Bounded Depth Circuits with Weighted Symmetric Gates: Satisfiability, Lower Bounds and Compression

A Boolean function f:{0,1}^n -> {0,1} is weighted symmetric if there exist a function g: Z -> {0,1} and integers w_0, w_1, ..., w_n such that f(x_1, ...,x_n) = g(w_0+sum_{i=1}^n w_i x_i) holds. In this paper, we present algorithms for the circuit satisfiability problem of bounded depth circuits with AND, OR, NOT gates and a limited number of weighted symmetric gates. Our algorithms run in time super-polynomially faster than 2^n even when the number of gates is super-polynomial and the maximum weight of symmetric gates is nearly exponential. With an additional trick, we give an algorithm for the maximum satisfiability problem that runs in time poly(n^t)*2^{n-n^{1/O(t)}} for instances with n variables, O(n^t) clauses and arbitrary weights. To the best of our knowledge, this is the first moderately exponential time algorithm even for Max 2SAT instances with arbitrary weights. Through the analysis of our algorithms, we obtain average-case lower bounds and compression algorithms for such circuits and worst-case lower bounds for majority votes of such circuits, where all the lower bounds are against the generalized Andreev function. Our average-case lower bounds might be of independent interest in the sense that previous ones for similar circuits with arbitrary symmetric gates rely on communication complexity lower bounds while ours are based on the restriction method

Improved Exact Algorithms for Mildly Sparse Instances of Max SAT

We present improved exponential time exact algorithms for Max SAT. Our algorithms run in time of the form O(2^{(1-mu(c))n}) for instances with n variables and m=cn clauses. In this setting, there are three incomparable currently best algorithms: a deterministic exponential space algorithm with mu(c)=1/O(c * log(c)) due to Dantsin and Wolpert [SAT 2006], a randomized polynomial space algorithm with mu(c)=1/O(c * log^3(c)) and a deterministic polynomial space algorithm with mu(c)=1/O(c^2 * log^2(c)) due to Sakai, Seto and Tamaki [Theory Comput. Syst., 2015]. Our first result is a deterministic polynomial space algorithm with mu(c)=1/O(c * log(c)) that achieves the previous best time complexity without exponential space or randomization. Furthermore, this algorithm can handle instances with exponentially large weights and hard constraints. The previous algorithms and our deterministic polynomial space algorithm run super-polynomially faster than 2^n only if m=O(n^2). Our second results are deterministic exponential space algorithms for Max SAT with mu(c)=1/O((c * log(c))^{2/3}) and for Max 3-SAT with mu(c)=1/O(c^{1/2}) that run super-polynomially faster than 2^n when m=o(n^{5/2}/log^{5/2}(n)) and m=o(n^3/log^2(n)) respectively

Large N Analysis of TTˉT\bar{T}-deformation and Unavoidable Negative-norm States

We study non-perturbative quantum aspects of $T\bar{T}$-deformation of a free $O(N)$ vector model by employing the large $N$ limit. It is shown that bound states of the original field appear and inevitably become negative-norm states. In particular, the bound states can be regarded as the states of the conformal mode in a gravitational theory, where the Liouville action is induced with the coefficient proportional to the minus of central charge. To make the theory positive-definite, some modification is required so as to preserve diffeomorphism invariance due to the Faddeev-Popov ghosts with a negative central charge.Comment: 1+20 pages, 1 figur

Metallicity dependence of the Hercules stream in Gaia/RAVE data -- explanation by non-closed orbits

The origin of the Hercules stream, the most prominent velocity substructure in the Solar neighbour disc stars, is still under debate. Recent accurate measurements of position, velocity, and metallicity provided by Tycho Gaia Astrometric Solution (TGAS) and RAdial Velocity Experiments (RAVE) have revealed that the Hercules stream is most clearly seen in the metal-rich region ([Fe/H] > 0), while it is not clearly seen in lower metallicity region ([Fe/H] < -0.25). By using a large number of chemo-dynamical 2D test-particle simulations with a rotating bar and/or spiral arms, we find that the observed [Fe/H] dependence of the Hercules stream is a natural consequence of the inside-out formation of the stellar disc and the existence of highly non-closed orbits in the rotating frame of the bar or spiral arms. Our successful models that reproduce the observed properties of the Hercules stream include not only fast-bar-only and fast-bar+spiral models, but also slow-bar+spiral models. This indicates that it is very difficult to estimate the pattern speed of the bar or spiral arms based only on the observations of the Hercules stream in the Solar neighbourhood. As a by-product of our simulations, we make some predictions about the locations across the Galactic plane where we can observe velocity bimodality that is not associated with the Hercules stream. These predictions can be tested by the forthcoming Gaia data, and such a test will improve our understanding of the evolution of the Milky Way stellar disc

The Kurashiki Prehospital Stroke Scale Is a Prehospital Scale That Can Predict Long-Term Outcome of Patients with Acute Cerebral Ischemia

Background and Purpose: Our aim was to confirm the clinical relationship between the Kurashiki Prehospital Stroke Scale (KPSS) scored by paramedics and favorable outcomes in patients with modified Rankin scale (mRS) scores of 0–1 assessed 3 months after symptom onset. Methods: We enrolled patients with acute stroke and transient ischemic attack showing symptoms on admission. Paramedics transferred patients to our hospital after estimating stroke severity using the KPSS. After categorizing patients into either the mRS 0–1 group (favorable outcome) or the mRS 2–6 group (no favorable outcome), we compared the background data between the two groups. We assessed KPSS scores predictive of a favorable outcome. Multivariate regression modeling was conducted to identify factors independently associated with a favorable outcome. Results: The study cohort comprised 147 patients with a premorbid status of mRS 0–1: 69 patients (47%) of them were in the mRS 0–1 group and 78 (53%) in the mRS 2–6 group at the follow-up 3 months after symptom onset. The median KPSS score was lower in the mRS 0–1 group than in the mRS 2–6 group (1 vs. 4, p Conclusion: KPSS score <3 apparently presents a reasonable cutoff for predicting a favorable outcome in patients with acute cerebral ischemia

Analysis of Surface Crack Propagation Considering the Effect of Micro-separations

In the TMCP steels rolled at the finishing temperature in austenite-ferrite region by the Non-AcC type TMCP method, it becames clear that the fatigue fracture surface had a lot of micro-separations, and fatigue crack propagation was prevented by micro-separations in the case of propagation in the direction of plate thickness. In this study, the authors investigated on the behavior of surface crack propagation using three TMCP steels with different SImax, and proposed the method of crack propagation analysis considering the effect of micro-separations. The results may be summarized as follows: (1) As SImax increase, propagation pattern of surface crack become shallow. (2) The macroscopic fatigue crack propagation can be analyzed with the assumption that the fatigue crack propagation in the direction of plate thickness passes through between estimated microseparations. (3) This estimation method shows the good agreement with the experimental change of aspect ratio for surface crack