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# Composite Logconcave Sampling with a Restricted Gaussian Oracle

We consider sampling from composite densities on ℝ^d of the form dπ(x) ∝(-f(x) - g(x))dx for well-conditioned f and convex (but possibly non-smooth) g, a family generalizing restrictions to a convex set, through the abstraction of a restricted Gaussian oracle. For f with condition number κ, our algorithm runs in O (κ^2 d log^2κ dϵ) iterations, each querying a gradient of f and a restricted Gaussian oracle, to achieve total variation distance ϵ. The restricted Gaussian oracle, which draws samples from a distribution whose negative log-likelihood sums a quadratic and g, has been previously studied and is a natural extension of the proximal oracle used in composite optimization. Our algorithm is conceptually simple and obtains stronger provable guarantees and greater generality than existing methods for composite sampling. We conduct experiments showing our algorithm vastly improves upon the hit-and-run algorithm for sampling the restriction of a (non-diagonal) Gaussian to the positive orthant.

## Authors

• 8 publications
• 19 publications
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## 1 Introduction

We study the problem of approximately sampling from a distribution on , with density

 dπ(x)dx∝exp(−f(x)−g(x)). (1)

Here, is assumed to be “well-behaved” (i.e. has finite condition number), and is a convex, but possibly non-smooth function. This problem generalizes the special case of sampling from for well-behaved , simply by setting

to be uniformly zero. The existing (and extensive) literature on logconcave sampling, a natural problem family with roots in Bayesian statistics, machine learning, and theoretical computer science, typically focuses on the case when the log-density is well-behaved, and the distribution has support

. Indeed, even the specialization of (1) where indicates a convex set is not well-understood; existing bounds on mixing time for this restricted setting are large polynomials in [BDMP17, BEL18], and typically weaker than guarantees in the general logconcave setting [LV06b, LV06a], where no assumptions are made at all other than convexity of , and only access to a zeroth order oracle is assumed111Throughout, we refer to a first order oracle for function as returning on query , the pair , whereas a zeroth order oracle only returns . Typical methods developed for sampling in the well-conditioned log-density regime are based on interacting with first order oracles..

Sampling from logconcave distributions and optimization of convex functions have a close relationship, which has been extensively studied [BV04, LV06a]. However, the toolkit for first-order convex optimization has to date been much more flexible in terms of the types of problems it is able to handle, beyond optimizing well-conditioned functions. Examples of problem families which efficient first-order methods for convex optimization readily generalize to solving are

 minx∈Xf(x), where X⊆Rd is a convex set,

as well as its generalization

 minx∈Rdf(x)+g(x), where g:Rd→R is convex and admits a proximal oracle. (2)

The seminal work [BT09] extends accelerated gradient methods to solve (2) via proximal oracles, and has prompted many follow-up studies. Existence of an efficient proximal oracle is a natural measure of “simplicity” of in the context of composite optimization, which we now define.

###### Definition 1 (Proximal oracle).

is a proximal oracle for convex if it returns

 O(λ,v)←argminx∈Rd{12λ∥x−v∥22+g(x)}.

In other words, a proximal oracle minimizes functions which sum a quadratic and . It is clear that the proximal oracle definition implies they can also handle arbitrary sums of linear functions and quadratics, as the resulting function can be rewritten as the sum of a constant and a single quadratic. Definition 1 is desirable as many natural non-smooth composite objectives arising in learning settings, such as the Lasso [Tib96] and elastic net [ZH05], admit efficient proximal oracles.

### 1.1 Our contribution

Motivated by the success of the proximal oracle framework, we study sampling from the family (1) through the natural extension of Definition 1, which we term a “restricted Gaussian oracle”. Informally, the oracle samples from a Gaussian (with covariance a multiple of ) restricted by .

###### Definition 2 (Restricted Gaussian oracle).

is a restricted Gaussian oracle for convex if it returns

 O(λ,v)←samplefromthedistributionwithdensity∝exp(−12λ∥x−v∥22−g(x)).

The notion of a restricted Gaussian oracle has appeared previously [CV18, MFWB19], and its efficient implementation was a key subroutine in the fastest (zeroth-order) sampling algorithm for general logconcave distributions [CV18]. It was shown in [MFWB19] that a variety of composite distributions arising in practical applications, including coordinate-separable , and or group Lasso regularized densities, admit such oracles. Our main result is an algorithm efficiently sampling from (1), assuming access to a restricted Gaussian oracle for and the minimizer of .222This assumption is not restrictive, as efficient algorithms minimize in gradient queries to and proximal oracle queries to [BT09]. Proximal oracle access is typically a weaker assumption than restricted Gaussian oracle access. We discuss effects of inexactness in this minimization procedure in Appendix D.

###### Theorem 1.

Consider a distribution of the form (1), where has a condition number , and convex admits a restricted Gaussian oracle . Also, assume we know the minimizer of . Algorithm 1, Composite-Sample, samples from within total variation distance , in iterations. Each iteration queries and an expected constant number of times.

Recent work [MFWB19] also considered the problem of composite sampling via a restricted Gaussian oracle. However, their work also assumed access to the normalization constant of the restricted Gaussian, as well as Lipschitzness of , amongst other criteria. Our result, Theorem 1, holds with no additional assumptions other than the relevant oracle access, including in the absence of a warm start. While there remains a gap between our runtime333Throughout, the notation hides logarithmic factors in , , and . of and recent runtimes of in the non-composite setting [LST20], our algorithm substantially improves upon prior composite sampling work in both generality and runtime guarantees. We believe this provides evidence that the restricted Gaussian oracle is a useful abstraction in studying logconcave sampling with composite potentials.

Finally, we remark that although our method follows several reductions, each is conceptually lightweight (as discussed in the following section) and easily implementable via either a rejection sampling procedure or oracle calls. To demonstrate this empirically, we evaluate our method for the task of sampling a (non-diagonal) Gaussian restricted to the positive orthant in Section 4.

### 1.2 Technical overview

We now survey the main components in the development of our algorithm.

Reduction to the shared minimizer case. We first observe that we can without loss of generality assume that and share a minimizer. In particular, by shifting both functions by a linear term, i.e. , , where is the minimizer of , first-order optimality implies both and are minimized by . Moreover, implementation of a first-order oracle for and a restricted Gaussian oracle for

are immediate without additional assumptions. This modification becomes crucial for our later developments, and we expect this simple observation, reminiscent of “variance reduction” techniques in stochastic optimization

[JZ13], to be broadly applicable to improving algorithms for the sampling problem induced by (1).

Beyond Moreau envelopes: expanding the space. A typical approach in convex optimization in handling non-smooth objectives is to instead optimize its Moreau envelope, defined by

 gη(y)def=minx∈Rd{g(x)+12η∥x−y∥22}. (3)

Intuitively, the envelope trades off function value with proximity to ; a standard exercise shows that is smooth (has a Lipschitz gradient), with smoothness depending on , and moreover that computing gradients of is equivalent to calling a proximal oracle (Definition 1). It is natural to extend this idea to the composite sampling setting, e.g. via sampling from the density

 exp(−f(x)−gη(x)).

However, a variety of complications prevent such strategies from obtaining rates comparable to their noncomposite, well-conditioned counterparts, including difficulty in bounding closeness of the resulting distribution, as well as bias in drift of the sampling process due to error in gradients.

Our approach departs from this smoothing strategy in a crucial way, inspired by Hamiltonian Monte Carlo (HMC) methods [Kra40, Nea11]. Hamiltonian Monte Carlo can be seen as a discretization of the ubiquitous Langevin dynamics, on an expanded space. In particular, discretizations of Langevin dynamics simulate the stochastic differential equation , where is Brownian motion. HMC methods instead simulate dynamics on an extended space , via an auxiliary “velocity” variable which accumulates gradient information. This is sometimes interpreted as a discretization of the underdamped Langevin dynamics [CCBJ18]. HMC often has desirable stability properties, and the strategy of expanding the dimension via an auxiliary variable has been used in algorithms obtaining the fastest rates in the well-conditioned logconcave sampling regime [SL19, LST20]. Inspired by this phenomenon, we consider the density on

 d^πdz(z)def=exp(−f(y)−g(x)−12η∥x−y∥22) where z=(x,y). (4)

Due to technical reasons, the family of distributions we use in our final algorithms are of slightly different form than (4), but this simplification is useful to build intuition. Note in particular that the form of (4) is directly inspired by (3), where rather than maximizing over , we directly expand the space. The idea is that for small enough and a set on of large measure, smoothness of will guarantee that the marginal of (4) on will concentrate near , a fact we make rigorous. To sample from (1), we then show that a rejection filter applied to a sample from the marginal of (4) will terminate in constant steps. Consequently, it suffices to develop a fast sampler for (4).

Alternating sampling with an oracle. The form of the distribution (4) suggests a natural strategy for sampling from it: starting from a current state , we iterate

1. Sample .

2. Sample , via a restricted Gaussian oracle.

When and share a minimizer, taking a first-order approximation in the first step, i.e. sampling , can be shown to be a generalization of the Leapfrog step of Hamiltonian Monte Carlo updates. However, for very small (as in our setting), we observe that the first step itself reduces to the case of sampling from a distribution with constant condition number, which can be performed in gradient calls by e.g. Metropolized HMC [DCWY18, CDWY19, LST20]. Moreover, it is not hard to see that this “alternating marginal” sampling strategy preserves the stationary distribution exactly, so no filtering is necessary. Directly bounding the conductance of this random walk, for small enough , leads to an algorithm running in iterations, each calling a restricted Gaussian oracle once, and a gradient oracle for roughly times. This latter guarantee is by an appeal to known bounds [CDWY19, LST20] on the mixing time in high dimensions of Metropolized HMC for a well-conditioned distribution, a property satisfied by the -marginal of (4) for small .

Stability of Gaussians under bounded perturbations. To obtain our tightest runtime result, we use that is chosen to be much smaller than to show structural results about distributions of the form (4), yielding tighter concentration for bounded perturbations of a Gaussian (i.e. the Gaussian has covariance , and is restricted by -smooth for ). To illustrate, let

and let its mean and mode be , . It is standard that , by -strong logconcavity of . Informally, we show that for and not too far from the minimizer of , we can improve this to ; see Proposition 8 for a precise statement.

Using our structural results, we sharpen conductance bounds, improve the warmness of a starting distribution, and develop a simple rejection sampling scheme for sampling the variable in expected constant gradient queries. These improvements lead to our main result, an algorithm running in iterations. Our proofs are continuous in flavor and based on gradually perturbing the Gaussian and solving a differential inequality; we believe they may of independent interest.

### 1.3 Related work

The broad problem of sampling from a logconcave distribution (with no assumptions beyond convexity on the log-density) has attracted much interest in the theoretical computer science community, as it generalizes uniform sampling from a convex set. General bounds under zeroth-order query access imply logconcave distributions are samplable in polynomial time ( in the absence of a warm start [LV06a]). For more densities with more favorable structure, however, the first-order access model is attractive to exploit said structure.

Since seminal work of [Dal17], an exciting research direction has studied first-order random walks for distributions with well-behaved log-densities, developing guarantees under assumptions such as Lipschitz derivatives of different orders [CCBJ18, DR18, CV19, CDWY19, DCWY18, DM19, DMM19, LSV18, MMW19, SL19, LST20]. To our knowledge, when the log-density has a condition number of (with no other assumptions), to obtain total variation distance the best-known guarantee is calls to [LST20], and to obtain -Wasserstein distance444 is the scale-invariant effective diameter of a -strongly logconcave distribution. the best-known is oracle calls [SL19]. These results do not typically generalize beyond when the support of is , prompting study of a more flexible distribution family.

Towards this goal, recent works studied sampling from densities of the form (1), or its specializations (e.g. restrictions to a convex set). Several [Per16, BDMP17, Ber18] are based on Moreau envelope or proximal regularization strategies, and demonstrate efficiency under more stringent assumptions on the structure of the composite function , but under minimal assumptions obtain fairly large provable mixing times . Algorithms derived from proximal regularization have also been considered for non-composite sampling [Wib19]. Another discretization strategy based on projections was studied by [BEL18], but obtained mixing time . Finally, improved algorithms for special constrained sampling problems have been proposed, such as simplex restrictions [HKRC18].

Of particular relevance and inspiration to this work is the algorithm of [MFWB19]. By generalizing and adapting Metropolized HMC algorithms of [DCWY18, CDWY19], adopting a Moreau envelope strategy, and using (a stronger version of) the restricted Gaussian oracle access model, [MFWB19] obtained a runtime which in the best case scales as , similar to our guarantee. However, this result required a variety of additional assumptions, such as access to the normalization factor of restricted Gaussians, Lipschitzness of , warmness of the start, and various problem parameter tradeoffs. The general problem of sampling from (1) under minimal assumptions more efficiently than general-purpose logconcave algorithms is to the best of our knowledge unresolved (even under restricted Gaussian oracle access), a novel contribution of our method and mixing time bound.

Section 3 states our algorithm and subroutines, and provides a proof of Theorem 1 assuming various properties of our process. We demonstrate empirical performance of our method in Section 4. We defer proofs of technical ingredients to appendices, but give strategy overviews in the body.

## 2 Preliminaries

##### General notation.

For , denotes the set of naturals . We use the Loewner order on symmetric matrices,

to denote the identity matrix of appropriate dimension, and

to mean the Euclidean norm. is the Gaussian density with specified mean and covariance.

##### Functions.

We call differentiable -smooth if it has a Lipschitz gradient, i.e. for all . If is twice-differentiable, it is well-known this implies for all , . We say twice-differentiable is strongly convex if everywhere. When a function is -smooth and -strongly convex, we define its condition number . Strong convexity and smoothness respectively imply for all ,

##### Distributions.

We say distribution is logconcave if , for some convex function ; it is -strongly logconcave if its negative log-density is -strongly convex. It is known that -strong logconcavity implies -sub-Gaussian tails (e.g. [DCWY18], Lemma 1). For , ; we denote the complement by . We say distribution is -warm with respect to if everywhere. The total variation distance between two distributions and is . Finally, for a density on and function ,

## 3 Algorithm

In this section, we state the components of our method. Throughout, fix distribution with density

 dπdx(x)∝exp(−f(x)−g(x)),where % f:Rd→R is L-smooth, μ-% strongly convex, (5) and g:Rd→R admits a % restricted Gaussian oracle O.

Observe that distribution is -strongly logconcave. We assume that we have precomputed ; see discussion in Section 1.1. Our algorithm proceeds in stages following the outline in Section 1.2, which are put together in Section 3.4 to prove Theorem 1.

1. Composite-Sample is reduced to Composite-Sample-Shared-Min, which takes as input a distribution with negative log-density , where and share a minimizer; this reduction is given in Section 3.1, and the remainder of the paper handles the shared-minimizer case.

2. The algorithm Composite-Sample-Shared-Min

is a rejection sampling scheme built on top of sampling from a joint distribution

on whose -marginal approximates . We give this reduction in Section 3.2.

3. The bulk of our analysis is for Sample-Joint-Dist, an alternating marginal sampling algorithm for sampling from . To implement marginal sampling, it alternates calls to and a rejection sampling algorithm Sample-Y. We prove its correctness in Section 3.3.

### 3.1 Reduction from Composite-Sample to Composite-Sample-Shared-Min

Correctness of Composite-Sample is via the following properties, whose proofs are in Appendix A.

###### Proposition 1.

Let and be defined as in Composite-Sample.

1. The density is the same as the density .

3. and are both minimized by .

### 3.2 Reduction from Composite-Sample-Shared-Min to Sample-Joint-Dist

Composite-Sample-Shared-Min is a rejection sampling scheme, which accepts samples from subroutine Sample-Joint-Dist

in the high-probability region

defined in (6). We give a general analysis for approximate rejection sampling in Appendix A.3.1, and Appendix A.3.2 bounds relationships between distributions and , defined in (5) and (7) respectively (i.e. relative densities and normalization constant ratios). Combining these pieces proves the following main claim.

###### Proposition 2.

Let , and assume samples within total variation of the -marginal on (7). Composite-Sample-Shared-Min outputs a sample within total variation of (5) in an expected calls to Sample-Joint-Dist.

### 3.3 Implementing Sample-Joint-Dist

Sample-Joint-Dist alternates between sampling marginals in the joint distribution , as seen by definitions (9), (10). In Appendix A.4.1, we give a short proof that marginal sampling attains the correct stationary distribution. We bound the conductance of the induced walk on iterates by combining an isoperimetry bound with a total variation guarantee between transitions of nearby points in Appendix A.4.2. Finally, we give a simple rejection sampling scheme Sample-Y as Algorithm 4 for implementing the step (9). Since the -marginal of is a bounded perturbation of a Gaussian (intuitively, is -smooth and ), we show in a high probability region that rejecting from the sum of a first-order approximation to and the Gaussian succeeds in iterations.

###### Remark 1.

For simplicity of presentation, we were conservative in bounding constants throughout; in practice (cf. Section 4), we found that the constant in Line 4 is orders of magnitude too large (a constant sufficed). Several constants were inherited from prior analyses, which we do not rederive to save on redundancy.

We now give a complete guarantee on the complexity of Sample-Joint-Dist.

###### Proposition 3.

Sample-Joint-Dist outputs a point with distribution within total variation distance from the -marginal of . The expected number of gradient queries per iteration is constant.

### 3.4 Putting it all together: Proof of Theorem 1

We show Theorem 1 follows from the guarantees of Propositions 12, and 3. By observing the value of in Sample-Joint-Dist, we see that the number of total iterations in each call to Sample-Joint-Dist is bounded by Proposition 3 also shows that every iteration, we require an expected constant number of gradient queries and calls to , the restricted Gaussian oracle for , and that the resulting distribution has total variation from the desired marginal of . Next, Proposition 2 implies that the number of calls to Sample-Joint-Dist in a run of Composite-Sample-Shared-Min is bounded by a constant, the choice of is , and the resulting point has total variation from the original distribution . Finally, Proposition 1 shows sampling from a general distribution of the form (1) is reducible to one call of Composite-Sample-Shared-Min, and the requisite oracles are implementable.

## 4 Experiments

We test our algorithm on the problem of sampling from a Gaussian restricted to an orthant. Formally, for a Gaussian with mean and covariance , and where is a random orthant555This generalizes the case of the positive orthant by changing signs of appropriately. (coordinatewise sign restrictions on ), we consider sampling from the distribution666The indicator is if and otherwise.

 π∗(x)∼exp(−12(x−m)⊤Σ−1(x−m)−1x∈O).

This problem is motivated by applications in posterior estimation with side information that the variable of interest has sign constraints, e.g. in physics simulations

[NBD18]. For such distributions with nondiagonal covariances, sampling in the high-dimensional regime can be challenging, and to our knowledge no high-accuracy practical samplers exist for this fundamental problem.

We verify the correctness of our algorithm by using the output of naïve rejection sampling (accepting samples in

) on Gaussian distributions with random covariance and random mean in low dimensions, where we can meaningfully plot histograms. We defer this test to Appendix

E.

In high dimensions, we show our algorithm vastly improves upon the hit-and-run method [LV06b], the most efficient general-purpose logconcave sampler in practice. Hit-and-run has a mixing time of theoretically [LV06b] and empirically. We test our algorithm on randomly generated Gaussian distributions with dense covariance matrices. For fair comparison to hit-and-run (which works on well-rounded distributions), the condition numbers of all randomly generated Gaussian distributions are small constants and the smoothness parameters are . The main tunable parameter in our algorithm is the step size , which we chose so that both Sample-Y and Sample-Joint-Dist reject with probability at most .

In Figure 1, we compare the mixing times of our algorithm and hit-and-run and show the dependence on the dimension . The mixing criterion used was that the process has an effective sample size for all coordinates. To ensure a stable scaling of the mixing time, we use fixed mean . Our algorithm used step size for each . We show that our algorithm improves upon hit-and-run by a factor , which corroborates our theoretical analysis.

In Figure 2, we plot the autocorrelation of the two algorithms’ trajectories for , projected on a random unit direction. We show that in the very high-dimensional regime, our algorithm can converge significantly faster than hit-and-run. In this experiment, each coordinate of is chosen uniformly at random from , and for our algorithm. We include an autocorrelation plot of shorter trajectories showing mixing time of our algorithm in Appendix E.

### Acknowledgments

We thank Yair Carmon for suggesting the experiment in Section 4.

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## Appendix A Deferred proofs from Section 3

### a.1 Technical facts

We will repeatedly use the following facts throughout this paper.

###### Fact 1 (Gaussian integral).

For any and ,

 ∫exp(−12λ∥x−v∥22)dx=(2πλ)d2.
###### Fact 2 ([Har04], Theorem 1.1).

Let be a -strongly logconcave density. Let be the Gaussian density with covariance matrix . For any convex function ,

 Eπ[h(x−Eπ[x])]≤Eγμ[h(x−Eγμ[x])].
###### Fact 3 ([Dcwy18], Lemma 1).

Let be a -strongly logconcave distribution, and let minimize its negative log-density. Then, for and any , with probability at least ,

 ∥x−x∗∥2≤√dμ⎛⎝2+2max⎛⎝4√log(1/δ)d,√log(1/δ)d⎞⎠⎞⎠. (11)
###### Fact 4 ([Dm19], Theorem 1).

Let be a -strongly logconcave distribution, and let minimize its negative log-density. Then, .

### a.2 Deferred proofs from Section 3.1

See 1

###### Proof.

For and with properties as in (5), with minimizing , define the functions

 ~f(x)def=f(x)−⟨∇f(x∗),x⟩,~g(x)def=g(x)+⟨∇f(x∗),x⟩,

and observe that everywhere. This proves the first claim. Further, implementation of a first-order oracle for and a restricted Gaussian oracle for are immediate assuming a first-order oracle for and a restricted Gaussian oracle for , showing the second claim; any quadratic shifted by a linear term is the sum of a quadratic and a constant. We now show and have the same minimizer. By strong convexity, has a unique minimizer; first-order optimality shows that

 ∇~f(x∗)=∇f(x∗)−∇f(x∗)=0,

so this unique minimizer is . Moreover, optimality of for implies that for all ,

 ⟨∂g(x∗)+∇f(x∗),x∗−x⟩≤0.

Here, is a subgradient. This shows first-order optimality of for also, so minimizes . ∎

### a.3 Deferred proofs from Section 3.2

#### a.3.1 Approximate rejection sampling

We first define the rejection sampling framework we will use, and prove various properties.

###### Definition 3 (Approximate rejection sampling).

Let be a distribution, with . Suppose set has , and distribution with has for some ,

 p(x)^p(x)≤C for all x∈Ω, and ∫^p(x)dx∫p(x)dx≤1.

Suppose there is an algorithm which draws samples from a distribution , such that . We call the following scheme approximate rejection sampling: repeat independent runs of the following procedure until a point is outputted.

1. Draw via until .

2. With probability , output .

###### Lemma 1.

Consider an approximate rejection sampling scheme with relevant parameters defined as in Definition 3, with . The algorithm terminates in at most

 11−ϵ′C−2δ (12)

calls to in expectation, and outputs a point from a distribution with .

###### Proof.

Define for notational simplicity normalization constants and . First, we bound the probability any particular call to returns in the scheme:

 ∫x∈Ωp(x)C^p(x)d^π′(x) ≥∫x∈Ωp(x)C^p(x)d^π(x)−∣∣∣∫x∈Ωp(x)C^p(x)(d^π′(x)−d^π(x))∣∣∣ (13) =∫x∈ΩZC^Zdπ(x)−∣∣∣∫x∈Ωp(x)C^p(x)(d^π′(x)−d^π(x))∣∣∣ ≥1−ϵ′C−∫x∈Ω|d^π′(x)−d^π(x)|≥1−ϵ′C−2δ.

The second line followed by the definitions of and , and the third followed by triangle inequality, the assumed lower bound on , and the total variation distance between and . By linearity of expectation and independence, this proves the first claim.

Next, we claim the output distribution is close in total variation distance to the conditional distribution of restricted to . The derivation of (13) implies

 ∫x∈Ωp(x)C^p(x)d^π(x)≥1−ϵ′C,∣∣∣∫x∈Ωp(x)C^p(x)(d^π′(x)−d^π(x))∣∣∣≤2δ, (14) ⟹1−2δC1−ϵ′≤∫x∈Ωp(x)C^p(x)d^π′(x)∫x∈Ωp(x)C^p(x)d^π(x)≤1+2δC1−ϵ′.

Thus, the total variation of the true output distribution from restricted to is

 12∫x∈Ω∣∣ ∣ ∣∣dπ(x)1−ϵ′−p(x)C^p(x)d^π′(x)∫x∈Ωp(x)C^p(x)d^π′(x)∣∣ ∣ ∣∣ ≤12∫x∈Ω∣∣ ∣ ∣∣dπ(x)1−ϵ′−p(x)C^p(x)d^π′(x)∫x∈Ωp(x)C^p(x)d^π(x)∣∣ ∣ ∣∣+12∫x∈Ω∣∣ ∣ ∣∣p(x)C^p(x)d^π′(x)∫x∈Ωp(x)C^p(x)d^π(x)−p(x)C^p(x)d^π′(x)∫x∈Ωp(x)C^p(x)d^π′(x)∣∣ ∣ ∣∣ ≤12∫x∈Ω∣∣ ∣ ∣∣dπ(x)1−ϵ′−p(x)C^p(x)d^π′(x)∫x∈Ωp(x)C^p(x)d^π(x)∣∣ ∣ ∣∣+δC1−ϵ′=12∫x∈Ωdπ(x)1−ϵ′∣∣∣1−d^π′d^π(x)∣∣∣+δC1−ϵ′.

The first inequality was triangle inequality, and we bounded the second term by (14). To obtain the final equality, we used

 ∫x∈Ωp(x)C^p(x)d^π(x)=∫x∈ΩZC^Zdπ(x)=(1−ϵ′)ZC^Z ⟹p(x)C^p(x)d^π′(x)∫x∈Ωp(x)C^p(x)d^π(x)=p(x)Z⋅^Z^p(x)⋅11−ϵ′⋅d^π′(x)=dπ(x)1−ϵ′⋅d^π′d^π(x).

We now bound this final term. Observe that the given conditions imply that is bounded by everywhere in . Thus, expanding we have

 12∫x∈Ωdπ(x)1−ϵ′∣∣∣1−d^π′d^π(x)∣∣∣≤C2(1−ϵ′)∫x∈Ω|d^π(x)−d^π′(x)|≤δC1−ϵ′.

Finally, combining these guarantees, and the fact that restricting to loses in total variation distance, yields the desired conclusion by triangle inequality. ∎

###### Corollary 1.

Let

be an unbiased estimator for

, and suppose with probability 1 for all . Then, implementing the procedure of Definition 3 with acceptance probability has the same runtime bound and total variation guarantee as given by Lemma 1.

###### Proof.

It suffices to take expectations over the randomness of everywhere in the proof of Lemma 1. ∎

#### a.3.2 Distribution ratio bounds

We next show two bounds relating the densities of distributions and . We first define the normalization constants of (5), (7) for shorthand, and then tightly bound their ratio.

###### Definition 4 (Normalization constants).

We denote normalization constants of and by

 Zπ def=∫xexp(−f(x)−g(x))dx, Z^π def=∫x,yexp(−f(y)−g(x)−12η∥y−x∥22−ηL22∥x−x∗∥22)dxdy.
###### Lemma 2 (Normalization constant bounds).

Let and be as in Definition 4. Then,

 (2πη1+ηL)d2(1+ηL2μ)−d2≤Z^πZπ≤(2πη)d2.
###### Proof.

For each , by convexity we have

 (15) =(2πη)d2exp(−f(x)−g(x))exp(η2∥∇f(x)∥22−ηL22∥x−x∗∥22) ≤(2πη)d2exp(−f(x)−g(x)).

Integrating both sides over yields the upper bound on . Next, for the lower bound we have a similar derivation. For each , by smoothness

 ∫yexp(−f(y)−g(x)−12η∥y−x∥22−ηL22∥x−x∗∥22)dy =exp(−f(x)−g(x)−ηL22∥x−x∗∥2+η2(1+ηL)∥∇f(x)∥2)(2πη1+ηL)d2

Integrating both sides over yields

 Z^πZπ≥(2πη1+ηL)d2∫xexp(−f(x)−g(x)−ηL22∥x−x∗∥22)dx∫xexp(−f(x)−g(x))dx≥(2πη1+ηL)d2(1+ηL2μ)−d2.

The last inequality followed from Proposition 7, where we used is -strongly convex. ∎

###### Lemma 3 (Relative density bounds).

Let . For all , as defined in (6), . Here, denotes the marginal density of . Moreover, for all , .

###### Proof.

We first show the upper bound. By Lemma 2,

 dπd^π(x) (16) ≤exp(−f(x)−g(x))∫yexp(−f(y)−


text:
# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Jordan mappings of semiprime rings. II. (English) Zbl 0727.16027
The author refines results on Jordan (triple) homomorphisms proved in an earlier paper [Part I, J. Algebra 127, 218-228 (1989; Zbl 0691.16040)]. An additive mapping of rings T: $R\to S$ satisfying $T(ab+ba)=T(a)T(b)+T(b)T(a)$ is a Jordan homomorphism, and is a Jordan triple homomorphism if $T(aba)=T(a)T(b)T(a)$. The results assume that S is 2-torsion free and that the annihilator of any ideal in S is a direct summand of S. The first theorem shows that when T is a Jordan homomorphism, then $R=I+J$ for ideals of R with $I\cap J=Ker T$, S is the direct sum of ideals T(I) and T(J), and T is an associative homomorphism on I and an antihomomorphism on J. For the second main theorem, $R\sp 2=R$ and T is a Jordan triple homomorphism. In this case there are four ideals $U\sb 1,...,U\sb 4$ of R whose sum is R and whose pairwise intersections are all Ker T, so that S is the direct sum of the ideals $T(U\sb i)$, and T acting on each $U\sb i$ is either an associative homomorphism, antihomomorphism, or the negative of one of these.

##### MSC:
 16W20 Automorphisms and endomorphisms of associative rings 16N60 Prime and semiprime associative rings 16W10 Associative rings with involution, etc. 16D70 Structure and classification of associative ring and algebras
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# Tomcat Not Starting with startup.bat

Shiva Kumar
Greenhorn

Joined: Jan 03, 2008
Posts: 9
hi

but when i try to run the startup.bat batch file it giving the message:

I set My System environment variable as follows:

variables & values
----------------------------------------------
variable: CLASSPATH
value : c:\Program Files\Java\jdk1.6.0\jre\lib\rt.jar;

variable: JAVA_HOME
value : C:\Program Files\Java\jdk1.6.0;

variable: PATH
value : C:\Program Files\Java\jre1.6.0;C:\Program Files\Java\jdk1.6.0\bin;C:\Program Files\Java\jre1.6.0\bin;C:\apache-tomcat-6.0.14\bin;

variable: JRE_HOME
value : C:\Program Files\Java\jre1.6.0;

these are my environment variables

if it starts here i need to link it to the eclipse with SysduoTomcat Plugin
in the eclipse

with regards
Shiva
Raghu ram
Ranch Hand

Joined: Oct 19, 2007
Posts: 34
Hi,

You can start the tomcat server using tomcat.exe. In tomcat old version there is file called start.bat. But in this latest version there is no such file. I think they had modified start.bat into tomcat.exe
Just try it.
Shiva Kumar
Greenhorn

Joined: Jan 03, 2008
Posts: 9
Hi

Tomcat available as different types as a zip file in windows, linux..etc
and also as installation in windows i have both of the versions and its working from .exe if i use installaton from .exe file only but its not working
from eclipse plugin, and it worked well some days before with plugin also
and even with extracted ziparchive(i.e with out installation as win service)
and also it worked well with startup.bat
after few days i tried it to start again to learn servelet's but now it giving
that error message from command line startup.bat and

classLoaderException when tried to start from eclipse plugin

Thank you very much
Shiva
James Jiang
Greenhorn

Joined: Dec 13, 2007
Posts: 24
I think if you add tomcat-plugin to your eclipse, then you open eclipse,go to Window->Preferences, and you will see Tomcat in the left and click it,set the correct version of tomcat and select the location of Tomcat Home. I think if these are set correctly,your problem will be solved.Hope this helps you.
David O'Meara
Rancher

Joined: Mar 06, 2001
Posts: 13459

JAVA_HOME and JRE_HOME should not have a semi-colon on the end

eg
JAVA_HOME=/usr/lib/jvm/java-6-sun-1.6.0.03
David O'Meara
Rancher

Joined: Mar 06, 2001
Posts: 13459

"ShivaKumar Learner",
Welcome to the JavaRanch.

We're a friendly group, but we do require members to have valid display names.

Display names must be two words: your first name, a space, then your last name. Fictitious names are not allowed.

Please edit your profile and correct your display name since accounts with invalid display names get deleted.
Shiva Kumar
Greenhorn

Joined: Jan 03, 2008
Posts: 9
Originally posted by YI-Feng Jiang:
I think if you add tomcat-plugin to your eclipse, then you open eclipse,go to Window->Preferences, and you will see Tomcat in the left and click it,set the correct version of tomcat and select the location of Tomcat Home. I think if these are set correctly,your problem will be solved.Hope this helps you.

i added plugin correctly and i enabled it and set the correct ver Tomcat 6.0
appeared at right on tomcat tab at windows-->prefer-->Tomcat it's (ver 6.0) is the last radio buttton it work well before
problem occured few das before and it even not starting with startup.bat also

Shiva
Shiva Kumar
Greenhorn

Joined: Jan 03, 2008
Posts: 9
Originally posted by David O'Meara:
"ShivaKumar Learner",
Welcome to the JavaRanch.

We're a friendly group, but we do require members to have valid display names.

Display names must be two words: your first name, a space, then your last name. Fictitious names are not allowed.

Please edit your profile and correct your display name since accounts with invalid display names get deleted.

sorry
i don't mean to use fictitious name but i did it when i have no choice
i need an account and some other account already exist with my first and last name so i used this name
i am sorry
i will change my name to as near as possible to my real name as this site allows

thank you
with regards

Shiva
Shiva Kumar
Greenhorn

Joined: Jan 03, 2008
Posts: 9
Originally posted by David O'Meara:
JAVA_HOME and JRE_HOME should not have a semi-colon on the end

eg
JAVA_HOME=/usr/lib/jvm/java-6-sun-1.6.0.03

hi
thank you very much for reply
your solution not only solved my problem, it also realized me
from an assumption that it's necessary to use ; at end of each & every value
of the Env variable

thanks again for solving my problem

now my Tomcat is running well with startup.bat

with regards
Shiva
Shiva Kumar
Greenhorn

Joined: Jan 03, 2008
Posts: 9
Originally posted by David O'Meara:
JAVA_HOME and JRE_HOME should not have a semi-colon on the end

eg
JAVA_HOME=/usr/lib/jvm/java-6-sun-1.6.0.03

now my eclipse plugin giving the error with classNotFoundException
stack trace:

also if carefully observed startup.bat also giving an error message even it starts well as:

should i need to any library to the build path to run tomcat properly
Shiva Kumar
Greenhorn

Joined: Jan 03, 2008
Posts: 9
hi

my problem solved just by adding rt.jar to the,

windows-->preferences-->Tomcat-->JVM settings
and
boot classpath "rt.jar" from one of jre's lib directory

now my Tomcat server is running from eclipse IDE with
Sysdeo Tomcat plugin.

So Please Don't waste time on my question
and i am posting my solution for reference to other people

thank you
with regards
Shiva
Maneesh Godbole
Saloon Keeper

Joined: Jul 26, 2007
Posts: 9748

6

sohils shah,
Your post was moved to a new topic.

subject: Tomcat Not Starting with startup.bat


text:
Asked • 07/29/19

How do I create a modal service with AngularJS?

I'm trying to make a service that's polymorphic based upon what mode is specified in the URL. If the char param in the route is set to 'p', I want to use a PresentMode service. If the char param is set to 'n', I want to use a NoteMode service. Each of these present the same interface, but I want to choose one at a time.So far the best solution I've come up with is something like this: var mod = angular.module('modeModule', []); mod.service('modeService', function($routeParams, presentMode, noteMode) { if ($routeParams.char === 'p') { this.mode = presentMode; } else if (\$routeParams.char === 'n') { this.mode = noteMode; } } mod.service('presentMode', function() {}); mod.service('noteMode', function() {});This works, but it requires that I append .mode to the end of every access (eg modeService.mode.blah(). Is there a better way to do this?

1 Expert Answer

By:

Alan Andres H. answered • 02/05/20

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text:
Display of pages per glossary item in glossaries-extra

Update: I've found an adequate answer - see below. I have many occurrences of certain glossary items spread out over a 100 page document. My output has two entries per line

GF Game force 14, 19,
42, 61,
65, 68,
75, 96,
99, 100,
107–109

and I want it to show e.g. five entries per line

GF Game force 14, 19, 42, 61, 65,
68, 75, 96, 99, 100,
107–109

I'm using a custom format My_Glossary but I don't see what parameter controls the maximum number of items per line.

Update: Changing widths gives enough space for more than two items, but still only two print i.e.

\setlength\glsdescwidth{0.3\linewidth}

Also changing another parameter gives me what I want though e.g.

\setlength\glspagelistwidth{0.6\linewidth}

The MWE seems to include all entries in a page range and that's different from my live system but I don't see why.

\documentclass{book}
\usepackage[xindy,automake=true]{glossaries-extra}
\usepackage{blindtext}

\newglossarystyle{My_Glossary}
{
\setlength\glsdescwidth{0.5\linewidth}
\renewcommand{\glossentry}[2]{%
\textbf{\glsentryitem{##1}\glstarget{##1}{\glossentryname{##1}}}
& \glossentrydesc{##1}
& ##2
\tabularnewline}%
}

\newglossaryentry{gf}{name={GF},  description={Game force}, first={First game forcing bid}}%

\makeglossaries

\begin{document}
\chapter{Ch One}

\gls{gf}
\blindtext
\gls{gf}

\Blindtext
\gls{gf}

\blindtext
\gls{gf}

\Blindtext
\gls{gf}

\Blindtext
\gls{gf}

\blindtext
\gls{gf}

\Blindtext
\gls{gf}

\printglossary[type=main,style=My_Glossary]

\end{document}

• When I try your example, I just get the page range 1--6 (since there's a reference on every page from 1 to 6). My TeX Live setup uses A4 paper by default. Adding \usepackage[letterpaper]{geometry} results in the page range 1--5 instead. So the difference you observe when you switch to TeX Live may be due to different default page settings. Incidentally, you have spurious spaces in your glossary style caused by uncommented end of line characters. These can cause unwanted spaces when you set the style, although I don't think it's noticeable in your particular case. – Nicola Talbot Jul 1 '18 at 11:30
• Thanks Nicola. My original is properly commented. I started with the misconception that there would be an explicit number of pages shown and finally in desperation read the documentation and discovered that setting the lengths did what I wanted :-) – DLyons Jul 1 '18 at 11:41
• Okay :-) If you've found the solution then you can self-answer otherwise the question will linger on the unanswered list. – Nicola Talbot Jul 1 '18 at 12:11


text:
# American Institute of Mathematical Sciences

November  2009, 24(4): 1409-1435. doi: 10.3934/dcds.2009.24.1409

## Variational principles of pressure

 1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  May 2008 Revised  December 2008 Published  May 2009

Given a topological dynamical system $(X, T)$, a Borel cover $\mathcal{U}$ of $X$ and a sub-additive sequence $\mathcal{F}$ of real-valued continuous functions on $X$, two notions of measure-theoretical pressure $P_\mu^- (T, \mathcal{U}, \mathcal{F})$ and $P_\mu^+ (T, \mathcal{U}, \mathcal{F})$ for an invariant Borel probability measure $\mu$ are introduced. When $\mathcal{U}$ is an open cover, a local variational principle between topological and measure-theoretical pressure is proved; it is also established the upper semi-continuity of P+$(T, \mathcal{U}, \mathcal{F})$ and P+$(T, \mathcal{U}, \mathcal{F})$ on the space of all invariant Borel probability measures. The notions of measure-theoretical pressure $P_\mu^- (T, X, \mathcal{F})$ and $P_\mu^+ (T, X, \mathcal{F})$ for an invariant Borel probability measure $\mu$ are also introduced. A global variational principle between topological and measure-theoretical pressure is also obtained.
Citation: Guohua Zhang. Variational principles of pressure. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1409-1435. doi: 10.3934/dcds.2009.24.1409
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2019 Impact Factor: 1.338


text:
Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets $${K\subset H}$$ of a Hilbert space H by the metric entropy of the set K where the covering numbers $${N(K, \varepsilon)}$$ of K by $${\varepsilon}$$ -balls of H satisfy the Lorentz condition $$\int\limits_{0}^{\infty} \left(\log N(K,\varepsilon) \right)^{r/s}\, d\varepsilon^{s} < \infty$$ for some fixed $${0 < r, s \le \infty}$$ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 < r < ∞, s = ∞. The integral here is an improper Stieltjes integral. Moreover, we obtain optimal estimates of Gelfand numbers of absolutely convex hulls if the metric entropy satisfies the entropy condition $$\sup_{\varepsilon >0 }\varepsilon \left(\log N(K,\varepsilon) \right)^{1/r}\left(\log(2+\log N(K,\varepsilon))\right)^\beta < \infty$$ for some fixed 0 < r < ∞, −∞ < β < ∞. Using inequalities between Gelfand and entropy numbers we also get optimal estimates of the metric entropy of the absolutely convex hull cov(K). As an interesting feature of the estimates, a sudden jump of the asymptotic behavior of Gelfand numbers as well as of the metric entropy of absolutely convex hulls occurs for fixed s if the parameter r crosses the point r = 2 and, if r = 2 is fixed, if the parameter β crosses the point β = 1. The results established in Hilbert spaces extend and recover corresponding results of several authors. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

, Volume 17 (1) – Jan 28, 2012
33 pages

/lp/springer_journal/gelfand-numbers-and-metric-entropy-of-convex-hulls-in-hilbert-spaces-ThMdMmw5Xf
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0157-9
Publisher site
See Article on Publisher Site

Abstract

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets $${K\subset H}$$ of a Hilbert space H by the metric entropy of the set K where the covering numbers $${N(K, \varepsilon)}$$ of K by $${\varepsilon}$$ -balls of H satisfy the Lorentz condition $$\int\limits_{0}^{\infty} \left(\log N(K,\varepsilon) \right)^{r/s}\, d\varepsilon^{s} < \infty$$ for some fixed $${0 < r, s \le \infty}$$ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 < r < ∞, s = ∞. The integral here is an improper Stieltjes integral. Moreover, we obtain optimal estimates of Gelfand numbers of absolutely convex hulls if the metric entropy satisfies the entropy condition $$\sup_{\varepsilon >0 }\varepsilon \left(\log N(K,\varepsilon) \right)^{1/r}\left(\log(2+\log N(K,\varepsilon))\right)^\beta < \infty$$ for some fixed 0 < r < ∞, −∞ < β < ∞. Using inequalities between Gelfand and entropy numbers we also get optimal estimates of the metric entropy of the absolutely convex hull cov(K). As an interesting feature of the estimates, a sudden jump of the asymptotic behavior of Gelfand numbers as well as of the metric entropy of absolutely convex hulls occurs for fixed s if the parameter r crosses the point r = 2 and, if r = 2 is fixed, if the parameter β crosses the point β = 1. The results established in Hilbert spaces extend and recover corresponding results of several authors.

Journal

PositivitySpringer Journals

Published: Jan 28, 2012

References

• Metric entropy of convex hulls in Hilbert spaces
Carl, B.
• Gelfand numbers and metric entropy of convex hulls in Hilbert spaces
Carl, B.; Edmunds, D.E.

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# Hydrometeorology Research Group

In [5]:
from IPython.display import HTML
HTML('''<script>
code_show=true;
function code_toggle() {
if (code_show){
$('div.input').hide(); } else {$('div.input').show();
}
code_show = !code_show
}
</script>

In [1]:
%matplotlib inline
from rain import *

In [2]:
%%capture

# Phoenix Rainfall¶

As a desert environment, the Phoenix rain gage network brings certain challenges. Rain events are much less common, so zero values dominate the datasets and the loss of a reading that happens during a storm can skew the whole gage record.

## Gage Locations¶

In [4]:
highlight = None
fig, ax = plt.subplots(figsize=(12,8))
#colors = list(df.index)
colors = df.start_date.apply(lambda x: x.year)
if highlight is not None:
colors = [0 if c==highlight else c for c in colors]
scat = ax.scatter(x=df.lon, y=df.lat, c=colors, s=70)
plt.title('Gages colored by start year (hover for gage name and start date)')
import mpld3
_df = df['start_date'].astype(str)
d = _df.to_dict()
labels = ['{i}\n start: {s}'.format(i=i, s=d[i]) for i in df.index]
tooltip = mpld3.plugins.PointLabelTooltip(scat, labels=labels)
fig.colorbar(scat)
mpld3.plugins.connect(fig, tooltip)
mpld3.display()

In [4]:
df = df.sort_values('start_date')
by_date = df.set_index(df.start_date)
by_date[3] = 1
by_date[3] = by_date[3].cumsum()
tot = by_date[3]
tot = tot.resample('1D', how='max').dropna().reindex(wet.index, method='ffill')
tot[tot.isnull()]=1
counts = pd.DataFrame(wet.count(axis=1)).join(tot)
counts.columns = ['wet', 'tot']
counts['ratio']=(counts.wet/counts.tot)
counts.columns = ['Gages with rain', 'Total number of gages', 'ratio']

This dataset in particular is difficult to wrangle because there are up to 313 gages at any given time and the data stretch back to 1980. The gages have been added a couple at a time since then. The plots below treat rainfall as a binary operation by just showing nuber of gages with rain vs the total number of gages that had been deployed up to that point.

In [5]:
fig, axes = plt.subplots(nrows=2, figsize=(14,8), sharex=True)
axes[0].set_title('Gages and gages observing rainfall')
axes[0].set_axis_bgcolor('white')
axes[1].set_axis_bgcolor('white')
counts.plot(y=['Gages with rain', 'Total number of gages'], ax=axes[0])
axes[0].set_ylabel('Number of gages')
counts['ratio'].plot(ax=axes[1], color='g')
axes[1].set_ylabel('Percentage of gages recording rainfall')
plt.xlabel('')
plt.show()

## Rain rate maps¶

One way to get an overview of the data is to map aggregated statistics at the gage locations.

#### Rain rate in mm/hr over period of record¶

In [4]:
s = rg.rate.mean(axis=0)
s.name = 'Mean Rain Rate for 1980-2014'
fig, axes = plt.subplots(ncols=2, nrows=1, figsize=(16,6))
Event(rg.ll.join(s)).map_rain(latlon=True, ax=axes[0], sharec=False, basemap=True, hide_title=True)
k = Event(rg.ll.join(s)).krige(res=False, latlon=True, basemap=True, ax=axes[1])

From that map there seems to be a strong north-eastern amplification in rainfall, but I was wondering if that might be dominated by a few big storms, or perhaps the gages were added in a patterned way. To try to get at this, I plotted the mean rain rate for each decade.

In [6]:
q = rg.rate['1980':'1989'].mean()
q.name = 'the 1980s'
w = rg.rate['1990':'1999'].mean()
w.name = 'the 1990s'
e = rg.rate['2000':'2009'].mean()
e.name = 'the 2000s'
r = rg.rate['2010':].mean()
r.name = 'the 2010s'
Event(rg.ll.join((q,w,e,r))).map_rain(latlon=True, figsize=(16, 14), basemap=True, hide_title=True)

The pattern holds up! Now I wanted to try plotting the mean rain rate given that it is raining. For this plot I used a threshold of 1 tip per 15min which is equivalent to 0.01 inches, .254mm, or 1mm/hr. Next to this plot is a map of the time of the day at which peak rainfall occurs.

In [12]:
rg.get_wet_rate()
fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(16,6))
s = rg.wet_rate.mean()
s.name = 'Mean rain rate given that it is raining'
scat = Event(rg.ll.join(s)).map_rain(latlon=True, basemap=True, hide_title=True, sharec=(6,9), ax=axes[0])
fig.colorbar(scat, ax=axes[0])
lm = rg.get_max_lowess(example_plot=False)
lm.columns = ['Time of max rainfall (using lowess)']
scatter = Event(rg.ll.join(lm)).map_rain(latlon=True, basemap=True, hide_title=True, sharec=(0,24), cmap='hsv',ax=axes[1])
fig.colorbar(scatter, ax=axes[1])
plt.show()

## Standard descriptive plots¶

By inspecting these plots we can get a sense of the temporal and spatial patterns of the rain

In [4]:
df = rg.rate.groupby(rg.rate.index.month).mean().T
df.columns = ['January', 'February', 'March','April', 'May', 'June','July', 'August','September','October','November','December']
Event(rg.ll.join(df)).movie(basemap=True)

Once Loop Reflect


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Encyclopedia > Dollar sign
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Punctuation Image File history File links Merge-arrows. ... CifrÃ£o on 2. ... \$, the dollar sign, is primarily used to represent currencies: Many different dollars Many different pesos Different escudos The Brazilian real The Tongan paanga The Nicaraguan cÃ³rdoba \$ may also be: \$ (film), also known as Dollars A sigil (computer programming) Â·\$ is a colour code for MSN Messenger. ... Emoticons originated with text representations. ... The term punctuation has two different linguistic meanings: in general, the act and the effect of punctuating, i. ...

apostrophe ( ' )
brackets (( )), ([ ]), ({ }), (< >)
colon ( : )
comma ( , )
dashes ( , , , )
ellipsis ( , ... )
exclamation mark ( ! )
full stop/period ( . )
guillemets ( « » )
hyphen ( -, )
question mark ( ? )
quotation marks ( ‘ ’, “ ” )
semicolon ( ; )
slash/stroke ( / )
solidus ( )

Interword separation

spaces ( ) ( ) ( )
interpunct ( · )
This article or section is in need of attention from an expert on the subject. ... A space is a punctuation convention for providing interword separation in some scripts, including the Latin, Greek, Cyrillic, and Arabic. ... An interpunct Â· is a small dot used for interword separation in ancient Latin script, being perhaps the first consistent visual representation of word boundaries in written language. ...

General typography

ampersand ( & )
at sign ( @ )
asterisk ( * )
backslash ( )
bullet ( )
caret ( ^ )
currency ( ¤ ) ¢, \$, , £, ¥, ,
dagger/obelisk ( ) ( )
degree ( ° )
inverted exclamation point ( ¡ )
inverted question mark ( ¿ )
not sign ( ¬ )
number sign ( # )
numero sign ( )
percent and related signs
( %, ‰, )
pilcrow ( )
prime ( )
section sign ( § )
tilde/swung dash ( ~ )
umlaut/diaeresis ( ¨ )
underscore/understrike ( _ )
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Uncommon typography

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A specimen of roman typefaces by William Caslon Typography is the art and techniques of type design, modifying type glyphs, and arranging type. ... In typography, an asterism is a rare symbol consisting of three asterisks placed in a triangle, used to call attention to a passage or to separate sub-chapters in a book. ... The symbol â˜ž is a rare punctuation mark, called an index or fist. ...   In a mathematical proof, the therefore sign is a symbol that is sometimes placed before a logical consequence, such as the conclusion of a syllogism. ... For other uses, see Interrobang (disambiguation). ... The irony mark or irony point (ØŸ) (French: point dâ€™ironie; also called a snark or zing) is a punctuation mark that purports to indicate that a sentence should be understood at a second level. ... This page lists Japanese typographic symbols which are not included in kana or kanji. ... A sarcasm mark, also called a sarcasm point, helps the reader identify certain messages as being derogatory or ironic. ...

The dollar sign or peso sign (\$) is a symbol primarily used to indicate a unit of currency.

History

The sign's ultimate origins are not certain,[3] though it is widely accepted that it comes from the Spanish coat of arms, which carries the two Pillars of Hercules and the motto Plus Ultra in the shape of an "S". Coat of Arms of Spain (Official model) The current Coat of arms of Spain was approved by law [1] in 1981, when the present established replaced the interim version which, in turn, replaced the official arms of Francoist Spain. ... This article does not cite any references or sources. ... For other uses, see Motto (disambiguation). ...

Spanish Coat of Arms

The Pillars of Hercules with "S"-shaped ribbon in the Town Hall of Seville, (Spain) (16th century)

The most widely accepted explanation is that the dollar sign derives from the Spanish coat of arms engraved on the Spanish colonial silver coins "Real de a Ocho" ("piece of eight") or Spanish dollar under circulation in the Spanish colonies of America and Asia, as well as in the English Thirteen Colonies and later the U.S. and Canada. Image File history File links Size of this preview: 567 Ã— 599 pixelsFull resolution (1053 Ã— 1113 pixel, file size: 2. ... Image File history File links Size of this preview: 567 Ã— 599 pixelsFull resolution (1053 Ã— 1113 pixel, file size: 2. ... This article does not cite any references or sources. ... For other uses, see Seville (disambiguation). ... (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... The Spanish dollar or peso (literally, weight) is a silver coin that was minted in the Spanish Empire after a Spanish currency reform in 1497. ... For other uses, see Asia (disambiguation). ... For other uses, see United States (disambiguation) and US (disambiguation). ...

The Spanish coat of arms has two columns (||), which represent the Pillars of Hercules and an "S"-shaped ribbon around each, with the motto "Non Plus Ultra" originally, and later "Plus Ultra".[4] This article does not cite any references or sources. ...

The Pillars of Hercules around the modern coat of arms of Spain.

There is also another explanation that makes the sign derive from where "\$" is a corruption of the letters "PS" or PS, used as an abbreviation for pesos.[6][7] Image File history File links Escudo_de_EspaÃ±a. ... Image File history File links Escudo_de_EspaÃ±a. ... This article does not cite any references or sources. ... The peso is a unit of currency. ...

Alternative origin hypotheses

There are a number of alternative origin theories, with several degrees of verifiability and academic acceptance.

From 'US'

That \$ is a monogram of U and S (United States AKA Uncle Sam), which was used as a mark on money bags issued by the United States Mint. The letters U and S superimposed resemble the historical double stroke "\$" sign: the bottom of the 'U' disappears into the bottom curve of the 'S', leaving two vertical lines. This double-stroke dollar sign has been used to refer to the U.S. currency. Thus, the one-stroke design may have been modified from the double-stroke design to represent United States currency. This idea was largely popularized by the novel Atlas Shrugged by philosopher Ayn Rand[citation needed]. It does not consider the fact that the symbol was already in use in the time of the British Colonies, when the term 'United States' did not exist yet. The Chi-Rho, a monogram of the first two letters in the Greek word for Christ E and L embroider for clothes and bedding, for a wife by the initials E L or L E A monogram is a motif made by overlapping or combining two or more letters or... This article is about the national personification of the USA. For other uses, see Uncle Sam (disambiguation). ... Seal of the U.S. Mint Denver United States mint building The United States Mint primarily produces circulating coinage for the United States to conduct its trade and commerce. ... For the film, see Atlas Shrugged (film). ... Ayn Rand (IPA: , February 2 [O.S. January 20] 1905 â€“ March 6, 1982), born Alisa Zinovyevna Rosenbaum (Russian: ), was a Russian-born American novelist and philosopher. ...

From a symbol used on the Roman sestertius

That the dollar sign goes back to the most important Roman coin, the Sestertius, which had the letters 'HS' as its currency sign. When superimposed, these letters form a dollar sign with two vertical strokes (the horizontal line of the 'H' merging into the 'S'). This explanation is widely discarded, in spite of the tendency of neo-classic Roman Republic influences in styles evident in other early US government designs, such as the Capitol and Senate buildings. The sestertius was an ancient Roman coin. ... This article is about the state which existed from the 6th century BC to the 1st century BC. For the state which existed in the 18th century, see Roman Republic (18th century). ... The United States Capitol is the capitol building that serves as the seat of government for the United States Congress, the legislative branch of the U.S. federal government. ... Type Upper House President of the Senate Richard B. Cheney, R since January 20, 2001 President pro tempore Robert C. Byrd, D since January 4, 2007 Members 100 Political groups Democratic Party Republican Party Last elections November 7, 2006 Meeting place Senate Chamber United States Capitol Washington, DC United States...

The two pillars in the temple of Solomon

That the two vertical lines represent the two cult pillars Boaz and Jachin in the original Temple of Solomon at Jerusalem. This is based on the idea that Masonic symbols, such as the All Seeing Eye of God, appear on U.S. currency; however, they did not in 1785. Boaz and Jachin were the name of the two pillars that stood on the eastern porch of Solomons Temple, the first temple in Jerusalem (1 Kings 7:21; 2 Kings 11:14; 23:3). ... Solomons Temple was the first Jewish temple in Jerusalem which functioned as a religious focal point for worship and the sacrifices known as the korbanot in ancient Judaism. ... Freemasons redirects here. ... The Eye of Providence floating above an unfinished pyramid on the reverse of the Great Seal of the United States. ...

From a sign used on the German Thaler

That it derives from the symbol used on a German Thaler. According to Ovason (2004), on one type of thaler, one side showed the crucified Christ, and the other side showed a serpent hanging from a cross, and near the serpent's head the letters NU, and on the other side of the cross the number 21. This refers to the Bible, Numbers, chapter 21 (see Nehushtan). Examples of German and Austrian Thalers compared to a US quarter piece (bottom center) The Thaler (or Taler) was a silver coin used throughout Europe for almost four hundred years. ... Icon of Christ in a Greek Orthodox church This page is about the title, office or what is known in Christian theology as the Divine Person. ... For other uses, see Snake (disambiguation). ... Also known as the Latin cross or crux ordinaria. ... For other uses, see Bible (disambiguation). ... Moses lifts up the brass snake, curing the Isrealites from Snake Bites. ...

Unit of Silver

The dollar symbol was in use in colonial times before the American Revolution. Prices were often quoted in units of silver, as the Spanish "piece of eight" was in common use for payment of goods and services. When a price was quoted the capital 'S' was used to indicate silver with a capital 'U' written on top to indicate units. Eventually, the capital 'U' was replaced by double vertical hash marks.

Other theories

Another possibility is that it derives from the British notation 8/ for eight shillings, referring to the Spanish 8 reales coin ("piece of eight"), which later became the USA dollar. Others derive it from the Portuguese Cifrão sign $mathrm{S}!!!Vert$.[8] This article is about coinage. ... The Spanish dollar or peso (literally, heavy, or pound) is a silver coin which was minted in Spain after a Spanish currency reform in 1497. ... CifrÃ£o on 2. ...

Image of 1768 Spanish Colonial Real silver coin, showing PTSI (\$) mint mark in lower right quadrant. It also shows the columns around the hemispheres (this time displayed one at the side of the other).

A common explanation is that the symbol is derived from the numeral eight with a slash through it denoting "pieces of eight". The Oxford English Dictionary prior to 1963 held that this was the most probable explanation, though later editions have placed it in doubt. Image File history File linksMetadata Potosi_Real. ... Image File history File linksMetadata Potosi_Real. ... The Spanish dollar or peso (literally, weight) is a silver coin that was minted in the Spanish Empire after a Spanish currency reform in 1497. ...

Still another explanation holds that the dollar sign is derived from (or at least inspired by) the mint mark on Spanish colonial silver coins ("real" or "piece of eight") that were minted in Potosí (in present day Bolivia). The mint mark was composed of the letters "PTSI" superimposed on one another, and bears an undeniable resemblance to the single-stroke dollar sign (see picture). The Potosí mine is generally accepted as having been the largest single silver strike in history. Silver coins minted in Potosí would have been in common use in colonial America, and its mint mark widely recognized. Poorly documented explanations sustains that the symbol represents an snake, guarding a bag, and that it was used in Tironian notes. A mint mark is an inscription on a coin indicating the mint at which the coin was produced. ... The real was a unit of currency in Spain for several centuries. ... The Spanish dollar or peso (literally, weight) is a silver coin that was minted in the Spanish Empire after a Spanish currency reform in 1497. ... PotosÃ is a city, the capital of the department of PotosÃ in Bolivia. ... A nugget of silver Silver mining refers to the resource extraction of the precious metal element silver, mostly through mines. ... Tironian notes (notae Tironianae) is a system of shorthand invented by Ciceros scribe Marcus Tullius Tiro. ...

However, the dollar sign may have also originated from the Greek God of bankers, thieves, messengers, and tricksters: Hermes. Besides the crane, one of his symbols was a peeled hasel stick from which white ribbons dangled. Foolish people often mistook these for snakes. So, the vertical line (|) may be the hasel stick and the S may be the ribbons. Greek mythological characters (Most of the gods and goddesses had Roman equivalents. ... For other uses, see Hermes (disambiguation). ... Genera Grus Anthropoides Balearica Bugeranus Cranes are large, long-legged and long-necked birds of the order Gruiformes, and family Gruidae. ...

First cast dollar symbol

The plaque on the bookshop on the corner of South Street and Church Street in St Andrews.

Image File history File linksMetadata Download high-resolution version (2048x1536, 711 KB)  Summary Plaque in St Andrews, Fife, Scotland reading: Here stood the house of Bailie Bell, who, before 1744, was an eager co-worker with Alexander Wilson, the father of Scottish type-founder, and John Baine, in whose... Image File history File linksMetadata Download high-resolution version (2048x1536, 711 KB)  Summary Plaque in St Andrews, Fife, Scotland reading: Here stood the house of Bailie Bell, who, before 1744, was an eager co-worker with Alexander Wilson, the father of Scottish type-founder, and John Baine, in whose...

Use in computer programming

As the dollar sign is one of the few symbols that is on the one hand almost universally present in computer character sets, but on the other hand rarely needed in its literal meaning within programming languages, the \$ character has been used on computers for many purposes not related to money, including: A character encoding is a code that pairs a set of characters (such as an alphabet or syllabary) with a set of something else, such as numbers or electrical pulses. ... This article is about the machine. ...

• \$ was used as a string terminator in CP/M and subsequently also in all versions of 86-DOS, PC-DOS, MS-DOS and derivatives (Int 21h with AH=09h)
• \$ signifies the end of a line or the file in text editors ed, ex, vi and derivatives, and consequently:
• \$ matches the end of a line or string in sed, grep, and POSIX and Perl regular expressions.
• \$ was used to define string variables in older versions of the BASIC programming language ("\$" was often pronounced "string" instead of "dollar" in this use).
• \$ is used to define hexadecimal constants in Pascal-like languages such as Delphi, as well as in certain variants of assembly language.
• \$ is used to define variables in the PHP programming language and scalar variables in the Perl programming language (see Sigil (computer programming)).
• In the AutoIt automation script language, any variable is required to have a \$ at the beginning of its name.
• In most shell scripting languages, \$ is used to interpolate environment variables, special variables, arithmetic computations and special characters, and to perform translation of localised strings.
• In UNIX-like systems the \$ is often part of the command prompt, depending on the user's shell and environment settings. For example, the default environment settings for the bash shell specify \$ as part of the command prompt.
• \$ is used in the TeX typesetting language to delimit mathematical regions.
• \$ is used by the `prompt` command in DOS to insert special sequences into the DOS command prompt string.
• Formulas in Microsoft Excel and other spreadsheets use \$ to indicate an absolute cell reference.

Currencies that use the dollar or peso sign

In addition to those countries of the world that use dollars or pesos, a number of other countries use the \$ symbol to denote their currencies, including, but not limited, to: This article is about the type of currency, for the U.S. Dollar see United States dollar. ... The peso is a unit of currency. ...

Except the Philippine peso, whose sign is written as . ISO 4217 Code NIO User(s) Nicaragua Inflation 9. ... The paanga (or Tongan dollar) is the currency of the Kingdom of Tonga. ... ISO 4217 Code PHP User(s) Philippines Inflation 2. ... Image File history File links PhilippinePeso. ...

Some currencies use the cifrão $(mathrm{S}!!!Vert )$, similar to the dollar sign, but always with two strokes: CifrÃ£o on 2. ...

The cifrão is also currently used to account for over 130,000,000 domestic standard US Mint (1986+) bullion US silver dollars as one dollar per one troy ounce fine (99.9%), thereby avoiding confusion with debased US trade dollar-denominated tokens and Federal Reserve Notes. ISO 4217 Code BRL User(s) Brazil Inflation 3. ... The official currency of Cape Verde (Portuguese: Cabo Verde), a former Portuguese colony, is called Escudo. ... Chilean notes currently in circulation: 1,000; 2,000; 5,000; 10,000; 20,000 pesos The peso is the currency of Chile. ... The escudo was the official currency of Portugal prior to the introduction of the euro in 1 January 1999 (euro coins and notes were not introduced until 2002). ... The United States Mint is responsible for producing and circulating coinage for the United States to conduct its trade and commerce. ... The Federal Reserve System is headquartered in the Eccles Building on Constitution Avenue in Washington, DC. The Federal Reserve System (also the Federal Reserve; informally The Fed) is the central banking system of the United States. ...

The euro (€; ISO 4217 code EUR) is the currency of twelve of the twenty-five nations that form the European Union (and four outside it, as well as Montenegro and Kosovo), which form the Economic and Monetary Union (EMU). ... This article is about the currency symbol. ...

References

1. ^ Lawrence Kinnaird: "The Western Fringe of Revolution", The Western Historical Quarterly. Vol. 7, No. 3 (Jul., 1976), page 259 [1]
2. ^ Pub. L. No. 109-145, 119 Stat. 2664 (Dec. 22, 2005).
3. ^ "Note on Our Dollar Sign", Bulletin of the Business Historical Society, Vol. 13, No. 4 (Oct., 1939), pp. 57-58 [2]
4. ^ Nussbaum, Arthur: A history of the dollar. New York : Columbia University Press, 1957.
5. ^ Earl Rosenthal: "Plus Ultra, Non plus Ultra, and the Columnar Device of Emperor Charles V", Journal of the Warburg and Courtauld Institutes, Vol. 34, 1971 (1971), pp. 204-228
6. ^ Arthur S. Aiton; Benjamin W. Wheeler: "The First American Mint", The Hispanic American Historical Review. Vol. 11, No. 2 (May, 1931), pp. 198-215. [3]
7. ^ "Origin of the \$ Sign", US Bureau of Engraving and Printing website
8. ^ Florian Cajori: "New Data on the Origin and Spread of the Dollar Mark", The Scientific Monthly. Vol. 29, No. 3 (Sep., 1929), pp. 212-216 [4]
• Cajori, Florian (1993). A History of Mathematical Notations. New York: Dover (reprint). ISBN 0-486-67766-4.  - contains section on the history of the dollar sign, with much documentary evidence supporting the "pesos" theory.
• Ovason, David (2004-11-30). The Secret Symbols of the Dollar Bill. Harper Paperbacks (reprint). ISBN 0-06-053045-6.

Results from FactBites:

 Dollar sign - definition of Dollar sign in Encyclopedia (463 words) The name Spanish dollar was used for a Spanish silver coin, the peso, an eight-real coin, which was widely circulated during the 18th century in the Spanish colonies in the New World. The use of the Spanish dollar and the Maria Theresa thaler as legal tender for the early United States is the reason for the name of that nation's currency. The dollar was also in use in Scotland during the 17th century, and there is a claim that it was invented at the University of St Andrews.
 Dollar sign - Wikipedia, the free encyclopedia (1010 words) The dollar sign is a symbol primarily used to indicate a unit of currency. The sign is attested in business correspondence between British North America and Mexico in the 1770s as referring to the Spanish-Mexican piastre. The piastre was known as "Spanish dollar" in British North America, and in 1785, it was adopted as U.S. currency, together with both the term "dollar" and the \$ sign.
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text:
## Forget Me Not

View as PDF

Points: 4
Time limit: 2.0s
Memory limit: 256M

Author:
Problem type
Allowed languages
Ada, Assembly, Awk, Brain****, C, C#, C++, COBOL, CommonLisp, D, Dart, F#, Forth, Fortran, Go, Groovy, Haskell, Intercal, Java, JS, Kotlin, Lisp, Lua, Nim, ObjC, OCaml, Octave, Pascal, Perl, PHP, Pike, Prolog, Python, Racket, Ruby, Rust, Scala, Scheme, Sed, Swift, TCL, Text, Turing, VB, Zig

The black foxen are the arch enemies of Tushan: they corrupt the love of reincarnated soulmates by absorbing their memories. This allows the black foxen to steal their pledge of love and the resulting spirit power.

The black foxen would like to prioritize a victim's memories, starting with the most important one. They have converted each memory to an integer, with up to 1 million digits (this will not fit in int data type). For this problem, the black foxen have hired you to compare two memories and print the one which is higher.

Given 2 non-negative integers and on different lines, where both and are less than . The number may contain leading zeros. If is larger than output S, if is larger than output T otherwise, output E.

#### Sample Input

123456765434567652143783748374938274932798478329749328
0432874832787589275324893729562839478

#### Sample Output

S

• commented on Sept. 17, 2020, 11:00 p.m.

Umm... Why is it that it says that I will get 4 points but I only got 2?

• commented on Sept. 18, 2020, 12:12 a.m.

• commented on Oct. 19, 2018, 11:10 p.m. edited

How do you convert S and T into an integer?

Is there a command that inputs integers with up to 1 million digits?

Nevermind-fixed

• commented on Oct. 5, 2019, 9:17 a.m.

s=int(input()) t=int(input())

• commented on Sept. 25, 2018, 3:55 p.m. edit 4

This comment is hidden due to too much negative feedback. Click here to view it.

• commented on Sept. 25, 2018, 4:22 p.m. edited

The constraints on this problem state the numbers in the input can be up to a hundred thousand digits long. Find a different way to store them, because they will certainly not fit in an int, long or float. Or a double, which I see you attempted to use after I commented.

• commented on Oct. 19, 2018, 11:23 p.m.

I'm using python 3 and my code worked by inputing an integer normally with int(input()). It seems that the test cases do fit in an int.

Am i wrong?

• commented on Oct. 20, 2018, 5:06 a.m.

Python int is a big int, in most languages though this isn't the case (c++, c, java, etc.)

• commented on Sept. 23, 2018, 6:01 p.m.

y am i getting IRs

• commented on Sept. 23, 2018, 7:45 p.m.

The number can be bigger than an int or a long

• commented on Sept. 27, 2018, 6:32 p.m.

is java BigInteger allowed?

• commented on Sept. 27, 2018, 9:28 p.m.

Yes


text:
## On the local structure of Dirac manifolds.(English)Zbl 1142.53063

Summary: We give a local normal form for Dirac structures. As a consequence, we show that the dimensions of the pre-symplectic leaves of a Dirac manifold have the same parity. We also show that, given a point $$m$$ of a Dirac manifold $$M$$, there is a well-defined transverse Poisson structure to the pre-symplectic leaf $$P$$ through $$m$$. Finally, we describe the neighborhood of a pre-symplectic leaf in terms of geometric data. This description agrees with that given by Vorobjev for the Poisson case.

### MSC:

 53D17 Poisson manifolds; Poisson groupoids and algebroids 58H05 Pseudogroups and differentiable groupoids

### Keywords:

normal forms; Poisson structure; Dirac structure; foliations
Full Text:


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I want to use Free Power 3D printer to create the stator and rotors. This should allow Free Power high quality build with lower cost. Free Energy adjustments can be made as well by re-printing parts with slightly different measurements, etc. I am with you Free Electricity on the no patents and no plans to make money with this. I want to free the world from this oppression. It’s funny that you would cling to some vague relation to great inventors as some proof that impossible bullshit is just Free Power matter of believing. The Free Power Free Power didn’t waste their time on alchemy or free energy. They sought to understand the physical forces around them. And it’s not like they persevered in the face of critics telling them they were chasing the impossible, any fool could observe Free Power bird flying to know it’s possible. You will never achieve anything even close to what they did because you are seeking to defy the reality of our world. You’ve got to understand before you can invent. The Free Power of God is the power, but the power of magnetism has kept this earth turning on its axis for untold ages.
Both sets of skeptics will point to the fact that there has been no concrete action, no major arrests of supposed key Deep State players. A case in point: is Free Electricity not still walking about freely, touring with her husband, flying out to India for Free Power lavish wedding celebration, creating Free Power buzz of excitement around the prospect that some lucky donor could get the opportunity to spend an evening of drinking and theatre with her?
Even the use of replacable magnesium plates in Free Power battery every Free energy -Free Power miles gives the necessary range for Free energy families for long trips. Magnet-only motors are easy to build. There are plans around. They are cheap to build. Trouble is no one knows how to get them to spin unaided. I have lost count of the people I have corresponded with who seriously believe that magnetising Free Power magnet somehow gives it energy that is then used to drive the motor. Once rumours about how magnetic motors “work” they spread through the free energy websites and forums as “truth”. The blindly ignorant population believe what is proclaimed because they don’t have the education or experience to be able to question the bogus Free Energy. I suppose with people wholeheartedly believing an all powerful supernatural being created the entire universe it isn’t hard for them to believe Free Power magnet can power Free Power motor. Both thoughts demonstrate ignorance. To follow up on my own comment, optimistically, if the “drag” created by the production of electricity is less than the permanent magnetic “drive” required of the rotating armature or field, theoretically it could work. Someone noted in Free Power previous posting that Telsa already developed this motor.
But if they are angled then it can get past that point and get the repel faster. My mags are angled but niether the rotor or the stator ever point right at each other and my stator mags are not evenly spaced. Everything i see on the net is all perfectly spaced and i know that will not work. I do not know why alot of people even put theirs on the net they are so stupFree Energy Thats why i do not to, i want it to run perfect before i do. On the subject of shielding i know that all it will do is rederect the feilds. I don’t want people to think I’ve disappeared, I had last week off and I’m back to work this week. I’m stealing Free Power little time during my break to post this. Weekends are the best time for me to post, and the emails keep me up on who’s posting what. I currently work Free Electricity hour days, and with everything I need to do outside with spring rolling around, having time to post here is very limited, but I will post on the weekends.

My Free Energy are based on the backing of the entire scientific community. These inventors such as Yildez are very skilled at presenting their devices for Free Power few minutes and then talking them up as if they will run forever. Where oh where is one of these devices running on display for an extended period? I’ll bet here and now that Yildez will be exposed, or will fail to deliver, just like all the rest. A video is never proof of anything. Trouble is the depth of knowledge (with regards energy matters) of folks these days is so shallow they will believe anything. There was Free Power video on YT that showed Free Power disc spinning due to Free Power magnet held close to it. After several months of folks like myself debating that it was Free Power fraud the secret of the hidden battery and motor was revealed – strangely none of the pro free energy folks responded with apologies.
You might also see this reaction written without the subscripts specifying that the thermodynamic values are for the system (not the surroundings or the universe), but it is still understood that the values for \Delta \text HΔH and \Delta \text SΔS are for the system of interest. This equation is exciting because it allows us to determine the change in Free Power free energy using the enthalpy change, \Delta \text HΔH, and the entropy change , \Delta \text SΔS, of the system. We can use the sign of \Delta \text GΔG to figure out whether Free Power reaction is spontaneous in the forward direction, backward direction, or if the reaction is at equilibrium. Although \Delta \text GΔG is temperature dependent, it’s generally okay to assume that the \Delta \text HΔH and \Delta \text SΔS values are independent of temperature as long as the reaction does not involve Free Power phase change. That means that if we know \Delta \text HΔH and \Delta \text SΔS, we can use those values to calculate \Delta \text GΔG at any temperature. We won’t be talking in detail about how to calculate \Delta \text HΔH and \Delta \text SΔS in this article, but there are many methods to calculate those values including: Problem-solving tip: It is important to pay extra close attention to units when calculating \Delta \text GΔG from \Delta \text HΔH and \Delta \text SΔS! Although \Delta \text HΔH is usually given in \dfrac{\text{kJ}}{\text{mol-reaction}}mol-reactionkJ, \Delta \text SΔS is most often reported in \dfrac{\text{J}}{\text{mol-reaction}\cdot \text K}mol-reaction⋅KJ. The difference is Free Power factor of 10001000!! Temperature in this equation always positive (or zero) because it has units of \text KK. Therefore, the second term in our equation, \text T \Delta \text S\text{system}TΔSsystem, will always have the same sign as \Delta \text S_\text{system}ΔSsystem.

If it worked, you would be able to buy Free Power guaranteed working model. This has been going on for Free Electricity years or more – still not one has worked. Ignorance of the laws of physics, does not allow you to break those laws. Im not suppose to write here, but what you people here believe is possible, are true. The only problem is if one wants to create what we call “Magnetic Rotation”, one can not use the fields. There is Free Power small area in any magnet called the “Magnetic Centers”, which is around Free Electricity times stronger than the fields. The sequence is before pole center and after face center, and there for unlike other motors one must mesh the stationary centers and work the rotation from the inner of the center to the outer. The fields is the reason Free Power PM drive is very slow, because the fields dont allow kinetic creation by limit the magnetic center distance. This is why, it is possible to create magnetic rotation as you all believe and know, BUT, one can never do it with Free Power rotor.