perform in rounds, and the time-complexity of a distributed algorithm is classically defined as the number of rounds before all computing nodes have output. Hence, this complexity measure captures the running time of the slowest node(s). In this paper, we are interested in the running time of the ordinary nodes, to be compared with the running time of the slowest nodes. The node-averaged time-complexity of a distributed algorithm on a given instance is defined as the average, taken over every node of the instance, of the number of rounds before that node output. We compare the node-averaged time-complexity with the classical one in the standard LOCAL model for distributed network computing. We show that there can be an exponential gap between the node-averaged time-complexity and the classical time-complexity, as witnessed by, e.g., leader election. Our first main result is a positive one, stating that, in fact, the two time-complexities behave the same for a large class of problems on very sparse graphs. In particular, we show that, for LCL problems on cycles, the node-averaged time complexity is of the same order of magnitude as the slowest node time-complexity. In addition, in the LOCAL model, the time-complexity is computed as a worst case over all possible identity assignments to the nodes of the network. In this paper, we also investigate the ID-averaged time-complexity, when the number of rounds is averaged over all possible identity assignments. Our second main result is that the ID-averaged time-complexity is essentially the same as the expected time-complexity of randomized algorithms (where the expectation is taken over all possible random bits used by the nodes, and the number of rounds is measured for the worst-case identity assignment). Finally, we study the node-averaged ID-averaged time-complexity.Comment: (Submitted) Journal versio

Expectancies, working alliance, and outcome in transdiagnostic and single diagnosis treatment for anxiety disorders: an investigation of mediation

Patients’ outcome expectancies and the working alliance are two psychotherapy process variables that researchers have found to be associated with treatment outcome, irrespective of treatment approach and problem area. Despite this, little is known about the mechanisms accounting for this association, and whether contextual factors (e.g., psychotherapy type) impact the strength of these relationships. The primary aim of this study was to examine whether patient-rated working alliance quality mediates the relationship between outcome expectancies and pre- to post-treatment change in anxiety symptoms using data from a recent randomized clinical trial comparing a transdiagnostic treatment (the Unified Protocol [UP]; Barlow et al., Unified protocol for transdiagnostic treatment of emotional disorders: Client workbook, Oxford University Press, New York, 2011a; Barlow et al., Unified protocol for transdiagnostic treatment of emotional disorders: Patient workbook. New York: Oxford University Press, 2017b) to single diagnosis protocols (SDPs) for patients with a principal heterogeneous anxiety disorder (n = 179). The second aim was to explore whether cognitive-behavioral treatment condition (UP vs. SDP) moderated this indirect relationship. Results from mediation and moderated mediation models indicated that, when collapsing across the two treatment conditions, the relationship between expectancies and outcome was partially mediated by the working alliance [B = 0.037, SE = 0.05, 95% CI (.005, 0.096)]. Interestingly, within-condition analyses showed that this conditional indirect effect was only present for SDP patients, whereas in the UP condition, working alliance did not account for the association between expectancies and outcome. These findings suggest that outcome expectancies and working alliance quality may interact to influence treatment outcomes, and that the nature and strength of the relationships among these constructs may differ as a function of the specific cognitive-behavioral treatment approach utilized.This study was funded by grant R01 MH090053 from the National Institutes of Health. (R01 MH090053 - National Institutes of Health)First author draf

The use of a non-absorbable membrane as an occlusive barrier for alveolar ridge preservation: A one year follow-up prospective cohort study

The aims of this study were to obtain preliminary data and test the clinical efficacy of a novel nonporous dense-polytetrafluoroethylene (d-PTFE) membrane (permamem®, botiss) in alveolar ridge preservation (ARP) procedures with a flapless approach. A traumatic extraction was performed in the premolar maxillary area, and a d-PTFE membrane was used to seal the alveolar cavity: no biomaterial was used to graft the socket and the membrane was left intentionally exposed and stabilized with sutures. The membrane was removed after four weeks and dental implants were placed four months after the procedure. The primary outcome variables were defined as the dimensional changes in the ridge width and height after four months. A total of 15 patients were enrolled in this study. The mean width of the alveolar cavity was 8.9 ± 1.1 mm immediately after tooth extraction, while four months later a mean reduction of 1.75 mm was experienced. A mean vertical reduction of 0.9 ± 0.42 mm on the buccal aspect and 0.6 ± 0.23 mm on the palatal aspect were recorded at implant placement. Within the limitations of this study, the d-PTFE membrane proved to be effective in alveolar ridge preservation, with the outcomes of the regeneration not affected by the complete exposure of this biomaterial

Parameterized Rural Postman Problem

The Directed Rural Postman Problem (DRPP) can be formulated as follows: given a strongly connected directed multigraph $D=(V,A)$ with nonnegative integral weights on the arcs, a subset $R$ of $A$ and a nonnegative integer $\ell$, decide whether $D$ has a closed directed walk containing every arc of $R$ and of total weight at most $\ell$. Let $k$ be the number of weakly connected components in the the subgraph of $D$ induced by $R$. Sorge et al. (2012) ask whether the DRPP is fixed-parameter tractable (FPT) when parameterized by $k$, i.e., whether there is an algorithm of running time $O^*(f(k))$ where $f$ is a function of $k$ only and the $O^*$ notation suppresses polynomial factors. Sorge et al. (2012) note that this question is of significant practical relevance and has been open for more than thirty years. Using an algebraic approach, we prove that DRPP has a randomized algorithm of running time $O^*(2^k)$ when $\ell$ is bounded by a polynomial in the number of vertices in $D$. We also show that the same result holds for the undirected version of DRPP, where $D$ is a connected undirected multigraph