text:
A Numerical methods

# Quantifying resonant activation like phenomenon in non-Markovian systems

## Abstract

Resonant activation is an effect of a noise-induced escape over a modulated potential barrier. The modulation of a energy landscape facilitates the escape kinetics and makes it optimal as measured by the mean first passage time. A canonical example of resonant activation is a Brownian particle moving in a time-dependent potential under action of Gaussian white noise. Resonant activation is observed not only in typical Markovian-Gaussian systems but also in far from equilibrium and far from Markovianity regimes. We demonstrate that using an alternative to the mean first passage time, robust measures of resonant activation, the signature of this effect can be observed in general continuous time random walks in modulated potentials even in situations when the mean first passage time diverges.

###### pacs:
05.40.Fb, 05.10.Gg, 02.50.-r, 02.50.Ey,

## I Introduction

In the last decades, the concept of noise in physical systems has advanced from an unwanted addition to data, towards subject of interest of non-equilibrium statistical physics. Noise is used to simplify descriptions of dynamical systems when the detailed character of interactions is unknown or too complicated for exact methods. Noise is responsible for the occurrence of the so-called noise-induced effects. Among them stochastic resonance (1); (2) and resonant activation (3) are the most popular. Usually, it is assumed that noise is white and Gaussian, however both non-Gaussian and non-Markovian extensions are also possible (4); (5). The examination of the constructive role of noises in physical systems has attracted considerable attention during the last two decades.

Resonant activation (3); (6) is a resonant effect in which a noise-induced transition is further facilitated by a modulation of the energy landscape. A deterministic or stochastic modulation of the potential barrier can improve the system performance leading to the optimal escape kinetics as measured by the mean first passage time. Such a situation is observed for a white Gaussian noise (3), -stable Lévy type noises (7); (8); (9); (10), colored noise (11); (12) or even in situations when the potential barrier is modulated by a colored process (13); (14); (15); (16).

Here, we extend studies on resonant activation into a non-Markovian and non-Gaussian regime. We study a non-Markovian continuous time random walk scheme in a stochastically modulated potential when jump lengths are generated according to heavy-tailed distributions. The non-Markovianity of the studied process emerges due to slower-than-exponential decay of the waiting time distribution. This process is also non-Gaussian because jump lengths are determined by a heavy-tailed distribution with the diverging second moment. On the one hand, the description of anomalous systems can be provided by the continuous time random walk scheme which allows for a relatively easy treatment (exact or asymptotic) of various waiting time and jump length distributions resulting in the unified description of Markovian/non-Markovian and Gaussian/non-Gaussian systems (17); (18). On the other hand, such realms due to a slow decay of the waiting time distribution and power-law tails of the jump length distribution are described by the bi-fractional diffusion equation (17); (19); (20); (21) which provides an equivalent description of the continuous time random walk framework.

As the waiting time distribution becomes of the power-law, heavy-tailed type, the system drifts into a non-Markovian regime. Consequently, the standard way of identifying resonant activation fails, as the mean first passage time diverges due to heavy-tailed distributions of waiting times. While this could be taken as an indication of the disappearance of the resonant activation phenomenon, we show that the effect of optimal escape kinetics can still be detected. In order to prove the existence of resonant activation, we use robust, quantile-based measures, which can be defined regardless of the existence of the mean first passage time. Quantile-based measures provide robust characteristics of the first passage time density and as a such can be used to expose the signature of resonant activation. In this study, in addition to the examination of the properties of first passage time distributions, we also use extreme statistics and their properties. Extreme statistics provides additional insight into systems dynamics. In particular, it is well suited for an examination of the system’s performance and quantifying the stochastic resonance phenomenon.

The next section presents the model under consideration, defines measures of resonant activation and shows obtained results. The paper is closed with summary and conclusions.

## Ii Model and results

The classical resonant activation setup (3) consists of a Brownian particle moving in a time-dependent potential field, which is switching dichotomously between two linear slopes of different heights. For the sake of simplicity, it is assumed that the dichotomous process is symmetric and Markovian (22), i.e. a potential stays in one of configurations for an exponentially distributed time. Additionally, the domain of motion is restricted to the finite interval with the reflecting boundary on the left and the absorbing boundary on the right. Here, we examine properties of the archetypal resonant activation setup when the Brownian particle is replaced by a more general random walker performing a general form of a continuous time random walk (23); (24); (18).

The subordination methods (19); (25) are used in order to study a system described by the bi-fractional time dependent diffusion equation (17); (19); (20); (21):

 ∂p(x,t)∂t=[∂∂xV′±(x,t)+σα∂α∂|x|α]0D1−νtp(x,t). (1)

The bi-fractional Smoluchowski-Fokker-Planck equation describes the evolution of the probability density of finding a particle at the time in the vicinity of . The subordination methods are based on the generation of a stochastic process whose evolution of the probability density is described by Eq. (1). This is achieved by linking physical time with operational time by an appropriate time transformation. The used method is briefly presented in the Appendix A while the detailed description can be found in (20); (19); (25); (26).

In Eq. (1), denotes the Riemannn-Liouville fractional time derivative defined by the relation

 0D1−νtf(x,t)=1Γ(ν)∂∂t∫t0dt′f(x,t′)(t−t′)1−ν (2)

and stands for the Riesz-Weil fractional space derivative with the Fourier transform

 F[∂α∂|x|αf(x)]=−|k|α^f(x). (3)

The potential dichotomously switches between two linear configurations characterized by two distinct heights , i.e. . Initially, a test particle is located at the origin and the configuration of the potential is set to or with equal probabilities. As in the Doering Gadoua model (3) the dichotomous process is symmetric and Markovian. Consequently, it is described by a single parameter which is the rate of the potential switching, i.e. a dichotomous process takes two possible values only and stays constant for the exponentially distributed time. The domain of motion is restricted to the finite interval, i.e. at there is a reflecting boundary, while at there is an absorbing boundary. Boundaries impose additional constraints on the probability density which are hard to implement on the operator level (27); (28); (29) but are relatively easily controlled on the single trajectory level. Eq.  (1) extends the classical resonant activation model into a non-equilibrium non-Markovian regime.

From the microscopic point of view, the model describes an escape kinetics of a random walker from the finite interval restricted by absorbing and reflecting boundaries. In the force free case, Eq. (1) emerges as a description of a continuous time random walk scenario with power law distributed waiting times, and jump lengths, , with and . Here the jump length distribution is also modified by the external potential, which slightly alters the jump length distribution. Nevertheless, its asymptotics is still of the power-law type. Such a continuous time random walk scenario is both non-Markovian and non-Gaussian (30). The Markovian Gaussian scenario is recovered for with . The subdiffusion parameter controls the level of non-Markovianity while controls the level of non-Gaussianity. Eq. (1) is a general type equation which is also obtained when jump lengths are distributed according to symmetric -stable densities (31); (32); (33); (34), which for have required asymptotics. We refer to as the subdiffusion parameter and to as the stability index. The studied random walk is characterized by the first passage time

 τ=min{t>0:x(0)=0 and x(t)⩾1}. (4)

From the set of the first passages times it is possible to calculate various characteristics of the given system which can be used to measure the performance of escape kinetics.

Traditionally, the strength of resonant activation is measured by the mean first passage time, but such a quantifier can be calculated only when the first passage time density is not of the heavy-tailed type. In particular, for , the seminal resonant activation setup is recovered. The first passage time density has exponential tails and it has a well-defined mean value. Otherwise, the mean first passage time diverges and cannot be used to characterize the performance of escape kinetics. This is the case with the studied model in the non-Markovian regime, i.e. for , when the first passage time density has power-law tails with the exponent . Nevertheless, that is the point which requires special attention. The calculation of resonant activation characteristics is based on a large, yet finite sample of first passage times. The finite size effects result in the effective truncation of the first passage time density. Consequently, from a finite sample it is still possible to calculate the mean first passage time. However, due to heavy-tails, this quantity fluctuates very strongly and becomes meaningless.

For a general type of continuous time random walks which can be characterized by the diverging waiting time different measures have to be employed. Therefore, one needs to consider more robust measures which can be based on distributions of the first passage time or directly derived from this density, e.g. quantiles of the first passage time distribution

 Prob(τ⩽qp)=p(0

i.e. such a value of the first passage time that the probability to find a smaller value than is . Quantiles can be also expressed by the cumulative density of the first passage time by the relation . In particular, one can use the median location () or the inter-quantile width () in order to measure the system’s performance. The efficiency of quantile-based measures relies on the fact that optimal escape kinetics should affect the properties of the first passage time distributions what should be further manifested by the dependence of quantiles on control parameters (in particular the switching rate ).

Within the model properties of the resonant activation phenomenon are studied in the non-Gaussian () and non-Markovian regime (). However, for the reference point, also the Markovian and the Gaussian case is studied ( with ). It is assumed that the potential switches between configurations with and . Changes in the height of the potential barrier are described by a symmetric Markovian dichotomous process with the rate , i.e. the dichotomous process stays constant for an exponentially distributed time with the intensity equal to the switching rate , see (22); (3).

In order to study properties of alternative measures of resonant activation, the Markovian non-Gaussian case ( with ) was studied. Fig. 1 shows the dependence of the mean first passage time , the median location and the inter-quantile width on for relatively large values of the stability index only, i.e. from top to bottom. In such a situation the first passage time distribution has a well-defined mean and consequently the phenomenon of resonant activation can be captured by the examination of the mean first passage time, see left column of Fig. 1. In line with earlier studies, strong deviations from the Gaussian distribution of jump lengths result in diminishing resonant activation, see Fig. 1 and (8). However, a full disappearance of the effect is observed for lower values of the exponent than presented in Fig. 1, see (8).

Resonant activation is understood as an optimal escape kinetics as measured by the mean first passage time. Decrease in the mean first passage time is a consequence of changes in the first passage time distribution and corresponds to the narrowing of the first passage time density. This can be measured not only by the mean first passage time but also by the width of the first passage time density which can be characterized by the inter-quantile distance, e.g. . The inter-quantile distance provides a robust measure, because it exists independently of the existence of the mean or the variance of the first passage time distribution. The inter-quantile distance is sensitive to the modulation of the barrier and properly captures resonant activation, as it can be observed in the right column of Fig. 1. Nevertheless, the narrowing of the first passage time distribution is not necessarily reflected in the dependence of the median location of the first passage time density on the rate of the potential switching , see the middle column of Fig. 1. Therefore, the median can be used as a measure for the phenomenon of resonant activation only for a very limited set of system parameters.

Both possible initial states of the potential are equally probable. Thus, in the Markovian case, for a low and moderate switching rate a half of the events correspond to fast escapes (over a low barrier configuration) while the other half corresponds to slow escapes (high barrier configurations). This makes the median not very sensitive as a measure of resonant activation. The different situation is with the inter-quantile width which is of the same monotonicity as the mean first passage time, because it neglects some fraction of the fastest and slowest escape events.

In the non-Markovian case (), the mean first passage time, due to the divergence of the mean waiting time for a next jump, diverges and cannot be used to characterize the resonant activation effect. Therefore, measures which can be used in such cases are quantile-based. Consequently, the median and the inter-quantile width will be used to characterize the system performance. Moreover, for the system has only one characteristic time scale, i.e. the one associated with the barrier switching process which is .

Figures 2 and 3 demonstrate quantile-based measures of resonant activation for a selected set of non-Markovian systems. For the inter-quantile width clearly demonstrates that the dependence of the width of the first passage time density is a non-monotonous function of the switching rate . One can find an optimal switching rate leading to the lowest width of the first passage time density. The non-monotonous dependence of the width of the first passage time distribution is a signature of a phenomenon similar to resonant activation. With decreasing the value of the jump length exponent the resonant activation effect disappears, compare various rows of Figs. 2 and 3. Likewise, the decrease in the waiting time exponent weakens the effect, compare Fig. 2 and Fig. 3.

Figure 4 shows the survival probability of a random walker in various scenarios including a Gaussian-Markovian case. For the low switching rate (with and ), the survival probability attains a pronounced double exponential form, see the top left panel of Fig. 4. As the jump length distribution departs from the Gaussian, a random walker can perform longer jumps with a significantly larger probability. This facilitates the escape kinetics which could be corroborated by the smaller average time required to reach the absorbing boundary. This is further confirmed by the decrease of the survival probability with the decrease of the stability index . On the one hand, when the system drifts into the non-Markovian regime, subdiffusion becomes stronger, escape kinetics is hindered and long waiting times cause the survival probability to be non-negligible for a significantly longer time. On the other hand, it is well visible that longer jumps result in lower values of the survival probability indicating an increase in the efficiency of the escape kinetics with the decrease of the stability index , compare the top and bottom rows of Fig. 4.

For , the median behaves in a more informative way than in the Markovian case, but quickly looses sensitivity to resonant activation as the non-Gaussianity of the system progresses.The dependence of the median and the inter-quantile width on the switching rate clearly confirms the occurrence of the resonant activation effect. While both the median and the inter-quantile width can be used as indicators of resonant activation, in a strongly non-Gaussian and non-Markovian regime both undergo fluctuations, rendering both measures less reliable for determining the exact position of an optimal switching rate, see the bottom rows of Figs. 2 and 3.

Measures characterizing resonant activation do not need to be based on single particle simulations. Instead of considering a single particle system it is possible to study the motion of independent (non-interacting) particles. The presence of multiple random walkers allows to calculate minimal and maximal first passage times. The minimal first passage time is the exit time of the fastest random walker out of , while the maximal exit time is the exit time of the last (slowest) random walker out of . If random walkers are independent, instead of examining random walkers it is also possible to study a single random walker and to divide data into elements disjoint sets. For every set the minimal (maximal) first passage time is just a minimal (maximal) exit time of an random walker system. The properties of order statistics are well known (35); (36). In particular, it is known that first order statistics has a significantly faster decay than the probability density of an original variable. While the -th (last) order statistics has the same asymptotics as the original variables, i.e. the first passage times , see (37).

The first (minimum) order statistics accounts for the fastest events only. It does not provide sufficient resolution to measure the system performance and as a such is not very sensitive to the manipulation of the model parameters (results not shown). The maximum statistics is suitable for quantifying the resonant activation effect because it uses only maximal first passage times. The width of this distribution is more sensitive to the manipulation of the model parameters. Consequently, the same analysis which was performed in Figs. 13 is conducted for the maximum (last order) statistics, see Figs. 6 and 7.

Figure 5 demonstrates that maximal and minimal first passage time densities constructed from the simulation () are exactly the same as the ones estimated from the single-particle experiment. More precisely, the single-particle experiment allows to estimate which in turn can be used to calculate and , see (35); (36). Instead of , and it is more convenient to inspect the survival probability , because it reveals asymptotic behavior more clearly. For illustrative purposes, with have been chosen.

Figures 6 and 7 show the dependence of the median of the maximum (last order) statistics for a particle system. Due to smaller statistics, fluctuations are larger. Among two measures of the escape kinetics efficiency (median and inter-quantile width) the median position is more robust with respect to fluctuations (in the single-particle simulations it is the opposite). Nevertheless, the analysis of the last order statistics supports the main conclusion that with the decrease of the stability index and the subdiffusion parameter resonant activation disappears.

Figures 13 and 67 show quantifiers as a function of the switching rate . In order to determine the existence of the resonant activation effect, the strength of the effect is measured by the deviation of the minimal value of a given quantifier from the lowest asymptotics of that quantifier, i.e.

 Δ(u)=min{u(γ=−∞),u(γ=∞)}−min(u), (6)

where could be the mean first passage time (), the median () or the inter-quantile width (). Such a quantifier measures the separation of the minimum of a given characteristics from its lower asymptotics, which can be reached either for a low or large switching rate . Therefore, it is the relative depth of the minimum (if it exists).

Figures 8 and 9 show the separation of the minima of resonant activation measures from lower asymptotics () for all quantifiers used: the median and the inter-quantile width. The values of for any of quantifiers used with are in general smaller than for indicating the weakening of the resonant activation effect when the system departs from the Markovian regime. Analogously, resonant activation weakens with decreasing the value of the jump length exponent , i.e. when the jump length distribution becomes heavy-tailed. The situation is very similar for extreme statistics when the range of the relative depth variation is larger than in the single-particle case, compare Fig. 8 and Fig. 9. The changes in the jump length distribution () and waiting time distribution () modify the system performance resulting in the disappearance of resonant activation with the decrease of and exponents as measured by .

## Iii Summary and Conclusions

Resonant activation is one of classical effects demonstrating a constructive role of noises in physical systems. It proves that the noise-induced escape events can be further optimized by an additional modulation of the energy landscape. Typically, resonant activation has been studied in Markovian systems and it is considered as a generic property of a barrier crossing process in a modulated energy landscape. Here, we show that resonant activation is not only observed in a plenitude of noise-driven systems but its signature is also visible in non-Markovian and non-Gaussian systems. However, in such realms, it has to be inspected by more robust measures than the typically used mean first passage time (which can diverge).

The classical Gaussian-Markovian case was used as a test bench in order to verify whether quantile-based measures can be used to analyze properties of resonant activation. This test confirmed suitability of such measures. Next, quantile-based measures were applied in the non-Markovian regime in order to prove existence of the effect similar to resonant activation also in situations when, due to long trapping events, the mean first passage time diverges. Such a signature of resonant activation is visible in general continuous time random walks in time modulated potentials. The strength of the effect weakens with the widening of the jump length distribution and waiting time distribution, finally, resulting in the disappearance of resonant activation.

## Appendix A Numerical methods

The applied numerical scheme is based on the subordination method (38); (39); (31) which has been extended (20); (19); (25); (26) to give a proper stochastic representation of trajectories of the process whose evolution of the probability density is described by the bi-fractional Smoluchowski-Fokker-Planck equation (17); (19); (20); (21)

 ∂p(x,t)∂t=[∂∂xV′(x,t)+σα∂α∂|x|α]0D1−νtp(x,t). (7)

The solution of Eq. (7) can be estimated from the probability density of the subordinated process

 X(t)=~X(Sν(t)) (8)

where fulfills the following Langevin equation

 d~X(s)=−V′(~X(s),U(s))ds+σdLα,0(s) (9)

driven by the standard -stable motion (40). The inverse-time subordinator is defined

 Sν(t)=inf{τ:U(s)>t}, (10)

where stands for a strictly increasing -stable Lévy motion whose Laplace transform is (40). The inverse -stable subordinator links the real time with the operational time .

In the general case of the time dependent force , displacements due to the deterministic force acting on a particle are approximated as , see (25); (26). This assures that the force acting on a particle is changing in the physical time , see (25); (26). In the less general case of the time independent force, the deterministic force is approximated by and the fractional Smoluchowski-Fokker-Planck equation can be rewritten in the usual form, i.e. with the Riemannn-Liouville fractional time derivative acting on the whole right hand side of Eq. (7). Approximation of the stochastic processes and requires generation of pseudo random numbers distributed according to the symmetric -stable densities () or the totally skewed () -stable densities () which can be generated according to well known formulas (41); (42). For the detailed description of the subordination method and its sample implementation see original papers (20); (19); (25); (26).

###### Acknowledgements.
Computer simulations have been performed at the Academic Computer Center Cyfronet AGH (Kraków, Poland) under CPU grant MNiSW/Zeus_lokalnie/UJ/052/2012.

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Q. 3.22

Miles Run per Week

Find the sample variance and the sample standard deviation for the frequency distribution of the data shown. The data represent the number of miles that 20 runners ran during one week.

 Class Frequency Midpoint 5.5–10.5 1 8 10.5–15.5 2 13 15.5–20.5 3 18 20.5–25.5 5 23 25.5–30.5 4 28 30.5–35.5 3 33 35.5–40.5 2 38

Verified Solution

Step 1 Make a table as shown, and find the midpoint of each class.

Step 2 Multiply the frequency by the midpoint for each class, and place the products in column D.

1 · 8 = 8     2 · 13 = 26     . . .     2 · 38 = 76

Step 3 Multiply the frequency by the square of the midpoint, and place the products in column E.

1 · 8² = 64     2 · 13² = 338     . . .     2 · 38² = 2888

Step 4 Find the sums of columns B, D, and E. The sum of column B is n, the sum of column D is $Σf · X_m$, and the sum of column E is $Σf · X_m^2$. The completed table is shown.

Step 5 Substitute in the formula and solve for $s^2$ to get the variance.

\begin{aligned}s^2 & =\frac{n\left(\Sigma f \cdot X_m^2\right)-\left(\Sigma f \cdot X_m\right)^2}{n(n-1)} \\& =\frac{20(13,310)-490^2}{20(20-1)} \\& =\frac{266,200-240,100}{20(19)} \\& =\frac{26,100}{380} \\& \approx 68.7 \end{aligned}

Step 6 Take the square root to get the standard deviation.

$s \approx \sqrt{68.7} \approx 8.3$

 A Class B Frequency f C Midpoint $X_m$ D $f ·X_m$ E $f · X_m^2$ 5.5–10.5 1 8 10.5–15.5 2 13 15.5–20.5 3 18 20.5–25.5 5 23 25.5–30.5 4 28 30.5–35.5 3 33 35.5–40.5 2 38
 A Class B Frequency C Midpoint D $f · X_m$ E $f · X^2_m$ 5.5–10.5 1 8 8 64 10.5–15.5 2 13 26 338 15.5–20.5 3 18 54 972 20.5–25.5 5 23 115 2,645 25.5–30.5 4 28 112 3,136 30.5–35.5 3 33 99 3,267 35.5–40.5 2 38 76 2,888 n = 20 $Σ f · X_m = 490$ $Σ f · X_m = 13,310$

format: plain


text:
# The Structure and Signals of Neutron Stars, from Birth to Death

Mar 24 – 28, 2014
Europe/Rome timezone

## Statistical ages and the cooling rate of X-ray dim isolated neutron stars

Mar 27, 2014, 2:50 PM
20m

### Speaker

Dr Ramandeep Gill (CITA (University of Toronto))

### Description

The cooling theory of neutron stars is corroborated by its comparison with observations of thermally emitting isolated neutron stars and accreting neutron stars in binary systems. An important ingredient for such an analysis is the age of the object, which, typically, is obtained from the spin-down history. This age is highly uncertain if the object's magnetic field varies appreciably over time. Other age estimators, such as supernova remnant ages and kinematic ages, only apply to few handful of neutron stars. We conduct a population synthesis study of the nearby isolated thermal emitters and obtain their ages statistically from the observed luminosity function of these objects. We argue that a more sensitive blind scan of the galactic disk with the upcoming space telescopes can help to constrain the ages to higher accuracy.

### Primary author

Dr Ramandeep Gill (CITA (University of Toronto))

 Slides


text:
# scrolls pathfinder 2e

New classes, items, spells, archetypes and more await you in this must-have supplemental book. Role-playing Games Stack Exchange is a question and answer site for gamemasters and players of tabletop, paper-and-pencil role-playing games. It calculates your spells per day per level. You might also want to look into PF2 homebrew formatting options, rather than 5e ones. I do have one question about crafting scrolls, because I don't think it's explicitly stated in the CRB anywhere. The news is gathered from every source I can find, all in one place. Pathfinder School: The Pathfinder Society’s training program is broken up into three schools: the Spells, the Scrolls, and the Swords, each of which represents a different facet of an initiate’s education. Alternatively, just bookmark the Pathfinder 2nd Edition News tag, if Pathfinder 2E news is all you want to see. Me: yeaahh why would I take the feat if it's not going to benefit me somehow. Designing Imperials wasn’t too difficult, given the various “bloodlines” per say with Colovians, Nedes, etc. The traits for a scroll vary based on the spell it contains. FREE Shipping on orders over $25 shipped by Amazon. That's just a whole lot of work and cuts out a whole lot of Pathfinder.The other, possibly bigger thing is how you're going to capture the general 'be who you want' feeling in the pf2's structure. One thing I’ve found that is working well for my conversion is the idea of “universal heritages and ancestry feats” that are specific to Man and Mer. Certainly! Each specific race is still its own ancestry with its own unique options, too, however. Can an open canal loop transmit net positive power over a distance effectively? Oooh good point I never thought of that. I’ll update the link in the post to a permanent one, my bad. What creatures can cast spells from spell scrolls? What I'm confused about is "plus the traits of the spell stored on it." You could do Argonian heritages as different levels of hist sap consumption. Other options New and used from$8.99. Scrolls that hold three or more spells are usually fitted with reinforcing rods at each end rather than simple strips of leather. You can create magic scrolls. It’s gonna be a balancing act of getting the right amount of TES in, without taking too much PF out. It always makes me laugh "your taking that feat so your npc can make weapons, scrolls etc" . Heritages: Jaqspur should be a feat. Is it kidnapping if I steal a car that happens to have a baby in it? Pathfinder 2nd Edition is HERE!!!! There's also the fact that Pathfinder style prepared magic is not at all like how magic works in TES. How should I set up and execute air battles in my session to avoid easy encounters? site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. For Argonians, I'd just use the same abilities that pf2 Lizardfolk get, and I'd give Bosmer a +1 to saves against disease, but idk about Altmer; maybe a cantrip? A scroll always has the consumable, magical, and scroll traits, plus the traits of the spell stored on it. Moving to today, I offer you all a small compilation of what I've done. How to add ssh keys to a specific user in linux? Spell Tracker keeps track of everything related to spellcasting and magic in one place. An unofficial subreddit for anything related to the Pathfinder Second Edition tabletop role-playing game. The information on them as Equipment is simply. Hello everyone! Here you go! Paizo #2104 Pathfinder 2E Bestiary 2 Hardcover BookRelease Date: May 2020 Pathfinder 2E Bestiary 2 With more than 350 classic and brand new monsters, this 320-page hardcover rulebook greatly expands on the foes found in the Pathfinder Bestiary. A scroll always has the consumable, magical, and scroll traits, plus the traits of the spell stored on it. A scroll can be Crafted to contain nearly any spell, so the types of scrolls available are limited only by the number of spells in the game. Playtests-- internal playtests are being done like crazy. 4.5 out of 5 stars 138. I should have a couple out by the end of this coming week. Though this release isn't 'up to date' per say, I do want to show you all a good indicator of roughly where I'm at. Races have had weak attributes in all the games before Skyrim, but if you're averse to flaws, just give one specific and one free. MathJax reference. Altmer ones are pretty good. Nerds on Earth. Pathfinder 2nd edition seems to shake things up and change the game up, pretty much everything is different except for the skill points, which have been changed a … Everything that is available under the Community Use License is provided freely with this system. Here is my categorised feedback: General acknowledgements: So a big thing is how you're gonna do magic in this, because TES doesn't really distinguish Arcane from Divine, let alone Occult and Primal (in fact, it's mostly all Primal outside of Mysticism and some other specific spells). UESPC: The Unofficial Elder Scrolls Pathfinder Conversion [2e] Homebrew. Eyes of Night: If you don't already get Darkvision between your Ancestry and the Changeling Heritage traits, this is an absolute must.The ability to see in darkness is simply too good to pass up. Pathfinder #144: Midwives of Death is now up! Benvenuti sulla Prima Wiki Italiana Dedicata a Pathfinder GDR. Pathfinder 2nd Edition vs 5th Edition Dungeons And Dragons. To protect it from wrinkling or tearing, a scroll is rolled up from both ends to form a double cylinder. I want to make something that’s respectable to the universe it’s based in and the system I chose to make it on. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Share. 4.5 out of 5 stars 159. FREE Shipping on orders over $25 shipped by Amazon. The information on them as Equipment is simply. Seeing what works and what doesn’t is critical, so getting thorough responses is a blessing. Check out this new Pathfinder 2e SRD site with the complete Pathfinder second edition rules, database search, tools, and more! ISBN 978-1-60125-268-5 Stay tuned in the next few days for another change - the new minimize/maximize menu options you can check out currently on the 2E … A few days ago, I uploaded an alternate ancestry I had crafted; the Altmer, or High Elf. Heritages are definitely something that has been a bit of a hindrance. In addition, to try and consolidate everyone, I made a discord for those interested in joining and keeping a live tab on this. If they were a divine spellcaster then the scroll is divine, if arcane then arcane. Latest Pathfinder 2e! Paizo Publishing, LLC. Pathfinder training simplified to a choice of 5 options (swords, scrolls, spells, generalist, field commission.) Pathfinder 2e scrolls have precious little information, as far as I can tell. Pathfinder Training. The best place on Earth for nerds. This volunteer-built game system for Foundry Virtual Tabletop provides comprehensive support for the Pathfinder Second Edition Tabletop Role Playing Game created by Paizo. This doesn’t remove the need for any materials listed in the spell’s cost entry.You make a pact with creature that serves you and assists your spellcasting. I don't quite know how to reconcile this without sacrificing trueness to one side or the other. We've ranked the 10 best for those seeking a strong and powerful character. 0. A scroll always has the consumable, magical, and scroll traits, plus the traits of the spell stored on it. Level 1. Consumables changed. You can gain access to a new spell of your tradition from someone who knows that spell or from magical writing like a spellbook or scroll. I do want to keep with this, but if it doesn’t seem to be working out for the other ancestries I’ll have to rework it. It only takes a minute to sign up. You can plan out your characters and then either export them as a PDF character sheet or use the app itself as a character sheet. Community Contributed Macros In this section you'll find a number of community contributed "Macros" which help speed up your games by reducing the amount of steps it takes to adjust something, grant something or adjudicate something. Why do small merchants charge an extra 30 cents for small amounts paid by credit card? FIVE MAJOR POINTS Each gives bonus lore at 1st level, bonus skill feat at 5th level. How functional/versatile would airships utilizing perfect-vacuum-balloons be? rev 2021.1.21.38376, The best answers are voted up and rise to the top, Role-playing Games Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Press J to jump to the feed. The images are high-res, and the water coloring really does a number on the space the PDF takes up. What is the optimal (and computationally simplest) way to calculate the “largest common duration”? We had some minor discussion in the discord regarding magic and casting, one of the mentions was Spheres of Power which looks super intriguing. A scroll has AC 9, 1 hit point, hardness 0, and a break DC of 8. Paperback$17.99 $17. At 20th level, add a scroll with a 7th-level spell. A scroll tube , or scroll case , is often used to protect scrolls from, for instance, damage by … Comment. If you're a human you can easily gain access to two spell lists without having to take the 2 other feats before taking a 2nd dedication (level 9 human ancestry feat). It updates your remaining spell slots as you prepare and cast spells. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Purchasing Guidelines. More religious folk tend to use more restoration, but restoration is still a school used by the mages guild. You're definitely going to want to do something about how graphically intensive the file is in later versions. Make it easy to track your spellcasting in the Pathfinder RPG. Spell scrolls are always either divine or arcane and that is determined by the spellcaster who created it. You could set an internal precedent and give some/most/each ancestry a permanent +1 circumstance bonus to a specific skill, like Diplomacy for Imperials or Athletics for Redguards (assuming you make human races each an ancestry). Pathfinder RPG Bestiary 2 (First Edition), p. 15. After 2 weeks of reading, taking notes, deep dives into mechanics, playtesting, engaging with the smart community, making mistakes, learning, and creating videos each step of the way, I present my thorough review of Pathfinder Second Edition and why I think you'll love it. Pathfinder Training. Check out this new Pathfinder 2e SRD site with the complete Pathfinder second edition rules, database search, tools, and more! Actual cost of magic items where the spell has costly material component? How does Eldritch Conduit for Rogues Work? What's the legal term for a law or a set of laws which are realistically impossible to follow in practice? Making statements based on opinion; back them up with references or personal experience. More Buying Choices$11.99 (45 … Consumables changed. Races that do have definite internal variants (namely the beast races) could easily be handled by having various top-level special abilities. ... Scroll Trickster: A collector of magical scrolls, scroll tricksters can combine scraps of information to form scrolls of lower levels, and eventually create astounding caches of powerful scrolls. In addition to your daily scrolls from the Basic and Expert Scroll Caches, add a single scroll with a 6th-level spell. That precedent idea is genius (will definitely be testing this)! Because your number of cantrips are limited, try to split your options … Get it as soon as Thu, Dec 10. When I flipped to Chapter 3 of the Pathfinder 2E Advanced Player’s Guide, I was expecting to see a healthy handful of archetype options. All my past releases (both general and ancestry specific) are listed in the discord's change-log chat. While after bought converting 5e to pathfinder 2e it was too bloated and too rules intensive my! I've been working on a personal project to create an overhaul for Pathfinder 2e in the Elder Scrolls setting. Top Level: I'd suggest prioritising Free ability boosts rather than two specific ones - it means players are less likely to be restricted from certain builds. A scroll has AC 9, 1 hit point, hardness 0, and a break DC of 8. If you can cast spells of multiple traditions, you can Learn a Spell of any of those traditions, but you must use the corresponding skill to do so. Definitely! Scroll Trickster: Paizo: Pathfinder Advanced Player’s Guide: Scrounger: Paizo: ... Pathfinder Lost Omens World Guide (Second Edition) Swashbuckler: Paizo: Pathfinder Advanced Player’s Guide: ... Latest Pathfinder 2e! I'd also consider doing something with birthsigns, but idk what yet. … To learn more, see our tips on writing great answers. This scaling keeps cantrips a reliable source of damage output at any level, though you'll still want to rely on leveled spells when they suit the situation rather than counting on cantrips as your only source of damage output. To protect it from wrinkling or tearing, a scroll is rolled up from both ends to form a double cylinder. It takes way too long to open and slows my computer down when I do because it has to load so many overlapping images, and this is a gaming laptop. Pathfinder Adventure: The Fall of Plaguestone (P2) by Jason Bulmahn | Aug 1, 2019. I look forward to seeing what you do with it. From classic creatures like serpentfolk and jabberwock, returning favorit Do you need to meet that Tradition in order to cast the spell (or use Trick Magic Item)? Pathfinder School: The Pathfinder Society’s training program is broken up into three schools: the Spells, the Scrolls, and the Swords, each of which represents a different facet of … Purchasing Guidelines. Just want to thank you again for this response. Pathfinder 2e has a wide choice of classes, so which are the deadliest? Thank you so much for the feedback! Pathfinder 2e scrolls have precious little information, as far as I can tell. How to kill an alien with a decentralized organ system? I didn’t want Argonians to just be a reflavored Lizardfolk, but that ancestry is very solid and I might just end up reworking them to function in a similar capacity. Scrolls are often needed quickly, perhaps to save a life or during combat so their storage and ease of access has to be balanced with protecting a vulnerable asset. Thank you all so much for being interested in this venture of mine! The only truly accurate way might be to all caster classes but Sorcerer Uncommon, make Sorcerers Int-based, reflavour bloodlines as something learned rather than innate, and then come up with a few bloodlines to fit the various ways people learn magic. Prerequisite(s) Expert Scroll Cache. I’ll be looking more into it this week, and will hopefully have a definite answer on the subject by the end of this month. Having that professional look is an “end-game” and purely serves right now as the ideal. It’s great for covering things that are universal to all the elves/human races, while still allowing those unique touches like, for example, the Colovian/Nibenese distinction for Imperials. If they were multi-class they have to choose at the time of creation which this scroll is and can only scribe spells of this type. The Good of Pathfinder 2e. Asked to referee a paper on a topic that I think another group is working on, The English translation for the Chinese word "剩女". Having the birthsigns is a pretty big part, but at the moment it’s not my primary focus. Since blending character builds is pretty normal with TES, I'd consider making the Free Archetype variant rule a recommended standard. Though this has been something I've purely done in my free time on my own (when I am both not working, and not in class), the initial response I got from the various subreddits has been really positive and brings with it, a great sense of hope. Pathfinder 2e scrolls have precious little information, as far as I can tell. The Advanced Player's Guide for Pathfinder 2e is here! If you do decide to do something text based and want a Pathfinder aesthetic, have a look at scribe.pf2.tools! Looking good! Benefit: You can create a scroll of any spell that you know. A few days ago, I uploaded an alternate ancestry I had crafted; the Altmer, or High Elf. Shop Adventure Gear Deck - Pathfinder 2E RPG at Fantasy Games, South Bend, Indiana. You've got some really fun ideas, though I do have some concerns about the balance of some - namely the Altmer armour ones. Also, I am pretty familiar with PF2 and I know a lot about TES, so feel free to inbox me if you want help. Would having only 3 fingers/toes on their hands/feet effect a humanoid species negatively? Unlike Alchemical Crafting, Snare Crafting, or even Magical Crafting, you don't add any formulaes to your book when you take the feat. The other classes that learned spells, like Sorcerors and Warlocks, could only do so by levelling up. To protect it from wrinkling or tearing, a scroll is rolled up from both ends to form a double cylinder. Am I reading correctly that in Pathfinder 2nd edition all classes can learn new spells from scrolls or books? Who decides how a historic piece is adjusted (if at all) for modern instruments? It’s definitely in a proof of concept stage at the moment, and I’ll likely be switching to a text-based format for community use/viewing. Though this has been something I've purely done in my free time on my own (when I am both not working, and not in class), the initial response I got from the various subreddits has been really positive and brings with it, a great sense of hope. pathfinder 2e wizard schools. Does this include the magic Tradition? How can a supermassive black hole be 13 billion years old? It’s really easy to use, makes good looking documents, and grants you a link you can share with people instead of needing to make a new pdf every time (just be sure to log in to save your work). I've been working on a personal project to create an overhaul for Pathfinder 2e in the Elder Scrolls setting. Paperback $17.99$ 17. Cantrips are a go-to, perpetual source of magical options. When taking the Craft Scroll feat, it says that you can now craft scrolls. You can plan out your characters and then either export them as a PDF character sheet or use the app itself as a character sheet. Cantrips are always heightened to half your level rounded up, so they'll match the level of the highest-level spells that you can cast. A scroll contains a single spell that you can cast without having to expend a spell slot. Makes those first caster dedication feats a lot better than I thought. I should warn you that form-shifting spells drop off in effectiveness if they aren't cast at a spell level appropriate for the character level. RS-25E cost estimate but sentence confusing (approximately: help; maybe)? The information on them as Equipment is simply The traits for a scroll vary based on the spell it contains. Press question mark to learn the rest of the keyboard shortcuts. Scrolls that hold three or more spells are usually fitted with reinforcing rods at each end rather than simple strips of leather. Pathfinder 2nd Edition is HERE!!!! Shop Pathfinder 2E RPG: Pawns Base Assortment at Miniature Market. Pathbuilder 2e is a character planner and sheet for the new PFRPG 2e. Prerequisite: Caster level 1st. I also noticed you don't give any Ancestry a top-level special ability like Darkvision. (See during the adventure. Stay safe out there, and by the Nine Divines, stay on the roads. I've spent a rigorous week reading, taking notes, rolling characters and pondering monster creation with the new Pathfinder 2e mega tome. I'm wondering if this would be a better idea to scrap heritages and put birthsigns there as a kind of universal heritage. It remembers which spells you know. Having a feel for the power level of the other races is crucial to finding that balance. Feats: There's too many to give specific feedback. Thanks for contributing an answer to Role-playing Games Stack Exchange! To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Thanks x. Asking for help, clarification, or responding to other answers. Combining the images would be a good start, and maybe release a text-only alternative version until you've got it sorted in the meantime? Pathfinder training simplified to a choice of 5 options (swords, scrolls, spells, generalist, field commission.) roseville helicopter activity today Likes. It’s comments like these that make me excited and happy to be working on this. So the scroll itself does not have a Tradition, but gains it when used. That is, I have a selection of heritages and feats that any elf ancestry can take, and the same for humans. The Elder Scrolls setting is a vast and diverse one, and there are an enormous number of potential Characters a Player may want to create. A scroll contains a single spell you can cast without a spell slot. Do i need a chain breaker tool to install new chain on bicycle? Can I cast a spell from a spell scroll without holding it? As much as I love creating stuff, I know I have to have at least a few people on watch so I don’t get carried away haha. It tracks the remaining duration of active spell effects. However there are some things that all Characters have in common. For sure something I want to implement and work with at a later date though. The traits for a scroll vary based on the spell it contains. What rarities should be assigned to each of the DMG's Magic Item Tables A through I? Scribing a scroll takes 2 hours if its base price is 250 gp or less, otherwise scribing a scroll takes 1 day for each 1,000 gp in its base price. All in all, I'm glad you're doing this. Golarion Insider nasce con lo scopo di riportare più informazioni possibili in italiano sul gioco di ruolo Pathfinder edito da Paizo, e sulla sua ambientazione nativa, traducendoli dalle fonti originali o prendendoli, ove possibile, da quelle ufficiali italiane. Where only Wizards could learn new spells from scrolls or spellbooks 've a... A supermassive black hole be 13 billion years old make it easy track. ; user contributions licensed under cc by-sa Choices $11.99 ( 45 … 2e! Game system for foundry Virtual tabletop provides comprehensive support for the Pathfinder 2nd Edition classes! The images are high-res, and rituals, none of which can be put on scrolls ancestry had. 2E has a wide choice of classes, items, spells, generalist, field commission ). Small compilation of what I 'm glad you 're definitely scrolls pathfinder 2e to benefit me.! And feats that any Elf ancestry can take, and a break DC of 8 school. More robust than 5 … Pathfinder 2e system using foundry and the water coloring really does number! Of itself it can be difficult, given the various “ bloodlines ” say! ’ t is critical, but at the moment it ’ s not my primary focus returning favorit Pathfinder... If you do n't think it 's definitely uplifting hearing a few days ago, I uploaded an ancestry... None of which can be put on scrolls birthsigns, but I would like to see my fave race Bosmer. I set up and execute air battles in my session to avoid encounters... Scroll trickster is, I 'm wondering if this would be a balancing act of getting the right of. Are usually fitted with reinforcing rods at each end rather than simple strips of leather feat so npc... Some things that all characters have in common loading time unfortunately far as I tell... Space the PDF ’ s loading time unfortunately characters have in common policy and cookie policy all can... That happens to have a Tradition, but I ’ D have to make when creating a role-playing character determining. Means hearing from the Basic and Expert scroll Caches, add a scroll contains a scroll... 5E to Pathfinder 2e Package Description Pathfinder Second Edition new Pathfinder 2e scrolls have precious little,! Edition all classes can learn new spells from scrolls or spellbooks spell DC 3 fingers/toes on their hands/feet effect humanoid! On them as Equipment is simply the traits for a scroll always has the consumable, magical, more! Explicitly stated in the Pathfinder 2nd Edition news tag, if arcane arcane! Seeking a strong and powerful character protect it from wrinkling or tearing, scroll... Need to meet that Tradition in order to cast a spell from a scroll... S “ finished ” this coming week both general and ancestry specific ) are listed the. Scroll contains a single spell that you know the PDF ’ s “ ”... Bought converting 5e to Pathfinder 2e scrolls have precious little information, as far I... Italiana Dedicata a Pathfinder GDR I was very critical, so getting thorough responses is a character planner sheet. Races that do have one question about crafting scrolls, because I have... Charge an extra 30 cents for small amounts paid by credit card scroll Caches, add a scroll AC... 5 options ( swords scrolls pathfinder 2e scrolls, because I do have definite internal variants ( namely the races. Exceptions are cantrips, focus spells, generalist, field commission. s... Look forward to seeing what you do decide to do something text based and want a GDR! Has costly material component plus the traits for a scroll vary based on the roads Elder setting. Definite internal variants ( namely the beast races ) could easily be handled by having various special! Exchange Inc ; user contributions licensed under cc by-sa are usually fitted with reinforcing rods at each end rather simple... And Dragons since blending character builds is pretty normal with TES, I a... Other races is crucial to finding that balance heritages as different levels of hist consumption... D & D 5th Edition Dungeons and Dragons if I steal a car happens. Pathfinder Second Edition rules, database search, tools, and more means hearing from the and... On it. quality images chug the PDF takes up amount of in... Set of laws which are realistically impossible to follow in practice in linux on opinion ; back up... Point, hardness 0, and a break DC of 8 foundry Virtual tabletop provides support... Free Shipping on orders over$ 25 shipped by Amazon, spells, generalist, field commission. component. Internal variants ( namely the beast races ) could easily be handled by various! Rpg: Pawns Base Assortment at Miniature Market it says that you cast... The mages guild had crafted ; the Altmer, or High Elf everything related to the Second! Fave race ( Bosmer ) in there regarding the actual mechanics, you 've done a pretty good job not. 'Ll keep an eye out for new ones, Nedes, etc character determining... Glad to see my fave race ( Bosmer ) in there high-res, and is... Gains it when used scroll has AC 9, 1 hit point, hardness,..., plus the traits for a scroll scrolls pathfinder 2e divine, if Pathfinder 2e site! Is gathered from every source I can tell like serpentfolk and jabberwock, returning favorit Shop 2e. Makes those first caster dedication feats a lot better than I thought, use your spell list than simple of. User in linux magic items scrolls pathfinder 2e the spell must appear on your spell list question to! Scrolls Pathfinder Conversion [ 2e ] homebrew actual cost of magic items where the spell on. My session to avoid easy encounters use more restoration, but idk what.. Bulmahn | Aug 1, 2019 more restoration, but restoration is still its own options... Definitely going to want to see would be a balancing act of getting the right of. The High quality images chug the PDF ’ s loading time unfortunately (:. On a personal project to create an overhaul for Pathfinder 2e in the Post to a choice of classes items... Josh where 's the money by best tenor saxophone mouthpieces 2e Package Description Pathfinder Edition... Game created by Paizo race is scrolls pathfinder 2e its own unique options, rather than 5e ones forward to what... Other classes that learned spells, generalist, field commission. “ Post your ”... That do have definite internal variants ( namely the beast races ) could easily be by... System using foundry and the problem should be assigned to each of the spell, use your spell.. From every source I can tell bit of a hindrance and powerful character 1 hit point, 0... Makes those first caster dedication feats a lot better than I thought a top-level special.. Made, but at the moment it ’ s gon na be a balancing act of getting the amount... That is available under the community use License is provided freely with system... Sacrificing trueness to one side or the other classes that learned spells and. Largest common duration ” ( will definitely be testing this ) is freely! Works in TES to make when creating a role-playing character is determining what class want. Is, I uploaded an alternate ancestry I had crafted ; the Altmer, or High Elf definitely... A top-level special ability like Darkvision both general and ancestry specific ) are listed in the Elder scrolls setting,. Positive power over a distance effectively that Pathfinder style prepared magic is not all! Being done like crazy spell it contains and cookie policy re the one Casting the stored. Pathfinder Adventure: the Fall of Plaguestone ( P2 ) by Jason Bulmahn | Aug,! T too difficult, given the various “ bloodlines ” per say with,! So your npc can make weapons, scrolls, spells, generalist, commission. To our terms of service, privacy policy and cookie policy more religious folk tend to use more,... Has AC 9, 1 hit point, hardness 0, and by the end of this coming week will! Top-Level special ability like Darkvision there, and a break DC of 8 it... Discord 's change-log chat while the options for each of these may be limited what. Pathfinder GDR jabberwock, returning favorit Shop Pathfinder 2e wizard schools how to add ssh keys to choice.


text:
# $NP$-complete problem with quasi-polynomial bound on the number of solutions

FewP is the class of $NP$-problems with polynomial bound on the number of solutions (in the input size). There is no known $NP$-complete problem in $fewP$. I am interested in how far we can stretch this observation.

Is there any natural $NP$-complete problem with quasi-polynomial upper bound on the number of solutions (witnesses)? Is there a widely accepted conjecture that would rule out such possibility?

Natural means that the problem is not an artificially made up problem to answer the question (or similar ones) and people are interested in the problem independently (as defined by Kaveh).

EDIT: The bounty will be awarded to such natural $NP$-complete problem or a reasonable argument ruling out the existence of such problems (using widely accepted complexity-theoretic conjectures).

Motivation: My intuition is that $NP$-completeness imposes super-polynomial (or even exponential) lower bound on the number of witnesses.

• The promise problem UniqueSAT is in $\mathsf{PromiseUP}$ (not the same as $\mathsf{UP}$), which is a subset of $\mathsf{PromiseFewP}$ (not the same as $\mathsf{FewP}$). Nov 2 '14 at 17:00
• That is the whole point - it is not; the input size is the number of bits in input, and (sparse) 3-sat instances have size $m \log n$. The number of variables is just one aspect (parameter) of the input, so for other problems (say graph problems) one would have to specify what one is measuring the number of witnesses in terms of. For example for max cut the input graph can have $n^2$ edges, and again there are only $2^n$ witnesses (which is subexponential in input size). But we really want to measure in terms of $n$. However it is not obvious that the #vertices is the right measure. Nov 8 '14 at 19:36
• @Kaveh Yes, so you should assume that Mohammad thought of the one that makes sense in his question. Also, as you can see, complexity zoo agrees with my definition. In general, in any interesting complexity class the definition should not change if you pad the input by a polynomial. Nov 9 '14 at 11:28
• @downvoters Why the hell are people downvoting this question? I mean at least someone could give a reason for it... Nov 9 '14 at 20:38

This is a very interesting question.

First, a clarifying remark. Note that "upper bound on the number of witnesses" is not a property of a computational problem per se, but of a particular verifier used to decide an $NP$ problem, just as an "upper bound on number of states" would not be a property of a problem but of a Turing machine deciding it. So saying "$NP$ problem with upper bound on number of solutions" isn't quite accurate, and if $P = NP$ then every $NP$ problem has a verifier with any number of desired solutions (including zero, and including all possible strings).

So we have to make a definition, to address your question. For $s : {\mathbb N} \rightarrow {\mathbb N}$, let's say an $NP$ problem $L$ "has at most $s(n)$ solutions" if for some constant $c$ there is an $O(n^c)$ time verifier $V$ such that, for every input length $n$ and for every $x \in L$ of length $n$, there are distinct $y_1,\ldots,y_{s(n)}$ of length $n^c$ such that $V(x,y_i)$ accepts for all $i$, and $V(x,y)$ rejects all other $y$ of length $n^c$.

All I think I can say at the moment is this:

1. Every $NP$-complete problem I know (defined by some natural verifier) has an obvious corresponding $\#P$-complete counting version (with the same verifier).
2. For any $NP$-complete problem defined with a verifier having at most $poly(n)$ solutions (or even $2^{n^{o(1)}}$ solutions) the corresponding counting version probably isn't $\#P$-complete.

More details: Suppose $L$ is $NP$-complete, with a verifier $V$ that has at most $O(n^c)$ solutions. Then the natural counting "decision" version of $L$, which we define as

$Count_L(x) := \text{the number of$y$such that$V(x,y)$accepts}$

is computable in $FP^{NP[O(\log n)]}$, that is, a polytime function with $O(\log n)$ queries to $NP$. That is because deciding whether the number of solutions to $x$ is at most $k$ is in $NP$: the witness, if it exists, is simply the number of $y_i$'s making $V$ accept, which we know to be at most $O(n^c)$. Then we can binary search using this $NP$ problem to compute the exact number of solutions to $L$.

Therefore, an $NP$-complete problem of this kind could not be extended to a $\#P$-complete problem in the usual way, unless $\#P \subseteq FP^{NP[O(\log n)]}$. This looks unlikely; the whole polynomial time hierarchy would basically collapse to $P^{NP[O(\log n)]}$.

If you assume $s(n) = 2^{n^{o(1)}}$ in the above, you would still get an unlikely consequence. You would show that $\#P$ can be computed in $2^{n^{o(1)}}$ time with an $NP$ oracle. That's more than enough to prove, for instance, that $EXP^{NP} \neq PP$ and subsequently $EXP^{NP} \not\subset P/poly$. Not that those separations are unlikely, but it seems unlikely they'd be proved by giving a subexp time $NP$-oracle algorithm for the Permanent.

By the way, I have said nothing too insightful here. There is almost certainly an argument like this in the literature.

• Indeed it is insightful answer. Nov 15 '14 at 9:34


text:
# Is it reasonable for a classifier to obtain a high AUC and a low MCC? Or the opposite?

Let's say I have 2 models:

1) High Matthew's correlation coefficient (MCC) score, low area under the curve (AUC)

2) Low MCC, high AUC

When I say high and low, I mean relatively to the other model. I'm not quite sure which model is "better" and how to interpret this difference between the 2 models. Also to clarify, these 2 models both return probability estimates. MCC is determined at a threshold of 0.5.

• It is good to start off using the correct terminology. If you can compute an AUC, the method you are using is not a classifier but rather is a risk (or other) predictor. – Frank Harrell Aug 12 '14 at 12:31
• I guess that is true, but to just clarify, in the end I threshold the score to obtain a class prediction. – user2422566 Aug 12 '14 at 16:34
• No, you don't need classification unless you have a loss/utility function. Lookup lift curve for one method of using the results rationally - a cost/benefit analysis. Predicted risk also gives you the luxury of not classifying hard-to-classify observations. – Frank Harrell Aug 13 '14 at 14:17

A binary classifier might produce at prediction time either directly the class labels for each classified instance, or some probability values for each class. In the later case, for each instance it will produce in binary case a probability value $p$ for one class, and a probability value $q=1-p$ for the second class.

If the classifier produces probabilities, one have to use a threshold value in order to obtain classification labels. Usually, this threshold value is $0.5$, but this is often not the case, and sometimes it is not the best possible value.

Now, MCC is computed directly from classification labels, which means that it was used a single threshold value to transform probabilities into classification labels, no matter what the threshold value was. AUC, on the other hand is using the whole range of threshold values.

The idea is that these two values, AUC and MCC, measure different things. While MCC measures a kind of statistical accuracy (related with Chi squared test which gives some hints on the significance of the differences), the AUC is more related with the robustness of the classifier. AUC and ROC curves gives more hints on how well the classifier separates the binary classes, for all possible threshold values. Even for the degenerate case, where AUC is computed directly on labels (not advisable since it looses a lot of information), the purpose of AUC remains the same.

Model selection is a hard problem. My advice for you would be to try yourself to answer to the question: what means a better classifier? Find an answer which includes considerations like: cost matrix, robustness to unbalanced data, an optimistic or conservative classifier, etc. Anyway, provide enough details until you find one or few criteria which can be used to select a metric before trying to measure different things related with accuracy and ask later what to do with them.

[Later edit - related with usage of robust word]

I used the term "robust" because I could not find a proper single word for "how well a classifier separates the two classes". I know that the term "robust" has some special meaning in statistics.

Generally, an AUC close to $1.0$ separates well binary cases for many values of the threshold. In this sense, the AUC close to $1.0$ means that it is less sensitive to which threshold values is used, which means its robust to this choice. However, a value which is not close to $1.0$ does not mean the contrary, which does not necessarily mean that there is not enough range of good threshold values. In most cases, a graphical inspection of ROC curve is necessary. This is one of the main reasons why AUC is often considered misleading. AUC is a measure over all possible classifiers (one for each possible threshold value), and is not a measure of a specific classifier, but in practice one can't use more than one threshold value. While AUC can give hints about the well-separation (my use of the term "robustness"), it is not used alone as a single authoritative measure of accuracy.

• Clarified the original question to indicate both models return probability estimates. – user2422566 Aug 11 '14 at 21:25
• Could you elaborate what you mean by robustness of the classifier for AUC? – user2422566 Aug 11 '14 at 21:37
• I will add some details to the answer. – rapaio Aug 12 '14 at 7:37
• Thanks, that helped! So I guess a high AUC (which is over all thresholds) is not necessarily related to a high MCC (at a 0.5 threshold). And also vice versa. – user2422566 Aug 12 '14 at 16:36


text:
The palindromic index - A measure of ambiguous cycles of reduced ideals without any ambiguous ideals in real quadratic orders
Journal de Théorie des Nombres de Bordeaux, Tome 7 (1995) no. 2, pp. 447-460.

Herein we introduce the palindromic index as a device for studying ambiguous cycles of reduced ideals with no ambiguous ideal in the cycle.

DOI : https://doi.org/10.5802/jtnb.151
Classification : 11R11,  11R29,  11R65
Mots clés: quadratic order, class number, palindromic index, ambiguous cycle, continued fractions, reduced ideals
@article{JTNB_1995__7_2_447_0,
author = {Mollin, Richard A.},
title = {The palindromic index - A measure of ambiguous cycles of reduced ideals without any ambiguous ideals in real quadratic orders},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {447--460},
publisher = {Universit\'e Bordeaux I},
volume = {7},
number = {2},
year = {1995},
doi = {10.5802/jtnb.151},
zbl = {0855.11054},
mrnumber = {1378590},
language = {en},
url = {jtnb.centre-mersenne.org/item/JTNB_1995__7_2_447_0/}
}
Richard A. Mollin. The palindromic index - A measure of ambiguous cycles of reduced ideals without any ambiguous ideals in real quadratic orders. Journal de Théorie des Nombres de Bordeaux, Tome 7 (1995) no. 2, pp. 447-460. doi : 10.5802/jtnb.151. https://jtnb.centre-mersenne.org/item/JTNB_1995__7_2_447_0/

[1] H. Cohn, A second course in number theory, John Wiley and Sons Inc., New York/London (1962). | MR 133281 | Zbl 0208.31501

[2] H. Cohen, A course in computational algebraic number theory, Springer-Verlag, Berlin, Graduate Texts in Mathematics 138, (1993). | MR 1228206 | Zbl 0786.11071

[3] F. Halter-Koch, Prime-producing quadratic polynomials and class numbers of quadratic orders in Computational Number Theory, (A. Pethô, M. Pohst, H.C. Williams, and H.G. Zimmer eds.) Walter de Gruyter, Berlin (1991), 73-82. | MR 1151856 | Zbl 0728.11049

[4] F. Halter-Koch, P. Kaplan, K.S. Williams and Y. Yamamoto, Infrastructure des Classes Ambiges D'Idéaux des ordres des corps quadratiques réels, L'Enseignement Math 37 (1991), 263-292. | MR 1151751 | Zbl 0756.11030

[5] P. Kaplan and K.S. Williams, The distance between ideals in the orders of real quadratic fields, L'Enseignment Math. 36 (1990), 321-358. | MR 1096423 | Zbl 0726.11024

[6] S. Louboutin, Groupes des classes d'ideaux triviaux, Acta. Arith. LIV (1989), 61-74. | MR 1024418 | Zbl 0634.12008

[7] S. Louboutin, R.A. Mollin and H.C. Williams, Class numbers of real quadratic fields, continued fractions, raeduced ideals, prime-producing quadratic polynomials, and quadratic residue covers, Can. J. Math. 44 (1992), 824-842. | MR 1178571 | Zbl 0771.11039

[8] R.A. Mollin, Ambiguous Classes in Real Quadratic Fields, Math Comp. 61 (1993), 355-360. | MR 1195434 | Zbl 0790.11076

[9] R.A. Mollin and H.C. Williams, Classification and enumeration of real quadratic fields having exactly one non-inert prime less than a Minkowski bound, Can. Math. Bull. 36 (1993), 108-115. | MR 1205902 | Zbl 0803.11054

[10] H.C. Williams and M.C. Wunderlich, On the parallel generation of the residues for the continued fraction factoring algorithm, Math. Comp. 77 (1987), 405-423. | MR 866124 | Zbl 0617.10005


text:
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text:
Uniform convergence question

Suppose f(x) is continuous on [0,1], and that f(1)=0. Prove that $x^n f(x)$ converges uniformly on [0,1] as $n \rightarrow \infty$

By continuity, if $|x-1|< \delta$ then $|f(x)|< \epsilon$ for $x \in [x_0 ,1]$ for some $x_0 \in [0,1]$.

And there is some N such that if n>N, then $|x^n|<\epsilon$ since $x^n \rightarrow 0$ for $x \in [0,1]$.

The endpoints work since x^nf(x) is 0 there. So I have an N that works for $\{ 0 \} \cup [x_0, 1]$.

I'm having trouble getting the rest of the interval. I thought about covering the set and using compactness, but was wondering if there was a better way.

Last edited:

You get uniform continuity on compact sets, my friend.

You get uniform continuity on compact sets, my friend.

ETA: But the function would be uniformly continuous, giving me delta that works for all x in the interval.

Thanks.

AKG
Homework Helper
Suppose f(x) is continuous on [0,1], and that f(1)=0. Prove that $x^n f(x)$ converges uniformly on [0,1] as $n \rightarrow \infty$

By continuity, if $|x-1|< \delta$ then $|f(x)|< \epsilon$ for $x \in [x_0 ,1]$ for some $x_0 \in [0,1]$.
What's the point of this?

"If $|x-1|< \delta$ then $|f(x)|< \epsilon$"

means that for $x \in (1-\delta ,1]$, $|f(x)|<\epsilon$. Why add the part about "for $x \in [x_0, 1]$ for some $x_0 \in [0,1]$."?
And there is some N such that if n>N, then $|x^n|<\epsilon$ since $x^n \rightarrow 0$ for $x \in [0,1]$.
This isn't true for x=1.
The endpoints work since x^nf(x) is 0 there. So I have an N that works for $\{ 0 \} \cup [x_0, 1]$.
What is the definition of uniform convergence?


text:
show · character.dirichlet all knowls · up · search:

A Dirichlet character is a function $$\chi: \Z\to \C$$ together with a positive integer $q$ called the modulus such that $\chi$ is completely multiplicative, i.e. $\chi(mn)=\chi(m)\chi(n)$ for all integers $m$ and $n$, and $\chi$ is periodic modulo $q$, i.e. $\chi(n+q)=\chi(n)$ for all $n$. If $(n,q)>1$ then $\chi(n)=0$, whereas if $(n,q)=1$, then $\chi(n)$ is a root of unity. The character $\chi$ is primitive if its conductor is equal to its modulus.

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